The sign ‘⊃’ (or ‘→’ or ‘C’) functions in many logical systems in a way which precludes its interpretation as either strict or material implication. For example, in the systems of Heyting, Johansson, Fitch and Bernays (positive logic), the following are theorems:

Now if ‘⊃’ were interpreted as strict implication, ⊃2 would mean ‘if p is true, then p is strictly implied by every proposition’, i.e. ‘if p is true, it is necessarily true’, which is false for contingently true p. If on the other hand ‘⊃’ were interpreted as material implication, ⊃1 would reduce to ‘~p ∨ p’, i.e. to the law of excluded middle, which is conspicuously lacking in the systems mentioned. The reader is likely in practice to veer between these two interpretations. Thus in Fitch or Heyting on realizing that ‘~p⊃▪ p⊃q’ is a theorem, one thinks of it as meaning ‘a false proposition implies everything’ and regards the implication as material; but the presence of ‘p⊃p’ as a theorem, even for choices of p which do not satisfy excluded middle, inclines one again to the strict interpretation. This vacillation, while it need not lead to the commission of any formal fallacies, tends to hamstring one's intuition and thus waste time. The purpose of this paper is to suggest an interpretation of ‘⊃’ which will prevent such havering.

Let two formulae A and B be called interdeducible if A ⊢ B and B ⊢ A.