Logical Form, Origins of

When philosophers use the expression "the Logical Form of a sentence," they refer to a linguistic representation whose desirable properties the surface grammatical form of the sentence either masks or lacks completely. Because philosophers have found ever so many desirable properties hidden or absent in grammatical form, there are ever so many different notions of Logical Form.

According to one tradition, which one might call the "descriptive conception" of Logical Form, the Logical Form of a sentence is something like the "deep structure" of that sentence (e.g., Harman 1972), and we may discover that the "real" structure of a natural language sentence is in fact quite distinct from its surface grammatical form. Talk of Logical Form in this sense involves attributing hidden complexity to natural language, complexity that may be revealed by philosophical, and indeed empirical, inquiry (see LOGICAL FORM IN LINGUISTICS). However, perhaps more common in the recent history of philosophy is what one might call the "revisionary conception" of Logical Form. According to it, natural language is defective in some fundamental way. Appeals to Logical Form are appeals to a kind of linguistic representation which is intended to replace natural language for the purposes of scientific or mathematical investigation (e.g., Frege 1879, preface; Whitehead and Russell 1910, introduction; Russell 1918, 58).

Nineteeth-century debates concerning the foundations of the calculus are one source of the revisionary flavor of some contemporary conceptions of Logical Form (e.g., Quine 1960, 248ff.). Perhaps the most vivid example is the overthrow of the infinitesimal calculus, which began with the work of Cauchy in the 1820s. Cauchy took the notation of the infinitesimal calculus, which contained explicit reference to infinitesimals, and reanalyzed it in terms of a notation that exploited the limit concept, and made no reference to infinitesimals. It subsequently emerged that the limit concept was analyzable in terms of logical notions, such as quantification, together with unproblematic numerical concepts. Thus progress in nineteenth-century mathematics involved replacing a notation that made explicit reference to undesirable entities (viz., infinitesimals) with a notation in which reference to such entities was replaced by logical and numerical operations on more acceptable ones (noninfinitesimal real numbers).

Bertrand Russell (e.g., 1903) was clearly affected by the developments in nineteenth-century mathematics, though his proposals (1905) have both revisionary and descriptive aspects. In them Russell treated the problem of negative existential sentences, such as "Pegasus does not exist." The difficulty with this sentence is that its grammatical form suggests that endorsing its truth commits one to the existence of a denotation for the proper name "Pegasus," which could be none other than that winged horse. Yet what the sentence seems to assert is precisely the nonexistence of such a being.

According to Russell, the problem lies in taking the grammatical form to be a true guide to the structure of the proposition it expresses, that is, in taking the surface grammatical form to exhibit the commitments of its endorsement. In Russell's view, the structure of the proposition expressed by "Pegasus does not exist" differs markedly from the grammatical form. Rather than being "about" Pegasus, its structure is more adequately reflected by a sentence such as "It is not the case that there exists a unique thing having properties F, G, and H," where F, G, and H are the properties we typically associate with the fictional winged horse. Once this is accepted, we may endorse the truth of "Pegasus does not exist" without fear of admitting Pegasus into the realm of being .

Russell took himself not as proposing a special level of linguistic representation, but rather as proposing what the true structure of the (nonlinguistic) proposition expressed by the sentence was. Russell's early writings are therefore no doubt the origin of the occasional usage of "Logical Form" as referring to the structure of a nonlinguistic entity such as a proposition or a fact (e.g., Wittgenstein 1922, 1929; Sainsbury 1991, 35). However, Russell's proposal could just as easily be adopted as a claim about a special sort of linguistic representation, one that allows us to say what we think is true, while freeing us from the absurd consequences endorsement of the original grammatical form would entail (Russell, after his rejection of propositions, himself construed it in this way). Thus arises a conception of Logical Form as a level of linguistic representation at which the metaphysical commitments of the sentences are completely explicit. This conception of Logical Form was later to achieve full articulation in the works of Quine (e.g., 1960, chaps. 5, 7) .

