The Baroque Labyrinth of Meaning: Carnap, Montague, and What Compositionality Does Not Require — Epoche C2
Buenos Aires, 12 December 2023 The proposition at issue is Richard Montague's requirement, stated in "Universal Grammar" of 1970, that the assignment of meanings to expressions be a homomorphism from the syntactic algebra of a language into its semantic algebra — the technical form in which the principle of compositionality entered linguistics — and the question is whether Rudolf Carnap's notion of a semantical system offers a looser framework, better suited to the context-dependence that pervades ordinary discourse. I have argued in print for the affirmative. I now think the argument was built on three mistakes, each of them mine, and that correcting them leaves a thesis that is more modest, more Carnapian, and considerably more defensible than the one I started with. The first mistake was the word isomorphism . Montague required a homomorphism, and the difference is not a technicality — it is the whole of the matter. The second was chronological: the meaning postulates on which my argument turned are not in the 1947 Meaning and Necessity . The third was a misattribution of position: Montague, far from excluding context, is the person who introduced contextual coordinates into model-theoretic semantics in the first place. Each correction is set out below, and each removes a supposed obstacle that was never there. What the homomorphism requirement actually demands The requirement is easiest to state and hardest to misread in its algebraic form, which is how Montague gave it. A language is presented as a syntactic algebra $\mathfrak{A} = \langle A, (F_\gamma)_{\gamma \in \Gamma} \rangle$, where $A$ is the set of (disambiguated) expressions and each $F_\gamma$ is a syntactic operation of some fixed arity. Alongside it stands a semantic algebra $\mathfrak{B} = \langle B, (G_\gamma)_{\gamma \in \Gamma} \rangle$, similar to the first in the algebraist's sense: the same index set $\Gamma$, with $G_\gamma$ matching $F_\gamma$ in arity. Compositionality is then the demand that the meaning assignment $h : A \to B$ satisfy $$h\big(F_\gamma(a_1, \ldots, a_n)\big) = G_\gamma\big(h(a_1), \ldots, h(a_n)\big)$$ for every operation and every tuple of arguments. That is all. Consider what the condition does not impose. It does not require $h$ to be injective, so distinct expressions and distinct derivations may perfectly well receive the same meaning; synonymy and paraphrase are therefore no embarrassment at all. It does not require $h$ to be surjective, so the semantic algebra may contain meanings that no expression of the language expresses. It does not require the two algebras to be isomorphic, since a homomorphism that is neither injective nor surjective is still a homomorphism. And it does not require syntactic categories to correspond one-to-one with semantic types. The last point is not a bare possibility left open by the formalism; Montague built the failure of one-to-one correspondence into his own fragment. In "The Proper Treatment of Quantification in Ordinary English" of 1973, the map $f$ from syntactic categories to semantic types is defined so that $f(A/B) = f(A//B) = \langle \langle s, f(B) \rangle, f(A) \rangle$ — the two slashes are collapsed. The category of intransitive verb phrases is $t/e$ and the category of common nouns is $t//e$, and both are therefore assigned the single type $\langle \langle s, e \rangle, t \rangle$. The two slashes exist precisely because the syntax needs a distinction (verb phrases and common nouns combine with different things) that the semantics does not need. A framework in which category and type stood in bijection could not have been written that way. What the requirement does impose is worth stating positively, since the entire dispute is about its strength. It demands that for each syntactic operation there exist a corresponding semantic operation, and consequently that the meaning of a complex expression depend on nothing beyond the meanings of its immediate constituents and the identity of the operation applied. It is a constraint on the format of a semantic theory. It is not a thesis about the metaphysics of meaning, and it is not, as I once wrote, a Procrustean bed. That characterisation gets the polarity exactly backwards, as the next section shows. Compositionality on its own has no content: Zadrozny's theorem The reason the format constraint cannot be too tight is that, taken by itself, it is not a constraint at all. Wlodek Zadrozny proved this in 1994, and the result deserves to be better known outside the formal-semantics literature, because it changes what an argument about compositionality can possibly be about. The statement is this. Let $m$ be any function whatever from the expressions of a language to a set of meanings — arbitrary, unsystematic, chosen out of malice. Then there exists a function $\mu$ such that, first, meanings compose by function application over concatenation, $$\mu(x \cdot y) = \mu(x)\big(\mu(y)\big),$$ and second, the original meaning is recoverable from the new one, since $\mu(x)(x) = m(x)$. The construction turns an arbitrary assignment into a compositional one while losing nothing. The proof idea is that the two displayed conditions constitute a system of set-theoretic equations in which $\mu(x)$ occurs on both sides — a self-referential system that has no solution in ordinary well-founded set theory. Zadrozny works instead in a non-well-founded set theory, where Aczel's anti-foundation axiom guarantees a unique solution to systems of exactly this shape. The resulting $\mu(x)$ is a function that carries, packaged within itself, both the original meaning of $x$ and the instruction for combining with any argument. The moral is not that compositionality is false but that it is empty until something else is fixed. What carries empirical content is never the homomorphism condition on its own; it is that condition together with independent constraints — that the syntactic algebra be the one motivated by grammatical eviden