(From the Logic and Mathematics page.)
For the symbolic extension of Verbal Logic, see First-Order Predicate Logic.
This outline is dedicated to Katherine Brandl, Ph.D. - certainly one of David Hilbert's most illustrious and beautiful mathematical descendants.
The definitions used in this outline are designed to minimize the undefined vocabulary as much as possible, where practicable, resulting in a vocabulary where all terms are ultimately some combination of only three, general, and essentially undefined, primitive terms:
- Existence - or terms synonymous with existence, such as "to be," "is manifest," "is expressed," "occurs," etc.,
- Negation - or terms synonymous with negation, such as "not" and the prefixes "non-," "un-," or "im-," or negations of previously defined terms, such as "neither" for the negation of "either," etc., and
- Relation - in this case we select inclusive disjunction as our most primitive use of a relational term, or terms synonymous with disjunction, such as "or," "otherwise," "else," "either," or "whether," etc.
The philosophy underlying this theory of logic is essentially an intersection of Frege's Objectivism, Kant's Rationalism and a non-Finitist version of Brouwer's Intuitionism.
The indefinite articles ("a" or "an") are also undefined and constitute structural, syntactic elements in the English grammar, denoting only the existence of an indefinite object, and are thus classed with the terms that are purely existential. In addition, no term used in this outline will be defined in terms that have not been previously defined in the order of definition of terms stated.
Although the terms used in this outline are selected to be "intuitively obvious," they remain "terms of art" and their meanings are necessarily idiosyncratic to this outline in a manner designed to generalize and eliminate as much nuance as possible and where reasonable. Of course, much grammar is assumed. Defining the nuanced and contextual meanings of grammatical terms as they might be used in ordinary English is beyond the scope of the present outline.
For the purposes of this outline, after a term has been defined in its most general form, a later part of the outline may refine that definition and thereby make it less general by the use of other previously defined terms.
Existence and Non-Existence
-  An object, entity, unit, individual, body, item, referent, occurrence, appearance, arrival, instance, event, fact, self, expression, thing, manifestation, actualization, presence, state, action, or construct (x) is manifest, expressed, created, referred to, signified, present, evident, constructed, built, made, conveyed, moved, transferred, put or placed, stated, exists, occurs, comes, arrives, appears, acts or does (∃).
- There, this, that, which, what, each, it, one, some, those, whom, or these is an object or objects.
- Nullity, nothing, naught, void, zero, empty, negation, elimination, absence, or never (∅, ¬) is non-existence.
- Objects that are absent, without, subtracted, removed, voided, emptied, negated, eliminated, missed, or that leave, depart or disappear, do not exist.
Identification, Differentiation, and Relationship
- An other is not a self.
- Identification with  an object (=, :=, ≡) is not an other.
- An object identified with an other or its self is identical, duplicative, redundant, repetitive, synonymous, equivalent, such, congruent, in common with or like an other, similar, self-similar, or the same.
- An object not identified with its self or an other is different, separate, apart, or distinct (. , ; ≠, ¬≡).
- Different objects are objects for/about/in/of/from/as/to/under which, where or whereby, when, if, or than an other object exists.
- A relation or correspondence of/for/as to each or in regards to each is an object that is identical to or different from an other.
Uniqueness and Definition
- An object is exactly one, precisely one, sole, single, mere, pure, the object, or unique (∃! ) where no other is identical.
- A definition, identity introduction, specification, particularization, valuation, extension, projection, representation, or assignment is a unique identification of an object.
- defining, calling, saying, naming, locating, marking, particularizing, specifying, valuing, projecting, representing, assigning, identifying, or describing are unique identification.
- A defined object is a definiendum.
- An expression or expressions that define, identify, call, say, name, locate, mark, particularize, specify, value, project, represent, assign, identify, or describe a definiendum are definiens.
The separation, division, disjunction, differentiation, or disunion (Axy, ∨, +, || ) of objects is a defined relation of an object to another or to its self. The related objects are disjuncts.
Or, otherwise, except, instead, alternatively, else, either, whether, but or but for, however, than, whereas, for/about/in/of/from/as/to/under which, where or whereby, when, or if, or represent a separation, division, disjunction, differentiation, or disunion.
An object that is a separation, division, disjunction, differentiation, or disunion of another object is a member, point, element, part, place, aspect, segment, portion, or constituent of the other object.
An object that is separated, divided, disjoined, differentiated, or disunited into other objects is composed of, comprised of, made of, consists of, contains, possesses, owns, includes, adds, appends, conjugates, conjoins, combines, or has those other objects as members, points, elements, parts, places, aspects, segments, portions, or constituents.
A combination, collection, compound, association, group, molecule, class, set, space, integration, union, construction, product, conjugation, or conjunction (Kxy, ·, ∧, &, &&) is the identification of the members, points, elements, parts, places, aspects, segments, portions, or constituents of an object. The objects related by conjunction are conjuncts.
And, yet, still, moreover, unless, nonetheless, also, together, as, so that, such that, depending on, lies within, is part of, by, or both represent a conjunction.
Analysis, determination, evaluation, computation, interpretation, or resolution is the combined or separate relations of integration and differentiation.
Any, every, all, or the totality or universe (∀) of objects is a collection of objects where no such other objects exist without the collection.
A combination that is identical with an other but for the presence of one or more than one different object or objects contained within the other, and where the combination otherwise possesses all the same constituents as the other, has fewer or less constituent objects than the other.
A combination that is identical with an other but for the presence of one or more than one different object or objects contained within it, and which does not have fewer or less objects than the other, has greater or more constituent objects than the other.
- The amount or quantity of an object is a determination as to whether the object or set of objects is singular, plural, contains fewer or more objects than an other object, or is non-existent.
An object that consists of nothing is empty.
Order and Timing
A first, beginning, or prime object is stated if no other related objects are yet stated.
A subsequent, successive, future, next, or anticipated object is stated if all other related objects are stated first.
After or will be represent a subsequent object.
A previous, preceding, former, past, or prior object is stated if all other related objects are subsequently stated.
Already, before, was, or has been represent a previous object.
A last, ultimate, ending, or final object is stated if all other related objects have been stated.
An object of a combination that is first or last is called a terminal or end.
An object or collection of objects in combination with more than one terminal or end point is in the middle, between, over, through, or within those terminals or end points, and the combination, including the terminal or end points, is called a segment.
An order, sequence, series, or time line of objects is a determination of whether an object or objects exist before or after another object.
Timing, ordering, or sequencing is a determination of an order, sequence, or series of events.
A determined order, sequence, or series of events is temporal, ordered, or sequenced.
- A period is a segment of time.
A moment is an atomic instance of time.
The present is a moment of time that is neither past nor future.
An object that exists in the past, present, and future always or forever occurs.
An object that always or forever occurs is eternal.
The Objects of Mind and Language
- An intention or purpose is an expression of self.
- An awareness or consciousness is a present manifestation of intention or purpose.
- Sentience, deliberation, intentionality, or thinking is an awareness of self and the relationship of self and an other.
- A being is a conscious object; a thinker is a sentient being.
- A mind, psyche, or intelligence is an objectification of a sentient being or thinker's consciousness.
- A thought, idea, cognition, or concept is an object related by a mind, psyche, or intelligence.
- A meaning or intensionality is a thought, idea, cognition, or concept put in relation to another or to one's self and is thereby conveyed or communicated.
- A thought, idea, cognition, or concept that is meaningful is semantic.
- A structural relation is a relation that is not semantic.
- Information is a thought, idea, cognition, or concept when conveyed or communicated.
- A symbol or signifier is an object or collection of objects that represent a thought, idea, cognition, or concept.
- A language is a symbolic expression of meaning or information.
- A grammar is a definition of a language.
- A syntax is a defined collection of structural relations of grammatical combination.
- Rhetoric is a conveyance or communication of information to another or one's self by a language.
- A word or string is an element of grammar.
- An alphabet is a collection of symbols, called letters or characters, of which words or strings are composed. A word, string, symbol, or signifier composed of only one letter is called an atomic or elemental word, string, symbol, or signifier. A word, string, symbol, or signifier composed of no letters is empty.
- A lexicon or vocabulary is a collection of words or strings that compose a language.
- A sentence or formula is a grammatical and syntactic combination of words, strings, letters, characters, symbols, or other signifiers.
- A phrase or sentence fragment is any part of a sentence.
- A syntactic and grammatical relation of words, strings, letters, characters, symbols, or other signifiers is well formed.
A proposition or expression is meaning intended or information conveyed by a sentence, statement, formula, or message.
The substance, content, or material of an object is a definition of that object.
A condition, predicate, context, circumstance, characteristic, attribute, property, state, structure, quality, design, figure, pattern, or form is a definition of an object's meaning apart from its substance.
- Objects share a condition or have a common condition where each of the objects possesses a same condition.
- A difference in a condition represents a change or variable in that condition.
- A condition that does not change is constant or remains the same.
- A condition is true, verum, a truth, positive, affirmative, manifest, actual, evident, solved, satisfied, found, applicable, valid, correct, or holds true where that condition exists in relation to an object. Where it does not so exist, the condition is false, falsum, a falsehood, negative, invalid, inconsistent, incorrect, inchoate, inert, fictitious, unsolved, or inapplicable.
- The opposite, complementary, or contradictory ( ¬, ~, !, N, -, \, "co-" prefix, or a bar over the term) truth value of true is false, and the opposite, complementary, or contradictory truth value of false is true.
- A condition that is not certain may/might or may/might not occur.
- A condition that exists so that contradictory conditions may or may not occur is possible ("sometimes").
- A possibility that always may occur can occur.
- A possibility that can occur is able to occur.
- A possibility that always may not occur cannot occur.
- A possibility that cannot occur is not able to occur.
- A possibility that cannot not occur must, will, or shall occur.
- An elemental, atomic, or singular object, member, point, element, part, place, aspect, segment, portion, or constituent is an object that cannot be separated, divided, disjoined, differentiated, or disunited into other objects.
- An impossible ("never") condition is a condition that cannot exist for truthfulness to occur.
- A necessary, required, inherent, essential, or intrinsic ("always") condition is a condition that must exist for truthfulness to occur.
- A sufficient ("enough") condition is a condition that is not necessary for truthfulness but for which, where it exists, truthfulness will always occur.
- A whole, entire, or complete object is one that possesses all its necessary parts.
- A well-formed sentence or formula is a sentence or formula that is whole, entire, or complete.
- Observation is a being knowing an other.
An assumption or presumption ( | or : ) is a condition that is not empirically proved but exists for the purpose of making a proposition or other statement.
A condition, the truthfulness or falsity of which can be determined is reasonable or within reason.
A rational condition is a condition that is both reasonable and true.
A rigorous condition is one that is both reasonable and certain.
- A memory or recollection is an awareness of a past event or events.
- A prescience, prediction, or foretelling is a belief in the possibility or certainty of a future event or events.
A term is a symbol, word, phrase, sentence, formula or other expression for a well-defined condition or combination of conditions.
- A term is proper when it is well formed.
- A term that cannot be known without definition or proof is explicit or express. Such a term is called a posteriori.
- A term that may be known without explicit definition or proof is intuitive or implicit. Such a term is called a priori.
- Terms which can only be known intuitively are primitive.
- Axiom: There are only three primitive terms in verbal logic: the collective existential terms, disjunction, and negation.
A subject term ( x ) of a statement is a term about which a predicate ( P ) term conveys a meaning.
Example: "x is over the age of eighteen" is the same as stating "Over the age of eighteen is x" because, regardless of word order, x is the subject for which being over the age of eighteen is the predicate.
- A copula is a relationship between the subject and predicate terms of a statement.
- A particular or specific condition is a condition applicable to one or more terms.
- To state there exists, there is, or for some (∃) is to state a particular or specific condition about a term or group of terms that is not necessarily unique.
- To state there is exactly one, precisely one, or only one (∃!) is to state a particular or specific condition about a term or group of terms so that the term or group of terms are necessarily unique.
- A general or universal condition is necessarily applicable to all terms of a sentence, statement, formula, or message.
- To state for all, every, any, or each (∀) is to state a general or universal condition for a term or terms.
A simple term or statement is a phrase or sentence composed of only one term or statement.
A compound term or statement is a phrase or sentence composed of more than one term or statement.
- substitution or replacement ( / ) occurs when the same conditions are applicable to a term as another term and the term stands in the same relation as the other term.
- A term that is true under some possible interpretation is satisfiable.
- A term that is true under every possible interpretation is validated.
- A term that is false under every possible interpretation is unsatisfiable.
- A term that is false under some possible interpretation is invalidated.
- Axiom: A validated substitution of a validated term in any well-formed statement, sentence, or formula always results in a valid expression.
- A validated substitution of a validated term is justifiable or truth preserving.
- A term that may only be substituted by certain other terms and result in a valid expression is bound by the conditions of the validation.
- A term that may be substituted by any other terms and result in a valid expression is free of any conditions of validation.
- The reversal or switching of terms occurs when, for both terms, one term is substituted for or replaces the other term.
- A term that stands in consistent and valid relation to another follows, derives, draws from, or depends on ( ⊢ or ∴ ) the other.
- Therefore, hence, as such, or how represent a consistent and valid relation.
- A statement has existential import (∃, ∃!, ∀) if its truth depends on the existence of an object or relation of objects.
Antecedents and Consequences
An antecedent, protasis, condition precedent, or premise is a condition from which another condition follows that is called a consequence, consequent, condition subsequent, apodosis, result, yield, conclusion, or derivation.
Axiom: A condition is necessarily either an antecedent or a consequence.
A condition is true, verum, a truth, positive, affirmative, manifest, actual, evident, solved, satisfied, found, applicable, valid, justified, correct, or holds true where it is impossible for the truth value of a conclusion to be different from the truth value of a sufficient premise; in this case, the premises and conclusions are said to be consistent or non-contradictory.
- Can, could, is able, or is represent a truth.
A condition is false, falsum, a falsehood, negative, invalid, unjustified, inconsistent, incorrect, inchoate, inert, fictitious, unsolved, or inapplicable where it is possible for the truth value of a conclusion to be different from the truth value of a sufficient premise; in this case, the premises and the conclusion are said to be inconsistent or contradictory.
- Cannot, could not, is unable, or is not represent a falsity.
Claims, Arguments, and Conclusions
A claim, conjecture, hypothesis, question, problem, belief, or allegation is a condition or combination of conditions of unknown truthfulness.
To purport, allege, surmise, hypothesize, posit, propose, believe, or make a case is to state a claim.
A passage or pericope is one or more terms or statements that together may or may not contain an argument. A passage or pericope contains an argument where it purports to prove the truthfulness of a relation or condition; otherwise it does not contain an argument.
A statement, proposition, or argument is unambiguous or unequivocal where it purports to prove one and only one possible conclusion.
A statement, proposition, or argument is ambiguous or equivocal where it purports to prove more than one possible conclusion.
The explanandum states the proposition to be explained; the explanans is the statement or group of statements that purport to explain the explanandum.
Sets and Their Members
A member of a set, element of a set, point of a set, instance of a set, or constituent of a set is an object that satisfies the conditions required for combination or collection (i.e., membership) in a set.
- A set contains, consists of, or is otherwise composed of, its members.
A member of a set is inside or contained within the set. An object that is not a member of a set is outside the set.
Axiom: A set is a kind of object and may therefore be a member of another set.
A case is a set of conditions.
- A categorical proposition is a case made for a category of objects.
An inference is an application of reason to the relations of terms of a statement, proposition, or argument.
An inference is valid, justified, or truth-preserving if a conclusion is true whenever any sufficient premise, or all necessary premises, are also true.
An inferential rule, principle, or criterion is a statement of a generally valid inference.
- A particular inference complies with, is permissible, consistent with, or obeys a rule or set of rules where any conclusion drawn from an application of the rule or set of rules is valid.
- A cause is any antecedent that is necessary and/or sufficient for the truth of an inference.
- An effect is any true result of a causal inference.
Logic is a reasoned analysis, determination, evaluation, computation, interpretation, or resolution of the satisfiability or validity of conditions, terms, or inferences contained within a statement, proposition, or argument.
A condition, term, or inference is logical if it is determined to be satisfiable or valid.
A logical statement, logical proposition, logical expression, or logical argument ("truth-bearer") is a statement, proposition, or argument constructed entirely from logical conditions, terms, or inferences.
A formal or abstract analysis is an analysis that occurs in regard to the meanings of predicate terms but without regard to the meanings of subject terms. Such an analysis is content neutral.
An informal, meta, concrete, or reified analysis is an analysis that occurs in regard to the meanings of both subject and predicate terms. Such an analysis is not content neutral.
A logical form, argument form, or test form is obtained by formal abstraction of the inferential relationships apart from the meanings of the subject terms.
Informal verbal logic is the logic of statements, propositions, expressions, or arguments and the terms they contain in the meta-language.
A logical object is a logical term, statement, proposition, expression, or argument, and any associated conditions or inferences, for which a purported truth value is claimed.
Any logical terms, statements, propositions, expressions, or arguments are logically equivalent (≡) if a term, statement, proposition, expression, or argument may be substituted for another term, statement, proposition, expression, or argument with no change in the logical conditions or truth value of the term, statement, proposition, expression, or argument in which the substitution occurs.
A logical object is self-consistent where it is true by its own conditions, terms, and inferences.
A logical object is self-contradictory where it is false by its own conditions, terms, and inferences.
An axiom or postulate is a logical object that is both self-consistent and self-evident.
Axiom: A logical object is always a reasonable object.
An axiom schema is an axiom that contains a condition of general applicability to the members of a specific set of objects.
Axiom: Truth of Axioms: All proper axioms are true.
Corollary: Logical Equivalence of Axioms: All proper axioms are logically equivalent.
Corollary: Logical Equivalence of Axioms and Corollaries: All corollaries of proper axioms are true.
Axiom: Logical Equivalence of Objects and Corollaries: A logical object and its corollary are logically equivalent.
Corollary: Complementarity of Bivalent Truth Values: For every affirmation there corresponds exactly one negation, and every affirmation and its negation are necessarily 'opposed' such that one and only one of them must be true, and the other false.
- A logical possibility is any statement that may or may not be true, depending on the choice of premises.
- A logical impossibility is any statement that cannot be true, regardless of the choice of premises.
- A logical necessity or logical truth is a statement that is true under any possible interpretation.
- A tautology or tautological truth ( ⊨ ) is a statement that is true regardless of the truth values of the premises.
- Axiom: The logical complement of a tautology is a contradiction.
- A vacuous truth is a logically valid statement that is devoid of content because it asserts something about all members of a class or set that is empty.
- Theorem: The negation of a tautology can never be proved without a contradiction.
Assume that x ≡ y is a tautology. Because, by definition, this statement must be true regardless of whether x is true or y is true then it is impossible to choose any premise for which the statement will be false. Therefore, we can never state/prove the negation (i.e., falsity) of a tautology. Since a logical tautology and a contradiction are logical complements of each other, the negation of a tautology can never be stated.
- Theorem: Every tautology is also a logical necessity.
Because a tautology is true regardless of our choice of premises, it must also be a logical necessity since it is always true.
- Theorem: Not every logical necessity is also a tautology.
It is a logical necessity that the Set of All Sets must contain itself as a member. However, such a set must also contain the Set of All Sets That Do Not Contain Themselves as Members, resulting in a contradiction. Therefore, although the premise is a logical necessity, it is not tautological because the conclusion is false.
- A logical object is tautologically equivalent ( ≡ or := ) to another logical object if both always result in the same truth values, given the truth values of all possible premises.
- Theorem: Every tautological equivalence is also a logical necessity.
Because a tautology is true regardless of our choice of premises, it must also be a logical necessity since it is always true. Because two statements that are equivalent must also share the same truth value, where the premises must be true then the equivalent statements must also be true.
- Theorem: Not every logical necessity is also a tautological equivalence.
It is a logical necessity that the Set of All Sets must contain itself as a member. However, such a set must also contain the Set of All Sets That Do Not Contain Themselves as Members, resulting in a contradiction. Therefore, although the premise is a logical necessity, it is not tautological because the conclusion is false. Since a logical necessity can result in a false conclusion, and because a tautological equivalence can only have a true result, not every logical necessity is also a tautological equivalence.