Nineteenth-century mathematics not only provided successful examples of notational revision; it also provided many examples of notational confusion (Wilson 1994 argues that this was not necessarily a tragic situation). One of Frege's central purposes (1879, Preface) was to provide a notation free of such confusion, devoid of vagueness, ambiguity, and context dependence, whose purpose was "to provide us with the most reliable test of the validity of a chain of inferences and to point out every presupposition that tries to sneak in unnoticed . . . ." Frege's remarks here suggest a test of the validity of arguments in terms of the syntax of his idealized language. According to this criterion, a chain of reasoning is logically valid if and only if its translation into his idealized "Begriffsschrift" proceeds by transitions, each of which is of the right syntactic form. Thus arises a conception of Logical Form as a linguistic representation for which it is possible to designate certain syntactic forms such that all and only basic logical transitions are translatable into instances of those forms (see LOGIC).

The purpose of formalization into Logical Form is to replace one notation by another, which has desirable properties which the original notation lacked. In the case of the Quinean conception of Logical Form, the purpose of formalization is to replace a notation (in the usual case, natural language) by one which more accurately reveals ontological commitments. In the case of the Fregean conception, the purpose of formalization is to replace notations which obscure the logical structure of sentences by one which makes this structure explicit (these are not necessarily conflicting goals). Many other conceptions of Logical Form have been proposed. For example, Richard Montague's favored way of giving an interpretation to a fragment of natural language involved translating that fragment into an artificial language, for example, the language of "tensed intensional logic" (Montague 1973), and then giving a formal semantic interpretation to the artificial language. This produces a conception of Logical Form as a level of linguistic representation at which a compositional semantics is best given (see COMPOSITIONALITY and SEMANTICS).

Strictly revisionary conceptions of Logical Form involve abstracting from features of the original notation that are problematic in various ways. Because notations may be problematic in some ways and not in others, in such a use of "Logical Form," there is no issue about what the "correct" notion of Logical Form is. Descriptive conceptions, on the other hand, involve claims about the "true" structure of the original notation, structure that is masked by the surface grammatical form. Someone who makes a proposal, in the purely descriptive mode, about the "true" Logical Form of a natural language sentence thus runs the risk that her claims will be vitiated by linguistic theory. To avoid this danger, most philosophers vascillate between a revisionary and descriptive use of the expression "Logical Form."

The tension between descriptive and revisionary conceptions of Logical Form mirrors a tension in the cognitive sciences generally. According to some, cognitive science should be interested in explaining the possession of abstract cognitive abilities such as thinking and language use in humans, and we should be interested, not in the most ideal representations of thought and language, but rather in how humans think and speak. If so, we should be interested in Logical Form only insofar as it is plausibly associated with natural language on some level of empirical analysis. According to others, cognitive sciences should be interested in the abstract properties of thinking and speaking, and we should be interested in arriving at an ideal representational system, one that may abstract from the defects of human natural language. Among such thinkers, the revisionary project of replacing natural language by a representation more suitable for scientific purposes is fundamental to the aims of cognitive science.

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-- Jason C. Stanley


Frege, G. (1879). Begriffsschrift: A formula language, modeled upon that of arithmetic, for pure thought. In Van Heijenoort, Ed., From Frege to Godel: A Sourcebook in Mathematical Logic, 1879-1931. Cambridge: Harvard University Press, 1967, pp. 5 - 82.

Harman, G. (1972). Deep structure as logical form. In D. Davidson and G. Harman, Eds., Semantics of Natural Language. Dordrecht: Reidel.

Montague, R. (1973). The proper treatment of quantification in ordinary English. In J. Hintikka, H. Moravcsik, and P. Suppes, Eds., Approaches to Natural Language: Proceedings of the 1970 Stanford Workshop on Grammar and Semantics. Dordrecht: Reidel, pp. 221-242.

Quine, W. V. O. (1960). Word and Object. Cambridge, MA: MIT Press.

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Russell, B. (1905). On denoting. Mind 14:479-493.

Russell, B. (1918/1985). The Philosophy of Logical Atomism. LaSalle: Open Court.

Sainsbury, M. (1991). Logical Forms: An Introduction to Philosophical Logic. London: Routledge and Kegan Paul.

Whitehead, A., and B. Russell. (1910). Principia Mathematica. Cambridge: Cambridge University Press.

Wilson, M. (1994). Can we trust Logical Form? Journal of Philosophy October 91:519-544.

Wittgenstein, L. (1922). Tractatus Logico-Philosophicus. Translated by C. K. Ogden. London: Routledge and Kegan Paul.

Wittgenstein, L. (1929). Some remarks on Logical Form. Vol 9, Proceedings of the Aristotelean Society .