formal fallacy an invalid inference pattern that is described in terms of a formal logic. There are three main cases: (1) an invalid (or otherwise unacceptable) argument identified solely by its form or structure, with no reference to the content of the premises and conclusion (such as equivocation) or to other features, generally of a pragmatic character, of the argumentative discourse (such as unsuitability of the argument for the purposes for which it is given, failure to satisfy inductive standards for acceptable argument, etc.; the latter conditions of argument evaluation fall into the purview of informal fallacy); (2) a formal rule of inference, or an argument form, that is not valid (in the logical system on which the evaluation is made), instances of which are sufficiently frequent, familiar, or deceptive to merit giving a name to the rule or form; and (3) an argument that is an instance of a fallacious rule of inference or of a fallacious argument form and that is not itself valid.
The criterion of satisfactory argument typically taken as relevant in discussing formal fallacies is validity. In this regard, it is important to observe that rules of inference and argument forms that are not valid may have instances (which may be another rule or argument form, or may be a specific argument) that are valid. Thus, whereas the argument form (i) P, Q; therefore R (a form that every argument, including every valid argument, consisting of two premises shares) is not valid, the argument form (ii), obtained from (i) by substituting P&Q for R, is a valid instance of (i): (ii) P, Q; therefore P&Q. Since (ii) is not invalid, (ii) is not a formal fallacy though it is an instance of (i). Thus, some instances of formally fallacious rules of inference or argument-forms may be valid and therefore not be formal fallacies. Examples of formal fallacies follow below, presented according to the system of logic appropriate to the level of description of the fallacy. There are no standard names for some of the fallacies listed below. Fallacies of sentential (propositional) logic. Affirming the consequent: If p then q; q / , p. ‘If Richard had his nephews murdered, then Richard was an evil man; Richard was an evil man. Therefore, Richard had his nephews murdered.’ Denying the antecedent: If p then q; not-p / , not-q. ‘If North was found guilty by the courts, then North committed the crimes charged of him; North was not found guilty by the courts. Therefore, North did not commit the crimes charged of him.’ Commutation of conditionals: If p then q / , If q then p. ‘If Reagan was a great leader, then so was Thatcher. Therefore, if Thatcher was a great leader, then so was Reagan.’ Improper transposition: If p then q / , If not-p then not-q. ‘If the nations of the Middle East disarm, there will be peace in the region. Therefore, if the nations of the Middle East do not disarm, there will not be peace in the region.’ Improper disjunctive syllogism (affirming one disjunct): p or q; p / , , not-q. ‘Either John is an alderman or a ward committeeman; John is an alderman. Therefore, John is not a ward committeeman.’ (This rule of inference would be valid if ‘or’ were interpreted exclusively, where ‘p or EXq’ is true if exactly one constituent is true and is false otherwise. In standard systems of logic, however, ‘or’ is interpreted inclusively.) Fallacies of syllogistic logic. Fallacies of distribution (where M is the middle term, P is the major term, and S is the minor term). Undistributed middle term: the middle term is not distributed in either premise (roughly, nothing is said of all members of the class it designates), as in
Some P are M ‘Some politicians are crooks.
Some M are S Some crooks are thieves. ,Some S are P. ,Some politicians are thieves.’ Illicit major (undistributed major term): the major term is distributed in the conclusion but not in the major premise, as in
All M are P ‘All radicals are communists.
No S are M No socialists are radicals. ,Some S are ,Some socialists are not not P. communists.’ Illicit minor (undistributed minor term): the minor term is distributed in the conclusion but not in the minor premise, as in
All P are M ‘All neo-Nazis are radicals.
All M are S All radicals are terrorists. ,All S are P. ,All terrorists are neo- Nazis.’ Fallacies of negation. Two negative premises (exclusive premises): the syllogism has two negative premises, as in
No M are P ‘No racist is just.
Some M are not S Some racists are not police. ,Some S are not P. ,Some police are not just. Illicit negative/affirmative: the syllogism has a negative premise (conclusion) but no negative conclusion (premise), as in
All M are P ‘All liars are deceivers.
Some M are not S Some liars are not aldermen. ,Some S are P. ,Some aldermen are deceivers.’ and
All P are M ‘All vampires are monsters.
All M are S All monsters are creatures. ,Some S are not P. ,Some creatures are not vampires.’
Fallacy of existential import: the syllogism has two universal premises and a particular conclusion, as in All P are M ‘All horses are animals. No S are M No unicorns are animals. ,Some S are not P. ,Some unicorns are not horses.’ A syllogism can commit more than one fallacy. For example, the syllogism Some P are M Some M are S ,No S are P commits the fallacies of undistributed middle, illicit minor, illicit major, and illicit negative/affirmative. Fallacies of predicate logic. Illicit quantifier shift: inferring from a universally quantified existential proposition to an existentially quantified universal proposition, as in (Ex) (Dy) Fxy / , (Dy) (Ex) Fxy ‘Everyone is irrational at some time (or other) /, At some time, everyone is irrational.’ Some are/some are not (unwarranted contrast): inferring from ‘Some S are P’ that ‘Some S are not P’ or inferring from ‘Some S are not P ‘ that ‘Some S are P ‘, as in (Dx) (Sx & Px) / , (Dx) (Sx & -Px) ‘Some people are left-handed / , Some people are not left-handed.’ Illicit substitution of identicals: where f is an opaque (oblique) context and a and b are singular terms, to infer from fa; a = b / , fb, as in ‘The Inspector believes Hyde is Hyde; Hyde is Jekyll / , The Inspector believes Hyde is Jekyll.’ See also EXISTENTIAL IMPORT , LOGICAL FORM , MODAL LOGIC , SYLLOGIS. W.K.W.