Horsehoes and Fishhooks

From an oline source quoting

"Definitions", from Lewis, 1918:293

"Strict Implication"

p => q

=_def

~(p /\ ~q)

"Lewis uses the fishhook implication symbol"

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A good precis is at

http://philosophy.ucdavis.edu/mattey/phi134/strict.htm

--- but the hypertext did not quite read well with my brower so it may need some editing!

"Lewis introduced a new symbol to extend Russell's logic, with the intention of making the symbol stand for the kind of "strict implication" which is the basis for proof in logic. The new symbol is in the shape of a fish-hook, "p -< q" [strict implication] and is interpreted as meaning that it is not possible that p be true and q false. As with strict implication, Lewis allowed the semantical notion of possibility to be reflected in the symbolic language: "p -< q iff <>~(p & ~q)
Thus, possibility is invoked to clarify the notion of implication."

"Indeed, students first introduced to the notion of a deductively valid argument are given a similar formulation: that it is not possible that all the premises be true and the conclusion false. Necessity arises through another equivalence: what cannot possibly not be the case is necessarily true. Thus we have, "p is equivalent to p"
and with substitution and suppression of double negation we have, "[(p q)] is equivalent to [(p & q)]. Therefore, [(p q)] is equivalent to [p q]. So it turns out that Lewis's strict implication is the necessary truth of Russellian material implication. Lewis recognized that there are some remnants of the "paradoxes of material implication" in his system. For example, we have a "paradox of strict implication" which is a modal version of the original paradox we looked at. [p (q p)] a necessary proposition is strictly implied by any proposition. Given that p is necessarily true, it cannot be false, and so it cannot be the case that q is true and p is false. (This assumes the validity of the inference from (p & q) to p, which in fact does not hold for Lewis's system, S1, which he developed some time later.) It has been objected that the fact that the a necessary proposition is said to be implied by another, irrelevant, proposition precludes the relation between the two from being one of implication. How can the connection be one of meaning or content, when the antecedent and consequent have nothing in common? Lewis defended the consequences of his logic as not paradoxical when correctly understood. We shall not describe his defense here, but some people remain unconvinced by his and subsequent arguments. For a lively discussion of these issues, see Richard Routley et al, Relevant Logics and their Rivals 1 (Atascadero, CA, 1982)."

(Editing needed for the copied and pasted formulae which did not _travel_).