- A logical object that must be false if any necessary premise is also false, and must be true if any sufficient premise is also true, is called a logical consequence or logical implication ( → ).
- A logical object that is always true because all possible antecedent premises are also always true is called a tautological consequence or tautological implication.
- Theorem: Every tautological consequence is also a logical consequence.
Since all possible antecedent are true for a tautological consequence, any necessary or sufficient conditions must also be true, which means that the consequence must also be true. Therefore any tautological consequence also satisfies the requirements of a logical consequence.
- Theorem: Not every logical consequence is also a tautological consequence.
A logical consequence may be false whenever a necessary condition is also false. A tautological consequence, on the other hand, can never be false because no antecedent condition may ever be false. Therefore, not every logical consequence is also a tautological consequence.
Example: A canine may or may not be a dog.
Example: A dog is not a dog.
Example: x ≡ x.
Example: "All cell phones in the room are turned off. Therefore, we will not hear any telephones ringing during the performance." If there were, in fact, no cell phones in the room to turn on at the time the argument was made then this argument states a vacuous truth.
The Logical Operators
An operator is a term or symbol that represents an inference.
An operand is any antecedent to which an operator is applied.
An operation is a combination of an operator and the operands on which the operator is applied that results in some consequence, also known as a result or solution.
A definition that contains an additional term or symbol in the definiendum that is also contained in a definiens, and where the definiendum does not appear in the definens, is a recursive definition.
An operation or inference that contains an additional term or symbol in the result that is also contained in one or more operands, and where the result does not appear as a term in any operand, is a recursive operation or inference.
Theorem: The Logical Validity of Recursion - Whereas the evaluation of a term stated in a conclusion might logically depend on an evaluation of the same term stated in an antecedent (e.g., x → [x → y], where x is known or assumed and y is unknown and unassumed), in a recursive inference the term in the antecedent does not depend for its truth on a proof of the statement's conclusion. This must be distinguished from a circular argument, which is not logically valid because, in such an argument, the evaluation of an antecedent term logically depends on an evaluation of the argument's conclusion (e.g., [x → y] → y, where x is known or assumed and y is unknown and unassumed).
Example: "If a duck has wings then it is a duck that can fly." Because we have assumed the fact that a duck exists as one of the premises, there is nothing wrong with including the fact of the duck within the conclusion - whether the duck can fly. This argument is not circular because it does not entirely assume the truth of the conclusion by assuming the truth of a premise but, instead, assumes a premise as part of the conclusion that is proved to be true. However, the statement: "if a duck has wings then a duck exists" is circular and logically invalid because the truth of the conclusion is entirely dependent on the truth of one of the statement's assumptions.
A negation ("not") operator ( ¬, ~, !, N, \, or a bar over the term ) states the opposite truth condition of an operand statement.
- A conditional operator ( → ), conditional statement ("if . . . then" statement, "because," "since"), or linguistic implication expresses an inference where a consequence follows from an antecedent or antecedents.
Example: If the animal has feathers then it is a bird.
- A conditional statement expresses a necessary inferential relationship or condition where the presence of the consequence implies the presence of the antecedent, but where the presence of the antecedent does not imply that the consequence will occur, hence with some ambiguity in the result.
- A conditional statement expresses a sufficient inferential relationship or condition where the presence of the antecedent implies the presence of the consequence, but where the presence of the consequence does not imply the presence of the antecedent, hence with some ambiguity in the cause.
- If the validity or invalidity of an antecedent is a necessary or sufficient condition for the occurrence of a consequence then the antecedent supports, implies, entails or is relevant to or evidence of (i.e., "tends to prove or disprove") the consequence, otherwise it is irrelevant.
- An inferential relationship is causal or material ( → ) if an unambiguous consequence or set of consequences necessarily result from the occurrence of a sufficient antecedent or set of antecedents.
- An inferential relationship is non-causal or immaterial if a consequence or set of consequences is ambiguous or does not necessarily result from the occurrence of an antecedent or antecedents.
- Theorem: Every term that is material is also relevant.
Proof: Since a material term must, by definition, be a sufficient antecedent for the truth of the consequent, it also satisfies the definition of a relevant antecedent, which can be either sufficient or necessary. Therefore, every material term is also a relevant term.
- Theorem: Not every relevant term is also material.
Proof: A relevant term may be a sufficient or necessary antecedent for the truth of the consequence. However, a material term, by definition may only be a sufficient term, not merely a necessary term, for the truth of the consequence. Therefore, not every relevant term is also a material term, as is the case with relevant necessary terms.
- A material implication, logical implication, or logical consequence ( → ) occurs where a conditional statement is false if and only if the antecedent is true and the consequence is false.
- Where a consequence is true if and only if both the antecedent and consequence are true, and false if and only if both are false, and both results are unambiguous, then the truth of the antecedent is a necessary and sufficient condition for the truth of the consequence, and vice-versa. This inferential relationship is also known as a bidirectional conditional statement ( ↔ ) and it is an alternative definition of logical equivalence (≡).
Example: If X is not legally an adult then X is not over the age of eighteen.
Example: If X is over the age of eighteen then X is legally an adult.
- A sound argument is an argument that contains both valid inferential claims and valid factual claims.
An unsound argument is an argument that has either an invalid inferential claim or an invalid factual claim, or both.
An induction is a predictive argument where a conclusion is arrived at by reasoning from the parts to a whole, from particulars to generals, or from the individual to the universal.
A syllogism is a deduction that is composed of at least two premises (the major and minor premises) and one conclusion, where each premise is stated in relation to the other premises so as to infer the conclusion.
Example: All birds have feathers. (the major premise) An ostrich is a kind of bird. (the minor premise) Therefore, an ostrich has feathers. (the conclusion)
The major term of a syllogism is shared by the major premise and the predicate of the conclusion.
The minor term of a syllogism is shared by the minor premise and the subject of the conclusion.
The middle term of a syllogism is shared by the major and minor premises.
A polysyllogism (also called multi-premise syllogism, climax, or gradatio) is a set of any number of syllogisms such that the conclusion of one is a premise for another. Each constituent syllogism is called a prosyllogism except the very last, because the conclusion of the last syllogism is not a premise for another syllogism.
Example: If one argues that a given number of grains of sand do not make a heap, and that an additional grain does not either, then to conclude that no additional amount of sand will make a heap is to construct a polysyllogism.
A sorites argument is a particular form of polysyllogism in which a set of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of another until the subject of the first is joined with the predicate of the last in the conclusion.
A theory or system is a class or set of theorems that prove an hypothesis.
A system or theory is completely/fully/entirely axiomatized when every aspect of the system or theory may ultimately be derived from a known set of axioms.
The scientific method is the expression of empirical evidence and inductive logic to posit the existence of universal laws or theories, called models, and the use of empirical evidence and deductive logic to test a model and thereby prove or disprove the truthfulness of the posited universal laws.
Example: If A then B; If B then C; If C then D; If D then E; Therefore, if A then E.
Manipulation of Terms
- The positive or original form of a term, statement, proposition, or argument is identical to the term, statement, proposition, or argument itself.
Conversion occurs when the subject and predicate terms of an inferential statement replace each other. The resulting statement is called the converse of the original statement. A statement and its converse are not logically equivalent statements unless the inferential relationship is bidirectional.
Example: If X is over the age of eighteen then X is legally an adult. Positive Statement If X is legally an adult then X is over the age of eighteen. Converse Statement
Contraposition occurs due to the complementation of the truth values of both the subject and predicate terms of an inferential statement and then their replacement of one term for the other. The resulting statement is called the contra-positive of the original statement.
Example: If X is over the age of eighteen then X is legally an adult. Positive Statement If X is not legally an adult then X is not over the age of eighteen. Contrapositive Statement
The inversion of an inferential statement occurs due to the complementation of the truth values of both the subject and predicate terms without substituting those terms for each other. The resulting statement is called the inverse of the original statement and, by definition, inversion necessarily contradicts the truth value of the original statement.
Example: If X is over the age of eighteen then X is legally an adult. Positive Statement If X is not over the age of eighteen then X is not legally an adult. Inverse Statement
The obversion of an inferential statement occurs due to the complementation of the truth value of the predicate term of the statement. The resulting statement is called the obverse of the original statement. Because of the complementation of the predicate term, obversion necessarily contradicts the truth value of the original statement since the predicate term of the obverse will always be the logical inverse of of the predicate term of the original statement.
Example: If X is over the age of eighteen then X is legally an adult. Positive Statement If X is over the age of eighteen then X is not legally an adult. Obverse Statement
Theorem: The Logical Equivalence of the Contraposition of Statements: The contraposition of the terms of an inferential statement creates a logically equivalent statement.
positive → p qp → q T T T F T T T F F F F T
contraposition → p q¬q → ¬p F F T F T T T F F T T T
Theorem: The Logical Equivalence of the Conversion and Inversion of Statements: The conversion of the terms of an inferential statement creates a statement that is logically equivalent to the inversion of the terms of that statement, and vice-versa.
conversion → p qq → p T T T T F F F T T F F T
inversion → p q¬p → ¬q F F T T F F F T T T T T
Theorem: The Principle of Bidirectionality: Because the subject and predicate terms of a bidirectional conditional statement are both necessary and sufficient conditions of each other, the subject and predicate terms are logically synonymous and either the contraposition, conversion, or inversion of the terms of an inferential statement are logically equivalent to the original, but not the obversion.
positive ↔ p qp ↔ q T T T F T F T F F F F T
conversion ↔ p qq ↔ p T T T T F F F T F F F T
contraposition ↔ p q¬q ↔ ¬p F F T F T F T F F T T T
inversion ↔ p q¬p ↔ ¬q F F T T F F F T F T T T
obversion ↔ p qp ↔ ¬q T F F F F T T T T F T F
The Classical Inferences
- Modus Ponens (Modus Ponendo Ponens or Implication Elimination) (A→B; A ∴ B)
- Modus Tollens (Modus Tollendo Tollens or Denying the Consequent) (A→B; ¬B ∴ ¬A)
- Modus Ponendo Tollens (¬[A∧B]; A ∴ ¬B)
- Conjunction Introduction (A & B ∴ A∧B)
- Simplification (A∧B ∴ A)
- Disjunction Introduction (A ∴ A∨B)
- Disjunction Elimination ([A∨B] & [A→C] & [B→C] ∴ C)
- Disjunctive Syllogism ([A∨B] & ¬A ∴ B)
- Hypothetical Syllogism (The Theory of Consequences or Logical Transitivity) ([A→B] & [B→C] ∴ A→C)
- Constructive Dilemma ([A→P] & [B→Q] & [A∨B] ∴ P∨Q)
- Destructive Dilemma ([A→P] & [B→Q] & [¬P∨¬Q] ∴ ¬A∨¬B)
- Biconditional Introduction ([A→B] & [B→A] ∴ A↔B)
- Biconditional Elimination (A↔B ∴ A→B)
This argument requires two premises. The first premise is the conditional ("if-then") claim, namely that A implies B. The second premise is that A, the antecedent of the conditional claim, is true. From these two premises it can be logically deduced that B, the consequent of the conditional claim, must be true as well. Modus ponens is both self-consistent and intuitively obvious. Therefore, we consider modus ponens to be axiomatic.
If a bird quacks then it is a duck.
A certain bird is quacking.
Therefore, the bird must be a duck.
This argument requires two premises. The first premise is the conditional ("if-then") claim, namely that A implies B. The second premise is that B is false. From these two premises, it can be logically deduced that A must be false.
If a bird quacks then it is a duck.
A certain bird is not a duck.
Therefore, the bird must not be quacking.
The proof for this proposition lies in the realization that modus tollens is simply stating the contra-position of the original statement which, we have seen earlier in this outline, always preserves the truth of the original statement.
If the negation of a conjunction and a conjunction of the negation with one of its conjuncts hold true, then the negation of its other conjunct also holds true.
We are told it is not true that a certain bird both quacks and is also a sparrow.
We then find that the bird does quack.
Therefore, the bird cannot be a sparrow.
The proof for this proposition can be seen from an evaluation of the sufficiency requirement for the antecedent, where both conditions of the antecedent cannot be true in order for the consequence to also be true. Therefore, where one of the antecedent conditions is true, the other antecedent must be false since otherwise both of the antecedents would be true, which would violate the sufficiency requirement of the truth condition for the antecedent.
If A is true, and B is true, then the conjunction of A and B is true.
If the bird quacks,
and if the bird is also a duck,
Then it is true that the bird both quacks and is also a duck.
This proposition is essentially axiomatic. If we have the presence of both antecedent conditions then, by definition, we have their conjunction since both conditions are present.
If the conjunction of A and B is true, then A must be true.
The bird both quacks and is a duck.
Therefore, it is true that the bird must quack.
(It is also true that the bird must be a duck.)
Simplification states that, where the disjunction of two antecedent conditions is permitted, the presence of either (or both) of those conditions satisfies the truth condition for the consequence. The utility of this proposition is readily observable from the symbolic statement of this proposition: it allows us to eliminate either term, at our discretion, and thus simply the statement by reducing the number of operative terms necessary to prove the truth of the proposition.
This argument has one premise, A, and an unrelated proposition, B. From the assumed truth of the premise, A, it can be logically concluded that either A or B is true, since we at least know the former.
The bird is quacking.
Therefore, it is true that the bird quacks or it is a duck (or both), since we know that it does indeed quack.
Disjunction introduction is essentially the converse statement of simplification, which itself is a bidirectional implication. Since either antecedent condition may be present for the sufficiency of the antecedent to be true then either or both terms may be present and the consequence will necesssarily be true. The utility of this proposition lies in the fact that it permits us to add additional desired terms to a logical proposition without contradiction so long as the relationship between the added terms is disjunctive (i.e., both antecedent terms need not be true in order for the consequence to be true so long as at least one of those antecedent terms is true).
If A or B is true, and A entails C, and B entails C, then we may justifiably infer C.
For all birds, either it quacks or it is a duck, or both.
If a bird quacks then we know it is a kind of water fowl.
If a bird is a duck then it is also a kind of water fowl.
Since all birds either quack or are ducks then all birds must be water fowl.
Here, the truth of the proposition lies in the fact that either of the first two antecedent conditions imply the truth of the third antecedent condition. Therefore, where either of those antecedent conditions is present (i.e., the disjunction of those first two terms), the third antecedent must also be present by reason of modus ponens.
In this argument, the disjunction tells us that at least one of two statements (A or B) is true. Then we are told that A is not true. Therefore, we must infer that B must be true.
For all birds, either it quacks or it is a duck, or both.
The bird does not quack.
Therefore, the bird must at least be a duck.
This proposition is similar to disjunction elimination in that we eliminate the disjunction but this time by way of modus tollens, rather than by modus ponens. Since either (or both) of the two antecedent conditions must be true for the consequence to be true, if one of those conditions is not true then the other must be true in order to preserve the truthfulness of the proposition.
This rule states the commonly known principle of transitivity. If A implies B, and B implies C, then A implies C.
If a bird quacks then it is a duck.
If a bird is a duck then it is water fowl.
Therefore, if the bird quacks then it is water fowl.
Beside modus ponens and modus tollens, this is probably the most important and useful of the classical inferences. It is transitivity that allows the causal linking of two antecedent conditions with a third so as to deductively prove a causal relationship between the first and last antecedent conditions in a series, which can be as long as the number of antecedents conditions used in the proposition. Its proof lies in the fact that each antecedent condition is the material cause of the next so that the "chain" of cause and effect ultimately links the first and last conditions via the chain of the others.
If two conditionals are true, and at least one of their antecedents is in fact true, then at least one of their consequents must also be true.
If a bird sings then it is a song bird.
If a bird quacks then it is a duck.
Either the bird is singing or it is quacking.
Therefore, the bird is either a song bird or it is a duck.
Here we have a syllogistic linking of terms without creating a chain of causality as we did in the hypothetical syllogism. The disjunction of the antecedent conditions for each material implication ensures that there is no "chaining" of all the antecedent conditions. However, the disjunction also ensures that, where either or both of the conditions that are antecedent to the two material implications are true then one or both of the consequences must also be true.
The destructive dilemma is the disjunctive version of modus tollens. The disjunctive version of modus ponens is the constructive dilemma.
If a bird sings then it is a song bird.
If a bird quacks then it is a duck.
Either the bird is not a song bird or it is not a duck.
Therefore either the bird is not singing or it is not quacking.
If B follows from A, and A follows from B, then A if and only if B.
If the bird quacks then it is a duck.
If the bird is a duck then it quacks.
Therefore, the bird quacks if and only if it is a duck.
This is essentially the very definition of bidirectionality and is an example of proof by definition where the definition is itself well-founded.
If ( A ↔ B ) is true then one may infer either direction of the biconditional - i.e., ( A → B ) and/or ( B → A ).
A bird quacks if and only if it is a duck.
Therefore, if a bird is quacking then it is a duck, and if a bird is a duck then it also quacks.
Every bidirectional implication may be constructed from two material implications that state the inferential relationship of necessary and sufficient conditions in each direction because, in the case of bidirectionality, both antecedents are both necessary and sufficient conditions, thereby permitting the bidirectionality to occur. Therefore, where bidirectionality exists then either or both of the component material implications can be essentially disjoined.
Forms of Logical Argument
Mathematical Argument (deductive or strong inductive): An argument in which the exclusive means of inferring a conclusion occurs by way of theorizing the logical relationships of quantitative operations.
Argument by Definition: An argument in which the exclusive means of inferring the conclusion occurs by way of the definition of some word or phrase.
Universal Instantiation: An argument wherein an inference is made from a truth about all members of a class of individuals to a truth about a particular individual of that class.
Universal Generalization: An argument wherein an inference is made from a truth about a particular member of a class of individuals to a truth about all members of that class.
Example: All birds have feathers. An ostrich is a bird. An ostrich has feathers.
Categorical Syllogism (Categorical Argument): A syllogism in which all the statements are categorical propositions.
Example: All birds have feathers. All ostriches are birds. All ostriches have feathers.
Hypothetical Syllogism (Hypothetical Argument): A syllogism having a conditional statement for at least one of its premises.
Example: All birds have feathers. If an ostrich exists then it is a kind of bird. All ostriches have feathers.
Disjunctive Syllogism: A syllogism having a disjunctive statement for at least one of its premises.
Example: Either a penguin has feathers or it has skin. All birds have feathers. A penguin is a kind of bird. Therefore, a penguin has feathers.
(Weak) Inductive Argument (Generalization): An argument where the conclusion is arrived at by reasoning from a part to a whole, from particulars to generals, or from the individual to the universal. With the exception of mathematical induction, inductive arguments are generally weak.
Prediction: An argument where one or more premises state a known or knowable proposition occurring in the present or the past but where the conclusion states a proposition that is inferred to occur in the future.
Argument Based on Signs: A kind of weak inductive prediction in which the stated signs (indications) are inferred to be predictive of the stated conclusion. A statistical syllogism occurs where this prediction is based on a statistical, mathematical model.
Argument by Analogy: An argument that depends on the inferred similarity between two or more propositions.
Argument by Authority: An argument in which the conclusion is inferred from a statement by a presumed authority.
Causal Inference: An argument in which the knowledge of an effect is inferred from the knowledge of a cause, or vice versa.
Example 1: I left the ice cream cone on the hot pavement. Therefore, it is probably melted.
Example 2: The ice cream cone is melted. This is probably because I left it on the hot pavement.
Plausability Argument (Educated Guess): An argument that hypothesizes a theory based on experience and similar results in analogous circumstances.
The quality of a categorical proposition refers to whether the proposition affirms or denies the inclusion of an object to the class of the predicate. The two qualities of a categorical proposition are therefore either affirmation or negation of the copula.
The quantity of a categorical proposition refers to the amount of objects in one class that are included in another class. The three possible quantities of a categorical proposition in informal verbal logic are either all, some, or none.
The distribution of a categorical proposition refers to the logically permissible inferences that may be drawn from a particular combination of quality and quantity terms.
Types of Categorical Propositions
The Universal Affirmative (Latin mnemonic A)
General Statement: All S are P. ("All" is the quantifier; S is the subject; "are" is the copula; P is the predicate.)
Distribution: only the subject term is distributed (the predicate term is not distributed).
Example: All dogs are mammals, but not all mammals are dogs.
The Universal Negative (Latin mnemonic E)
General Statement: No S is P.
