A- The definition of strict implication given at the beginning of the article is the definition adopted by scholars and must be respected as such.
(84.100.243.150 (talk) 15:11, 8 September 2012 (UTC)) (84.100.243.132 (talk) 08:29, 14 August 2012 (UTC)) In logic, a strict conditional is a material conditional that is acted upon by the necessity operator from modal logic. For any two propositions p and q, the formula (p → q) says that p materially implies q while L (p → q) says that p strictly implies q.
The definition of strict implication adopted by the contributors of wikipedia and by John Lyons says that p strictly implies q ,if we have L (p → q) or ~M (p & ~q), the latter expression being to be read It is im-possible to have together p and not-q.
L (p → q)and ~M (p & ~q) are equivalent expressions. They represent the strict implication of q by p not only in the articles of wikipedia but also in excellent authors like the John Lyons of Semantics 1. For this reason, I subscribe without hesitation to what Arthur Rubin writes below: It may be that you have a different definition of the strict conditional, but, unless it is published, it has no place in the Wikipedia article. — Arthur Rubin (talk) 09:54, 4 September 2012 (UTC) on, search.
L (p → q) and ~M (p & ~q) are equivalent expressions. According to De Morgan's laws, p → q means ~ ( p & ~q). In effect, the material implication of q by p, p → q signifies that one cannot have together p and not-q.
p → q signifies therefore that one of two things, either p&q or ~p. For all three conjunctions: p&q, ~p & q, ~p & ~q obviously exclude p & ~q. p&q contains ~ ( p & ~q) and therefore p → q, ~p&q also contains ~ ( p & ~q) and therefore p → q, ~ p & ~q also contains ~ ( p & ~q)and therefore p → q.
In wikipedia and John Lyons, L (p → q) as as well as p ⇒ q symbolizes the strict implication , that is, the material implication p → q acted upon by L the necessity operator from modal logic. If in L (p → q), we replace p → q by the equivalent ~ ( p & ~q), we first obtain L ~( p & ~q) to be read It is necessary not to have the conjunction of p and not-q. It is clear that the necessity not to have is equivalent to the im-possibility to have.
Thus, instead of L ~( p & ~q) we can write ~M (p & ~q) to be read It is im-possible to have together p and not-q.
B- The definition of strict implication given at the beginning of the article is perhaps insufficient
First remark: to define the strict implication by saying that it is equivalent to ~ M (p & ~q) is deficient in that the impossibility to have both the fact p and the fact not-q may result from the fact that p is im-possible and not from the fact that p is the cause of its effect q.
If we have ~Mp i.e L~p, if p is im-possible, in other words if not-p is "necessary", if not-p is certain, it is im-possible to have p & q as well as p & ~q.
~Mp may be represented by the combination: ~ M (p & q) & ~ M (p & ~q). Therefrom, it clearly appears that ~ M (p & q) and ~ M (p & ~q) are not at all in-compatible. Both are true propositions. if p is im-possible. Hence the necessity of adding Mp to ~ M (p & ~q)to eliminate the spectre of ~Mp.
But if the first two elements ~ M (p & ~q) and Mp are necessary, they are not sufficient.
Associated with ~ M (p & ~q), the second ingredient Mp eliminates the direful spectre of ~Mp im-possibility of p, that is, L~p certainty of not-p. To be able to say that a fact p is the cause of a fact q, it is evident that the fact p must be thought possible. How could we think that p has an effect q if p is said to be im-possible from the start ?
It is no less clear that if the fact q is certain in any case, whether p is the case or not-p is the case, it is absolutely im-possible to think that the certainty of the fact q is the effect resulting from the fact p exclusively.
~p → M~q is an expression saying that not-p implies M~q the possibility of not-q.
If associated with the conjunction ~ M (p & ~q) & Mp, the expression ~p → M~q efficiently eliminates the second state of things, namely, Lq & Mp. M~p.
Lq & Mp. M~p is incompatible with that strict implication symbolized in good authors like John Lyons by p ⇒ q wrongly held to be equivalent to ~ M (p & ~q).
