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If we were to prove that an argument is a logical truth only in S5 logic out of (K, T, S4, and S5). does that make S5 the weakest of these four logics in which the argument is a logical truth?

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First, a little correction. It is best to think of propositions as being logical truths (or not) and arguments as being valid (or invalid). An argument is not a logical truth.

If a proposition is a logical truth of S5, and not a logical truth of K, T or S4, then of those four logics, S5 is the only one in which it is a logical truth. Hence it is the weakest, the strongest, the most fun and the most anything.

So, I'm not sure whether that is really what you meant to ask. If a proposition is not a logical truth of K, T or S4, then out of all possible modal logics, S5 might be the weakest in which it is a logical truth, but there are modal logics intermediate in strength between S4 and S5, so it might not.

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  • Sorry for the incorrect wording I meant a proposition not an argument. However, thanks a lot that is exactly what I asked! Appreciate it.
    – Raghad
    Sep 27 at 2:16

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