Distribution: all objects in the subject and predicate terms distribute bidirectionally.
Example: No beetles are dogs and no dogs are beetles.
The Particular Affirmative (Latin mnemonic I)
General Statement: Some S are P.
Distribution: neither term is entirely distributable in the other term.
Example: Some flowers have scent (it is not possible to say that all flowers have scent or that all scent comes from flowers).
The Particular Negative (Latin mnemonic O)
General Statement: Some S are not P.
Distribution: only the predicate term is distributed (the subject term is not distributed).
Example: Some mammals are not dogs, but all dogs are mammals.
The Square of Opposition
The modern Square of Opposition states that corresponding universal affirmatives and universal negatives always necessarily contradict each other, and that corresponding particular affirmatives and particular negatives always necessarily contradict each other.
Classification of Categorical Syllogisms
Because, by the definition of a syllogism, S is the subject of both the minor premise and of the conclusion, P is the predicate of both the major premise and of the conclusion, and M is the middle term, the major premise links M with P, and the minor premise links M with S. However, the middle term can be either the subject or the predicate of each premise in which it appears. The logical equivalence of the middle term in each premise gives rise to the four basic types of syllogisms, known as The Four Figures:
Therefore, if S then P.
If M then P.
If S then M.
Therefore, if S then P.
If P then M.
If S then M.
Therefore, if S then P.
If M then P.
If M then S.
Therefore, if S then P.
If P then M.
If M then S.
Identification of Categorical Syllogisms
Because there are four types of categorical proposition within each syllogism, and there are four types of categorical syllogisms, there are four to the fourth power (256) combinations of possible syllogisms. Each syllogism can be identified by combining the Latin mnemonic for each type (class) of categorical proposition for each premise, followed by the number of the figure of the syllogism structure.
Example: All dogs are mammals. Universal Affirmative - I No beetles are dogs. Universal Negative - A Therefore, no beetles are mammals. (Universal Negative - A)
Therefore, this syllogism can be identified as type IAA-1.
Validity of Categorical Syllogisms
Although there are 256 possible combinations of terms in categorical syllogisms, only some of those syllogisms are logically valid. During medieval times, the valid combinations were given the following designations:
Barbara (AAA-1) Example: All men are animals. All animals are mortal. Therefore, all men are mortal.
Celarent (EAE-1) Example: No reptiles have fur. All snakes are reptiles. Therefore, no snakes have fur.
Darii (AII-1) Example: All kittens are playful. Some pets are kittens. Therefore, some pets are playful.
Ferio (EIO-1) Example: No homework is fun. Some reading is homework. Therefore, some reading is not fun.
Cesare (EAE-2) Example: No healthy food is fattening. All cakes are fattening. Therefore, no cakes are healthy.
Camestres (AEE-2) Example: All horses have hooves. No humans have hooves. Therefore, no humans are horses.
Festino (EIO-2) Example: No lazy people pass exams. Some students pass exams. Therefore, some students are not lazy.
Baroco (AOO-2) Example: All informative things are useful. Some websites are not useful. Therefore, some websites are not informative.
Darapti (AAI-3) Example: All fruit is nutritious. All fruit is tasty. Therefore, some tasty things are nutritious.
Disamis (IAI-3) Example: Some mugs are beautiful. All mugs are useful. Therefore, some useful things are beautiful.
Datisi (AII-3) Example: All the industrious boys in this school have red hair. Some of the industrious boys in this school are boarders. Therefore, some boarders in this school have red hair.
Felapton (EAO-3) Example: No jug in this cupboard is new. All jugs in this cupboard are cracked. Therefore, some of the cracked items in this cupboard are not new.
Bocardo (OAO-3) Example: Some cats have no tails. All cats are mammals. Therefore, some mammals have no tails.
Ferison (EIO-3) Example: No tree is edible. Some trees are green. Therefore, some green things are not edible.
Bramantip (AAI-4) Example: All apples in my garden are wholesome. All wholesome fruit is ripe. Therefore, some ripe fruit are apples in my garden.
Camenes (AEE-4) Example: All coloured flowers are scented. No scented flowers are grown indoors. Therefore, no flowers grown indoors are coloured.
Dimaris (IAI-4) Example: Some small birds live on honey. All birds that live on honey are colourful. Therefore, some colourful birds are small.
Fesapo (EAO-4) Example: No humans are perfect. All perfect creatures are mythical. Therefore, some mythical creatures are not human.
Fresison (EIO-4) Example: No competent people are people who always make mistakes. Some people who always make mistakes are people who work here. Therefore, some people who work here are not competent people.
Occurring in Any Syllogism Type
Fallacy of Four Terms (Syllogistic Equivocation)
Also called quaternio terminorum, this fallacy occurs when a categorical syllogism has four terms but only two premises.
Valid categorical syllogisms always have no more than three terms for two premises.
|Example:||All fish have fins.||(major premise)|
|All goldfish are fish.||(minor premise)|
|All goldfish have fins.||(conclusion)|
Here, the three terms are: "goldfish," "fish," and "fins." However, using four terms and only two premises invalidates the syllogism.
|Example:||All fish have fins.||(major premise)|
|All goldfish are fish.||(minor premise)|
|All humans have fins.||(conclusion)|
The premises don't connect "humans" with "fins," so the reasoning is invalid.
Occurring in Categorical Syllogisms
- Affirmative Conclusion from a Negative Premise
This fallacy occurs when a categorical syllogism has a positive conclusion but one or more negative premises. For example: No fish are dogs, and no dogs can fly, therefore all fish can fly. This is a fallacy because any valid form of categorical syllogism that asserts a negative premise must have a negative conclusion.
- Existential Fallacy
In traditional Aristotelian logic, this fallacy occurs where a categorical syllogism has two universal premises and a particular conclusion. In other words, for the conclusion to be true, at least one member of the class must exist, but the premises do not establish this condition. In modern logic, the existential fallacy is obviated by the use of conditional premises. Example #1: All inhabitants of other planets are friendly, and all Martians are inhabitants of another planet; therefore, there are friendly Martians. The conclusion assumes the existence of Martians, the factual invalidity of which can be rectified by using the premise if there are inhabitants of other planets then they are friendly. Example #2: All unicorns are animals; therefore, some animals are unicorns. The conclusion assumes the existence of unicorns, the factual invalidity of which can be rectified by using the conditional premise if unicorns exist then they are animals.
- Fallacy of Exclusive Premises
This fallacy occurs when both of the premises of a categorical syllogism are negative. Example: No mammals are fish. Some fish are not whales. Therefore, some whales are not mammals. This syllogism is not valid because at least one premise of any given syllogism must be affirmative.
- Fallacy of the Undistributed Middle
This fallacy occurs when the middle term in a categorical syllogism isn't distributed. Example: All Zs are Bs. Y is a B. Therefore, Y is a Z. It mayor may not be the case that "all Zs are Bs," but in either case it is irrelevant to the conclusion. What is relevant to the conclusion is whether it is true that "all Bs are Zs," which is ignored in the argument. The fallacy is similar to affirming the consequent and denying the antecedent in that, if the terms were swapped around in either the conclusion or the first co premise, there would no longer be a fallacy.
- Illicit Major Premise
This fallacy occurs in a categorical syllogism that is invalid because its major term is undistributed in the major premise but distributed in the conclusion. Example: All dogs are mammals. No cats are dogs. Therefore, no cats are mammals. In this argument, the major term is "mammals." This term is distributed in the conclusion because we are making a claim about a property of all mammals: that they are not cats. However, it is not distributed in the major premise (the first statement) where we are only talking about a property of some mammals: only some mammals are dogs. This error occurs because we are assuming that the converse of the first statement (that all mammals are dogs) is also true, which is never logically possible.
- Illicit Minor Premise
This fallacy occurs in a categorical syllogism that is invalid because its minor term is undistributed in the minor premise but distributed in the conclusion. This fallacy has the following argument form: All A are B. All A are C. Therefore, all C are B. Example: All cats are felines. All cats are mammals. Therefore, all mammals are felines. The minor term here is "mammal," which is not distributed in the minor premise "All cats are mammals" because this premise is only defining a property of possibly some mammals (i.e., that they're cats.) However, in the conclusion "all mammals are felines," mammal is distributed by stating that all mammals will be found to be felines. It is shown to be false by any mammal that is not a feline; for example, a dog.
- Fallacy of Necessity
This fallacy occurs when a degree of unwarranted necessity is asserted in the conclusion. Example: Bachelors are necessarily unmarried. John is a bachelor. Therefore, John cannot marry. The major premise (bachelors are necessarily married) is a tautology and therefore valid on its face. The minor premise (John is a bachelor) is a statement of fact about John which makes him subject to the major premise; that is, the minor premise declares John a bachelor, and the major premise states that all bachelors are unmarried. Because the conclusion presumes the minor premise will be valid in every case, this presumption creates a fallacy of necessity. John, of course, is always free to stop being a bachelor, simply by getting married; if he does so, the minor premise is no longer true and thus not subject to the tautology of the major premise. In this case, the conclusion has an unwarranted necessity by assuming, incorrectly, that John cannot stop being a bachelor.
Occurring in Disjunctive Syllogisms
- Affirming a (non exclusive) Disjunct
Also known as the fallacy of the alternative disjunct, this fallacy occurs when a deductive argument takes either of the two following forms: A or B A Therefore, it is not the case that B A or B B Therefore, it is not the case that A The fallacy lies in concluding that one disjunct must be false because the other disjunct is true; in fact they may both be true (unless an exclusive disjunctive forms the major premise). A similar form that is valid has the second premise (rather than the conclusion) be a negation. The valid form is known as disjunctive syllogism. The following argument is a clear case of this fallacy: It will rain somewhere tomorrow or the sun will shine somewhere tomorrow. It will rain somewhere tomorrow. (It will rain here according to the weather forecast). Therefore, it is not the case that the sun will shine somewhere tomorrow. This inference is obviously invalid. The sun is almost always shining somewhere on earth.Both of the premises are clearly true while the conclusion is clearly false. The following example is trickier: Two is an even number or two is an odd number. Two is an even number. Therefore, it is not the case that two is an odd number. This argument seems to be valid because there is another use of the word "or" in ordinary language that would seem more appropriate. If the disjunction is exclusive, that is to say, the "or" implies that only one of the disjuncts is perfectly true, then the argument is valid. However, the meaning of"or" used in ordinary language is different than its use in informal verbal logic where it is defined as an operator that avoids equivocation, and therefore this argument is invalid. In this case, the "or" is said to be inclusive, in that it stipulates that one or both of the disjuncts is true. A similar argument that is in fact valid will have the implied assumption explicitly stated, as follows: Two is an even number or two is an odd number. Two is an even number. No number can be both even and odd. Therefore, it is not the case that two is an odd number.
Occurring in Statistical Syllogisms
Also called destroying the exception or dicto simpliciter ad dictum secundum quid, this fallacy occurs in statistical syllogisms (a kind of argument based on a generalization) when an exception to the generalization is ignored. It is one of the thirteen fallacies originally identified by Aristotle. The fallacy occurs when one attempts to apply a general rule to an irrelevant situation. Related inductive fallacies include the overwhelming exception and the hasty generalization. For instance: Cutting people with a knife is a crime. Surgeons cut people with knives. Surgeons are criminals. It is easy to construct fallacious arguments by applying general statements to specific incidents that are obviously exceptions. Generalizations that are weak generally have more exceptions(the number of exceptions to the generalization need not be a minority of cases) and vice versa. This fallacy may occur when we confuse particular generalizations ("some") for universal categorical statements ("always and everywhere"). It may be encouraged when no qualifying words like"some," "many," "rarely," etc., are used to mark the generalization. For example: All Germans were Nazis. The premise above could be used in an argument concluding that all Germans or current Germans should beheld responsible for the crimes of the Nazis. Qualifying the first term corrects the argument: Some Germans were Nazis. This premise makes the weakness of the generalization more obvious, rather than appearing to be the statement of a categorical rule.
- Converse Accident
Also called reverse accident, destroying the exception or dicto secundum quid ad dictum simpliciter, this fallacy is the deductive version of the hasty generalization, and occurs in a statistical syllogism when an exception to a generalization is wrongly called for. For example: Every swan I have seen is white, so it must be true that all swans are white. This fallacy is similar to the slippery slope, where the opposition claims that if a restricted action under debate is allowed (i.e., allowing people with glaucoma to use medical marijuana) then the action will by stages become acceptable in general (i.e., eventually everyone will be allowed to use marijuana). The two arguments imply there is no difference between the exception and the rule and, in fact, fallacious slippery slope arguments often use the converse accident to the contrary as the basis for the argument. However, a key difference between the two is the point and position being argued. The above argument using converse accident is an argument for full legal use of marijuana given that glaucoma patients use it. The argument based on the slippery slope argues against medicinal use of marijuana because it will lead to full use. Whereas a slippery slope argument is not necessarily fallacious, a converse accident is always a formal fallacy.
Methods of Proof
Direct Proof (Proof of Consequence or Entailment)
Given any set of one or more propositions, statements, or arguments, the set will entail another proposition, statement, or argument if the conjunction of the elements of the set is inconsistent with the negation of any of the set's elements [(Γ → γ) ↔ ¬(θ ∧ ¬Ψ)].
Our theory tells us that all ducks have webbed feet.
We look for a duck without webbed feet.
However, every duck we observe has webbed feet.
Without a contradiction, our theory holds.
Indirect Proof (Proof of Non-Consequence or Disentailment)
Indirect Proof, Disentailment, Proof by Contradiction ("counter-example"), Reductio Ad Impossible, or Reductio Ad Absurdum occurs when, by assuming the negation of a proposition, statement, or argument, another proposition, statement, or argument that would otherwise logically follow is shown to be contradicted [¬P → (Q ∧ ¬Q)].
We assume, for the sake of argument, if a bird quacks then it is not a duck.
We have found a duck and it is quacking.
However, according to our definition, a quacking bird is not a duck.
The contradiction proves that the assumption does not hold in all cases.
Therefore, some ducks do quack.
Proof by Transposition (Proof by Contraposition)
Proof by transposition or proof by contraposition establishes the conclusion "if p then q" by proving the equivalent contrapositive statement "if not q then not p".
No birds quack except ducks.
Therefore, if the bird is not a duck then it does not quack.
Hence, if a bird quacks then it is a duck.
Non-Constructive Proof (Existence "Proof" or "Pure" Existence Theorem)
In a non-constructive proof, existence "proof," or "pure" existence theorem, we assume the non-existence of a thing whose existence is required to be proved and then deduce a logical contradiction without producing an empirical example. The non-existence of the thing has therefore been shown to be logically impossible, and yet an actual example of the thing has not been determined.
Mathematical Existence Theorems
Proof by Construction
Proof by Exhaustion
In proof by exhaustion, the conclusion is established by dividing it into a finite number of all possible cases and proving each one separately.
A probabilistic proof is one in which an example is shown to exist, with certainty, by using methods of Probability Theory. This is not to be confused with an argument that a theorem is 'probably' true.
A combinatorial proof establishes the equivalence of different expressions by showing that they count the same object in different ways.
In proof by mathematical induction, first a "base case" is proved, and then an "induction rule" is used to prove a (potentially infinite) series of other cases. Since the base case is true, the infinity of other cases must also be true, even if all of them cannot be proved directly because of their infinite number. As such, mathematical induction, also known as strong induction is in fact an application of deductive reasoning. For proof of the logical validity of mathematical induction, see Set Theory.
Fallacies of Relevance
Fallacies of relevance are attempts to prove a conclusion by offering considerations that simply don't bear on its truth. In order to prove that a conclusion is true, one must offer evidence that supports it. Arguments that commit fallacies of relevance don't do this; the considerations that they offer in support of their conclusion are irrelevant to determining whether that conclusion is true. The considerations offered by such are usually psychologically powerful, however, even if they don't have any evidentiary value, making such arguments appear to be persuasive, even if logically fallacious.
- Ad Hominem Attack
It is important to note that the label ad hominem is ambiguous, and that not every kind of ad hominem argument is fallacious. In one sense, an ad hominem argument is a valid argument in which the arguer offers premises that the arguer doesn't accept, but which the arguer knows the listener does accept, in order to show that his position is incoherent (as in, for example, the Euthyphro dilemma of Plato). There is nothing logically wrong with this type of ad hominem argument. The other type of ad hominem argument is a form of genetic fallacy. Arguments of this kind focus not on the evidence for a view but on the character of the person advancing it; they seek to discredit positions by discrediting those who hold them. It is always important to attack arguments, rather than arguers, and this is where ad hominems fall down. Example: William Dembski argues that modern biology supports the idea that there is an intelligent designer who created life. Dembski would say that because hes religious. Therefore, modern biology doesn't support intelligent design . This argument rejects the view that intelligent design is supported by modern science based on a remark about the person advancing the view, not by engaging with modern biology. It ignores the argument, focusing only on the arguer; it is therefore a fallacious argument ad hominem.
- Bandwagon Fallacy
The bandwagon fallacy is committed by arguments that appeal to the growing popularity of an idea as a reason for accepting it as true. They take the mere fact that an idea is suddenly attracting adherents as a reason for us to join in with the trend and become adherents of the idea themselves. This is a fallacy because there are many other features of ideas than truth that can lead to a rapid increase in popularity. Peer pressure, tangible benefits, or even mass stupidity could lead to a false idea being adopted by lots of people. A rise in the popularity of an idea, then, is no guarantee of its truth. The bandwagon fallacy is closely related to the appeal to popularity; the difference between the two is that the bandwagon fallacy places an emphasis on current fads and trends, on the growing support for an idea, whereas the appeal to popularity does not. Example: Increasingly, people are coming to believe that Eastern religions help us to get in touch with our true inner being. Therefore, Eastern religions help us to get in touch with our true inner being. This argument commits the bandwagon fallacy because it appeals to the mere fact that an idea is becoming more fashionable as evidence that the idea is true. Mere trends in thought are not reliable guides to truth, though; the fact that Eastern religions are becoming more fashionable does not logically imply that they are true.
- Fallacist's Fallacy
The fallacist's fallacy involves rejecting an idea as false simply because the argument offered for it is fallacious. Having examined the case for a particular point of view, and found it wanting, it can be tempting to conclude that the point of view is false. This, however, would be to go beyond the evidence. It is possible to offer a fallacious argument for any proposition, including those that are true. One could argue that 2+2=4 on the basis of an appeal to authority: "Simon Singh says that 2+2=4". Or one could argue that taking paracetamol relieves headaches using a post hoc: "I took the paracetamol and then my headache went away; it worked!" Each of these bad arguments has a true conclusion. A proposition therefore should not be dismissed because one argument offered in its favour is faulty.
- Fallacy of Composition
The fallacy of composition is the fallacy of inferring from the fact that every part of a whole has a given property that the whole also has that property. This pattern of argument is the reverse of that of the fallacy of division. It is not always fallacious, but we must be cautious in making inferences of this form. Examples: A clear case of the fallacy of composition is this: Every song on the album lasts less than an hour. Therefore, the album lasts less than an hour. Obviously, an album consisting of many short tracks may itself be very long. Not all arguments of this form are fallacious, however. Whether or not they are depends on what property is involved. Some properties, such as lasting less than an hour, may be possessed by every part of something but not by the thing itself. Others, such as being bigger than a bus, must be possessed by the whole if possessed by each part. One case where it is difficult to decide whether the fallacy of composition is committed concerns the cosmological argument for the existence of God. This argument takes the contingency of the Universe (i.e. the alleged fact that the universe might not have come into being) as implying the existence of a God who brought it into being. The simplest way to argue for the contingency of the Universe is to argue from the contingency of each of its parts, as follows: Everything in the Universe is contingent (i.e. could possibly have failed to exist). Therefore, the Universe as a whole is contingent (i.e. could possibly have failed to exist). It is clear that this argument has the form of the fallacy of composition; what is less clear is whether it really is fallacious. Must something composed of contingent parts itself be contingent? Or might it be that the universe is necessarily existent even though each of its parts is not? Another controversial example concerns materialistic explanations of consciousness. Is consciousness just electrical activity in the brain, as mind brain identity theory suggests, or something more? Opponents of mind brain identity theory sometimes argue as follows: The brain is composed of unconscious neurons. Therefore, the brain itself is not conscious. It is certainly difficult to see how consciousness can emerge from purely material processes, but the mere fact that each part of the brain is unconscious does not entail that the whole brain is the same.
