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I understand that relevant logic deals with a natural-language interpretation of implication, but it seems too restrictive. It does seem a bit of a reach to say that there is a conceptual link between “the moon being made of cheese” and “1+1=2,” but let me try to motivate it. Assuming that 1+1=2 is necessary, then it holds under any and all conditions. So, doesn’t that mean that anything is relevant to whether 1+1=2? That is, you can’t speak of some sentence U that is seemingly unrelated to 1+1=2 without having access to that 1+1 in fact is equal to 2. So, U implies 1+1=2.

Still, classical and intuitionistic conditionals seem to produce odd results. For example, suppose ~A=“The axiom of choice is unprovable” and C=“The continuum hypothesis is provable.” It seems that ~A->(C->~A) is unwarranted since, though the continuum hypothesis is unprovable if the axiom of choice is unprovable, it seems to be the case that C->~A isn’t an example of an actual implication, even if it is valid in a given proof system. But, I am willing to accept that in such a proof system any attempt at refuting C->~A leads to absurdity. What do you think are some properties of a natural-language/intuitive interpretation of implication?

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    Natural language interpretation of implication is not to seek truth in the twinkling of an eye or prove some serious argument constructively, they're more pragmatics and rhetoric like conversational implicatures studied by Grice where in most cases there's no actual relevance between A/C, and highly hypothetical (same issue as you observed in FOL) or counterfactual. Natural language implication is casual implication and perhaps its currently mundanely commonly accepted concept or name needs to be rectified on a mass level in this postmodern era... Commented Apr 9, 2023 at 3:55
  • "What do you think" questions are not a good fit for SE format since they invite users' personal opinions. Conditionals of the natural language are called indicative conditionals (as opposed to the material conditional of classical logic) and there is a long SEP article on their properties and modeling them formally. There is also a long article on relevance logic. Could you please read them and make the question more specific and less opinion-based.
    – Conifold
    Commented Apr 9, 2023 at 4:06
  • @Conifold I don’t see the problem as I was testing others’ intuitions. I’ve looked over at least the relevant logic article, and can’t remember if I’ve looked over the one on indicative conditionals. I am more interested in testing the intuitions about implication of people on SE.
    – PW_246
    Commented Apr 9, 2023 at 10:43
  • Antecedent and consequent my be entirely independent of one another. See my recent posting here: math.stackexchange.com/questions/4703222/… Commented May 27, 2023 at 2:34
  • In daily discourse, we rarely if ever consider implications when their antecedents are false, but so-called vacuous truth (with false antecedents) is a widely used method of proof in very technical arguments, e.g. in mathematical proofs. Commented May 27, 2023 at 2:43

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You are right that relevance logic is motivated by a desire to obtain a more natural account of conditionals and to exclude as true those where there is no relevant connection between the antecedent and consequent.

A fairly standard approach to the semantics of the conditional of relevance logic is to understand it in an information-theoretic way. It can be thought of as expressing an information channel between two information sites, or as an implication between situations under various restrictions, or as a conditional that is subject to pragmatic maxims, or as a relation that combines information to yield a result.

The sentence, "if the moon is made of cheese then 2+2=4" holds true for the material conditional and also for the strict conditional. In relevance logic, the antecedent is irrelevant to the consequent, so it does not hold. We might understand this by observing that the information expressed by "the moon is made of cheese", whether it holds or not at a particular world, contributes no information to the fact that 2+2=4 holds, so it is redundant.

In order to understand what 'information' and 'contributing information' mean in this context, you would need to go into the details of Routley-Meyer semantics and the varieties of relevance logic that arise based on the restrictions imposed by its accessibility relation.

The conditionals of classical and intuitionistic logics do indeed produce many odd results. So much so that they cannot be considered general accounts of the logic of conditionals. Natural language conditionals are often non-monotonic, context-dependent, non-truth valued, uncertain, etc. Also, it is important to remember that conditionals are not just used to conditionalise statements. We can conditionalise questions, commands, offers, threats, bets, promises, etc. A satisfactory account of conditionals should be able to include these also, though many make no attempt to do so.

There is a huge literature on conditionals that attempts to understand their logic, their semantics and their pragmatics. Some approaches are: the use of non-classical logics such as relevance logics, probability logic, connexive logic and default logics; the addition of stronger conditionals to classical logic such as strict and variably strict conditonals; theories of counterfactuals; non-truth valued accounts of conditionals; evidential theories of conditionals, linguistic treatment of conditionals as quantificational restrictors; dynamic understanding of conditionals in terms of belief revision.

If you are fairly new to conditionals, it is worth reading some of the articles in the Stanford Encyclopedia, particularly on Indicative Condiitonals, Counterfactuals Conditionals, and The Logic of Conditionals. A good introductory book is Jonathan Bennett's Conditionals: A Philosophical Guide (2003).

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  • I disagree with the relevant logic position that there is no information transfer from “the Moon is made of cheese” to “2+2=4” just because anything that can be assumed in either a possible or impossible world will satisfy that 2+2=4. It isn’t redundant, but rather saying that anything implies a necessary truth just because implication is about preserving truth and/or justification.
    – PW_246
    Commented Apr 9, 2023 at 2:54
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    Relevance logic is not about preserving truth but about information flow. In relevance logic it is not the case that anything implies a necessary truth. A common approach to the semantics is to allow for worlds that are inconsistent or where bivalence does not hold. This means you cannot rely on 2+2=4 holding in all worlds, which is why relevant conditionals differ from strict conditionals in this respect. If you are trying to establish or prove that 2+2=4 in a world, the information that the moon is made of cheese (whether true or false) will not help you.
    – Bumble
    Commented Apr 9, 2023 at 3:48

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