Semantics and the Theory of Meaning

Semantics is the study of meaning — what it is for a word, sentence, or symbol to mean something, and how the meaning of a complex expression depends on its parts. A theory of meaning is a philosophical account of what meaning is: what fact about a person, a community, or the world makes it the case that an expression has the content it does. This remark surveys the main answers and connects them to the formal, model-theoretic semantics used in logic and mathematics, where "semantics" has a precise technical sense.

The guiding tension: in logic, "semantics" means interpretation in a structure (assigning set-theoretic denotations and truth values — see Models, Truth, and Metalanguage); in the philosophy of language, "meaning" is a richer, partly normative and social phenomenon that formal semantics models only in part.

What a theory of meaning must explain

Any adequate account is constrained by several robust facts about meaning:

• Compositionality. The meaning of a complex expression is determined by the meanings of its parts and how they are combined. This is what lets us understand sentences we have never heard before, and it is the feature that formal semantics captures best (Tarski's recursive satisfaction clause is compositionality made precise).
• Productivity / systematicity. Finite speakers command an unbounded range of meaningful sentences; meaning must be generated by finite rules.
• The relation to truth. To understand a declarative sentence is, at least, to know what the world would have to be like for it to be true.
• The relation to use. Meaning guides and is manifested in use — assertion, inference, instruction-following — and is publicly learnable and shareable.
• Reference and aboutness. Words latch onto things in the world; a theory must explain this intentional directedness.

The major theories differ on which of these is taken as fundamental and which as derived.

Reference and truth-conditions

Frege: sense and reference

The modern subject begins with Frege's distinction between reference (Bedeutung — what an expression stands for: an object for a name, a truth value for a sentence) and sense (Sinn — the "mode of presentation," the way the referent is given). "The morning star" and "the evening star" share a referent (Venus) but differ in sense, which is why "the morning star is the evening star" is informative rather than trivial. Sense explains the cognitive significance of identity statements and the behaviour of expressions in belief contexts. Frege's principle of compositionality — and his context principle ("only in the context of a sentence does a word have meaning") — set the agenda for everything after.

Truth-conditional semantics

The dominant program in formal semantics identifies the meaning of a (declarative) sentence with its truth-conditions — the conditions under which it would be true. To know the meaning of $\phi$ is to know what it is for $\phi$ to be true. Davidson proposed that a theory of meaning for a natural language could take the form of a Tarski-style truth theory: a finite set of axioms generating, for each sentence $s$, a T-sentence

$$s\text{ is true}⟺p$$

(e.g. "'snow is white' is true iff snow is white"). The recursive structure delivers compositionality and productivity automatically. This is the point of closest contact with logic: the model-theoretic semantics of a formal language is a truth-conditional theory, with "true" relativized to a structure.

A standard objection: truth-conditions are too coarse-grained. Necessarily-equivalent sentences (all mathematical truths, say) share truth-conditions but plainly differ in meaning — the hyperintensionality problem, which pushes toward sense, structured propositions, or finer notions of content.

Truth and meaning: which explains which?

Truth and meaning are distinct but coupled. Meaning is what a sentence says — its content, independent of how the world is ("snow is black" is meaningful yet false); a sentence must already mean something before its truth can even arise. Truth is whether what is said obtains — a relation between content and world (or, formally, between a sentence and a structure). Meaning is trivially prior in that a truth-value presupposes a content; the substantive dispute is the direction of explanation — which notion is conceptually basic.

• Truth-first (Frege → Tarski → Davidson): take truth as the better-understood notion and define meaning as truth-conditions. A Tarskian truth theory doubles as a theory of meaning, since in a T-sentence the used right-hand side gives the meaning while the mentioned left-hand side ascribes truth. Truth grounds meaning.
• Meaning-first (verificationism, Dummett, inferentialism): reverse it. Verification-transcendent classical truth is the suspect notion; start from use / assertibility / proof — epistemically accessible and manifestable — and let truth be derived or dispensable. Dummett's manifestation argument (truth-conditions for undecidable sentences may not be manifestable in use) is precisely the meaning-theoretic case for intuitionistic over classical logic, and dovetails with the BHK reading on which a statement's meaning just is what counts as a proof.