- Fallacy of Division
The fallacy of division is the reverse of the fallacy of composition. It is committed by inferences from the fact that a whole has a property to the conclusion that a part of the whole also has that property. Like the fallacy of composition, this is only a fallacy for some properties; for others, it is a legitimate form of inference. Example: An example of an inference that certainly does commit the fallacy of division is this: Water is liquid. Therefore, H2O molecules are liquid. This argument, in attributing a macro property of water, liquidity, to its constituent parts, commits the fallacy of division. Though water is liquid, individual molecules are not. Note, however, an argument inferring from the fact that a computer is smaller than a car that every part of the computer is smaller than a car would not be fallacious; arguments with this logical form need not be problematic.
- Gambler's Fallacy
The gamblers fallacy is the fallacy of assuming that short term deviations from probability will be corrected in the short term. Faced with a series of events that are statistically unlikely, say, a serious of nine coin tosses that have landed heads up, it is very tempting to expect the next coin toss to land tails up. The past series of results, though, has no effect on the probability of the various possible outcomes of the next coin toss. Example: This coin has landed heads up nine times in a row. Therefore, it will probably land tails up next time it is tossed. This inference is an example of the gamblers fallacy . When a fair coin is tossed, the probability of it landing heads up is 50%, and the probability of it landing tails up is 50%. These probabilities are unaffected by the results of previous tosses. The gambler's fallacy appears to be a reasonable way of thinking because we know that a coin tossed ten times is very unlikely to land heads up every time. If we observe a tossed coin landing heads up nine times in a row we therefore infer that the unlikely sequence will not be continued, that next time the coin will land tails up. In fact, though, the probability of the coin landing heads up on the tenth toss is exactly the same as it was on the first toss. Past results don't bear on what will happen next.
- Genetic Fallacy
The genetic fallacy is committed when an idea is either accepted or rejected because of its source, rather than its merit. Even from bad things, good may come; we therefore ought not to reject an idea just because of where it comes from, as ad hominem arguments do. Equally, even good sources may sometimes produce bad results; accepting an idea because of the goodness of its source, as in appeals to authority, is therefore no better than rejecting an idea because of the badness of its source. Both types of argument are fallacious. Examples: My mommy told me that the tooth fairy is real. Therefore, the tooth fairy is real. Eugenics was pioneered in Germany during the war. Therefore, eugenics is a bad thing. Both of these arguments commit the genetic fallacy. Each judges an idea by the goodness or badness of its source, rather than on its own merits.
- Naturalistic Fallacy
Assume there are two fundamentally different types of statement: statements of fact which describe the way that the world is, and statements of value which describe the way that the world ought to be. The naturalistic fallacy is the alleged fallacy of inferring a statement of the latter kind from a statement of the former kind. To understand how this is so, consider arguments that introduce completely new terms in their conclusions. The argument, (1) All men are mortal, (2) Socrates is a man, therefore (3) Socrates is a philosopher is clearly invalid; the conclusion obviously doesn't follow from the premises. This is because the conclusion contains an idea that of being a philosopher that isn't contained in the premises; the premises say nothing about being a philosopher, and so cannot establish a conclusion about being a philosopher. Arguments that commit the naturalistic fallacy are arguably flawed in exactly the same way. An argument whose premises merely describe the way that the world is, but whose conclusion describes the way that the world ought to be, introduce a new term in the conclusion in just the same way as the above example. If the premises merely describe the way that the world is then they say nothing about the way that the world ought to be. Such factual premises cannot establish any value judgment; you cannot get an ought from an is. Examples: Feeling envy is only natural. Therefore, there's nothing wrong with feeling envy. This argument moves from a statement of fact to a value judgment, and therefore commits the naturalistic fallacy. The arguments premise simply describes the way that the world is, asserting that it is natural to feel envious. To describe the way that the world is, though, is to say nothing of the way that it ought to be. The arguments conclusion, then, which is value judgment, cannot be supported by its premises. It is important to note that much respectable moral argument commits the naturalistic fallacy. Whether arguments of the form described here are fallacious is controversial. If they are, then the vast majority of moral philosophy commits a basic logical error.
- Moralistic Fallacy
The moralistic fallacy is the opposite of the naturalistic fallacy. The naturalistic fallacy moves from descriptions of how things are to statements of how things ought to be, the moralistic fallacy does the reverse. The moralistic fallacy moves from statements about how things ought to be to statements about how things are; it assumes that the world is as it should be. This, sadly, is a fallacy; sometimes things arent as they ought to be. Examples: Have you ever crossed a one way street without looking in both directions? If you have, reasoning that people shouldn't be driving the wrong way up a one way street so there's no risk of being run over from that direction, then you've committed the moralistic fallacy. Sometimes things aren't as they ought to be. Sometimes people drive in directions that they shouldn't. The rules of the road don't necessarily describe actual driving practices.
- Red Herring Argument
The red herring is as much a debate tactic as it is a logical fallacy. It is a fallacy of distraction, and is committed when a listener attempts to divert an arguer from his argument by introducing another topic. This can be one of the most frustrating, and effective, fallacies to observe. The fallacy gets its name from fox hunting, specifically from the practice of using smoked herrings, which are red, to distract hounds from the scent of their quarry. Just as a hound may be prevented from catching a fox by distracting it with a red herring, so an arguer maybe prevented from proving his point by distracting him with a tangential issue. Example: Many of the fallacies of relevance can take red herring form. An appeal to pity, for example, can be used to distract from the issue at hand: You may think that he cheated on the test, but look at the poor little thing!How would he feel if you made him sit it again?
- Weak Analogy
Arguments by analogy rest on a comparison. Their logical structure is this: A and B are similar. A has a certain characteristic. Therefore, B must have that characteristic too. For example, William Paley's argument from design suggests that a watch and the universe are similar (both display order and complexity), and therefore infers from the fact that watches are the product of intelligent design that the universe must be a product of intelligent design too. An argument by analogy is only as strong as the comparison on which it rests. The weak analogy fallacy (or false analogy, or questionable analogy) is committed when the comparison is not strong enough. The example of an argument by analogy given above is controversial, but is arguably an example of a weak analogy. Are the similarities in the kind and degree of order exhibited by watches and the universe sufficient to support an inference to a similarity in their origins?
Irrelevant appeals attempt to sway the listener with information that, though it may be generally relevant, is not specifically irrelevant to the matter at hand. There are many different types of irrelevant appeal - i.e., many different ways of influencing what people think without using evidence. Each is a different type of fallacy of relevance.
- Appeal to Antiquity
An appeal to antiquity is the opposite of an appeal to novelty. Appeals to antiquity assume that older ideas are better, that the fact that an idea has been around for a while implies that it is true. This, of course, is not the case; old ideas can be bad ideas, and new ideas can be good ideas. We therefore cant learn anything about the truth of an idea just by considering how old it is. Example: Religion dates back many thousands of years (whereas atheism is a relatively recent development). Therefore, some form of religion is true. This argument is an appeal to antiquity because the only evidence that it offers in favor of religion is its age. There are many old ideas, of course, that are known to be false: e.g. that the Earth is flat, or that it is the still center of the solar system. It therefore could be the case that the premise of this argument is true (that religion is older than atheism) but that its conclusion is nevertheless false (that no religion is true). We need a lot more evidence about religion (or any other theory) than how old it is before we can be justified in accepting it as true. Appeals to antiquity are therefore fallacious.
- Appeal to Authority
An appeal to authority is an argument from the fact that a person judged to be an authority affirms a proposition to the claim that the proposition is true. Appeals to authority are always deductively fallacious; even a legitimate authority speaking on his area of expertise may affirm a falsehood, so no testimony of any authority is guaranteed to be true. However, the informal fallacy by way of induction occurs only when the authority cited either (a) is not an authority, or (b) is not an authority on the subject on which he is being cited. If someone either isn't an authority at all, or isn't an authority on the subject about which they are speaking, then that undermines the value of their testimony. Example: Marilyn vos Savant says that no philosopher has ever successfully resolved the problem of evil. Therefore, no philosopher has ever successfully resolved the problem of evil. This argument is fallacious because Marilyn vos Savant, though arguably an authority, is not an authority on the philosophy of religion. Her judgment that no philosopher has ever successfully resolved the problem of evil therefore carries little evidential weight; if there were a philosopher somewhere that had successfully resolved the problem then there's a good chance that Marilyn vos Savant wouldn't know about it. Her testimony is therefore insufficient to establish the conclusion of the argument.
- Appeal to Consequences
An appeal to consequences is an attempt to motivate belief with an appeal either to the good consequences of believing or the bad consequences of disbelieving. This may or may not involve an appeal to force. Such arguments are clearly fallacious. There is no guarantee, or even likelihood, that the world is the way that it is best for us for it to be correct. Belief that the world is the way that it is best for us for it to be, absent other evidence, is therefore just as likely to be false as true. Examples:
- Appeal to Good Consequences:
If you believe in God then you'll find a kind of fulfillment in life that you've never felt before. Therefore, God exists. Appeal to Bad Consequences: If you don't believe in God then you'll be miserable, thinking that life doesn't have any meaning. Therefore, God exists. Both of these arguments are fallacious because they provide no evidence for their conclusions; all they do is appeal to the consequences of belief in God. In the case of the first argument, the positive consequences of belief in God are cited as evidence that God exists. In the case of the second argument, the negative consequences of disbelief in God are cited as evidence that God exists. Neither argument, though, provides any logical evidence for the actual existence of God. The consequences of a belief are rarely a good guide to its truth. Both arguments are therefore fallacious. Each of the arguments above features in real world discussions of God's existence. In fact, they have been developed into an argument called Pascals Wager, which openly advocates belief in God based on its good consequences, rather than on evidence that it is true. Example: People argue that there must be an afterlife because they just can't accept that when we die that's it. This is an appeal to consequences; there is no life after death. Another example occurs in the film The Matrix. There Neo is asked whether he believes in fate; he says that he doesn't. He is then asked why, and replies, I don't like the thought that I'm not in control. This is not an appeal to evidence, but to the unpleasantness of believing in fate: fate would imply that the world is a way that I don't want it to be, therefore there is no such thing.
- Appeal to Force
An appeal to force is an attempt to persuade using threats. Its Latin name, argumentum ad baculum, literally means argument with a cudgel. Disbelief, such arguments go, will be met with sanctions, perhaps physical abuse; therefore, you'd better believe. Appeals to force are thus a particularly cynical type of appeal to consequences, where the unpleasant consequences of disbelief are deliberately inflicted by the arguer. Of course, the mere fact that disbelief will be met with sanctions is only a pragmatic justification of belief;it is not evidence that the resultant belief will be true. Appeals to force are therefore fallacious. Example: If you don't accept that the Sun orbits the Earth, rather than the other way around, then you=ll be excommunicated from the Church. Therefore, the Sun orbits the Earth, rather than the other way around. This argument, if it can properly be called an argument, makes no attempt to provide evidence for its conclusion; whether or not you=ll be excommunicate d for disbelieving the geocentric model has no bearing on whether the geocentric model is true. The argument therefore commits the appeal to force fallacy.
- Appeal to Novelty
An appeal to novelty is the opposite of an appeal to antiquity. Appeals to novelty assume that the newness of an idea is evidence of its truth. They are thus also related to the bandwagon fallacy. That an idea is new certainly doesn't entail that it i s true. Many recent ideas have no merit whatsoever, as history has shown; every idea, including those that we now reject as absurd beyond belief, were new at one time. Some ideas that are new now will surely go the same way. Examples: String theory is the most recent development in physics. Therefore, string theory is true. Religion is old fashioned; atheism is a much more recent development. Therefore, atheism is true. Each of these arguments commits the appeal to novelty fallacy. The former takes the newness of string theory to be evidence that string theory is true; the latter takes the newness of atheism to be evidence that atheism is true. Merely being a new idea, of course, is no guarantee of truth. The newness of string theory and atheism alone, then, should not be taken to be evidence of the truth of these two positions.
- Appeal to Pity
An appeal to pity attempts to persuade using emotion, specifically sympathy, rather than evidence. Playing on the pity that someone feels for an individual or group can certainly affect what that person thinks about the group; this is a highly effective, and so quite common, fallacy. This type of argument is fallacious because our emotional responses are not always a good guide to truth; emotions can cloud, rather than clarify, issues. We should base our beliefs upon reason, rather than on emotion, if we want our beliefs to be true. Examples: Pro life campaigners have recently adopted a strategy that capitalizes on the strength of appeals to pity. By showing images of aborted foetuses, anti abortion materials seek to disgust people, and so turn them against the practice of abortion. A BBC News article, Jurors shown graphic 9/11 images, gives another clear example of an appeal to pity: A US jury has been shown graphic images of people burned to death in the 11 September 2001 attack on the Pentagon. The jurors will decide whether al Qaeda plotter Zacarias Moussaoui should be executed or jailed for life... Prosecutors hope such emotional evidence will persuade the jury to opt for the death penalty.
- Appeal to Popularity
Appeals to popularity suggest that an idea must be true simply because it is widely held. This is a fallacy because popular opinion can be, and quite often is, mistaken. Hindsight makes this clear: there were times when the majority of the population believed that the Earth is the still center of the universe, and that diseases are caused by evil spirits; neither of these ideas was true, despite its popularity. Example: Most people believe in a god or higher power. Therefore, God, or at least a higher power, must exist. This argument is an appeal to popularity because it suggests that God must exist based solely on the popularity of belief in God. An atheist could,however,accept the premise of this argument (the claim that belief in God is widespread) but reject its conclusion without inconsistency.
- Appeal to Poverty
The appeal to poverty fallacy is committed when it is assumed that a position is correct because it is held by the poor. The opposite of the appeal to poverty is the appeal to wealth. There is sometimes a temptation to contrast the excesses, greed, and immorality of the rich with the simplicity, virtue, and humility of the poor. This can give rise to arguments that commit the appeal to poverty fallacy. The poverty of a person that holds a view, of course, does not establish that the view is true; even the poor can sometimes err in their beliefs. Example: The working classes respect family and community ties. Therefore, respect for family and community ties is virtuous. This argument is an appeal to poverty because it takes the association between a position and poverty as evidence of the goodness of that position. There is, however, no necessary connection between a position being associated with poverty and its being true, and so the argument is fallacious.
- Appeal to Wealth
The appeal to wealth fallacy is committed by any argument that assumes that someone or something is better simply because they are wealthier or the thing is more expensive. It is the opposite of the appeal to poverty. In a society in which we often aspire to wealth, where wealth is held up as that to which we all aspire, it is easy to slip into thinking that everything that is associated with wealth is good. Rich people can be thought to deserve more respect than poorer people; more expensive goods can be thought to be better than less expensive goods solely because of their price. This is a fallacy. Wealth need not be associated with all that is good, and all that is good need not be associated with wealth. Examples: My computer cost more than yours. Therefore, my computer is better than yours. Warren is richer than Wayne. Therefore, Warren will make a better dinner guest than Wayne. Each of these arguments takes an association with money to be a sign of superiority. They therefore both commit the appeal to wealth fallacy.
Fallacies of Ambiguity
Fallacies of ambiguity appear to support their conclusions only due to their imprecise use of language. Once terms are clarified, fallacies of ambiguity are exposed. It is to avoid fallacies of this type that philosophers often carefully define their terms before launching into an argument.
- Accent Fallacies/Equivocation
Accent fallacies are fallacies that depend on where the stress is placed in a word or sentence. The meaning of a set of words may be dramatically changed by the way they are spoken, with out changing any of the words themselves. Accent fallacies are a type of equivocation. Example: Suppose that two people are debating whether a rumor about the actions of a third person is true. The first says, I can imagine him doing that; it's possible. The second replies, Yes, its possible to imagine him doing that. This looks like agreement. If however, the second person stresses the word imagine, then this appearance vanishes; Yes, its possible to imagine him doing that. This now sounds like a pointed comment meaning that though it may just about be possible to imagine him doing that, there's no way that he would actually do it.
- Straw Man Arguments
The fallacy of equivocation is committed when a term is used in two or more different senses within a single argument. For an argument to work, words must have the same meaning each time they appear in its premises or conclusion. Arguments that switch between different meanings of words equivocate, and so don't work. This is because the change in meaning introduces a change in subject. If the words in the premises and the conclusion mean different things, then the premises and the conclusion are about different things, and so the former cannot support the latter. Example: The church would like to encourage theism. Theism is a medical condition resulting from the excessive consumption of tea. Therefore, the church ought to distribute tea more freely. This argument is obviously fallacious because it equivocates on the word theism. The first premise of the argument is only true if theism is understood as belief in a particular kind of god; the second premise of the argument is only true if theism is understood in a medical sense. Real World Examples: Christianity teaches that faith is necessary for salvation. Faith is irrational, it is belief in the absence of or contrary to evidence. Therefore, Christianity teaches that irrationality is rewarded. This argument, which is a reasonably familiar one, switches between two different meanings of faith. The kind of faith that Christianity holds is necessary for salvation is belief in God, and an appropriate response to that belief. It does not matter where the belief and the response come from; someone who accepts the gospel based on evidence (e.g. Doubting Thomas) still gets to heaven, according to Christianity. For the kind of faith for which (1) is true, (2) is therefore false. Similarly, for the kind of faith for which (2) is true, (1) is false. There is no one understanding of faith according to which both of the arguments premises are true, and the argument therefore f ails to establish its conclusion.
Fallacies of Presumption
Fallacies of presumption are not errors of reasoning in the sense of logical errors, but are nevertheless commonly classed as fallacies. Fallacies of presumption begin with a false (or at least unwarranted) assumption, and so fail to establish their conclusion.
- Affirming the Consequent
The fallacy of affirming the consequent is committed by arguments that have the form: "if A then B; B, therefore A." The first premise of such arguments notes that if a state of affairs A obtained then a consequence B would also obtain. The second premise asserts that this consequence B does obtain. The faulty step then follows: the inference that the state of affairs A obtains. Examples: If Fred wanted to get me sacked then he'd go and have a word with the boss. There goes Fred to have a word with the boss. Therefore, Fred wants to get me sacked. This argument is clearly fallacious; there are any number of reasons why Fred might be going to have a word with the boss that do not involve him wanting to get me sacked: e.g. to ask for a raise, to tell the boss what a good job I'm doing,etc. Fred's going to see the boss there fore doesn't show that he's trying to get me fired. If Zeus was a real, historical figure, but the Catholic Church covered up his existence, then we wouldn't have any evidence of a historical Zeus today. We don't have any evidence of a historical Zeus today. Therefore, Zeus was a real, historical figure, but the Catholic Church covered up his existence.
- Argument from Ignorance
Arguments from ignorance infer that a proposition is true from the fact that it is not known to be false. Not all arguments of this form are fallacious; if it is known that if the proposition were not true then it would have been disproven, then a valid argument from ignorance may be constructed. In other cases, though, arguments from ignorance are fallacious. Example: No one has been able to disprove the existence of God. Therefore, God exists. This argument is fallacious because the non-existence of God is perfectly consistent with no one having been able to prove God's non existence.
- Begging the Question (the "Circular" Argument)
An argument is circular if its conclusion is among its premises - i.e, if it assumes (either explicitly or not) what it is trying to prove. Such arguments are said to "beg the question." A circular argument fails as a proof because it will only be judged to be sound by those who already accept its conclusion. This is because, anyone who rejects a circular argument's conclusion should also reject at least one of its premises (the one that is the same as its conclusion), and so they should also reject the argument as a whole. Anyone who accepts all of the argument's premises already accepts the argument's conclusion, so they can't be said to have been persuaded by the argument. In neither case, then, will the argument be successful.
Example: The Bible affirms that it is inerrant. Whatever the Bible says is true. Therefore, the Bible is inerrant. This argument is circular because its conclusion, that the Bible is inerrant, is the same as its second premise - whatever the Bible says is true. Anyone who would reject the argument's conclusion should also reject its second premise and, along with it, the argument as a whole.
