conventionalism the philosophical doctrine that logical truth and mathematical truth are created by our choices, not dictated or imposed on us by the world. The doctrine is a more specific version of the linguistic theory of logical and mathematical truth, according to which the statements of logic and mathematics are true because of the way people use language. Of course, any statement owes its truth to some extent to facts about linguistic usage. For example, ‘Snow is white’ is true (in English) because of the facts that (1) ‘snow’ denotes snow, (2) ‘is white’ is true of white things, and (3) snow is white. What the linguistic theory asserts is that statements of logic and mathematics owe their truth entirely to the way people use language. Extralinguistic facts such as (3) are not relevant to the truth of such statements. Which aspects of linguistic usage produce logical truth and mathematical truth? The conventionalist answer is: certain linguistic conventions. These conventions are said to include rules of inference, axioms, and definitions.
The idea that geometrical truth is truth we create by adopting certain conventions received support by the discovery of non-Euclidean geometries. Prior to this discovery, Euclidean geometry had been seen as a paradigm of a priori knowledge. The further discovery that these alternative systems are consistent made Euclidean geometry seem rejectable without violating rationality. Whether we adopt the Euclidean system or a non-Euclidean system seems to be a matter of our choice based on such pragmatic considerations as simplicity and convenience.
Moving to number theory, conventionalism received a prima facie setback by the discovery that arithmetic is incomplete if consistent. For let S be an undecidable sentence, i.e., a sentence for which there is neither proof nor disproof. Suppose S is true. In what conventions does its truth consist? Not axioms, rules of inference, and definitions. For if its truth consisted in these items it would be provable. Suppose S is not true. Then its negation must be true. In what conventions does its truth consist? Again, no answer. It appears that if S is true or its negation is true and if neither S nor its negation is provable, then not all arithmetic truth is truth by convention. A response the conventionalist could give is that neither S nor its negation is true if S is undecidable. That is, the conventionalist could claim that arithmetic has truth-value gaps.
As to logic, all truths of classical logic are provable and, unlike the case of number theory and geometry, axioms are dispensable. Rules of inference suffice. As with geometry, there are alternatives to classical logic. The intuitionist, e.g., does not accept the rule ‘From not-not-A infer A’. Even detachment – ‘From A, if A then B, infer B’ – is rejected in some multivalued systems of logic. These facts support the conventionalist doctrine that adopting any set of rules of inference is a matter of our choice based on pragmatic considerations. But (the anti-conventionalist might respond) consider a simple logical truth such as ‘If Tom is tall, then Tom is tall’. Granted that this is provable by rules of inference from the empty set of premises, why does it follow that its truth is not imposed on us by extralinguistic facts about Tom? If Tom is tall the sentence is true because its consequent is true. If Tom is not tall the sentence is true because its antecedent is false. In either case the sentence owes its truth to facts about Tom. See also MANY -VALUED LOGIC , PHILOSO – PHY OF LOGIC , PHILOSOPHY OF MATHEMATICS , POINCARÉ . C.S.