Three things sharpen the coupling:

• Tarski presupposes meaning to define truth. Tarski's satisfaction definition fixes truth only relative to an already-interpreted language — the object language's meaning is supplied in the metalanguage. So the formalism does not show truth is prior to meaning; it shows truth is definable once meaning is fixed. The truth-first program is a philosophical bet run on top of that machinery, not a consequence of it.
• Your theory of truth constrains your theory of meaning (deflationism). If truth is deflationary — "$p$ is true" says no more than "$p$" — it is too thin to explain anything, so a truth-conditional theory of meaning loses its ground and meaning must come from use. A substantive truth-conditional semantics therefore needs an inflationary notion of truth: the two theories cannot be chosen independently.
• In formal/model-theoretic semantics, meaning comes first. There, "truth" always means truth-in-a-structure, $\mathcal{M}\models \phi$, and the structure $\mathcal{M}$ — the interpretation of the symbols — is the meaning. One must supply the interpretation before any sentence has a truth-value, so pure model theory realizes the interpretation-first order (interpretation → truth-value). The Davidsonian truth-first picture is thus a claim about natural-language epistemology (recovering meanings from observed truth-judgments), not about the formal order of definition.

Both notions are equally metalinguistic: just as truth-in-$V$ cannot be defined object-internally (Tarski undefinability), the interpretation that confers meaning is supplied from the metalanguage — and Skolem / Löwenheim–Skolem shows the formalism underdetermines which structure is the intended one. That underdetermination is exactly the gap a formal truth-definition cannot close, and is what gets handed back to the philosophy of language below.

Use, inference, and the social dimension

A rival tradition locates meaning not in word–world reference but in use.

• Later Wittgenstein: "the meaning of a word is its use in the language." Meaning is constituted by the practice of a linguistic community — the language game — and there is no private, use-transcendent fact of meaning (the private language and rule-following considerations). This stresses the normative and social character of meaning that truth-conditional theories underplay.
• Inferential-role (conceptual-role) semantics: the meaning of an expression is its role in inference — the conditions under which one is entitled to assert it and what may be inferred from it. For the logical connectives this is the proof-theoretic account (introduction and elimination rules fix the meaning of $\wedge$, $\vee$, $\to$), the natural-language cousin of intuitionistic logic's Curry–Howard reading. Brandom's inferentialism generalizes this into a full theory of conceptual content.
• Verificationism: the meaning of a sentence is its method of verification (logical positivism; and, for mathematics, the intuitionistic / BHK reading, where a statement's meaning is what counts as a proof of it). Dummett turned this into a sustained argument that meaning must be manifestable in use, hence given by assertibility-conditions rather than potentially-verification-transcendent truth-conditions — a meaning-theoretic case for intuitionistic over classical logic.

Reference fixed by the world and the community

A third cluster concerns how words connect to things — and argues the connection is not purely descriptive:

• Causal–historical theory of reference (Kripke, Putnam): names and natural-kind terms refer via a causal chain back to an initial "baptism," not via descriptions a speaker associates. "Gödel" would still refer to Gödel even if everything you believe about him were false. Hence rigid designation and the possibility of necessary a posteriori truths ("water is H₂O").
• Externalism / "meaning ain't in the head" (Putnam's Twin Earth): the contents of our terms depend partly on the environment and on the division of linguistic labour (deference to experts). Meaning is not fully fixed by individual psychology.

How formal/mathematical semantics relates

Formal semantics is a precise model of one strand of meaning — the truth-conditional, compositional strand — and deliberately brackets the rest:

• What it captures well. Compositionality, productivity, truth-conditions, entailment (as model-theoretic consequence, $\Gamma \models \phi$), scope, quantification, and the logical vocabulary. Montague's slogan that there is "no important theoretical difference between natural languages and the artificial languages of logicians" is the high-water mark of this approach.
• What it brackets. The constitutive question — what makes a structure the intended one, why these symbols mean these things — is pushed into the metalanguage and ultimately into use and community practice. As the logic remarks stress, an interpretation is conferred from outside; the formalism does not fix its own meaning. This is the philosophical residue of Skolem's paradox and the Löwenheim–Skolem phenomenon: the formal theory underdetermines its intended interpretation, so "meaning" must come from somewhere the formal semantics does not supply.
• A division of labour. Model-theoretic semantics answers "given an interpretation, what are the truth-conditions and entailments?"; a philosophical theory of meaning must answer the prior question "in virtue of what does the expression have that interpretation at all?" The former is mathematics; the latter is irreducibly about minds, use, and the world.

Summary

There is no settled winner; the live options pair a truth-conditional core (which formal/model-theoretic semantics makes rigorous) with a use-based and externalist account of how interpretations get fixed (which the formalism presupposes but cannot supply). Mathematical semantics is thus best read not as a complete theory of meaning but as an exact treatment of its compositional, truth-conditional skeleton — with the question of whence the interpretation handed back to the philosophy of language.

Related discussions: Models, Truth, and Metalanguage and The Metalanguage Hierarchy and Semantics in Mathematics (the technical sense of "semantics"), Model Theory (interpretation in a structure), and Intuitionistic Logic (meaning-as-proof / verificationism).