Barbara H Partee
University of Massachusetts Amherst
There have been centuries of study of logic and of language. Some philosophers and logicians have argued that natural language is logically deficient, or even that “natural language has no logic”. And before the birth of formal semantics in the late 1960’s, most linguists and philosophers were agreed that there was a considerable mismatch between the syntactic structure of natural language sentences and their “logical form”. The logician and philosopher Richard Montague argued that natural languages do have a very systematic semantic structure, but that it can be understood only if one uses a rich enough logic to mirror the rich syntactic structure of natural languages. This essay briefly sketches the history of arguments about the relation between natural language syntax and logical structure, concentrating on the period from Frege to Montague, roughly 1880 to 1970, illustrating the issues with sentences containing quantifiers.
1. Semantics in American Linguistics before 1970
Semantics tended to be rather neglected in early and mid-20th century American linguistics for several reasons. There had been rather little semantics in early American anthropological linguistics, which focused on linguistic fieldwork. The behaviorists viewed meaning as an unobservable aspect of language, not fit for scientific study, which influenced the American structuralists. And Quine had strong skepticism about the concept of meaning, and had some influence on Chomsky.
At the same time there was great progress in semantics in logic and philosophy of language, but that was relatively unknown to most linguists.
In 1954, Yehoshua Bar-Hillel wrote an article in Language (Bar-Hillel 1954) inviting cooperation between linguists and logicians, arguing that advances in both fields made the time ripe for combining forces to work on syntax and semantics together. But Chomsky (1955) rebuffed the invitation, arguing that the artificial languages invented by logicians were too unlike natural languages for any methods the logicians had developed to have any chance of being useful for developing linguistic theory.
Various linguists proposed ways of adding semantics to Chomsky’s syntactic theories starting in the 1960’s (see Partee 2011); Chomsky himself ranged from ambivalent (1965) to skeptical (later) about semantics. His 1965 syntactic theory (Chomsky 1965) posited a notion of “Deep Structure” which was hypothesized to form the input to semantic interpretation, as proposed by (Katz and Postal 1964); then transformational rules would map that structure onto “Surface Structure”, which in turn formed the input to the phonogical component. On this approach syntax was the central mediator of the mapping between sound and meaning.
Examples like (1) led the Generative Semanticists (Lakoff, Ross, McCawley, and others) to abandon Chomsky’s 1965 view of Deep Structure, since on that theory, the Deep Structure of (1) was something like (2). The resulting “Linguistic Wars” have been well documented (Harris 1993, Newmeyer 1980).
(1) Everyone wants to win.
(2) Everyone wants everyone to win.
What were the early linguistic notions of “logical form”? In Generative Semantics (Lakoff, Ross, McCawley, Postal, and others) (see Newmeyer 1996), the belief was that in order for deep structure to capture semantics, it needed to be deeper, more abstract, more like “logical form”, which for linguists meant first-order logic. In fact, both linguists and philosophers who worried about the semantics of quantified sentences before Montague’s work thought of “logical form” in terms of first-order logic. But given that generalized quantifiers were only developed starting in the late 1950’s, that could hardly have been otherwise.
2. Semantics in Logic and Philosophy before 1970
When Aristotle invented logic, he focused on quantification; operators like and and or were added by the Stoics. Implicit in Aristotle’s syllogistic is a semantics for the quantifiers. Each of the four quantifier expressions can be seen as standing for a binary relation between properties:
(3) (i) all: (A,B) ⇔ A ⊆ B
(ii) some: (A,B) ⇔ A ∩ B ≠ ∅
In hindsight, this is close to the idea of Generalized Quantifiers. But the idea of giving a semantic value to the quantifiers themselves was not explicitly developed until much later, really not until Frege.
Frege. The greatest foundational figure for formal semantics is Gottlob Frege (1848-1925). His crucial ideas include the Compositionality Principle and the idea that function-argument structure is the key to semantic compositionality (see Janssen 2012, Krifka 1999, Partee 1984, Pelletier 1994, Szabó 2008).
The Principle of Compositionality: The meaning of a complex expression is a function of the meanings of its parts and of the way they are syntactically combined.
One of Frege’s great contributions was the logical structure of quantified sentences. That was part of the design of the Begriffschrift (Frege 1879), a “logically perfect language” to satisfy Leibniz’s goals; he did not see himself as offering an analysis of natural language, but a tool to augment it, as the microscope augments the eye.
Does ordinary language “have no logic”? As Russell, Carnap, and Tarski were making advances in logic and the philosophy of language, a war began within philosophy of language, the “Ordinary Language” vs “Formal Language” war (Cocchiarella 1997). Ordinary Language Philosophers rejected the formal approach, and urged more attention to ordinary language and its uses. Strawson said in ‘On referring’ (Strawson 1950): “The actual unique reference made, if any, is a matter of the particular use in the particular context; …Neither Aristotelian nor Russellian rules give the exact logic of any expression of ordinary language; for ordinary language has no exact logic.” Russell (1957) replied, “I may say, to begin with, that I am totally unable to see any validity whatever in any of Mr. Strawson’s arguments.” But near the end of the paper, he adds, “I agree, however, with Mr. Strawson’s statement that ordinary language has no logic.”
Russell was not the first logician to complain about the illogicality of natural language. One of his favorite complaints was that English puts phrases like “every man”, “a horse”, “the king” into the same syntactic category as proper names. He considered the formulas of his first-order logic a much truer picture of “logical form” than English sentences.