Real World Examples: The above argument is a straightforward, real world example of a circular argument. Other examples can be a little more subtle. Typical examples of circular arguments include rights claims - e.g.,I have a right to say what I want, therefore you shouldn't try to silence me; Women have a right to choose whether to have an abortion or not, therefore abortion should be allowed; The unborn has a right to life, therefore abortion is immoral. Having a right to X is the same as other people having an obligation to allow you to have X, so each of these arguments begs the question, assuming exactly what it is trying to prove.
However, it should be noted that, while a circular argument is not logically valid, a recursive argument is. An argument is recursive if one of its premises is contained within its conclusion. Such arguments are not circular because they do prove what they assume. This is because, anyone who rejects a circular argument's conclusion must also reject the premise that is also contained in the conclusion and, thereby, may logically reject the argument as a whole. Anyone who accepts all of the argument's premises logically accepts the argument's conclusion and, since they accept the premise already stated in the conclusion, there is nothing logically incorrect with accepting the conclusion.
Example: If a duck has wings then it is a duck that can fly. Because we have assumed the fact that a duck exists as one of the premises, there is nothing wrong with including the fact of the duck within the conclusion. This argument is not circular because it does not assume the conclusion among one of the premises but, instead, assumes a premise as part of the conclusion. However, we cannot say "if a duck has wings then a duck exists" because the truth of the conclusion must be assumed as a premise of the argument and is therefore an example of circular reasoning.
- Complex Question
The complex question fallacy is committed when a question is asked (a) that rests on a questionable assumption, and (b) to which all answers appear to endorse that assumption. Examples "Have you stopped beating your wife?" This is a complex question because it presupposes that you used to beat your wife, a presupposition that either answer to the question appears to endorse. "Are you going to admit that you=re wrong?" Answering yes to this question is an admission of guilt. Answering no to the question implies that the accused accepts that he is in the wrong, but will not admit it. No room is left to protest ones innocence. This is therefore a complex question, and a subtle false dilemma.
- Argument Cum Hoc
The cum hoc fallacy is committed when it is assumed that because two things occur together, they must be causally related. This, however, does not follow; correlation is possible without causation. This fallacy is closely related to the post hoc fallacy. Real World Example: Nestle, the makers of the breakfast cereal Shredded Wheat, once ran an advertising campaign in which the key phrase was this: People who eat Shredded W heat tend to have healthy hearts. This is very carefully phrased. It does not explicitly state that there is any causal connection between eating Shredded Wheat and having a healthy heart, but it invites viewers of the advertisements to make the connection; the implication is there. Whether or not there is any such connection, the mere fact that the two things are correlated does not prove that there is such a connection. In tempting viewers to infer that eating Shredded Wheat is good for your heart, Nestle are tempting viewers to commit a fallacy.
- False Dilemma
The bifurcation fallacy is committed when a false dilemma is presented, i.e. when someone is asked to choose between two options when there is at least one other option available. Of course, arguments that restrict the options to more than two but less than there really are are similarly fallacious. Examples: Either a Creator brought the universe into existence, or the universe came into existence out of nothing. The universe didn't come into existence out of nothing (be cause nothing comes from nothing). Therefore, a Creator brought the universe into existence. The first premise of this argument presents a false dilemma; it might be thought that the universe neither was brought into existence by a Creator nor came into existence out of nothing, because it existed from eternity. Another example emerged when George W Bush launched the war on terror, insisting that other nations were either for or against America in her campaign, excluding the quite real possibility of neutrality. Complex questions are subtle forms of false dilemma. Questions such as: "Are you going to admit that you're wrong?" implicitly restrict the options to either being wrong and admitting it or being wrong and not admitting it, thus excluding the option of not being wrong.
- Hasty Generalization
A hasty generalization draws a general rule from a single, perhaps atypical, case. It is the reverse of a sweeping generalization. Example: My Christian / atheist neighbor is a real grouch. Therefore, Christians / atheists are grouches. This argument takes an individual case of a Christian or atheist, and draws a general rule from it, assuming that all Christians or atheists are like the neighbor. The conclusion that it reaches hasn't been demonstrated, because it may well be that the neighbor is not a typical Christian or atheist, and that the conclusion drawn is false.
- No True Scotsman Argument
The no true Scotsman fallacy is a way of reinterpreting evidence in order to prevent the refutation of ones position. Proposed counter examples to a theory are dismissed as irrelevant solely because they are counter examples, but purportedly because they are not what the theory is about. Example: If Angus, a Glaswegian, who puts sugar on his porridge, is proposed as a counter example to the claim "no true Scotsman puts sugar on his porridge," the No True Scotsman fallacy would run as follows: Angus puts sugar on his porridge. No (true) Scotsman puts sugar on his porridge. Therefore, Angus is not a (true) Scotsman. Therefore, Angus is not a counter example to the claim that no Scotsman puts sugar on his porridge. This fallacy is a form of circular argument, with an existing belief being assumed to be true in order to dismiss any apparent counter examples to it. The existing belief thus becomes unfalsifiable. Real World Examples: An argument similar to this is often arises when people attempt to define religious groups. In some Christian groups, for example, there is an idea that faith is permanent, that once one becomes a Christian one cannot fall away. Apparent counter examples to this idea, people who appear to have faith believe but subsequently lose it, are written of fusing the No True Scotsman fallacy: they didn't really have faith, they werent true Christians. The claim that faith cannot be lost is thus preserved from refutation. Given such an approach, this claim is unfalsifiable, there is no possible refutation of it.
- Argument Post Hoc
The Latin phrase post hoc ergo propter hoc means, literally, "after this therefore because of this." The post hoc fallacyis committed when it is assumed that because one thing occurred after another, it must have occurred as a result of it. Mere temporal succession, however, does not entail causal succession. Just because one thing follows another does not mean that it was caused by it. This fallacy is closely related to the cum hoc fallacy. Example: Most people who are read the last rites die shortly afterward. Therefore, priests are going around killing people with magic words! This argument commits the post hoc fallacy because it infers a causal connection based solely on temporal order. Real World Examples: One example of the post hoc flaw is the evidence often given for the efficacy of prayer. When someone reasons that as they prayed for something and it then happened, it therefore must have happened because they prayed for it, they commit the post hoc fallacy. The correlation between the prayer and the event could result from coincidence, rather than cause, so does not prove that prayer works. Superstitions often arise from people committing the post hoc fallacy. Consider, for example, a sportsman who adopts a pre-match ritual because one time he did something before a game he got a good result. The reasoning here is presumably that on the first occasion the activity preceded the success, so the activity must have contributed to the success, so repeating the activity is likely to lead to a recurrence of the success. This is a classic example of the post hoc fallacy in action.
- Slippery Slope Argument
Slippery slope arguments falsely assume that one thing must lead to another. They begin by suggesting that ifwe do one thing then that will lead to another, and before we know it we'll be doing something that we don't want to do. They conclude that we therefore shouldn't do the first thing. The problem with these arguments is that it is possible to do the first thing that they mention without going on to do the other things; restraint is possible. Example: If you buy a Green Day album, then next you'll be buying Buzzcock's albums, and before you know it you'll be a punk with green hair and everything. You don't want to become a punk. Therefore, you shouldn't buy a Green Day album. This argument commits the slippery slope fallacy because it is perfectly possible to buy a Green Day album without going on to become a punk; we could buy the album and then stop there. The conclusion therefore hasn't been proven, because the argument's first premise is false.
- Sweeping Generalization
A sweeping generalization applies a general statement too broadly. If one takes a general rule, and applies it to a case to which, due to the specific features of the case, the rule does not apply, then one commits the sweeping generalization fallacy. This fallacy is the reverse of a hasty generalization, which infers a general rule from a specific case. Example: Children should be seen and not heard. Little Wolfgang Amadeus is a child. Therefore, little Wolfgang Amadeus shouldn't be heard. No matter what you think of the general principle that children should be seen and not heard, a child prodigy pianist about to perform is worth listening to; the general principle doesn't apply.
- Overwhelming Exception
This is a logical fallacy similar to a hasty generalization. It is a generalization which is accurate, but which comes with one or more qualifications that eliminate so many cases that what remains is much less impressive than the initial statement might have led one to assume. Examples: "All right, but apart from the sanitation, the medicine, education, wine, public order, irrigation, roads, a fresh water system, and public health, what have the Romans ever done for us?" (The attempted implication (fallaciously false in this case) is that the Romans did nothing for us). This is a quotation from Monty Python's Life of Brian. "Our foreign policy has always helped other countries, except of course when it is against our National Interest..." (The false implication is that our foreign policy always helps other countries). "All Americans are useless at foreign languages. Ok, I'll make an exception for those who live in multi-ethnic neighborhoods, have parents who speak a foreign language, are naturally gifted in languages, have lived abroad or who went to a school with a good foreign language program, but the rest are absolutely useless at foreign languages." All dogs are black, except for those which are not black. (This is also a tautology).
- Argument Tu Quoque
The tu quoque fallacy is committed when it is assumed that because someone else has done a thing there is nothing wrong with doing it. This fallacy is classically committed by children who, when told off, respond with "so and so" did it too, with the implied conclusion that there is nothing wrong with doing whatever it is that they have done. This is a fallacy because it could be that both children are in the wrong, and because, as we were all taught, two wrongs dont make a right. Example: The Romans kept slaves. Therefore, we can keep slaves too. This argument commits the tu quoque fallacy because it assumes that if someone else does a thing then its okay for us to do it too. It does not follow, however, from the simple fact that, because the Romans kept slaves, that there is nothing wrong with keeping slaves. It is plausible to think that the Romans acted immorally in keeping slaves,and that we would act immorally if we followed their example. The conclusion of the argument therefore does not follow from its premise. Examples of the tu quoque fallacy occur all the time. For instance, in an article entitled "Manchester United defend ticket price rise," BBC Sport reported: "Manchester United have hit their fans with a 12.3% average rise in season ticket prices for the next campaign. A top price ticket will cost $38, and the cheapest $23 ... But United have defended the price rises, saying they compare favorably with the rest of the Premiership. 'We do not know what most of our rivals will charge next year, buy even a price freeze across the rest of the Premiership would mean that next year only seven clubs will have a cheaper ticket than $23, and nine clubs will have a top price over $39, in some cases almost double,' said Humby [Manchester United finance director]." Humby's argument was essentially this: Other Premiership clubs charge more, therefore our ticket prices are justified. This commits the tu quoque fallacy because it is quite possible that all clubs, including Manchester United, "overcharge" for their tickets.
- ↑ Although Liebniz originally called his predicate logic the "Propositional Calculus" (and some scholars continue to call it by this name), the term "propositional" has also become associated with the "logic" of ordinary, spoken language, where the meanings of the subject terms may be considered, as opposed to limiting study to the meanings of the predicate terms (i.e., the inferential relationships). In this outline, we refer to "verbal logic" exclusively as the relatively informal, meta-theoretical statement of logical principles and to "predicate logic" as the more abstract, symbolic and formal statement of those principles.
- ↑ Alternative designations might be "linguistic logic" or "verbal reasoning." The designation "Propositional Logic" is eschewed for the reasons given in the first footnote of this outline above.
- ↑ Because the communication of all understanding must begin with a common language that is generally understood in an intuitive manner, the informal expression of logical statements (i.e., in plain English or another language of common usage) may be considered the "meta-language" from which the formal, symbolic languages are derived.
- ↑ It is also apparent that, no matter how one might attempt to reduce the meanings of linguistic terms to a combination of only three, this reduction can only represent a gross approximation of the ordinary, semantic relationships between linguistic terms. Because the terms used in this Verbal Logic outline are taken from ordinary language, they inevitably connote their ordinary linguistic meanings which cannot be avoided but which are generally beyond the scope of this outline. In this outline, we are only concerned with the essential, intuitive, inferential relationships between these terms, as they are used in a manner that is peculiar to this outline.
- ↑ Not the Objectivism of Ayn Rand.
- ↑ In other words, although the idea of an infinite quantity may be illogical, there is still the logical possibility of a potentially infinite quantity, at least where a class (rather than a well-defined set) is concerned.
- ↑ A very brief description of the intersection of these philosophies could go as follows: One must assume the existence of an objective reality that is knowable. However, one must also assume that we cannot know this reality without the analysis of our observations by a rational mind that can make an abstract and recursive, albeit ultimately artificial (i.e., not a part of objective reality except as a product of the mind), distinction between form and substance for the purposes of analysis (and recursive because the form of some thing may also be analyzed itself as an object subject to formal conditions, and so on, until we reach some fundamentally basic, a priori object that cannot be understood in terms more formal or simpler than itself). Finally, although an objective reality is assumed to exist, one must assume that the purely logical and mathematical structures applied to an analysis or understanding of this reality are essentially products of the mind and do not exist as a part of objective reality except in the mind itself, where the mind itself is an object of reality. Of course, a pure intuitionist would reject any sort of linguistic meta-philosophical approach to an understanding of logic. In this regard, this outline is not purely intuitionist but represents an effort to bridge the gap between "ordinary," linguistic understanding and "pure" formalism.
- ↑ And therefore more nuanced.
- ↑ In this sense, all logical, rational thought, as defined in this outline, must ultimately be recursive in nature. See the Recursion Theory outline.
- ↑ In formal Predicate Logic, and as explained in the Axioms of Further Definability section of the First-Order Predicate Logic outline, we have defined our system so that all terms may be ultimately defined as alternative systems based on one or the other of the following combinations of certain pairs of inferences: (1) conjunction ("and") and negation ("not"); (2) disjunction ("or") and negation; or (3) material implication ("if-then") and negation. In this regard, the inference of conjunction is fundamentally analogous to the process of identification by combination; the inference of disjunction is fundamentally analogous to the process of identification by differentiation; and the inference of material implication is fundamentally analogous to the process of identification by implication. Although it is equally valid to build a system of logic on the basis of combinations of the operations of conjunction and negation or of implication and negation, the author of this outline has chosen to proceed with disjunction and negation as the two most "primitive" or entirely intuitive operations of logic, upon which every other logical term or inference may be constructed. This methodology was chosen primarily because the author believes that the operation of disjunction is the simplest operation that can be described in Boolean terms and, therefore, has the greatest informational entropy of all logical terms. See the Information Theory outline.
- ↑ This principle of "opposites" was first stated in purely logical terms by Heracleitus of Ephesus at the beginning of the Fifth Century B.C.E.
- ↑ Many items in this outline are presented as numbered listes. However, it should be noted that numeric quantity has not yet been defined (which occurs later in Set Theory). Therefore, the use of numbering in this outline is a meta-linquistic use of the numeric terms that are defined in this outline for the terms "more" or "fewer." Numeric ordering is not yet defined and should not be implied by the use of any numeric, subscript terms or by any line item numbering.
- ↑ Best represented linguistically by the interrogative "what".
- ↑ Best represented linguistically by the interrogatives "who" or "whom," which includes the later-defined concept of subjectivity, and which is distinguished further from all other kinds of objects in the next section of this outline.
- ↑ Italicized, lower-case, Roman letters are often used to denote an unspecified object.
- ↑ In this context, verbs such as "leaves," "departs," or "disappears" denote negation and therefore are not stated here as being existential.
- ↑ Although this outline is not concerned primarily with symbolic logic, the symbols of symbolic logic will be introduced within parentheses, when they first become relevant.
- ↑ In ordinary experience, an object is the manifestation of some entity. In such a case, we may say that the entity's existence is "true." However, this idea of truth, which accounts for all possible characteristics of an object (i.e., the object's "identity"), is distinct from the logical "truth value" of a proposition - a part of the proposition's logical "equivalence" - which accounts only for the logical values of "true" or "false" and the minimally sufficient conditions that must exist to cause those values, as explained more fully and later in this outline. Therefore, "truth" is not included in the definition of "object" and its synonyms and is defined more particularly later in this outline. See the footnotes to "Empirical Truth," below.
- ↑ In this definition, the predicate terms: "is manifest," "is expressed," "is created," "is referred to," "is signified," "is present," "is evident," "is constructed," "is built," "is made," "is conveyed," "is moved," "is transferred," "is put or placed," "is stated," "exists," "occurs," "comes," "arrives," "appears," "acts," and "does" are equivalent and essentially undefined. We call these predicate terms "existential" because they are all defined to be synonymous with the essentially undefined existential term "is" (albeit with the possibility of later-defined symantic nuances for each particular term, such as will occur later in this outline with the definition of the subject term "self").
- ↑ Although, at first glance, the definition of an object and its relation to the existential terms may appear indistinct or logically circular, in fact it is not. For the purposes of this outline, the predicated existential terms: "is manifest," "is expressed," "is created," "is referred to," "is signified," "is present," "is evident," "is constructed," "is built," "is made," "is conveyed," "is moved," "is transferred," "is put or placed," "is stated," "exists," "occurs," "comes," "arrives," "appears," "acts," and "does" are considered generally synonymous with each other and essentially undefined. However, the subject terms of any proposition - i.e., "object," "entity", "unit," "individual," "body," "item," "referent," "occurrence," "appearance," "arrival," "instance," "event," "fact," "self," "expression," "thing," "manifestation," "actualization," "presence," "state," "action," and "construct" - although essentially synonymous with each other, are not synonymous with the predicated existential terms because these subject terms are the objects upon which those existential terms operate; they are not the existential terms themselves. Therefore, although the existential terms are considered wholly undefined as a priori objects that can only be understood intuitively, the subject (bolded) terms introduced in this definition are considered generally synonymous with the subject term "object" and are therefore properly defined as already manifested into existence and not purely existential and a priori. This is the most general statement of Renee Descartes' famous axiom: "I think, therefore I am" and, as such, it constitutes the most primitive possible abstraction of form from substance. In regard to any objection that we have now defined "something" in terms of something else that is not itself expressly definable, all we can say is that we must first find some place to start, if we will reason at all about anything. In so doing, we have achieved our first instance of abstraction - i.e., distinguishing the thing from its creation, the noun from the verb.
- ↑ Unfortunately, the semantics of ordinary language cannot make the distinction between the act of existence and the thing that exists with complete effectiveness. Therefore, this entire outline rests on three fundamental assumptions: (1) that an objective reality exists apart from the observer; (2) that an objective reality can be known by the observer; and (3) that the act of coming into existence or knowing the truth of some thing's existence may be distinguished (i.e., abstracted) from the thing itself - e.g., that the act of giving birth can be distinguished from the fact of the birth - which simply states the relationship of assumptions (1) and (2) with each other. In addition, it should be noted that an act or event in itself may be an object for examination. Therefore, an object need not be material to qualify for this definition and, hence, the term "materialization" is not included with the other synonyms for the term "object." Therefore, although one might believe that, ultimately, "all is an illusion," this outline assumes the existence of a provable truth and leaves the possibility of an ultimate "nothingness" to more transcendental endeavors that are worthier of the question, such as theology.
- ↑ It should be noted that, even though an event is manifest, this does not mean the event has become known. The characteristics of knowledge are defined later in this outline.
- ↑ Although this outline assumes the existence of an objective reality, this definition does not denote an objective permanence; it only denotes that an object exists, even if only for a moment. The concept of time, or a sequencing of events or objects, is treated later.
- ↑ This definition relates the concept of objective existence to the use of the impersonal pronouns.
- ↑ Note that this definition connotes the plural, whereas the previous definition connotes only the singular. The more rigorous definition of plural ("more than one") is given later in this outline. However, it should be noted that Verbal Logic, or its extension First-Order Predicate Logic, has no more particular quantification of objects, except to say that an object is not unique or that a predication is universal. Numeric quantification is the subject of the extensions of Second-Order Predicate Logic, such as Set Theory.
- ↑ These terms include both the act of negation as well as the thing (i.e., referent) that is negated - both are "instances" of non-existence; either the "thing" is already naught or it is eliminated. The more specifically relational versions of these terms is provided in the next definition.
- ↑ As explained above, negation is a wholly undefined and primitive concept that must be accepted, intuitively, for the purposes of this outline, as a priori, and without further definition. Furthermore, the state of non-existence can only "exist" as an ideal conceptualization since, by referring to it as something that exists (in this instance, a "state of non-existence"), it is posited that something exists, even if the only thing that is described is simply the idea of the state of non-existence (which, itself, is simply a description of the absence of some thing).
