# Logical form

DOI
10.4324/9780415249126-X021-1
DOI: 10.4324/9780415249126-X021-1
Version: v1,  Published online: 1998
Retrieved March 01, 2021, from https://www.rep.routledge.com/articles/thematic/logical-form/v-1

## 1. Logical form in Aristotelian logic

Broadly speaking, the ‘logical form’ of a sentence of natural language is what determines both its logical properties and its logical relations to other sentences. Most notably, the logical form of a sentence S determines whether or not it is logically true, and the logical forms of the sentences in a set K together with the logical form of S jointly determine whether or not the latter is a logical consequence of the former.

Although particularly prominent in the twentieth century, due especially to the influence of Bertrand Russell, the idea of logical form can be traced back to Aristotle. In the so-called ‘traditional logic’ that stems from Aristotle’s work in the Prior Analytics, the central form of argument is the ‘syllogism’, that is, an argument consisting of two premises and a conclusion. For example:

• (1) Every whale is a mammal.

• Some carnivore is a whale.

• ∴ Some carnivore is a mammal.

Aristotle was the first to recognize explicitly that the validity of a syllogism (that is, its conclusion being a logical consequence of its premises) is entirely independent of the common noun phrases, or ‘terms’, in its constituent sentences. Rather, each sentence in a valid syllogism such as (1) above exhibits a certain form and, taken together, those forms constitute a pattern that, of itself, guarantees the truth of the conclusion given the truth of the premises.

To capture this idea systematically, Aristotle first restricted his attention to syllogisms the constituent sentences of which exhibit any of four basic sentential forms, known traditionally as A, E, I and O, respectively: ‘Every $\alpha \text{is a}\beta$ ‘, ‘No $\alpha \text{is a}\beta$ ‘, ‘Some $\alpha \text{is a}\beta$ ’ and ‘Some $\alpha \text{is not a}\beta$ ‘, where ‘α’ and ‘β’ represent the roles of the subject and predicate terms in such sentences. Thus, the first premise of (1) is of the sentential form A, and both the second premise and the conclusion are of the form I. These sentential forms alone, however, are not enough to characterize the pattern that (1) exhibits. A further logically relevant feature of (1) is the way in which the terms occur in its constituent sentences: ‘carnivore’, for instance, is the subject term of both the second premise and the conclusion but does not occur at all in the first premise. To capture this feature generally, Aristotle introduced schematic variables F, G, H,…. Call the result of replacing the terms in a sentence that exhibits one of the basic sentential forms with distinct schematic variables a ‘schematic’ form. The pattern exhibited by a given syllogism can then be represented by replacing its constituent sentences with schematic forms that preserve the arrangement of terms in the syllogism. For example, (1) exhibits the following pattern:

• (2) Every F is a G.

• Some H is an F.

• Some H is a G.

By sole means of representations such as (2), Aristotle proved, for each possible syllogistic pattern, whether or not it is valid; that is, for Aristotle, whether or not it is impossible for the premises of any instance of the pattern to be true and the conclusion false. Thus, the schematic forms of its premises and conclusion (relative to a given choice of schematic variables) completely determine the validity or invalidity of each pattern. Aristotle’s schematic forms are thus paradigms of logical forms.

The central goal of a theory of logical form is to explain the logical properties of (and relations between) the members of a broad class K of sentences of natural language in terms of the logical forms that the theory assigns to those sentences. With only four general types of logical form to choose from – the four basic sentential forms, the scope of Aristotle’s account as it stands is far too limited to count as a fully fledged theory of logical form. However, a perusal of any modern text with a section on traditional logic yields a variety of techniques for translating sentences with entirely different grammatical forms into instances of Aristotle’s four basic sentential forms. For example, a simple individual assertion such as

• (3) Matthew is a politician

is obviously not an instance of any of the basic sentential forms, since it does not begin with one of the quantifiers ‘Every’, ‘Some’ and ‘No’. But by introducing a term that picks out a class containing only Matthew, (3) can be translated into the intuitively equivalent, albeit somewhat stilted, sentence

• (4) Every person identical to Matthew is a politician,

which exhibits the sentential form A. Again, a sentence such as

• (5) Someone gave The Brothers Karamazov to Andrea,

which involves a transitive verb, can be put into the appropriate sentential form by replacing the verb with its nominal ‘-er’ counterpart followed by the preposition ‘of’, to produce a term that picks out exactly the class of individuals to which the original predicate applies. (5) then gives way to

• (6) Some person is a giver of The Brothers Karamazov to Andrea.

Supplemented by such techniques, Aristotle’s logic has a considerably broader application, as a much larger class of sentences can be assigned logical forms than would initially appear to be the case. So supplemented, then, Aristotle’s account is plausibly taken to be the first genuine theory of logical form (see Logic, ancient §§1–3).