Talk:Strict implication
Short comment on the definition of strict implication given in the entry
editI am not sure at all that the definition of strict implication as a material implication that is acted upon by the necessity operator from modal logic is sufficient and right. I invite the reader of these lines to read what I write in the article devoted by wikipedia to strict conditional and to peruse two papers to be found on the site http://www.grammar-and-logic.com: Traité de logique modale pour grammairiens et Les deux postulats du traité de logique modale. Both papers will be translated into English in a few weeks. L~(p & ~q) or ~ M (p & ~q) cannot by itself symbolize the strict implication of q by p. In effect, ~ M (p & ~q) It is im -possible to have p and ~q together is quite compatible with ~ M (p & q) It is im -possible to have p and q together.If one has ~ Mp , that is to say, if p is im-possible, it is im-possible to have p & q and it is also im-possible to have p & ~q.If ~ M p, then ~ M (p & q) & ~ M (p & ~q). Jean-François Monteil May 09
On strict implication : p ≡ Lq. Note concerning modal logic.
editCertain linguists, for instance John Lyons, affirm that the formula of strict implication p => q has been found. According to them, p strictly implies q, if one can pose ~ M (p & ~q) It is im -possible to have p and ~q together So it is not. ~ M (p & ~q) cannot by itself symbolize the strict implication of q by p. In effect, ~ M (p & ~q) It is im -possible to have p and ~q together is quite compatible with ~ M (p & q) It is im -possible to have p and q together.If one has ~ Mp , that is to say, if p is im-possible, it is im-possible to have p & q and it is also im-possible to have p & ~q.If ~ M p, then ~ M (p & q) & ~ M (p & ~q).Of itself, the proposition ~ M (p & ~q) cannot represent the strict implication of q by p, cannot represent the causal relation between a cause p and its effect q in so far as the impossibility of p & ~ 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. Hence, the necessity of adding the idea that p is possible to the content of ~ M (p & ~q), of adding Mp to ~ M (p & ~q). Hence, our formula of strict implication: ~ M (p & ~q) & Mp, which formula becomes p ≡ Lq p strictly implies q, if p is equivalent to the certainty of q.The developed form of p ≡ Lp is ( p & Lq) w (~ p & M~q ) One of two things: Either we have p and then certainly q or we have not p and in that case it is possible to have ~q. ( p & Lq) w (~p & M~q ), the developed form of p ≡ Lq, contains the two elements of ~ M (p & ~q) & Mp namely the idea that it is im-possible to have p and ~q on the one hand and the idea that p is possible on the other.In John Lyons (page 165, chapitre 6 Logical semantics, Semantics 1,Cambridge University Press, 1977), one can read:“Entailment can be defined in terms of poss and material implication as follows (19)(p => q) ≡ ~ poss (p &~q).That is to say, if p entails q, then it is not logically possible for both p to be true and not-q to be true and conversely…..” If I translate in my own terms, it reads “Strict implication can be defined in terms of possibility M and material implication as follows:19) (p => q) ≡ ~ M (p &~q). That is to say, if the fact p entails the fact q, if the fact p strictly implies the fact q, then it is not logically possible for the two facts p and not-q to coexist in reality and conversely if the facts p and not-q cannot coexist in reality, it means that the fact p entails the fact q, it means that the fact p strictly implies the fact q.” John Lyons is obviously wrong.Indeed, ~ M (p &~q) is a value implied by p => q the strict implication of q by p,but the converse is untrue for ~ M (p &~q) does not imply of itself p => q, does not imply our p ≡ Lq. ~M (p &~q) it is im-possible to have the conjunction of p and non-q is perfectly compatible,I repeat,with ~M (p & q) it is impossible to have the conjunction of p and q. In fact, when you have ~Mp, the impossibility of p, you have necessarily the conjunction of two impossibilities,for you can write
~M p ≡ ~M (p & q) & ~M (p &~q).