- ↑ Unlike the terms defined in the prior definition, the terms defined here are applied to an extant referent that is then described as standing in relation to the defined terms of non-existence. Therefore, in this case, it is the referent that is described to be missing. Whether non-existence itself can be described as a referent that is "missing" forms a linguistic paradox which arises from an attempt to realize the concrete expression of an ideal concept - the a priori concept of nothingness. It is a fundamental assumption of this outline that two paradoxes must exist that can never be eliminated from a "complete" theory of logic: an ideal nothingness and an ideal identification of two objects. Ideal identification is defined in the next section of this outline.
- ↑ Until now, the term "self" has been indistinguishable from other objects. Now it is distinguished by its essential relationship with the "other" that is uniquely characteristic of the self - an object whose existence can only be understood in terms of its relation to others.
- ↑ Note that, in each definition, the definiendum is only a sufficient condition for the definiens, whereas the definiens are necessary conditions for the definiendum. Therefore, an object is not necessarily a self and a self is always an object. Here, the importance of the definition is to put "self" and "other" into a relation, but not necessarily as purely complementary states, as well as to make the definition of "self" a more particular kind of object.
- ↑ The processes of identification and differentiation are fundamental to the development of any understanding. Note also that this definition does not necessarily denote a state of consciousness, which will be defined later in this outline; it merely denotes a distinction made between two objects (i.e., "a thing in itself" is not necessarily a conscious being).
- ↑ "itself"
- ↑ The preposition "with" is often omitted and implied.
- ↑ Note that the definition of identity given in the Verbal Logic Outline is greatly refined in the Second Order Predicate Logic Outline as the concept of "equality" so as to distinguish it entirely from the First-Order concept of equivalence.
- ↑ As with the other terms in this outline, the meanings of these symbols become more refined and precise, and thereby become distinguished from each other, later in this outline.
- ↑ In their a priori understanding, identification (in which two things are exactly the same in every way, including spatial-temporal location and energy) and negation (a perfect nothing which, by definition, can never exist because we still need some concept or symbol to represent it) are ideal states. However, since ideal identification results in no distinction, its symbolic representation can only be an approximation of the idea itself, as noted by Ludwig Wittgenstein during the early part of the Twentieth Century. An ideal "nothing" or "non-existence" is also an ideal concept we can only approximate since, to give it any symbolic representation, is to create "something."
- ↑ As with "identity" or "equality," this term obtains a much more precise meaning in the definition of "logical equivalence" below.
- ↑ The definition of "common," as distinguished from "in common," is provided for later in the section on Conditions in Sentences.
- ↑ This is the broadest possible definition for the synonyms of identification and will not necessarily comport with common usage, which is much more nuanced in its meanings. However, the meanings of the particular definienda in this definition may be refined later with other previously defined terms so as to provide the desired nuance.
- ↑ Of course, any two objects that are identical in all ways, including spatial-temporal location and energy, would not be distinguishable as two, different objects and, logically and in fact, could only be one object. Therefore, any well-defined definition of identity must be qualified in some way so that the term "identity" only reflects those qualities actually possessed by an object, not all possible qualities (see Russell's Paradox and the Problem of Universals).
- ↑ There is also "as to."
- ↑ The separation of these terms by the slash (/) symbol indicates that each term may substitute for another in combination with the term "which" to form the inference of differentiation. In cases where these terms appear by themselves, use of the term "which" is implied.
- ↑ Although this term is intimately associated with material implication ("if-then"), necessarily defined much later in this outline, the term "if" by itself may also denote the very general concept of differentiation without the necessity of implication and therefore is presented also in this much more primitive definition.
- ↑ Note that these terms are not prepositions in the spatial-temporal sense but have a meaning that is idiosyncratic to logic - for example, where a statement is valid "under every possible interpretation." In this manner, "under" states an inferential relationship, rather than a spatial-temporal relationship.
- ↑ "vis-à-vis"
- ↑ The concept of a relationship is the most generalized understanding of the nature of the self and the other.
- ↑ The reference to "each" is often omitted and implied.
- ↑ See the footnotes to the definitions of "identification" and "identical" above.
- ↑ The precise definition of numeric "one" depends on many antecedent concepts presented in this outline and the other logic outlines and remains to be defined in the Set Theory outline, where Set Theory is an extension of Second-Order Predicate Logic.
- ↑ In this outline, all definienda are bolded so as to distinguish them from the associated definiens.
- ↑ Anytime a relation exists (even when it is a thing put in relation with itself as an identity), a differentiation occurs between the two things that are put in relation to each other. In the case of an identification of objects, a differentiation occurs simply because perfect identity is an ideal state and any relation of identity that occurs must differentiate the operands, even if only symbolically.
- ↑ According to the theory of this outline, because every other logical operation can be reduced to a combination of disjunction and negation operations, every other logical operation can be considered a special case of disjunction and negation. Furthermore, this definition also reflects the primitive, a priori nature of disjunction assumed in this outline (i.e., disjunction cannot be defined in simpler terms), as well as the formal definition that any disjunction is true so long as at least one of its operands also exists.
- ↑ The same may be said for conjunction - i.e., that any time a thing is put in relation to another thing a conjunction exists. This is also true because, as we will see, any disjunction can be expressed as a combination of conjunction and negation operations and vice-versa. Therefore, whether we begin with disjunction or with conjunction as the most primitive relation in terms of which all other relations are defined (except negation), we will achieve the same result.
- ↑ A conditional statement ("if-then," "because," "since," etc.) also results in the differentiation or, alternatively, conjugation of a causal relation in regards to the components of cause and effect; its specifically logical form is more particularly defined later in this outline.
- ↑ When only one object is present, a self-referential "differentiation" or "combination" exists that is called a "singleton."
- ↑ This is not a strictly Intuitionist understanding of the relations of disjunction and conjunction since, according to the Intuitionists, those terms cannot be expressed in completely equivalent terms in regard to each other.
- ↑ "But for" also connotes implication whereas "but" alone does not.
- ↑ There is also "as to."
- ↑ The separation of these terms by the slash (/) symbol indicates that each term may substitute for another in combination with the term "which" to form the inference of disjunction. In cases where these terms appear by themselves, use of the term "which" is implied.
- ↑ Although this term is intimately associated with material implication ("if-then"), necessarily defined much later in this outline, the term "if" by itself may also denote the very general concept of disjunction without the necessity of implication and is therefore also stated as one of the definienda of this more primitive definition.
- ↑ Note that many of these terms might be more properly categorized as forms of exclusivity or implication, both of which are more specific forms of disjunction. But here we are only interested in any form of inference that might generally be described as disjunction without necessarily distinguishing any more specific inferential powers they might have. For this reason also, the general terms of differentiation presented earlier are also stated here.
- ↑ Best represented linguistically by the interrogative "where".
- ↑ Therefore, possession and membership necessarily imply the complementary existence of each other or of an other; see DeMorgan's Law.
- ↑ The possessive tense of a noun also expresses this concept.
- ↑ This definition is refined later in this outline to state that an elemental object cannot be differentiated in any way; however, at this stage, we have not defined the term "cannot" and therefore must satisfy ourselves with the present definition. In this sense, an object is deemed "elemental" until it can be shown to consist of more than one object.
- ↑ It is critically important to note that a singleton may be considered a "collection" of one object but is still atomic because there are no other objects collected with it from which it may be divided. However, as defined in a following definition, a singleton is never a plural object.
- ↑ The simple term "more," as distinguished from "more than one," is defined more precisely later in this section of this outline.
- ↑ Therefore, the terms "atomic" (and its synonyms) and "plural" are logically complementary to each other.
- ↑ The definition of "set" is further refined and distinguished from the term "class" later in this outline.
- ↑ Note that it is this process of identification that distinguishes conjunction from disjunction, which merely places the constituent objects in relation to each other but does not necessarily identify them as parts of the whole.
- ↑ Note that, in this outline, "such that" or "so that" denote the most general form of inclusive disjunction whereas, by themselves, "such" denotes identification and "so" denotes a conclusion.
- ↑ This language is awkward but necessary to avoid using undefined terms where possible. Also, a "universe" of objects must be well-defined so as to contain only objects that are specified for that universe and cannot contain all possible objects; see also, Russell's Paradox.
- ↑ It should be noted that Verbal Logic, or its extension First-Order Predicate Logic, has no more particular quantification of objects, except to say that an object is not unique or that a predication is universal. Numeric quantification is the subject of an extension of Second-Order Predicate Logic that we call Set Theory.
- ↑ Of course, an object that consists entirely of nothing is ideal and is therefore only approximated by the symbol ∅. In addition, an object may be "empty" for a particular condition where some specified aspect of the object contains nothing but where the object, as a whole, exists.
- ↑ For use of the term "yet," see the definition of "conjunction," above.
- ↑ It should be noted that, in this and the prior definitions, "object" may refer to a single object or to an individual set or other grouping of objects, where those objects are collectively identified as an individual set or grouping.
- ↑ See the previous definition for use of the terms "has been."
- ↑ Note that these terms are not prepositions in the spatial-temporal sense but have a meaning that is idiosyncratic to logic, such as where a statement is valid "over every possible interpretation." In this manner, these terms state an inferential relationship, rather than a spatial-temporal relationship.
- ↑ Note that this definition of the term "segment" provides a more refined or specific meaning than provided by the earlier use of the term in the definition of the separation, division, disjunction, differentiation, or disunion of another object, as stated in the above section on Combination.
- ↑ It is important to note that, unless stated otherwise, the contents of a set, or any other grouping, are not assumed to be ordered unless expressly stated otherwise.
- ↑ Of course, according to the theory of this outline, "event" is generally synonymous with object, referent, occurrence, instance, fact, self, expression, thing, manifestation, actualization, presence, or construct. Therefore, this outline's use of terms associated with "time" includes a sequence or order of any of those objects as well.
- ↑ Best represented linguistically by the interrogative "when".
- ↑ Unless the moment referred to is the beginning or ending moment of time, a moment is always a segment of time.
- ↑ The term "immortal" is not used since that denotes a living condition that is not defined as such in this outline, although it may be implied by the later definitions of consciousness and sentience.
- ↑ Note that this term is completely distinct from the concept of "intentionality," which is part of the philosophy of Utilitarianism.
- ↑ Although not explicit, this definition and the prior distinction drawn between the self and an other provide a contextual frame of reference for the term "self" that can be distinguished from other objects.
- ↑ We have chosen to define these terms according to their ordinary usage in the English language, which does not denote their usage as requiring any sentience for their expression but are commonly used to describe the mechanical action of an inanimate object or process - e.g., "a falling rock will follow its intended path to its target." In the sense of this outline's definition, a non-living object rolling down a hill does have an intention or purpose by the fact it is moving according to a thermodynamic tendency but without the necessity of consciousness or self awareness. In other words, the "intention" or "purpose" of the rolling, non-living object is provided by some other motivating force. Therefore, in our system, intention, like consciousness (see below), is not necessarily the possession of any particular object.
- ↑ Unlike awareness or consciousness, this definition does not necessarily connote a spontaneous self-motivation, as opposed to purely programmed behavior. The definition of existence itself is beyond the scope of this outline.
- ↑ Let's assume that an "awareness" or "consciousness" is defined to be merely a temporal manifestation of intention or purpose. Furthermore, let's define an "intention" or "purpose" as essentially a momentary expression of self that may connote a continuity of intention or purpose but which does not necessarily describe a self-awareness or sentience, and that self-awareness or sentience are prerequisites for knowledge. Therefore, according to our definition, a non-living object rolling down a hill, or a chemical substance that is crystallizing, may express an awareness or consciousness of its motion or self-ordering known to others who are self-aware or sentient and observing it but only insofar as this expression is not known to the rolling or crystallizing objects and must therefore be knowable by others. In this sense, awareness or consciousness would not be the possession of any particular object but would be a universal phenomenon that is known only by sentient beings - i.e., beings who are "self-aware"; whether or not it is known by some particular object is another matter entirely. Hence, such a definition of consciousness would be the most general definition possible and would therefore be consistent with the possibility of what has been called a "cosmic" or "universal" consciousness but does not necessarily require the existence of such a phenomenon. According to this definition, a log rolling down a hill or a chemical that is crystallizing might possess a temporal continuity that we have defined as "consciousness" or "awareness" while it expresses its thermodynamic "motivation"; however, such motion is not spontaneously self-ordered or self-motivated, is not necessarily (and, by our definition of "knowledge" stated later in this outline, is not at all) known by the log or crystal, and could therefore never be described as sentient or self-aware (i.e., by our definition of knowledge stated later in this outline, there must be the existence of a sentient being for the log or crystal's own existence to be known). Therefore, although we might assume the existence of an objective Universe, it does not exist for all practical purposes until we can observe it; see the "Anthropic Principle." Therefore, in our system, consciousness or awareness is not necessarily the possession of any particular object. The fact that a particular self may believe that consciousness is its own possession does not rule out the possibility it is not. Our definition of consciousness is merely the most general statement of this idea that is possible under all circumstances. It may be the possession of a particular object or it may be something global. We simply choose the more general definition for more general purposes and to distinguish this very general idea from sentience as that form of consciousness to which we can ascribe a more particular self-motivation or spontaneous self-ordering - that "something more" that makes us self-motivated, knowing beings.
- ↑ This understanding is similar to that of John Locke's, who defined the realization of "self" as the experience of a continuity of consciousness.
- ↑ Sentience connotes the necessity of a spontaneous self-motivation as essential to an expressed intention or purpose that is more than mere computation. What exactly that something "more" is that results in spontaneous self-motivation is beyond the scope of this outline; see "Fuzzy Logic."
- ↑ According to our definition, both self-awareness and the awareness of others are necessary for a finding of "sentience."
- ↑ In this context, "thinking" is a spontaneously self-motivated condition and therefore necessarily more than mere computation.
- ↑ Hence, the mere application of logic does not, by itself, constitute "thinking"; something more is needed to constitute true intelligence.
- ↑ As such, these objects may be purely structural (i.e., syntactic) and without semantic meaning, as well as meaningful and semantic. However, it should be considered that even syntax has some meaning in regards to its purely structural/relational purposes.
- ↑ Best represented linguistically by the interrogative "why".
- ↑ By self-reflection.
- ↑ Therefore, by our definition, the experience of a pure emotion is only "meaningful" once it has been put in relation to the self or another.
- ↑ Coincidentally, the understanding of language, meaning, and structure occur in separate regions of the human brain.
- ↑ As such, relations that are purely structural may have a kind of meaning vis-a-vis other structural elements but they do not have a meaning in regard to the observer outside of this structure.
- ↑ According to our earlier definitions, information must be conveyed to be meaningful (i.e., put in relation to something), even if only to one's own self via self-reflection. Furthermore, information that is expressed but not yet conveyed constitutes an inchoate meaning for the individual who may convey it, and meaning continues to exist for information that is conveyed but that meaning becomes particular to the recipient's understanding and may not match the precise meaning that was intended by the individual who conveyed it.
- ↑ As such, a symbol is a generalization or abstraction of the particular experience of thinking of an idea, cognition, or concept, whether or not that meaning is conveyed or communicated.
- ↑ Because a purely formal and symbolic language, such as Predicate Logic, may state the meaning of an inference with only one symbol, and because this inference need not be communicated or otherwise conveyed to be valid, this outline assumes that all meaning that is symbolically expressed fits within the definition of "language," even if this meaning is not communicated so that it becomes "information."
- ↑ As such, a grammar includes a syntax.
- ↑ This definition of "rhetoric" is very broad and would encompass any meaningful use of language, including linguistic constructions not often considered in the same category as classical rhetoric, such as poetry or self-reflective meditation.
- ↑ An idea, thought, cognition, or concept may be essentially structural and not semantic other than by any meaning inherent in the structure itself. As such, a word may be an essentially structural element of grammar and need not convey any meaning beyond the structure.
- ↑ As such, a word or string is a kind of symbolic expression, as is a gesture, which can also be considered a kind of signifier.
- ↑ As such, it is possible for a word or string to be composed of only one letter.
- ↑ As such a lexicon or vocabulary is composed of both structural and semantic words or strings.
- ↑ Therefore, to be a sentence as defined in this outline, the combination of words must be intentional.
- ↑ Later, these terms are refined to mean a "complete," well-formed expression.
- ↑ As defined above, grammar includes syntax, so the mention of both in this definition is redundant, albeit explicit.
- ↑ Although phrases are generally considered to be groups of more than one word, string, letter, character, symbol, or other signifier, individual words, strings, letters, characters, symbols, or other signifiers are also phrases by this definition.
- ↑ As this term is defined and used in the study of informal verbal logic. It should be noted that to "state a proposition" is entirely different from a "state of being," and therefore the term "state" has an entirely different meaning in those two contexts.
- ↑ As defined later in this outline, a logical statement concerns only the form of the statement, not its substance, and is generally declarative or descriptive in structure (as distinguished from interrogatory, exclamatory, suggestive, or imperative statements).
- ↑ For the purposes of this and later outlines, the term "sentence" or "formula" will be considered synonymous with the terms "statement" or "message."
- ↑ By this definition, all sentences, statements, formulas, or messages are necessarily well formed.
- ↑ As defined above, grammar includes syntax, so the mention of both in this definition is redundant, albeit explicit.
- ↑ In traditional Aristotelian logic, proposition and statement were identical terms and not distinguished from each other, whereas in modern Predicate Logic this distinction is fundamental. In this outline, however, whenever the term "statement" is used, it is assumed that a proposition is also present, unless stated otherwise.
- ↑ An object, without more, simply exists. Therefore, a substance is something more abstract since it is a definition of an object. However, although a definition might ultimately describe the meaning of an object in terms of its conditions, this present definition merely describes an object's substance as something different and a priori vis-à-vis other objects. Therefore, identification in this context does not connote an understanding of meaning or conditions associated with that definition but merely that something may exist and be identified as different from other objects.
- ↑ In this sense, the conditions of an object are an expression of the "idea" of that object. It is this non-Aristotelian (in fact Stoic, although ironically Platonic) abstraction of form from substance that is essential to the later development of the Predicate Calculus (symbolic logic) and the Principle of Extensionality. The Platonic connection is ironic because Aristotle, Plato's student, did not effectively distinguish subjective meaning from an objectively predicated inference. The distinction of form from substance is later fully developed and rigorously stated by Immanuel Kant in his seminal work, The Critique of Pure Reason. In that work, Kant states that what we consider a prior may also be some combination of purely a priori mental constructs with a posteriori experience. An example that he gives of such a combination is causality (and here the author is inclined to agree since causality is considered by him to be an extension of relation, which is a more primitive notion upon which causality must necessarily depend for its existence).
- ↑ This definition addresses a distinction that is necessary for the abstraction of a logical inference, also called a predicate, from the subject of the predicate. Without this distinction, there can be no separate, symbolic representation of an inference apart from its subject, and it is the study of the validity of the inferences, and not the subjects, which is the purpose of Predicate Logic.
- ↑ In this sense, the terms "substance" and "form" are relative and recursive terms; the particular form of an object is also the substance of an analysis of that form. In addition, and unless the object is ideally and entirely undifferentiated (which, beyond the mere idea of existence or a location in space-time, is utterly meaningless), forms must also be contained within an object's substance. A philosophical question exists as to whether the substance of an object can be known apart from its forms or conditions, or whether the "substance" of an object is nothing but forms and conditions. In this outline, we assume that some substance exists apart from forms and conditions, even if this is merely a mental construct used to articulate the concept of form by the creation of a distinction which is purely intellectual, as with the definitions of "identification" and "nothing." In this sense, although an ultimate substance is assumed or hypothecated to objectively exist (at least for the sake of argument), it is not assumed that an "ultimate substance" is knowable or can be proved (even though it may be assumed to objectively exist). For purposes of this outline, an "ultimate substance" is purely existential and is therefore an undefinable, a priori, conceptualization. This does not mean that such a conceptualization cannot exist in objective reality since, at the very least, it exists as an object of the mind, which itself is defined as an object of reality.
- ↑ We reserve the use of the term "consistent" for the definition of truth and distinguish the term "constant" for other purposes.
- ↑ Often the word "true" is omitted and implied when a valid condition is said to "hold."
