logical form the form obtained from a proposition, a set of propositions, or an argument by abstracting from the subject matter of its content terms or by regarding the content terms as mere placeholders or blanks in a form. In a logically perfect language the logical form of a proposition, a set of propositions, or an argument is determined by the grammatical form of the sentence, the set of sentences, or the argument-text expressing it. Two sentences, sets of sentences, or argument-texts are said to have the same grammatical form, in this sense, if a uniform one-toone substitution of content words transforms the one exactly into the other. The sentence ‘Abe properly respects every agent who respects himself’ may be regarded as having the same grammatical form as the sentence ‘Ben generously assists every patient who assists himself’. Substitutions used to determine sameness of grammatical form cannot involve change of form words such as ‘every’, ‘no’, ‘some’, ‘is’, etc., and they must be category-preserving, i.e., they must put a proper name for a proper name, an adverb for an adverb, a transitive verb for a transitive verb, and so on. Two sentences having the same grammatical form have exactly the same form words distributed in exactly the same pattern; and although they of course need not, and usually do not, have the same content words, they do have exactly the same number of content words. The most distinctive feature of form words, which are also called syncategorematic terms or logical terms, is their topic neutrality; the form words in a sentence are entirely independent of and are in no way indicative of its content or topic.
Modern formal languages used in formal axiomatizations of mathematical sciences are often taken as examples of logically perfect languages. Pioneering work on logically perfect languages was done by George Boole (1815–64), Frege, Giuseppe Peano (1858–1952), Russell, and Church. According to the principle of logical form, an argument is (formally) valid or invalid in virtue of logical form. More explicitly, every two arguments in the same form are both valid or both invalid. Thus, every argument in the same form as a valid argument is valid and every argument in the same form as an invalid argument is invalid. The argument form that a given argument fits (or has) is not determined solely by the logical forms of its constituent propositions; the arrangement of those propositions is critical because the process of interchanging a premise with the conclusion of a valid argument can result in an invalid argument.
The principle of logical form, from which formal logic gets its name, is commonly used in establishing invalidity of arguments and consistency of sets of propositions. In order to show that a given argument is invalid it is sufficient to exhibit another argument as being in the same logical form and as having all true premises and a false conclusion. In order to show that a given set of propositions is consistent it is sufficient to exhibit another set of propositions as being in the same logical form and as being composed exclusively of true propositions. The history of these methods traces back through non-Cantorian set theory, non-Euclidean geometry, and medieval logicians (especially Anselm) to Aristotle. These methods must be used with extreme caution in languages such as English that fail to be logically perfect as a result of ellipsis, amphiboly, ambiguity, etc. For example, ‘This is a male dog’ implies ‘This is a dog’ but ‘This is a brass monkey’ does not imply ‘This is a monkey’, as would be required in a logically perfect language. Likewise, of two propositions commonly expressed by the ambiguous sentence ‘Ann and Ben are married’ one does and one does not imply the proposition that Ann is married to Ben.
Quine and other logicians are careful to distinguish, in effect, the (unique) logical form of a proposition from its (many) schematic forms. The proposition (A) ‘If Abe is Ben, then if Ben is wise Abe is wise’ has exactly one logical form, which it shares with (B) ‘If Carl is Dan, then if Dan is kind Carl is kind’, whereas it has all of the following schematic forms: (1) If P then if Q then R; (2) If P then Q; (3) P. The principle of form for propositions is that every two propositions in the same logical form are both tautological (logically necessary) or both non-tautological. Thus, although propositions A and B are tautological there are non-tautological propositions that fit the three schematic forms just mentioned. Failure to distinguish logical form from schematic form has led to fallacies. According to the principle of logical form quoted above every argument in the same logical form as an invalid argument is invalid, but it is not the case that every argument sharing a schematic form with an invalid argument is invalid. Contrary to what would be fallaciously thought, the conclusion ‘Abe is Ben’ is logically implied by the following two propositions taken together, ‘If Abe is Ben, then Ben is Abe’ and ‘Ben is Abe’, even though the argument shares a schematic form with invalid arguments ‘committing’ the fallacy of affirming the consequent. See also AMBIGUITY, FORMAL LOGIC, LAWS OF THOUGHT, LOGICAL SYNTAX , TAUTOLOGY. J.Cor.