One way to appreciate Montague’s use of a higher-order typed logic, including generalized quantifiers, is to consider the question of where in Russell’s formula (4), symbolizing Every man walks, is the meaning of every man?
(4) ∀x (man(x) → walk(x) )
The answer is that it is distributed over the whole formula – in fact everything except the predicate walk in the formula can be traced back to every man.
3. Arguing that natural language is not “illogical”.
One way to answer Russell is to devise a logic in which the translation of every man is a constituent in the logical language. Terry Parsons did it with a variable-free combinatoric logic (Parsons 1968, Parsons 1972). Montague did it with a higher-order typed intensional logic (Montague 1973). Both were reportedly influenced by seeing how to devise algorithms for mapping from (parts of) English onto formulas of first-order logic, thereby realizing that English itself was not so logically unruly. (See also Lewis 1970.) First-order logic has many virtues, but similarity to natural language syntax is not one of them.
Montague, a student of Tarski’s, contributed greatly to the development of formal semantics with his development of intensional logic and his combination of pragmatics with intensional logic (Montague 1968, 1970, 1974). His higher order typed intensional logic unified modal logic, tense logic, and the logic of the propositional attitudes, extending the work of Carnap (1956), Church (1951), and Kaplan (1964), putting together Frege’s function‑argument structure with the treatment of intensions as functions to extensions.
The Fregean principle of compositionality was central to Montague’s theory and remains central in formal semantics. Montague showed that one could give a model-theoretic semantics for ordinary English, with a syntax rather close to surface structure; his higher-order typed logic was crucial for making that possible. The treatment of English noun phrases as uniformly denoting generalized quantifiers was one of the most vivid examples of that; that achievement made a big impression on linguists. (In my own case, it was one of the things that made me devote a good number of years to finding ways to integrate Montague’s work into linguistics. The enterprise started out as “Montague Grammar” (Partee 1973, Partee 1975, Partee 1976), and as more linguists and philosophers got involved, it developed into the more inclusive field of formal semantics.)
Montague and generalized quantifiers: According to Peters and Westerståhl (2006), the logical notion of quantifiers as second-order relations is “discernible” in Aristotle, full-fledged in Frege, then forgotten until rediscovered by model theorists like Mostowski (1957) and Lindström (1966). It was Lindström who introduced binary generalized quantifiers, without which one can express Most things walk, but not Most cats walk. What we are now accustomed to calling “generalized quantifiers”, e.g. the denotation of the NP most cats, represents the application of a Lindström binary quantifier most, syntactically a Determiner, to its first argument cats, syntactically a Noun, giving a unary generalized quantifier most cats.
Montague (1973) (and Lewis 1970) proposed that English NPs like every man, most cats can be treated categorematically, uniformly, and compositionally if they are interpreted as generalized quantifiers. This was a big part of the refutation of the point Russell and Strawson (and Chomsky) were agreed on, that there is no logic of natural language. That refutation opened a floodgate, and the next decade saw a great surge of work by linguists and philosophers, individually and together.
Generalized quantifiers and English syntax. Montague’s work showed how with a higher-typed logic and the lambda-calculus (or other ways to talk about functions), NPs could in principle be uniformly interpreted as generalized quantifiers (sets of sets). And Determiners could then be interpreted uniformly as functions that apply to common noun phrase meanings (sets) to make generalized quantifiers.
In Montague’s work, there is a semantic type, <<e,t>,t>, sets of sets of entities, to correspond to English NPs. In a simple sentence, the subject NP is the function, and the Verb Phrase (VP) is its <e,t>-type argument, as shown in (5). And although Montague treated the determiner every syncategorematically, that was inessential; every can be analyzed as in (5d).
(5) a. Every student λP[∀x[student(x) → P(x)]] type <<e,t>,t>
b. walks: walk type <e,t>
c. Every student walks: λP[∀x[student(x) → P(x)]] (walk) type t
≡ ∀x[student(x) → walk(x)]
d. every: λQλP[∀x[Q(x) → P(x)]] <<e,t>,<<e,t>,t>>
Montague’s interpretation of the sentence Every man walks is the same as Russell’s; the big difference is that Montague derives the interpretation compositionally; the semantic structure is homomorphic to the syntactic structure.
Martin Stokhof (2006) describes the model of Montague grammar as presented in (Montague 1973) in some detail, isolating “two core principles that are responsible for its remarkable and lasting influence”:
A. Semantics is syntax-driven, syntax is semantically motivated.
B. Semantics is model-theoretic.
Montague did not invent model-theoretic semantics; but it was through his work that the model-theoretic approach became more widely known and adopted among linguists, with far-reaching changes to the field of linguistic semantics. An important, if obvious, moral of the story is that the question of whether natural language is “logical”, which may be viewed as a version of the question of whether natural language semantics is compositional, is a theory-dependent matter. Contemporary formal semanticists tend to see challenges to compositionality as calls for enrichments to their logic and semantics or improvements to their syntactic theories.
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How to cite this post:
Partee, Barbara H. 2013. ‘On the history of the question of whether natural language is “illogical”.’ History and Philosophy of the Language Sciences. https://hiphilangsci.net/2013/05/01/on-the-history-of-the-question-of-whether-natural-language-is-illogical/