# How does modal logic address this equivalence?

I was picking where to eat dinner, and this logic question arose. Because I didn't want to dine at the same place twice in a single day.

Let me call the restaurant "Kim's". The following statement is obviously true:

Since I dined at Kim's once, I'd be able to dine at Kim's twice.

The following statement is also true, which seems to be some sort of "dual" to the statement above:

Since I dined at Kim's twice, I must've dined at Kim's once.

At my first glance, some sort of modality seems to be in action. I'd like to formally prove the equivalence between these statements. However, because of the presence of complex auxiliary verbs, it's hard to identify what kind of modal logic is this. As such, I can't even formally formulate these statements.

Note that this problem can be demonstrated in another language. Here's one instance in Korean, my native language:

내가 김가네에서 한 번 식사했으니, 김가네에서 두 번 식사할 수도 있겠다.

내가 김가네에서 두 번 식사했으니, 김가네에서 한 번 식사했었겠다.

• The "must've" in the second sentence is not modal. You could rewrite the sentence without a modal indicator: "I dinned at Kim's twice therefore I dined at Kim's once". As to the first sentence, there is no logical or mathematical connection between the first and second clause. Try a substitution: "Since Socrates died once, Socrates would be able to die twice". Commented Jul 10, 2022 at 11:20
• You can simply interpret "would be able to" as possibility and "must've" as necessity, and link your 2 primitive propositions using material implication such as p→◊q, q→□p... Commented Jul 10, 2022 at 19:20

Since I dined at Kim's once, I'd be able to dine at Kim's twice.

Since I dined at Kim's twice, I must've dined at Kim's once.

How does modal logic address this equivalence?

The two statements are not logically equivalent.

The first thing you would need to do would be to articulate the many reasons you have for believing that each of these statements is true.

Without going into that, we can look at two similar-looking statements which are however much simpler:

ϕ → ψ

ψ → ϕ

These statements are not logically equivalent either. They are called the converse of each other. However, this is all there is to it. There is no logical relation between them. One does not imply the other.

To get a logical relation between them, it would be necessary to assume first some logical relation between ϕ and ψ.

As Speakpigeon and Double Knot observed, your two statements come down to ϕ → ◊ψ and ψ → □ϕ, which other things being equal are both true as far as things go (but note David Gudeman's remark about the number of times Socrates has perished). That is, mathematically, a sequence that has reached a first stage can go on to reach a second, and one that has reached a second stage must have reached a first one.

However, this all is less a matter of equivalence than quasi-temporal implication, then. I imagine there is a tortuous path through some species of modal logic which might make a stricter relation go through, but this proposed relation would be suspicious (again on account of e.g. David Gudeman's observation), hence so would the encompassing modal logic. That being said, you are not going amiss in trying to find interesting modal relations between these propositions, and I wish I could read Korean to see how the syntax plays in to the manifestation of your question.