The classical material conditional is given a truth-functional definition that can be determined with truth-tables. Intuitionistic implication is a kind of strict implication that can be translated to S4 modal logic via □(A′→B′) where A′ and B′ are the translations of formulas A and B of Intuitionism to corresponding formulas in classical S4. The full translation can be found here: https://en.m.wikipedia.org/wiki/Modal_companion
Going with the same translation, if we add ¬¬A→A to Intuitionism, we get Classical Propositional Logic, even though the corresponding frame condition is Euclidean, as opposed to reflexive and functional. This is due to the fact that propositional atoms obey the heredity condition:
If w is a world and p a propositional atom, then for any world v such that w≤v, if w⊩p, then v⊩p. This corresponds to the fact that the translation of a propositional atom p of Intuitionism gets mapped to □p in S4.
If the heredity principle is removed, then it seems the only way to get a “strict” implication out of Classical Logic is to translate a formula A of Classical Propositional Logic to a formula A’ of the modal logic known as K=, also known as Triv.
I know that both S5 and Triv are strong systems, but technically one can interpret classical implication via one of these systems, depending on whether or not the heredity principle is assumed.
What are the philosophical motivations for discounting these modal companions from being seen as sufficient for formalizing strict implication? If strict implication simpliciter is just some implication of the form □(A→B), then it seems like one could just define material implication as a strict implication, and abandon truth tables for being too simplistic or for not being able to account for infinitely many valuations/worlds. I understand that strict implication was developed to prevent some of the oddities of material implication; I guess my question is whether or not it even makes sense to consider modal companions that validate the classical material conditional as a strict conditional, and if so, then where and why is the line drawn?