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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?

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    Considering that the strict conditional has the material conditional under □ in its definition it would be rather circular to define the latter in terms of the former. Moreover, the "modal companion" in this case just has a single possible world, the actual one, with a copy of classical logic. It does not even provide a novel perspective on the material conditional. In contrast to intermediate logics, which are extensionalized by modal companions.
    – Conifold
    Jun 22, 2023 at 6:14
  • The verificationist version of S5 can have more than one possible world. Still, I see your point.
    – PW_246
    Jun 22, 2023 at 14:22

2 Answers 2

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I would say that strict implication is indeed just an implication of the form □(A → B). I think some of the issues in your question are terminological, and it would help to unpack the history a little.

Russell coined the term 'material implication' in Principia Mathematica to refer to a truth functional connective. It is equivalent to ¬A ∨ B, also to ¬(A ∧ ¬B) and is also definable by the truth table T/F/T/T. Many have said that it is misleading to call this connective 'implication'. Quine, for example, described it as a use/mention error and proposed instead to refer to the connective as the 'material conditional'. I'm not sure whether it is really a use/mention error, but I agree it can be misleading to call it implication because it invites a confusion between object language and metalanguage.

The material conditional is simply a truth function within the object language, in the same way that conjunction and disjunction are. Being a truth function means it is a function from two truth valued operands to a truth valued result. It does not express any kind of relationship between the content of its operands and depends only on their truth values.

This contrasts with the logical consequence relation, which expresses a logical relationship. C I Lewis, the inventor of modal logic, was originally motivated to introduce strict implication as a way to express logical consequence. When speaking of logical consequence, or of a valid argument, it is natural to use modal language. If the premises of a valid argument are true, the conclusion must be true. If an argument is valid, it is impossible for the premises to be true and the conclusion false. Lewis' modal logics were intended to capture this modal relationship.

As things turned out, this original motivation was overtaken by developments in proof theory and model theory. We now use the turnstile symbols Γ ⊢ φ to express that Γ proves φ, and Γ ⊨ φ to express that Γ has φ as its semantic consequence. Unfortunately, the word 'implication' is equivocal between the connective → in the object language and the turnstile in the metalanguage. Peter Smith makes the point thus:

There is an unfortunate practice that - as we said - goes back to Russell of talking, not of the 'material conditional', but of 'material implication', and reading something of the form (α → γ) as α implies γ. If we also read α ⊨ γ as α (tauto)logically implies γ, this makes it sound as if an instance of (α → γ) is the same sort of claim as an instance of α ⊨ γ, only the first is a weaker version of the second. But, as we have just emphasized, these are in fact claims of a quite different status, one in the object language, one in the metalanguage. Talk of 'implication' can blur the very important distinction. (Even worse, you will often find the symbol ⇒ being used in informal discussions so that α ⇒ γ means either α → γ or α ⊨ γ, and you have to guess from context which is intended. Never follow this bad practice!)

Peter Smith, Introduction to Formal Logic 2nd edition, p. 159.

Although the original motivation for modal logic as a way to express logical consequence was overtaken, modal logics acquired a life of their own as a way to express modal flavours such as necessity, obligation, knowledge, etc. Kripke, von Wright and Hintikka later came along and supplied the possible world semantics.

The concept of modal companions, and of using modal logics to express implication relations in different systems of logic almost brings us full circle back to C I Lewis. It provides a way to map non-classical logics into modal extensions of classical logic. There is no need to abandon the material conditional once we understand its role as a connective: it definitely has its uses, but it needs to be kept in its proper place as a connective within the objective language.

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  • That makes sense. I guess my question is why can’t the material conditional be interpreted as a strict conditional since there are modal companions to Classical Propositional Logic.
    – PW_246
    Jun 22, 2023 at 2:07
  • The material conditional is not considered strict because it does not have any modal flavour. It can be written as A → B without a □. If you do choose to add a box operator, it is a trivial one, since it does not change the logic.
    – Bumble
    Jun 22, 2023 at 8:23
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Russell's definition of the material conditional as a truth function works as a definition of implication in classical logic works because it has the properties of a mathematical partial ordering relation (transitive, reflexive, antisymmetric). It is also possibly the simplest non-trivial example of such a relation. It expresses the notion that the consequent is not less true than the antecedent, or that the conclusion is not less true than the hypotheses. This is the most basic criterion of valid reasoning; that the process does not in itself introduce error that was not present in the hypotheses.

The material conditional does not work so well outside strict two valued logic, whether this is a rhetorical or natural language approach, symbolic modal logic as developed by Lewis, or multivalued logic. In each of these areas, there at least the suspicion that this conditional is not quite sufficient to express the relation between two statements. The concept of the strict conditional works better.

There are some differences between the definition of the strict conditional employed by Lewis in modal logic and a version suitable for multi-valued logic. The distinctions between the various conditionals appear to collapse if one retreats from non-classical logic back to the familiar territory of classical two-valued logic.

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  • I don’t see how it’s anti-symmetric. Surely a relation “->” can be symmetric for distinct formulas A and B without A and B being the same formula. They are logically equivalent, but that’s different.
    – PW_246
    Jun 30, 2023 at 3:14
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    An antisymmetric relation is one in which if (A -> B), then ( B -> A) if and only if A = B. The material conditional has this property.
    – Confutus
    Jul 20, 2023 at 17:36
  • I think we’re interpreting ‘=‘ differently. Surely (P->P) isn’t the same formula as (Pv~P), even though they’re classically equivalent. It is trivially true that (A->B)->((B->A)<->(A<->B)), but again, the material bi-conditional is different than straight up equality.
    – PW_246
    Jul 20, 2023 at 17:53
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    The differences are not important in classical propositional logic. The material biconditional has the properties of an equivalence relation, as does straight equality, and as far as truth values are concerned, acts just the same. In classical propositional logic, the meaning of propositions, or how or why they are true or false is irrelevant: only whether they are. If one is working with natural language, or nonclassical logic, distinctions such as equality versus the biconditional become important, but at the simplest level, the concepts coincide.
    – Confutus
    Jul 20, 2023 at 21:00
  • That seems fair enough.
    – PW_246
    Jul 20, 2023 at 21:06

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