Two Kinds of Consequential Implication

Abstract

The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least two different operators of decreasing strength (symbolized by $\rightarrow$ and $\Rightarrow$). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for $\Rightarrow$, CI$\Rightarrow$, is definitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for $\rightarrow$, CI. The main drawback ot the weaker operator $\Rightarrow$ is that it lacks unrestricted contraposition, but the final section of the paper argues that $\Rightarrow$ has some properties which make it a valuable alternative to $\rightarrow$, turning out especially plausible as a basis for the definition of operators representing synthetic (i.e. context-dependent) conditionals.