LOGICAL FORM Kirk Ludwig

1.3 LOGICAL FORM Kirk Ludwig Interest in logical form has been driven historically by interest in valid argument forms, that is, semantic entailment relations between the premises and conclusion of an argument underwritten by their semantic forms. The pressure to provide an ever deeper and more general account of valid argument forms has led to a generalization of the notion of logical form to semantic structure relevant to the evaluation generally of the truth or falsity of a sentence. It is a staple of the tradition since Frege that logical form differs, sometimes dra- matically, from the surface or superficial grammatical form of a sentence. (We return below to whether logical form may be associated with some further level of syntactical analysis.) For example, some sentences, like 2 and 3, have multiple readings (“someone is loved by everyone” and “everyone is such that there is someone he loves”; “relatives who are visiting can be boring” and “it can be boring visiting relatives”) which should be assigned different logical forms because the same words contribute differently to truth conditions on each. Some sentences with the same superficial form, such as 1 and 2, 4–6, 9 and 10, 11 and 12, and 13 and 14, are assigned different logical forms because they enter into different systematic entailment relations. And some sentences with superficially different forms, such as 6–8, are assigned the same logical form because they appear to enter into the same systematic entailments. 1. Mary loves John 2. Everyone loves someone 3. Visiting relatives can be boring 4. The hearth is warm 5. The weather is fine 6. The whale is a mammal 7. Whales are mammals 8. Every whale is a mammal 9. Felix does not bark 10. Pegasus does not exist 11. The president is 6' 1" inches tall 12. The average man is 5' 9" inches tall 13. Jack is a wolf 14. Jack ate a wolf KIRK LUDWIG In the early analytic tradition, the divergence of surface and logical form was associated with a divergence between the form of thoughts and the sentences expressing them. Wittgenstein puts this vividly in his Tractatus Logico-Philosophicus ([1921] 1961: 4.002). Language disguises thought. So much so, that from the outward form of the clothing it is impossible to infer the form of the thought beneath it, because the outward form of the clothing is not designed to reveal the form of the body, but for entirely different purposes. Bertrand Russell, in the second of his 1914 Lowell lectures, Our Knowledge of the External World, defined “philosophical logic” as that portion of logic that concerned the study of forms of propositions—or, as he called them, “logical forms.” He claimed 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. ([1914] 1993: 53) The most famous exemplar of this activity is Russell’s Theory of Descriptions (1905), according to which the logical form of “The King of France is bald” is not that of a subject–predicate sentence (or even a noun phrase + verb phrase construction—we return to this divergence later) but a quantificational sentence with internal truth-functional structure, “There is a king of France and whatever is a king of France is identical with it and it is bald,” in which the grammatical subject of the original is not treated as con- tributing any entity to the meaning of the sentence, in contrast to, say, “Louis XIV was bald” (5, 6, and 12 introduce additional complexities). In the early twentieth century, especially in the short-lived but influential theory of Logical Atomism, logical form was directly connected with metaphysics and the philos- ophy of mind (Russell [1918–19] 1985; Wittgenstein [1921] 1961). The logical form of sentences was taken to reveal both the forms of possible facts or reality and the thoughts expressed with them. This interest in logical form is clearly broader than an interest in inference patterns among sentences induced by the meanings of logical terms. For Russell, for example, a crucial question about logical form was whether belief sentences express relations to propositions, a view he held at one point but later rejected (Rus- sell [1918–19] 1985: 87–8). (Importantly, Russell’s and Wittgenstein’s interest in the structure of thought blurs the distinction Davidson drew later between investigation of logical form and conceptual analysis (Davidson [1980] 2001b: 105–6, [1984] 2001c: 31).) Even apart from Logical Atomism, investigation of logical form is an important com- ponent of what Strawson (1959) called “descriptive metaphysics,” the project of uncov- ering the ontology embedded in natural languages, what must exist if the sentences we endorse are true. Given Quine’s (1948: 33) criterion of ontological commitment, according to which “a theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true,” it follows that one goal of the recovery of logical form is to reveal the quantificational commitments of natural language sentences (see Davidson 30 LOGICAL FORM 1977; Higginbotham 1993). We will consider a dramatic example below in the event analysis of action sentences. One response to the divergence of surface and logical form is to treat propositions or possible facts as the primary bearers of logical form. Sentences then have the same logical forms if and only if the propositions or possible facts they express do. This gives us, though, only a temporary illusion of progress. As Ayer put it, This would be all very well, if we had any means of determining the logical forms . other than through the grammatical forms of the sentences which are used to state them. What happens, in practice, is that we decide, on other grounds, which forms of sentences convey this information most perspicuously, and that these are not always the grammatical forms in which the sentences are originally cast. (1984: 29–30) Ayer’s last remark points to a popular way of talking about logical form, according to which the logical form of a natural language sentence is the form of a sentence in a specially regimented, “ideal,” typically formal language, that translates it—or the forms of the sentences that translate it if it is structurally ambiguous (Kalish 1952; Harman 1972; Audi 1995: 511–12; Sainsbury 2001). These sentences are sometimes said to be, and sometimes to give, the logical forms of the originals. The ideal or regimented language is to have no ambiguities and to encode in its syntax all differences in the logical or semantic roles of terms in them. An interpreted formal language for a first-order logic, designed to make shared validity-relevant semantic features correspond to the construction of a sentence out of its parts, would be an example. Natural language sentences have the same form if and only if they are translated by sentences of the regimented language of the same form. This is not, however, a satisfactory account, for it leaves us with the question of what the form of the sentence in the ideal language is (Grandy 1974). If we think of it in terms of the pattern of terms in it, perhaps the pattern of logical terms, then this makes the notion of logical form relative to the choice the favored language, and there are many that would do equally well. Russell’s analysis of “The King of France is bald” can be rendered into a standard infix logical notation or into Polish or prefix notation (with “f(x)” for “x is a king of France,” “b(x)” for “x is bald,” and “i(x,y)” in prefix notation for “x = y”), as in 15 and 16. 15. (x)((f(x) & (y)(f(y) ʛx = y)) & b(x)) 16. ͚xKKf(x)͟yCf(y)i(x,y)b(x) The logical form of “the king of France is bald” cannot be identified with either, or with the patterns of the terms in them, since they are distinct and the choice arbitrary. We can only say what the logical form is relative to a system of regimentation. Yet, the original and its translations (if they are) share logical form. Sensitivity to this point explains why philosophers often talk of such renderings as giving the logical form of the original. But this leaves us with the question of what it gives and how it is doing it. The idea that logical form attaches first to propositions points to a resolution, for propositions (in the Frege–Russell tradition) were introduced to be simultaneously the objects of thought and the meanings of (declarative) sentences. This suggests 31 KIRK LUDWIG looking midway between the sentence and the proposition it expresses to a comprehen- sive semantic theory that details how words and structures contribute to fixing truth conditions for sentences. An influential suggestion of Donald Davidson’s along these lines is that the notion of logical form is best placed in the context of an interpretive truth theory for a natural language. What should we ask of an adequate account of the logical form of a sentence? Above all . such an account must lead us to see the semantic character of the sentence—its truth or falsity—as owed to how it is composed, by a finite number of applications of some of a finite number of devices that suffice for the language as a whole, out of elements drawn from a finite stock (the vocabulary) that suffices for the language as a whole.

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