Strong Boethius' thesis and consequential implication (Q1366766)

Consequential implication, \(p\to q\), is a form of implication that satisfies so-called Aristotle's Thesis, \(\neg(p\to\neg p)\), and Weak Boethius's Thesis, \((p\to q)\supset\neg(p\to\neg q)\). It can be formalized in its own right, or it can be defined in (weak) normal modal logics as \(L(p\supset q)\and(Lp\equiv Lq)\and(Mp\equiv Mq)\); i.e., that \(p\) strictly implies \(q\) and they have the same modal status. (Alternatively, \(Lp\) can be defined in a logic of consequential implication as \(\top\to p\).) This paper investigates relations between systems of consequential implication and normal modal logics, and shows that there is a one-one correspondence between them. Most notably, the weakest system containing Weak Boethius corresponds to the system KD. Strong Boethius is the thesis \((p\to q)\to\neg(p\to\neg q)\). In normal systems, in the presence of Weak Boethius, it generates a modal collapse; \(p\to q\) becomes equivalent to \(p\equiv q\). Nevertheless, Strong Boethius can be maintained non-trivially. The weakest system containing it corresponds to the normal modal logic \(\text{Kdf}\) \((=\text{K}+ LM\top+ LMp\supset LLp)\).