Reasoning

III · The semantic anchor

Truth

veritas · ‘that which is’

What premises and conclusions can have. Validity is a relation among propositions; soundness asks whether those propositions are true. The further question — what truth itself is — divides into five principal answers.

Orientation

Why truth belongs here

The vocabulary of argument presupposes truth at every step. Validity is the property of an argument such that, if its premises are true, its conclusion must be. Soundness adds that the premises are true. Cogency is graded probability of truth. Each definition leans on truth as a primitive — and what that primitive amounts to is a substantive question.

The page below is divided into two: five philosophical theories of what makes a proposition true, then a short note on how truth functions inside formal systems — where the issue is not metaphysics but the technical definition that lets a logic do its work.

The theories

Five answers

The five theories below are not exhaustive, but they cover the field. Each can be stated in a single sentence — the thesis — and each carries its own characteristic difficulty.

I

Correspondence

A proposition is true when what it asserts is the case.

Correspondence is the default theory — the one almost everyone accepts before they hear of the others. Truth consists in a relation between a proposition (or sentence, or belief) and the world: the proposition that snow is white is true if and only if snow is, in fact, white.

The technical work of correspondence theory is to specify the relation. Russell's logical atomism made truth a structural fit between a proposition and a fact; Wittgenstein's Tractatus took the same line. The picture has classical force and considerable difficulty: what is a fact? What does it mean for a sentence to 'picture' one?

Origin
Aristotle, Metaphysics IV.7 ("To say of what is that it is not, or of what is not that it is, is false; to say of what is that it is, or of what is not that it is not, is true"). Modern defenders: Russell, the early Wittgenstein, Austin.
Characteristic difficulty
Specifying the correspondence relation without circularity is hard; sentences about mathematics, modality, or ethics resist a fact-corresponding-to-them reading.

II

Coherence

A proposition is true when it coheres with a larger system of accepted propositions.

Coherence theories locate truth not in a relation to the world but in a relation among propositions. A proposition is true to the extent that it fits, supports, and is supported by the body of other propositions we accept.

The motivation is partly epistemic: we never compare a proposition to the world unmediated; we compare it to what else we accept. The objection — that two distinct, internally coherent systems could both be 'true' on this account — is the standard one. Coherence theorists reply by demanding maximal coherence of the totality of judgements, but the maximal totality remains hard to characterise.

Origin
Spinoza and the German idealists; refined in the early twentieth century by Bradley, Blanshard, and the Vienna positivists (Neurath, in particular).
Characteristic difficulty
Two incompatible coherent systems cannot both be true — but coherence alone does not decide between them.

III

Pragmatic

A proposition is true when accepting it works — when it leads to successful action and prediction.

Pragmatism makes truth a feature of the use of a proposition rather than its relation to the world or to other propositions. A belief is true if acting on it succeeds — if it survives experimental test, guides action well, and would be endorsed at the ideal limit of inquiry.

Peirce's version locates truth at the end of inquiry: truth is what an unbounded community of inquirers would converge on. James was more practical: a true belief is one that pays its way in lived experience. The latter formulation has been accused of confusing truth with utility; the former is closer to the realist correspondence view than it appears.

Origin
Charles Sanders Peirce (1878), William James, John Dewey. Revived in the late twentieth century by Putnam and Rorty (in different versions).
Characteristic difficulty
Distinguishing truth from utility — a belief can be useful and false, or true and useless.

IV

Deflationary

There is no substantive property of being true; the predicate 'is true' is merely a logical device.

Deflationary (or 'redundancy') theories deny that truth is a substantive metaphysical property in need of analysis. To say that the proposition that snow is white is true is to say nothing more than that snow is white. The truth predicate is useful — it lets us generalise ("everything Aristotle said is true") — but it adds no content.

The deflationary view is the natural fit for a working logician: truth, on this picture, is just whatever is governed by the T-schema. It is the closest thing to a contemporary consensus among philosophers of logic.

Origin
Frank Ramsey (1927); developed by Quine, Horwich, and others. The disquotational variant traces to Tarski's T-schema.
Characteristic difficulty
Explaining why truth seems to matter — to inquiry, to assertion, to objectivity — if it is merely a logical device.

V

Semantic (Tarski)

A formalised language can have a precise definition of truth for its sentences, given a model.

Tarski's contribution was not a metaphysical theory but a technical one: for any formal language with explicit syntax, one can define a predicate that satisfies the T-schema — '"P" is true if and only if P' — for every sentence of the language. The definition uses a higher-order meta-language and works through the notion of satisfaction by a model.

Tarski's result is what makes contemporary formal semantics possible. Model theory is the study of the relation between formal sentences and the structures that make them true. The semantic conception is neutral on the metaphysical debate above; it is what every formal system tacitly relies on.

Origin
Alfred Tarski, 'The Concept of Truth in Formalised Languages' (1933).
Characteristic difficulty
The Tarskian construction works inside a formal language; extending it to natural languages, with their context-sensitivity and self-reference, is hard.

Inside the systems

Truth in formal logic

Inside a formal system, the metaphysical question is set aside. Truth is defined operationally: a sentence is true in a model, where the model specifies what the non-logical symbols refer to and what the predicates apply to. The truth predicate is then governed by a small number of rules.

In propositional logic, truth is a function of the truth values of the atomic propositions, with the connectives defined by truth-tables:

pqp ∧ qp ∨ qp → q¬p
TTTTTF
TFFTFF
FTFTTT
FFFFTT

In predicate logic, the apparatus is heavier. Tarski's definition (1933) characterises truth by way of satisfaction: a formula is satisfied by an assignment of values to its free variables; a sentence (a formula with no free variables) is true in a model when it is satisfied by every assignment.

Tarski's T-schema is the load-bearing condition:

‘P’ is true if and only if P.

The schema is, in one sense, trivial — and that is its point. Whatever else truth is, it must satisfy the schema. The schema is what makes formal semantics possible; it is also the technical anchor of the deflationary theory.

Lineage

Where the theories come from

The correspondence formula is in Aristotle (MetaphysicsIV.7, c. 350 BC). Coherence is German idealist in lineage — Spinoza, Hegel, Bradley — and takes its modern form in the early twentieth century. Pragmatism is American: Peirce's ‘How to Make Our Ideas Clear’ (1878) is the founding text. Ramsey's 1927 paper is the modern source of deflationism. Tarski is 1933.

The theories are not, in every case, rivals. Tarski's construction is technical; deflationism is philosophical; correspondence and coherence make different metaphysical commitments. A working position can combine them: a Tarskian semantics for formal languages, deflationism about the truth predicate, and correspondence as the folk theory behind ordinary assertion.

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