- ↑ In regard to Proof Theory, this definition pertains to the "satisfiability" of a proposition, not its ultimate consistency or, in the language of Proof Theory, its "validity," despite the fact that we use the term "valid" in this outline as synonymous with "satisfied," so as to comport with ordinary usage. Of course, "inconsistent" also means "not consistent with truth" and, therefore, false, so we continue to use this term as a synonym for false, even though it is also used in regard to the certainty of an outcome, whether true or false. To distinguish the terms used in this outline from their Proof Theory counterparts, we use the terms "satisfiable" and "validated" when referring to the Proof Theory concepts. To avoid conflict with the definitions of Proof Theory, "consistent" is not one of the definienda for the definition of truth (although "inconsistent" is permissibly used for the definition of false without creating such a conflict).
- ↑ The symbol ∅ and the numeral 0 are generally considered to be logically synonymous terms.
- ↑ Also called the "logic value of the expression." We avoid the term "logical condition," which connotes a much broader meaning than the expression "logical value."
- ↑ In informal verbal logic, only two truth values exist: true and false. A two-valued system of logic is called "bivalent."
- ↑ In some systems, this symbol may represent bidirectionality of implication and therefore its use must be explicitly defined to avoid confusion.
- ↑ Note that, unlike the definition of truth given above, this definition does not require that a condition must exist, only that it consistently exist or not exist. According to this definition, although the condition may be either true or false, what matters is that it remains true or false. This corresponds to the principle of ultimate "validity" found in Proof Theory, whereas our use of the term "valid" above pertains to the idea of "satisfiability," as this term is used in Proof Theory. In Proof Theory, the terms "validity" and "satisfiability" are terms of art and peculiar to that doctrine, and their meanings in that regard do not comport with ordinary usage. To distinguish the terms used in this outline from their Proof Theory counterparts, we use the terms "satisfiable" and "validated" when referring to the Proof Theory concepts.
- ↑ Note that this definition does not require that a certain condition remain forever certain. A condition is certain only so long as it is not contradicted. Once contradicted, the certainty evaporates and the condition becomes once again "uncertain." As such, one of the premises of this outline is that absolute certainty on any question, like an absolutely universal proposition, is not provable. Because the validity of every question ultimately rests on the validity of its assumptions, any change in those assumptions may cause a contradiction to arise. Even the validity of 1 + 1 = 2 depends entirely on the assumptions upon which that hypothesis is based and a change in those assumptions may cause such a statement to become invalid (see Peano Arithmetic). Therefore, according to Godel's Incompleteness Theorem, the validity of "first assumptions" will always rest with the discipline of Philosophy, not Science.
- ↑ Therefore, as defined in this outline, "can" connotes a certain universality of existence whereas "may" only connotes the possibility of existence. Therefore, "can" and "may" are not essentially synonymous.
- ↑ This definition is important for inclusion of the intransitive form of the verb tense for "can."
- ↑ Here it is important to show that the verbs responsible for permissibility are also subject to negation.
- ↑ The complementary truth value of possible is impossible. However, the complementary truth values of never/impossible and always are ambiguous. This is because the complement of "always" is never or possible, while the complement of "never" is always or possible. This means that, philosophically speaking, we can only, truly prove with logic alone that something does not happen or does not exist; logic, by itself, can never prove the mere fact of existence - existence must be assumed at some point as a necessary antecedent to every logical proof. See below, "The Modern Square of Opposition."
- ↑ As seen later in this outline, necessity may be defined entirely in terms of the disjunction of all sufficient terms.
- ↑ As seen later in this outline, sufficiency may be defined entirely in terms of the conjunction of all necessary terms.
- ↑ We know from the definition of "sentence" or "formula" given in the preceding section that a well-formed sentence or formula must also be syntactic and grammatic.
- ↑ Note that, by this definition and the definition of "phrase" given above (which must, by that definition, be only a portion of a sentence or formula), a phrase must now be any syntactic expression that is not whole, entire, or complete but that cannot, by this definition, be "well-formed."
- ↑ Although a word or string is not, in itself, a well-formed sentence, a word or string may be considered a well-formed word or string where its spelling obeys the rules of common usage.
- ↑ In this outline, we have chosen to define knowledge in terms of sentience; i.e., knowledge is the special province of self-awareness. Therefore, although we may loosely say that the log rolling down a hill comes to "know" a rock when it collides with that rock, causing its trajectory to be altered by the collision, and although the terms "consciousness" or "awareness" have been very broadly defined as not identified with any particular entity but as merely the temporal expression of intention or purpose, it is the author's opinion that defining "knowledge" on such broad terms would make its usage relatively meaningless for its purpose, which is to relate sentient entities (the ultimate arbiters of what it means for some condition to be "true") to truth finding.
- ↑ Of course, the existence of an object "in itself" - i.e., apart from the conditions that describe it - can never be logically proved; it must be assumed.
- ↑ By virtue of the definition of knowledge, the knowing "object" must be sentient.
- ↑ For the purposes of this outline, the definition of "observation" is extremely broad and includes the act of knowing "objects of the mind" as well as tangible or material objects.
- ↑ It should also be noted that, because this definition is stated in terms of knowledge, and because knowledge is stated in terms of sentience, observation on these terms is solely the province of sentient beings. Therefore, although we can say that a camera might "observe" a subject, it is really not the camera that makes the observation but the person who is operating it that does so.
- ↑ An empirical truth is a condition that is believed to exist as a consequence of the immediate observation of that condition by one's senses. Note that "immediate observation" may occur by reading the measurement of an instrument that acts as an extension of the observer's senses where, without that instrument, ordinary and immediate observation of a particular property would not be possible. Of course, one must reasonably believe, again through logical analysis and empirical observation, that the measuring instrument itself is capable of transmitting measurements or other data that may be considered reasonably reliable for proving the truth of the property observed.
- ↑ Whether known empirically or intuitively, an object that is the subject of a proposition is only "true" if we may know the truth of its predicate conditions.
- ↑ An empirical truth is "proved" and "consistent," according to the terminology of Proof Theory
- ↑ In human law, a "presumption" has the additional property of shifting the burden of proof to the party that did not previously have the burden of proof wherever this condition is applicable.
- ↑ Best represented linguistically by the interrogative "how".
- ↑ Reason is more general than logic because logic, as defined later in this outline, relates more specifically to the analysis of the validity of inferences. Therefore, logic is a special case of reason.
- ↑ Defining this term to comport with ordinary English language usage, our definition of "reason" does not require sentience for its meaning, unlike knowledge. Therefore, a computer's calculations may be reasonable even if they cannot be known by the machine that is performing the process.
- ↑ Note that the terms of a condition may not be certain by definition and the existence of uncertain terms may be the quality that is determined.
- ↑ As such, a belief is an abstraction of the condition of determinability from the thing that is believed to be determined. Therefore, a belief can exist without actually knowing that a thing does in fact exist. For this reason, belief and knowledge are fundamentally different concepts.
- ↑ Because belief requires knowledge for its definition in this outline, according to our system only sentient beings possess the ability to "believe" a proposition may or may not be true.
- ↑ i.e., a rigorous condition may be consistently false.
- ↑ i.e. reasonable, valid, and certain.
- ↑ Note that the same can be said of the condition's definition or proof.
- ↑ This definition connotes no claims as to the knowledge of certainty or possibility.
- ↑ As such, a term may be a complete proposition or just part of a proposition, so long as a well-defined meaning exists for the term.
- ↑ And, as defined earlier, a term must also have well-defined conditions for its content. Therefore, all proper terms are also necessarily well-defined.
- ↑ i.e., without definition or proof
- ↑ Hence, to be truly primitive, such a term must be a priori.
- ↑ See above, Semiotics
- ↑ See the Introduction to this outline.
- ↑ Italicized, lower-case Roman letters are often used to denote an unspecified object. In this definition, the subject term is an object of the sentence, as is the predicate term.
- ↑ Predicates are often represented symbolically by italicized, upper-case roman letters.
- ↑ In this sense, the predicate term is some attribute, circumstance or other condition regarding the subject term, and the predicate and subject terms must therefore necessarily stand in relation to each other.
- ↑ A premise, conclusion, or conditional statement all contain both subject and predicate terms.
- ↑ See also the footnotes to the definition for condition or predicate above for an explanation of the distinction of form from substance that lies at the heart of the definition of "subject."
- ↑ Although the predicate inferentially "follows" (see below) from the subject, subject and predicate are not necessarily stated in any particular word order, except as provided by the rules of grammar for the language used.
- ↑ In an English-language unconditional statement, the subject term usually (although not necessarily) precedes the copula (see below) in word order and the predicate term usually (although not necessarily) follows the copula in word order.
- ↑ In a conditional statement, a subject term is synonymous with an antecedent; a predicate term is synonymous with a consequent (see below).
- ↑ In a logical statement, a copula is usually a linking verb with an accompanying subject complement or adverbial phrase. As such, the copula of a sentence is more properly considered a part of the predicate, rather than the subject. Also, formal (symbolic) logic is only concerned with the validity of predicates and assumes the validity of subjects as presumed premises of the sentence.
- ↑ Note that the term "one" is often omitted and is implied by the use of the terms "exactly," "precisely," and "only."
- ↑ As such, a general statement may also operate as the statement of a class.
- ↑ A purely intuitionist understanding of this concept would be that a valid substitution is "justifiable," not necessarily truth-preserving.
- ↑ Sometimes expressed by the Latin term "vice-versa."
- ↑ This is a slightly different meaning from the use of this term in law, where it more particularly means a condition the occurrence of which causes a duty that has previously arisen to be extinguished (as opposed to a condition precedent that must occur before a duty will arise).
- ↑ "Q.E.D.," "therefore," "as such," "wherefore," or "ergo"
- ↑ Therefore, an antecedent and a consequence must necessarily stand in relation to each other.
- ↑ As such, a condition may be particular or general, simple or compound, and conjunctive or disjunctive.
- ↑ Often the word "true" is omitted and implied when a valid condition is said to "hold."
- ↑ In other words, the consequence is a necessary result of the premise.
- ↑ The logical inference of implication follows from the sufficiency, not the necessity, of any antecedent premise, since there may be more than one sufficient premise which may or may not also be necessary to the occurrence of the consequence. However, it is important to remember that any sufficient premise might also be considered a conjunction of all its necessary parts and that, in the absence of any necessary condition, there will be no sufficiency for the occurrence of the consequence. Therefore, a sufficient premise always implies the occurrence of all necessary antecedents.
- ↑ It is important to note that a valid expression can result where both the antecedent and the consequence are false or where the antecedent is false but the consequence is true so long as that relation is consistent (and the latter relation may occur where the antecedent is sufficient but not necessary). However, an expression where a sufficient antecedent is true and the consequence is false is always false by virtue of the definition of sufficient conditions.
- ↑ In other words, the consequence is not a necessary result of a sufficient premise.
- ↑ As defined earlier, a "belief" is a reason to know the truth or that the truth can be known but it is not, in itself, knowledge of the truth.
- ↑ This refers to the truthfulness of an inference (see below).
- ↑ The condition may be either a premise or a conclusion.
- ↑ As distinguished from an argument, where the conclusion must be proved to be true before it will be accepted as factual.
- ↑ Sometimes the distinction between an explanation and an argument (which are both examples of passages) must be determined from the context in which the passage exists. Due to this similarity, arguments can be restated as explanations. In cases where the intention of the passage is truly ambiguous, the passage may be considered as either an argument or an explanation, depending on how the audience or author chooses to view it.
- ↑ Because an object is the most general form of entity, it may or may not be logically constructed. Therefore, an object may be well-defined (i.e., logically and internally consistent) or ill-defined (i.e., illogical or not internally consistent). Sets, on the other hand, must be well-defined if we want them to be a logically sound building block upon which the rest of mathematics may be constructed.
- ↑ A set may be empty and a non-empty, un-ordered set (the most general kind of set) disregards any order or repetition of the objects contained within it. Whether a well-defined set may be infinite is a subject addressed in the Set Theory outline.
- ↑ The un-ordered set is considered the simplest, logical definition of the concept of a collection that remains useful to set theorists for reasons that are revealed as one studies Set Theory.
- ↑ Note that this definition is distinguishable from the definition of "elemental object, member, or point" given in the Semiotics section of this outline. A "member of a set" may be another set, whereas an "elemental object" is something that is essentially indivisible, although it may be a set by itself when existing in the form of a singleton.
- ↑ And, therefore, this is an object for which the definition of membership in the set is true.
- ↑ In NBG Set Theory, the term "class" has the more specific meaning of any combination or collection of objects which share a common condition but for which the definition of the class is not necessarily well-defined, and therefore is not synonymous with set.
- ↑ The expression "categorical statement" implies a categorical proposition.
- ↑ As distinguished from a class, which treats of objects more generally.
- ↑ The logical operators "and", "or", "not," and "if-then" are all examples of logical inferences.
- ↑ A purely intuitionist understanding of this concept would be that a valid substitution is "justifiable," not necessarily truth-preserving.
- ↑ A rule may be made more particular by qualifying specific conditions for its application.
- ↑ The prepositions "by" and "with" used in this definition are generally implied and omitted.
- ↑ In the law, a cause that is necessary but not sufficient is often called a "substantial factor."
- ↑ Even without the explicit use of the term, "reason" (see above) is necessarily implied within the present definition.
- ↑ Formal arguments generally use symbolic methods of analysis, such as we find in Predicate Logic, since metalinguistic context is no longer directly determinative.
- ↑ Logical form should not be confused with the syntax used to represent it; there may be more than one set of symbols that represents the same logical form, depending on the language used.
- ↑ Usually used as a descriptive prefix to another term, such as "meta-language."
- ↑ i.e., by regarding the content terms as mere placeholders for any particular subject matter, like blanks on a form.
- ↑ For purposes of this outline, "statement" and "proposition" are considered generally equivalent terms.
- ↑ Logical equivalence is concerned with only two circumstances: (1) the truth value of the proposition and (2) the minimal set of conditions that are sufficient to produce that truth value. Therefore, logical equivalence is not the same thing as Second-Order logical identity, which is concerned with all possible conditions that define the object.
- ↑ The meaning of a logical statement can be either its truth value or a meta-logical description of a condition, logical inference, or term. Likewise, the meaning of a mathematical statement can be either its numeric value or the quantitative concept it expresses.
- ↑ Of course, any two objects that are identical in all ways, including position in time, space, and energy, would not be distinguishable as two objects and, logically, could only be, in fact, one object. Therefore, any well-defined definition of equality must be qualified in some way so that the relation only reflects specified qualities, not all possible qualities (see Russell's Paradox).
- ↑ Where such a state exists.
- ↑ The complement of an object is generally formed by the logical operation of negation.
- ↑ By itself or another.
- ↑ Knowledge is essential to this definition for, without it, an undefined object (i.e., one that is not identified in any manner) would be utterly without meaning.
- ↑ As such, not every axiom of general applicability is an axiom schema. An axiom schema must be an axiom that relates to a series of specific objects, even if that series is stated in the most general terms.
- ↑ Stands in a consistent and valid relation with the other object.
- ↑ Can be known without proof or definition.
- ↑ A principle of Bivalence or "Two-Valued" Logic.
- ↑ Applies only to informal verbal logic and bivalent First Order Predicate Logic, but not necessarily to other logical forms, such as Fuzzy Logic.
- ↑ A principle of Bivalence or "Two-Valued" Logic.
- ↑ Pursuant to the Law of Non-Contradiction stated below, one and only one of these states must be true at any particular moment.
- ↑ This axiom applies only to informal, verbal logic and bivalent, First Order, Predicate Logic, but not necessarily to other, more Intuitionist, logical forms, such as Fuzzy Logic.
- ↑ The Law of the Excluded Middle is not strictly Intuitionist since, where neither A or not-A have been proved or disproved then we cannot assume the truth of this axiom.
- ↑ A principle of Bivalence or "Two-Valued" Logic.
- ↑ Applies only to informal verbal logic and bivalent First Order Predicate Logic, but not necessarily to other logical forms, such as Fuzzy Logic.
- ↑ In other words, for every logical object there is exactly one complement, and each logical object is necessarily contradicted by its complement. Truth is always the complement of falsity, and falsity is always the complement of truth. Therefore, truth and falsity necessarily contradict each other.
- ↑ This axiom is only applicable to systems of bivalent logic.
- ↑ Logical Necessity and Tautological Truth are logically complementary to a Logical Impossibility.
- ↑ It is often disputed whether a particular claim can constitute a logical necessity under all circumstances. For instance, the proposition that "the set of all sets must contain itself as a member" is contradicted by some non-standard arithmetics created under various applications of the Peano Axioms. This is also known as the logical Problem of Universals and relates to Russell's Paradox.
- ↑ The essential difference between a logical necessity and a tautological truth is that the latter is necessarily well-defined, whereas the former is not necessarily so.
- ↑ Logical Necessity and Tautological Truth are logically complementary to a Logical Impossibility.
- ↑ A tautology is an explicit identification of the same logical objects.
- ↑ Although a pure tautology in informal verbal logic is both self-consistent and self evident (like an axiom), it is devoid of any real meaning (unlike an axiom or definition), and therefore should be avoided. The same is not true of predicate logic, where the meanings of the inferential terms, and not the subject terms, are generally the subject of examination.
- ↑ The essential difference between a logical necessity and a tautological truth is that the latter is necessarily well-defined, whereas the former is not necessarily so.
- ↑ This is also called an "arbitrary tautology" and is represented by the same symbol for truth. An arbitrary contradiction is represented by F or, more commonly, by an inverted T, ⊥.
- ↑ In formal logic, we do not determine the truthfulness of the subject terms; we only determine the truthfulness of the inference. Determining the truthfulness of the subject terms may be a worthy endeavor (i.e., so that we construct a sound argument), however it is not necessary if we are analyzing the inference, in which case we can assume the truthfulness of the subject terms for the "sake of argument."
- ↑ The proof of this theorem is so obvious that it can be properly classified as an axiom or corollary.
- ↑ It should be noted that bidirectional implication ("if and only if", symbolized by ↔) also expresses the relationship of equivalence.
- ↑ The latter symbol is more particularly used to denote that a definition follows.
- ↑ The term "equivalence," rather than the term "equal," is used to describe this relationship since the idea of equivalence does not necessarily connote the required condition in logic of identical form, or the required condition in Set Theory or mathematics of identical meaning and/or quantity. Because identity is a concept of Second-Order Predicate Logic and is not definable in first-order terms, the equals sign is not seen at all in First-Order Predicate Logic. As such, equivalence is only concerned with whether two expressions evaluate to the same truth value, which is why every equivalence in logic is essentially nothing more than a tautology (although one we find useful and desirable). Equality (identity), on the other hand, is the equivalence relation which every thing has to itself and to nothing else and which satisfies Leibniz's Law (a second-order expression): ∀x∀y[(x = y) ↔ ∀P(Px ↔ Py)] (entities x and y are identical if and only if any predicate possessed by x is also possessed by y and vice versa).
- ↑ "if-then"
- ↑ e.g., the operators of equivalence and implication are ≡ and →, respectively.
- ↑ Distinguish "Begging the Question" in Fallacies of Presumption.
- ↑ Distinguish "Begging the Question" in Fallacies of Presumption.
- ↑ In the first example, we see that the value of a known x in the antecedent does not depend on the result of the conclusion for an unknown y (and the intermediate conclusion x is merely a tautological implication by identity of the antecedent x). However, in the second example, we see that the value of an unknown y, stated in the antecedent x → y, depends on the value of the conclusion y which, of course, is also unknown, being the same term. Because the proof of an unknown conclusion must be based on known antecedents, we cannot prove the truth of an hypothesis where the truth of one of the antecedents must also be proved based on an evaluation of the conclusion.
- ↑ Distinguish "Begging the Question" in Fallacies of Presumption.
- ↑ Note that this does not necessarily mean an atomic, or even a simple, operand. Hence, a negation may operate on one operand that itself is composed of multiple terms, but it will negate the value of the operand as a whole.
- ↑ i.e., containing more than one antecedent.
- ↑ Note that this logical use of the word "or" as an inclusive disjunctive is actually equivalent to the grammatical expression "and/or."
- ↑ Hence, the operation of inclusive disjunction connotes existence.
- ↑ i.e., containing more than one antecedent.
- ↑ The remaining alternative premises must be complementary to the consequent.
- ↑ Hence, the operation of exclusive disjunction connotes uniqueness.
- ↑ The absence of any operator is also generally interpreted to be the operation of conjunction.
- ↑ i.e., containing more than one antecedent.
