Abstract

In this paper, we give an introduction to intuitionistic logic and a defense of it from certain formal logical critiques. Intuitionism is the thesis that mathematical objects are mental constructions produced by the faculty of a priori intuition of time. The truth of a mathematical proposition, then, consists in our knowing how to construct in intuition a corresponding state of affairs. This understanding of mathematical truth leads to a rejection of the principle, valid in classical logic, that a proposition is either true or false (put symbolically, a ∨ ~a). The rejection of this principle leads to a different system of formal logic. This logic has been critiqued as being three-valued in such a way that it is self-contradictory. That this is a misunderstanding of intuitionistic logic can be proven formally on the basis of Heyting's axioms and rules of inference for intuitionistic logic. A proposition that is neither true nor false does not, on an intuitionist view, have some third truth value, but lacks any truth value whatsoever. In the process of proving that this is the case we will also prove several other theorems which will give us some insight into the formal similarities and differences between intuitionistic and classical mathematics, specifically with regard to the validity of different proof techniques.