The law of excluded middle (LEM) says that every sentence of the form A∨¬A (‘A or not A’) is logically true. This law is accepted in classical logic, but not in intuitionistic logic. The reason for this difference over logical validity is a deeper difference about truth and meaning. In classical logic, the meanings of the logical connectives are explained by means of the truth tables, and these explanations justify LEM. However, the truth table explanations involve acceptance of the principle of bivalence, that is, the principle that every sentence is either true or false. The intuitionist does not accept bivalence, at least not in mathematics. The reason is the view that mathematical sentences are made true and false by proofs which mathematicians construct. On this view, bivalence can be assumed only if we have a guarantee that for each mathematical sentence, either there is a proof of the truth of the sentence, or a proof of its falsity. But we have no such guarantee. Therefore bivalence is not intuitionistically acceptable, and then neither is LEM.
A realist about mathematics thinks that if a mathematical sentence is true, then it is rendered true by the obtaining of some particular state of affairs, whether or not we can know about it, and if that state of affairs does not obtain, then the sentence is false. The realist further thinks that mathematical reality is fully determinate, in that every mathematical state of affairs determinately either obtains or does not obtain. As a result, the principle of bivalence is taken to hold for mathematical sentences. The intuitionist is usually an antirealist about mathematics, rejecting the idea of a fully determinate, mind-independent mathematical reality.
The intuitionist’s view about the truth-conditions of mathematical sentences is not obviously incompatible with realism about mathematical states of affairs. According to Michael Dummett, however, the view about truth-conditions implies antirealism. In Dummett’s view, a conflict over realism is fundamentally a conflict about what makes sentences true, and therefore about semantics, for there is no further question about, for example, the existence of a mathematical reality than as a truth ground for mathematical sentences. In this vein Dummett has proposed to take acceptance of bivalence as actually defining a realist position.
If this is right, then both the choice between classical and intuitionistic logic and questions of realism are fundamentally questions of semantics, for whether or not bivalence holds depends on the proper semantics. The question of the proper semantics, in turn, belongs to the theory of meaning. Within the theory of meaning Dummett has laid down general principles, from which he argues that meaning cannot in general consist in bivalent truth-conditions. The principles concern the need for, and the possibility of, manifesting one’s knowledge of meaning to other speakers, and the nature of such manifestations. If Dummett’s argument is sound, then bivalence cannot be justified directly from semantics, and may not be justifiable at all.
Hybrid Deduction-Refutation Systems for FDE-Based Logics