- ↑ Hence, the operation of conjunction connotes universality.
- ↑ As explained further in this outline, this definition is not "truth-functional" or "truth-preserving" because an ambiguity may exist as to the use of necessary and/or sufficient conditions. However, this ambiguity may be eliminated through a refinement of the definition (i.e., by considering a conditional statement false if and only if the antecedent is true and the consequence is false, see below), in which case the unambiguous form is called material implication. In contradistinction, the ambiguous form of the present definition is sometimes called linguistic implication.
- ↑ Aristotelian logic concerns itself with the meaning of the terms of a logical argument. Therefore, under the Aristotelian system, a conditional statement is true if and only if both the antecedent is true and the consequence is true, or if and only if both the antecedent is false and the consequence is false, making the Aristotelian form of the conditional statement effectively bidirectional (see below, "Bidirectional Conditional Statement"). However, under the Aristotelian system, and unlike a truly bidirectional conditional statement, the truth value of the antecedent in relation to that of the consequence in a strictly unidirectional material implication is not without ambiguity. This ambiguity occurs because, in the Aristotelian system, the truthfulness of the statement cannot be known with certainty where an antecedent is necessary but has unknown sufficiency, or where a consequence is sufficient but has unknown necessity. In a bidirectional conditional statement, by contrast, there is no such ambiguity; a bidirectional conditional statement is only true if both the antecedent and consequence are true or if they are both false. Also, unlike a bidirectional conditional statement, the converse of an Aristotelian unidirectional conditional statement is never necessarily also true.
TRUTH TABLE (where p is a necessary condition for q but with unknown sufficiency; U = unknown value):
→ p qp → q T T T F T F T F U F F T
TRUTH TABLE (where p is a sufficient condition for q but with unknown necessity; U = unknown value):
→ p qp → q T T T F T U T F F F F T
The Stoics eliminated the ambiguity in the Aristotelian system by disregarding the truthfulness of the meaning of the antecedent terms of a statement. Instead, they focused solely on the meanings of the inferential relationships contained within the logical structure of a statement. Hence, under the Stoic system, we assume the truthfulness or falsity of the meaning of the terms of a statement "for the sake of argument"; all that matters under this system is that the structure of a logical proposition is itself correct, and the actual truthfulness of the meanings of the subject, antecedent terms may be determined later. Therefore, under the Stoic system, the following argument would be considered true:
Example: If the animal does not have feathers then it is a bird. The animal has feathers. Therefore, the animal is not a bird.
Although the results of such an argument may not seem reasonable according to our everyday knowledge about the world, the Stoic method has been adopted by modern logicians because it allows for a completely unambiguous analysis of whether an argument is logically correct in form, as distinguished from whether or not the terms of an argument are empirically correct; the proof of an empirical truth for any of the terms may be conducted separately, and a substitution of the new, empirically true terms does not change the logical validity of the propositions - it only changes the meaning of the result. The more intuitive understanding, where both the subject and predicate terms must be true for the result to also be true, is distinguished as a "sound" argument. However, if we limit our interest to the validity of the inference and not the truthfulness of the subject terms, as under the Stoic method, a unidirectional conditional is false if and only if the antecedent is true and the consequence is false, and this results in the following, unambiguous, truth table:
→ p qp → q T T T F T T T F F F F T
Therefore, under the Stoic system, in order for a consequence to be true, the antecedents must be sufficient - to wit, they are either sufficient, albeit unnecessary, or they are necessary and sufficient - and the consequence must necessarily follow, even if insufficient by itself, for the occurrence of the antecedent.
- ↑ In this example, there is no claim that the animal of which we speak actually has feathers (although this fact might be assumed for the sake of understanding the inference) and, for that reason, there exists no statement that reaches the definite (or even probable) conclusion that the animal is actually a bird. Therefore, although the inference is sufficient to be logically true, it does not necessarily constitute a sound argument.
- ↑ States a condition where a consequence follows from an antecedent.
- ↑ Only a conjunction of all necessary conditions will achieve sufficiency for the existence of a conclusion.
- ↑ This occurs because another antecedent may also be required to cause the consequence to occur; therefore, the truth of the antecedent does not necessarily imply the truth of the consequence unless all necessary conditions are present.
- ↑ A disjunction of any sufficient conditions will always achieve necessity for the existence of a conclusion.
- ↑ This occurs because another antecedent may alternatively cause the consequence to occur; therefore, the truth of the consequence does not necessarily imply the truth of the antecedent.
- ↑ However, there may be some other antecedent, such as legal emancipation, that may permit a finding of adulthood even without reaching the age of legal majority. Therefore, the age of legal majority is a sufficient, but not necessary, condition for a finding of adulthood.
- ↑ This definition only pertains to a determination of relevance, not a certain statement of truth.
- ↑ The set is an unambiguous consequence, as opposed to other possible sets of consequences.
- ↑ The absence of ambiguity in this inference is the reason that the Stoics described the truth of the conditional statement as requiring a sufficient condition for the antecedent and called this inference a "material implication"; see the definition for "Material Implication" below.
- ↑ As defined earlier in this outline, a condition is true if and only if it is impossible for the truth value of a conclusion to be different from the truth value of a sufficient premise. Because it is unambiguous, this definition of implication, as opposed to the definition of linguistic implication given earlier, is used for the definition of the conditional statement in Predicate Logic.
↔ or ≡pq(p ↔ q) or (p ≡ q) T T T T F F F T F F F T
- ↑ In informal verbal logic, as opposed to formal Predicate Logic, for a passage to prove a sound conclusion, and therefore contain a sound argument, two conditions must be satisfied: (1) at least one of the statements must claim to know the existence of facts or reasons to believe that certain evidence is true, also known as a factual claim; and (2) there must be a claim that the facts or reasons to believe those facts support (or imply) the conclusion, also known as an inferential claim. A passage that does not satisfy both of these conditions, such as may typically occur in warnings or advisory statements, statements of unqualified belief or opinion, loosely associated statements, unsubstantiated reports, expository passages, or illustrations, does not contain an empirically and logically valid argument and cannot prove an empirically and logically valid conclusion. Note, however, that according to the Stoics (see above), it may still prove a logically valid, albeit unempirical, conclusion. Hence, in Predicate Logic, as opposed to informal verbal logic, the empirical validity of the factual premises is generally unimportant. See below, "Vacuous Truth." The best practice is to test the validity of the inferential claim by testing the validity of the logical relationships between premises and conclusion, by assuming that all the premises are true, before testing the validity of the factual claims since, if the inferential claim is false - i.e., if the supposed logical argument is faulty in its inferential method - the validity of any factual claim, although it may be interesting in itself, will have no importance to the empirical validity of the argument.
- ↑ i.e., valid logical relationships.
- ↑ i.e., all the factual premises are empirically true.
- ↑ Whether an argument is "sound" is only important to informal verbal logic and not to formal Predicate Logic, where the only concern is the validity of the argument and not the empirical truthfulness of the premises on which the conclusion is based. However, where science is concerned, all arguments must be sound.
- ↑ We have called this an axiom to remain consistent with the style of this outline and, although such a statement qualifies for the definition of an axiom, it is more properly a "proof by definition."
- ↑ A single conditional statement or other inferential claim may become an argument if the antecedent and/or consequence are restated to posit both factual and inferential claims. However, such constructions usually result in wordy and cumbersome statements and are therefore generally avoided.
Example: If the animal has feathers, and in this instance it actually does have feathers, the animal is a bird.
- ↑ A conditional statement or other inferential claim may serve as either (or both) a premise or conclusion of a statement, proposition, or argument.
Example: If the animal has feathers then it may be a bird. (inferential claim stated as a premise) The animal has feathers. (premise) The animal is a bird, but only if it can fly. (conditional conclusion)
- ↑ Therefore, the inclusion or omission of a false but irrelevant argument will not affect the argument's validity.
- ↑ As a matter of style and clarity, however, it is rhetorically ineffective to include irrelevant statements in a well-formed argument.
- ↑ Any inference that is invariably truth preserving is an example of deductive reasoning. Material implication is a specific example of deductive logic, but the definition also applies to the operation of any well-defined logical operator.
- ↑ As such, deductive reasoning tends to move from the more general case to the more specific - i.e., arriving at a particular conclusion by inference from one or more premises.
- ↑ Unlike deductive reasoning (which is the process of arriving at a particular conclusion by inference from one or more general or universal premises), a conclusion arrived at through inductive reasoning in informal verbal logic generally does not necessarily follow from the premises. This is not the case for mathematical induction, which does necessarily prove one, and only, conclusion. The case of mathematical induction is proved during the study of Second-Order Predicate Calculus.
- ↑ i.e., in the example above, the major term is "have/has feathers").
- ↑ i.e., in the example above, the minor term is "ostrich is a kind" of some class, in this case the class of all birds).
- ↑ i.e., in the example above, the middle term is "bird").
- ↑ A sorites argument is an example of the Principal of Transitivity.
- ↑ i.e., proved.
- ↑ Which themselves may be the conclusions of other theorems.
- ↑ A theory is not an unproven argument, as often assumed by lay persons who are misusing this term to mean a proposition that is stated hypothetically.
- ↑ A lemma (see below) is a theorem that is particularly useful because of its many uses for the proof of other theorems. There is no real, logical difference between theorems and lemmas and it is only a matter of custom that they are distinguished based on a perceived "usefulness" by the community of logicians that use them.
- ↑ Usually in the context of a particular theory.
- ↑ i.e., universal truths. See generally, the Problem of Universals.
- ↑ At first glance it appears that the statements are logically equivalent. However, this is not necessarily the case. If "legal adulthood" can be satisfied by a judicial finding of emancipation then a child may be an adult and still be under the age of 18, which is why this example was chosen. If this is the case then being over the age of 18 is a sufficient, but not a necessary, condition for adulthood. Thus, by replacing the subject and predicate terms with each other, we are not guaranteed a logically true statement since we could have someone who is under the age of 18 and still be an adult, which means that the converse is not necessarily true as stated. However, if legal emancipation is not an option and being over the age of 18 is both a necessary and sufficient condition for adulthood then the inferential relationship is bidirectional and the statement is logically equivalent to its converse.
- ↑ An inferential statement and its contraposition are always logically equivalent. This can be readily seen from the above example. For bidirectional inferences, if being over the age of eighteen is a necessary and sufficient condition for legal adulthood then, as we saw with conversion above, the replacement of subject and predicate terms with each other will not invalidate the truth of the statement due to the bidirectionality of the inferential relationship. Furthermore, inverting both the subject and predicate terms by taking their logical complements does not invalidate a statement that is bidirectional since, by definition, a bidirectional statement is true if both its subject and predicate terms are true or if both its subject and predicate terms are false.
- ↑ For material implications (i.e., inferences that are not bidirectional), the truth is also preserved for contraposition. If we suppose that being over the age of 18 is a sufficient but not a necessary condition for adulthood because of the option of legal emancipation then the original statement is true. If we replace the subject and predicate terms with each other and then invert both terms, we still have a true statement since, by definition, a material implication is only false if the truth value of the predicate term is different than the truth value of a sufficient subject term, and we don't care whether a necessary subject term is true or false since, in that case, the truth of the statement will be the same as the truth of the predicate term in any case. However, with contraposition, the truth is preserved because, even if the subject term is a sufficient but not necessary condition for the truth of the predicate term, replacing and negating both terms will preserve the truth value of the original statement. For a definitive proof of this proposition, see the truth tables given below.
- ↑ The subject of a sufficient conditional statement may be restated as the predicate of a necessary conditional statement by the contraposition of the original terms.
- ↑ The subject of the converse of a sufficient conditional statement may be restated as the predicate of the inverse of a necessary conditional statement, or the subject of the converse of a necessary conditional statement may be restated as the predicate of the inverse of a sufficient conditional statement, and the two statements will remain equivalent.
- ↑ Proofs of the following propositions are stated in the First-Order Predicate Logic outline.
- ↑ "The way that affirms by affirming."
- ↑ This is probably the most fundamental inference in all of logic.
- ↑ In artificial intelligence, modus ponens is called forward chaining.
- ↑ "The way that denies by denying."
- ↑ This is essentially the contra-positive form of Modus Ponens.
- ↑ "The way that affirms by denying."
- ↑ However, we cannot conclusively deduce that the bird is a duck, based solely on the information given.
- ↑ This is a good example of how the meaning of terms in informal verbal logic tends to obscure the actual logical inference that exists apart from the truth of the meaning of the terms. This occurs because the reader will tend to use their actual experience to evaluate the terms because of the ordinary meanings normally associated with them. If we know absolutely nothing about ducks or bird that quack then the example given rings true in every sense. Hence the value of a purely formal, symbolic logic, where the meaning of the subject terms is not important (those can be proved in separate empirical investigations) and what is really important are the meanings and values of the logical inferences.
- ↑ Even though the two conjuncts are also material implications, it is the conjunction of the terms, and not the material implications themselves, that are at issue here. In this sense, it might also be possible to write the problem as (A→B & A→C) ⊢ (A→B ∧ A→C), but this would not serve to illustrate the intended point as clearly.
- ↑ And B is also true, which simply restates the initial assumption.
- ↑ The second consequent (that the bird must be a duck) is not necessary to illustrate the inference in its most essential character, but it is included to make the point that the illustration need not be limited to only the first conjunct.
- ↑ Of course, it is not possible to assume that both are true, based on the information that is given.
- ↑ But, based on the information given, we cannot say with any certainty that it is in fact a duck.
- ↑ Since at least one of two statements (A or B) is true, and since either of them would be sufficient to entail C, C is certainly always true in these circumstances.
- ↑ The reason this is called "disjunctive syllogism" is that, first, it is a syllogism--a three-step argument--and second, it contains a disjunction, which means simply an "or" statement.
- ↑ Note that the disjunctive syllogism works whether 'or' is considered 'exclusive' or 'inclusive' disjunction.
- ↑ In this sense, we have our first inkling of mathematical induction.
- ↑ However, we simply don't know with any certainty which of the antecedents is in fact true.
- ↑ In symbolic logic, we never use a definition unless it is meets the criteria of eliminability and non-creativity, which are explained in the outlines for those subjects or, in the case of set theory, the proof of a justifying theorem which does not include the definition itself as a premise.
- ↑ Note that a style of argument does not necessarily constitute a method of proof and might even be logically invalid.
- ↑ i.e., if the premises are true then the conclusion cannot be false.
- ↑ As such, a deductive argument tends to move from the more general case to the more specific - i.e., by arriving at a particular conclusion by inference from one or more general or universal premises.
- ↑ As such, an argument by definition is essentially axiomatic in character, although a logically true axiom will not necessarily result; see above: Axiom.
- ↑ Therefore, universal instantiation is essentially an example of deductive reasoning.
- ↑ Without the use of valid inductive reasoning, universal generalizations are not generally valid.
- ↑ There exists a proof in Second-Order Predicate Calculus for why mathematical induction is as strong a form of reasoning as deductive logic.
- ↑ A statistical syllogism is a weak inductive argument and, with some very significant exceptions, generally does not qualify as true mathematical induction.
- ↑ Real or supposed.
- ↑ Or other expert witness.
- ↑ Usually weak inductive and predictive.
- ↑ This style of argument can be distinguished from an argument by analogy in that an educated guess emphasizes experience and past empirical observation.
- ↑ i.e., "is/are" and "is/are not."
- ↑ See below, "Types of Categorical Propositions").
- ↑ In the modern square of opposition there are no contraries, sub-contraries, or sub-alterns (as required in the square of opposition of traditional Aristotelian logic).
- ↑ The names of syllogism types that result in the existential fallacy are italicized.
- ↑ Excepting polysyllogisms and disjunctive syllogisms.
- ↑ Notice that there are four terms: "fish", "fins", "goldfish" and "humans." Two premises aren't enough to connect four different terms since there must be one term common to both premises to establish a connection.
- ↑ In everyday reasoning, the fallacy of four terms occurs most frequently by equivocation - i.e., using the same word or phrase in each statement but with a different meaning each time, creating a fourth term even though only three apparently distinct words or phrases are used.
Example: Nothing is better than eternal happiness. (major premise) A ham sandwich is better than nothing. (minor premise) A ham sandwich is better than eternal happiness. (conclusion)
The word "nothing" in the example above has two meanings: "nothing is better" means the thing being named has the highest value possible; "better than nothing" means the thing being described has only marginal value. Therefore, "nothing" acts as two different terms, creating the fallacy of four terms.
- ↑ A more tricky example of syllogistic equivocation is as follows:
Example: The hand touches the pen. (major premise) The pen touches the paper. (minor premise) Therefore, the hand touches the paper. (conclusion)
The fallacy is more clear if one uses "is touching" instead of "touches." It then becomes clear that "touching the pen" is not the same as "the pen," thus creating four terms: "the hand"; "touching the pen"; "the pen"; and "touching the paper." A valid form of this argument would then be as follows:
Example: The hand touches the pen. (major premise) All that touches the pen also touches the paper. (minor premise) Therefore, the hand touches the paper. (conclusion)
Now the term "the pen" has been eliminated, leaving three terms and correcting the logic of the syllogism.
- ↑ Dicto Simpliciter syllogisms.
- ↑ The truth of this proposition is proved in the First-Order Predicate outline.
- ↑ It should be noted that, unlike entailment, indirect proof does not prove the universality of any condition; it only proves the possibility of a contrary circumstance, thereby disproving the hypothesis. (needs proof).
- ↑ An existence theorem may be called "pure" if the statement given does not also indicate the construction of whatever kind of object for which existence is asserted. From a more rigorous point of view, this is a problematic concept. This is because, in these instances, "existence theorem" is merely a tag applied to a statement for which the "proof" is never unqualified. Hence, the term "pure" is used in a manner that violates the standard "proof irrelevance" rule of mathematical theorems. That is, these "theorems" are in fact unproven statements of truth, at least in the formal sense of the term "proof." Thus, many constructivist mathematicians who work in extended, predicate logics (such as intuitionistic logic, where pure existence statements are considered to be intrinsically weaker than their constructivist counterparts) generally do not utilize non-constructive proofs, except for meta-definitional purposes. Thus, the use of the term "proof" to describe these statements amounts to an informal misnomer.
- ↑ This is essentially the same thing as an indirect proof except that it generally finds its application in non-empirical, purely "philosophical" disciplines, such as Meta-Logic. The fundamental difference between an indirect proof and a non-constructive proof is that, whereas the possibility of the existence of a material thing (such as a quacking duck) may be proved directly by the observation of its existence, a non-constructive proof only results in the proof of the reality of an idea as a mental construct without the certainty of knowing that the idea actually exists in objective reality beyond the reality of the mental construct (see below, "Constructive Proof"). For example, we may assert the existence of the Axiom of Infinity, the Axiom of Choice, or the Law of the Excluded Middle, and we might "prove" the existence of all these ideas non-constructively or indirectly, but we cannot actually prove their existence as logical objects except in a purely a priori manner. Therefore, in the disciplines of math or symbolic logic, such concepts are usually posited as axioms requiring no other proof except the formal statement of their existence, usually by way of a non-semantic symbolism. Hence, their "proof" is provided merely by the construction of the symbolism that represents the concept to be utilized.
- ↑ The proof of the validity of these algorithms is the subject of formal Predicate Logic.
- ↑ Proof by construction, or proof by example, is the construction of a concrete example with a property to show that something having that property exists. Joseph Liouville, for instance, proved the existence of transcendental numbers by constructing an explicit example.
- ↑ See above, "Non-Constructive Proof."
- ↑ The number of cases sometimes can become very large. For example, the first proof of the "four color theorem" was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases were checked by a computer program, not by hand. The shortest known proof of the "four color theorem" today still has over 600 cases.
- ↑ The latter type of reasoning can be called a 'plausibility argument' and is not a proof; this is clearly seen in the case of the Collatz Conjecture. Probabilistic proof, like proof by construction, is one of many ways to state an existence theorem. Likewise, a "statistical proof" does not prove any proposition with certainty but only "proves" the proposition within a certain range of error or "certainty."
- ↑ Often a bijection between two sets is used to show that the expressions for their two sizes are equal. Alternatively, a double counting argument provides two different expressions for the size of a single set, again showing that the two expressions are equal.
- ↑ So as to distinguish it from predictive, or "weak," induction.