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I'm doing the body of exercises in Priest's Introduction to Non-classical logic, and got weirdly stuck on 2.12 (o). (I'm on Problems 2.12, if anyone has a link to the answers that would also be much appreciated; can't find them online.) The exercise is to "show" that it is not the case that it is not a logical truth that necessarily p entails p and the method is to construct a tableau.

I start by negating it. 1) Not (necessarily p entails p). 2) necessarily p, not p (from 1).

Then I can't get any further with tableaux methods at least.

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The first step is correct :

1) ¬(◻ P → P), i

2) ◻ P, i

3) ¬P, i

4) P, j --- from 2), where (i R j)

and we are stuck... there is no contradiction, provided that R is not reflexive, i.e. not-(i R i).

See Modal Axioms and Conditions on Frames :

the Axiom (M) □A → A corresponds to the Condition on Frames that R is Reflexive.

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