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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

2. A general characterization of logical form

Although differing considerably in content and detail, the basic components and characteristics of Aristotle’s theory of logical form (in this somewhat anachronistic rendering) are essentially the same as in contemporary accounts deriving from Bertrand Russell. In this section, these components and characteristics are made explicit.

In general, the notion of logical form is always relative to a theory T of logical form which is directed towards some broad class K of sentences of natural language. Call K the ‘target class’ of the theory. The logical forms of the members of K are represented by formulas in a ‘logically pure’ canonical language; that is, a language that contains (perhaps in addition to punctuation) only variables and logical constants – expressions with fixed and, according to the theory, distinctly logical meanings (see Logical constants §1). For Aristotle, this is the language of schematic forms the logical constants of which are ‘Every’, ‘Some’, ‘No’, ‘is a’ and ‘is not a’; and for Russell and his followers it is generally (some variant of) the language of Principia Mathematica (1910–13), with its now familiar quantifiers and propositional connectives.

The route from the sentences of the target class K to their logical forms according to T consists of two steps. First, and most important, is the translation of the members of K into an ‘impure’ hybrid language that consists of the canonical language supplemented by nonlogical constants: for Aristotle, ordinary terms such as ‘carnivore’ and also stilted terms such as ‘person identical with Matthew’; for Russellians, individual constants such as ‘Socrates’ and n-place predicates such as ‘mortal’ (or perhaps abbreviations thereof such as ‘s’ and ‘M‘). These nonlogical constants carry the same meaning as their informal counterparts and are used to construct the hybrid language translations of the sentences in which those counterparts occur. The translation of a sentence of K into a hybrid language is typically known as its ‘analysis’. (Sentences such as ‘Every boy danced with a girl’, which are logically ambiguous, should of course receive a distinct analysis for each possible reading.) The analysis of a sentence is a paraphrase of the sentence which displays its logical form overtly in its surface grammatical structure. Thus, (3) and (5) are translated as (4) and (6), respectively, while for the Russellian, (7), for example, is translated as (8):

  • (7) If every man is mortal and Socrates is a man, then Socrates is mortal.

  • (8) (∀y(man(y) → mortal(y)) & man(Socrates)) → mortal(Socrates).

In the second step of the process, the logical forms for the sentences of K are derived simply by uniformly replacing all the nonlogical constants in the analyses of those sentences with variables of the appropriate type. So, for example (assuming a choice of replacement variables for the nonlogical constants), the sentences in (1) – which in this case are identical with their analyses – yield the logical forms in (2), and (8) likewise yields (9) as the logical form of (7):

  • (9) (∀y (Fy →Gy) & Fx) →Gx.

As noted, the primary goal of a theory T of logical form is to account for the logical properties of the sentences in its target class K in terms of their logical forms. This is accomplished by means of a logical theory for the canonical language of T in which formal correlates of the ordinary logical notions – logical truth, logical consequence and so on – are defined for the formulas of the language. The apparent logical properties of the members of K are then explained (or explained away) by virtue of the fact that the formal correlates of those properties hold (or fail to hold) among the logical forms of the members of K. Thus, once again, the intuitive validity of (1) is explained by the formal validity of (2). Similarly, for the Russellian, the intuitive logical truth of (7) is explained by the formal logical truth of (9) – typically understood as truth in any domain under any interpretation (relative to that domain) of the free variables ‘F‘, ‘G’ and ‘x‘, following the work of Tarski (1933; see Model theory). In either case, the actual meanings of the nonlogical expressions in the natural language sentences in question play no role in determining the logical properties of those sentences. Rather, as the theories demonstrate, the logical properties are determined by their logical forms alone.

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Citing this article:
Menzel, Christopher. A general characterization of logical form. Logical form, 1998, doi:10.4324/9780415249126-X021-1. Routledge Encyclopedia of Philosophy, Taylor and Francis, https://www.rep.routledge.com/articles/thematic/logical-form/v-1/sections/a-general-characterization-of-logical-form.
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