Bivalence and Determinacy

The principle that every statement is bivalent (i.e. either true or false) has been a bone of philosophical contention for centuries, for an apparently powerful argument for it (due to Aristotle) sits alongside apparently convincing counterexamples to it. I analyse Aristotle’s argument (§§1-2), showing that it relies crucially on the logical laws of Excluded Middle and Proof by Cases. Even given these logical laws, however, the argument only shows that every determinate statement is true or false, where a determinate statement ‘says one thing’, i.e. has univocal truth-conditions. In the light of this analysis, I examine three sorts of problem case for bivalence. Future contingents, I contend, are bivalent (§3). Certain statements of higher set theory, by contrast, are not. Pace the intutionists, though, this is not because Excluded Middle does not apply to such statements, but because they are not determinate (§§4-6). Vague statements too are not bivalent, in this case because the law of Proof by Cases does not apply (§§7-8). I show how this opens the way to a solution to the ancient Paradox of the Heap or Sorites (§9) that draws on quantum logic.