A system of strict implication
P. G. J. Vredenduin · Journal of Symbolic Logic · 1939
Lewis has made an attempt to construct a logistic system containing an implication-relation ⊰ in such a way that p⊰q shall have the meaning: q is deducible from p. Lewis and Langford admit that “the one serious doubt which can arise concerning the equivalence of p⊰q to the relation of deducibility … arises from the fact that strict implication has its corresponding paradoxes: 19.74 ˜⋄ p ⊰ · p ⊰ q, ‘If p is impossible, then p strictly implies any proposition q’; and 19.75 ˜⋄˜p ⊰ · q ⊰ p, ‘If p is necessary, then any proposition q strictly implies p.’” Indeed, it is not obvious that 19.74 and 19.75 should hold. It is true that in many cases 19.75 (as well as 19.74) is valid; for in the system of Lewis and Langford p · ˜p · ⊰ q is provable, and the conformity of this law with real deduction is shown by them. In all cases in which a proposition of the form p·∼p can be derived from an impossible proposition, 19.75 holds. But it is not obvious that there are no other kinds of impossible propositions. Fortunately it is easy to see what is the origin of the paradoxes. They are introduced into the system by the definition 11.02 p ⊰ q · = ˜ ⋄(p · ˜q). Assume this definition. Then if q is necessary, p·∼q is impossible, and so, in accordance with 11.02, p⊰q is valid.