Abstract

By a valid inference is here understood an inference that succeeds in its aim to justify its conclusion given that its premisses are already justified. For an inference to be valid it is thus not enough that the sentence asserted in the conclusion is a logical consequence of the sentences asserted in the premisses. A proof is understood as a succession of valid inferences that is closed (i.e. all its assumptions are discharged and all its free variables are bound by some of the inferences). On the other hand, in mathematics the justification of an assertion has traditionally been taken to consist in a proof of it, which gives rise to an explanatory circle.In an attempt to avoid this circle, I have in previous works tried to explain the notion of ground for an assertion without referring to valid inferences. However, it turns out that the resulting notion of ground has unwanted consequences like the notion of proof in the so-called BHK-interpretation of intuitionistic logic. For instance, one is guaranteed a ground or a proof of an implication A → B as soon as there is in fact no ground or proof of A, regardless of whether one really knows this fact, and hence without having a real justification of the assertion of the implication.In the absence of a non-circular explanation of the notions of valid inference and proof, I suggest that we should try to formulate a number of principles of how these two epistemic notions depend on each other. A more careful explanation of the validity of an inference results in what I call a heuristic idea about the validity of inference, which is used as a guide to formulate precise principles. One such principle generalizes Gentzen’s idea that his introduction rules define the meaning of the logical constants. Another principle generalizes a previously noted property of Gentzen’s elimination rules that I have called that of satisfying the inversion principle. It results in a notion that I call non-creativity, which implies validity.