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What Is Logical Form? What is Logical Form? Ernest Lepore Center for Cognitive Science Rutgers University New Brunswick, NJ 08903 [email protected] Kirk Ludwig Department of Philosophy University of Florida Gainesville, FL 32611-8545 [email protected] in Logical Form and Language Ed. Gerhard Preyer 2001 Oxford University Press What is Logical Form? Ernest Lepore and Kirk Ludwig Philosophy, as we use the word, is a fight against the fascination which forms of expression exert upon us. ‘ Wittgenstein I Bertrand Russell, in the second of his 1914 Lowell lectures, Our Knowledge of the External World, asserted famously that ‘every philosophical problem, when it is subjected to the necessary analysis and purification, is found either to be not really philosophical at all, or else to be, in the sense in which we are using the word, logical’ (Russell 1993, p. 42). He went on to characterize that portion of logic that concerned the study of forms of propositions, or, as he called them, ‘logical forms’. This portion of logic he called ‘philosophical logic’. Russell asserted that ... some kind of knowledge of logical forms, though with most people it is not explicit, is involved in all understanding of discourse. It is the business of philosophical logic to extract this knowledge from its concrete integuments, and to render it explicit and pure. (p. 53) Perhaps no one still endorses quite this grand a view of the role of logic and the investigation of logical form in philosophy. But talk of logical form retains a central role in analytic philosophy. Given its widespread use in philosophy and linguistics, it is rather surprising that the concept of logical form has not received more attention by philosophers than it has. The concern of this paper is to say something about what talk of logical form comes to, in a tradition that stretches back to (and arguably beyond) Russell’s use of that expression. This will not be exactly Russell’s conception. For we do not endorse Russell’s view that propositions are the bearers of logical form, or that appeal to propositions adds anything to our understanding of what talk of logical form comes to. But we will be concerned to provide an account responsive to the interests expressed by Russell in the above quotations, though one clarified of extraneous elements, and expressed precisely. For this purpose, it is important to note that the concern expressed by Russell in the above passages, as the surrounding text makes clear, is a concern not just with logic conceived narrowly as the study of logical terms, but with propositional form more generally, which includes, e.g., such features as those that correspond to the number of argument places in a propositional function, and the categories of objects which propositional 1 functions can take as arguments. This very general concern with form is expressed above in the claim that all understanding of discourse involves some knowledge of logical forms. It is logical form in this very general sense, which is connected with an interest in getting clear about the nature of reality through getting clear about the forms of our thoughts or talk about it, with which we will be concerned.1 The conception we will champion dispenses with talk of propositions, reified sentence meanings, as a useless excrescence, and treats logical form as a feature of sentences. Consonant with Russell’s general interest in the form of propositions, we will treat talk about the logical form of a sentence in a language L to be essentially about semantic form as revealed in a compositional meaning theory for L. We do not, however, treat logical form itself as a sentence, or anything else. On our account, it is a mistake to think that logical forms are entities, or to think of logical form as revealed by what symbols occur in a sentence, either in its surface syntax, or in the syntax of its translation into an ‘ideal’ language. Rather, we will take the relation of sameness of logical form as basic. We will give a precise account of the notion of sameness of logical form between any two sentences in any two languages, first for declarative sentences, then for sentences in any sentential mood. Our account is inspired by remarks of Davidson, and we develop the account for declaratives in the context of a Davidsonian truth- theoretic semantics. We develop the account for non-declaratives in terms of a generalization of the notion of an interpretive truth theory, namely, that of an interpretive fulfillment theory. We will also be concerned to say something about the relation of this characterization of logical form to logic more narrowly conceived, that is, a study of the semantics of logical terms or structures. We will urge that these are distinct, and, to some degree, independent concerns. We will also suggest a criterion (essentially due to Davidson) for picking out logical terms or structures that is particularly salient from the standpoint on logical form we advance, though we make no claim for its being the only way of extending in a principled way the use of the notion beyond where it is currently well-grounded. (This discussion will show, incidentally, that no good basis exists, contrary to what has been relatively recently alleged (Etchemendy 1983; Etchemendy 1988; Etchemendy 1990; Lycan 1989), for denying that a principled distinction between the logical and non-logical terms of a language can be drawn.) 2 The program of our paper is as follows. In section II, we consider the origins of the notion of logical form in reflection on argument form, and criticize two traditional conceptions, one of which remains dominant. In section III, we introduce the notion of logical form we wish to develop, logical form as semantic form, and describe our conception of how to use a truth theory to give a compositional semantic theory for declarative sentences in a language as background for our development of this conception. In section IV, we give a precise characterization of sameness of logical form of sentences applicable across languages in terms of the notion of corresponding proofs of the T-sentences for them in interpretive truth theories for the languages. This allows us to clarify what could be meant by the expression ‘x is the logical form of y’. In section V, we employ examples from natural language semantics in illustration of the usefulness of the present approach. In section VI, we show how the basic approach can be extended to non- declarative sentences. (This extension is based on some work by one of the authors (Ludwig) which the other (Lepore) has some reservations about, so it is put forward here as a suggestion about how this desirable extension might be effected.) In section VII, we discuss the relation of the conception of logical form we advance to the project of identifying logical terms or structures, and contrast it with an alternative conception articulated in terms of an invariance condition traceable back to (Tarski 1986) and (Lindstrom 1966a). Section VIII is a brief summary and conclusion. II The origin of interest in logical form lies in the recognition that many intuitively valid natural language arguments can be classified together on the basis of common features, a form which guarantees their validity apart from their different content. We group together arguments which exemplify a pattern, and say that they share a form. Forms of arguments are represented by replacing (certain) of the expressions in their premises and conclusions with schematic letters ‘ thereby abstracting away from what the arguments are about. This gives rise to a common characterization of the logical form of a sentence, namely, that structure of a sentence that determines from which sentences it can be validly deduced, and which sentences can be validly deduced from it and other premises, where these sentences are in turn characterized in terms of their logical structures. 3 This loose characterization is far from satisfactory because it leaves unexplicated how ‘structure of a sentence’ is being used. Logical form cannot be just any schema that results from replacing one or more expressions within a sentence. There are too many, and not every such schema will be taken to reveal logical form. In addition, for sentences with more than one reading, such as [1], we associate more than one logical form, but they will generate the same schemas. [1] Everyone loves someone. Similarly, sentences we are intuitively inclined to assign distinct logical forms, such as the pairs [1]-[2], [3]-[4], and [5]-[6], yield the same schemas. Likewise, sentences, as might be urged for the pair [6]-[7], to which we wish to assign the same logical form (in the same or different languages) may yield distinct schemas. Examples can be multiplied endlessly. [2] John loves Mary. [3] Dogs bark. [4] Unicorns exist. [5] The President is a scoundrel. [6] The whale is a mammal. [7] Everything which is a whale is a mammal. Russell’s response, of course, was to bypass sentences and to take logical form to be a property of the propositions that sentences express (as above). This renders intelligible talk of similar sentences having distinct logical forms, and of different sentences, in the same or different languages, having the same logical form. Sentences on this view can be said in a derivative sense to have logical form: sentences have the same logical form when they express propositions with the same logical form. An alternative approach, more usual today, is to identify the logical form of a natural language sentence as the form of a sentence in a specially regimented, ‘ideal’, perhaps formal, language that translates it (or, in the case of an ambiguous sentence, the logical forms it can have are associated with the sentences that translate the various readings of it).2 A regimented language must contain no ambiguities and syntactically encode all differences in the logical (or semantical) roles of terms.
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