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We can easily prove that : $(\lnot p \lor q) \rightarrow (p \rightarrow q)$ holds intuitionistically with Natural Deduction. I'm quite unfamiliar with the intuitionistical version of tableau method; I've seen : Melvin Fitting, Intuitionistic logic Model theory and Forcing (1969), page 28 where the usual tableau rules are modified as follows : ...


There are Fitch systems for about twenty different modal logics described very briskly here http://projecteuclid.org/download/pdf_1/euclid.ndjfl/1093888133 More than enough to be going on with, I guess! More expansively, Garson's good and accessible book Modal Logic for Philosophers (CUP) introduces Fitch style systems for modal logics.


You have $\neg p$ back in $w_0$, and $p$ in $w_1$. But there's no contradiction between those two, even if $w_0Rw_1$! It is absolutely fundamental that a propositional variable can take different values at different worlds, even if those worlds are related by the accessibility relation.


See in SEP Modal Logic, Ch.6 : Possible Worlds Semantics : In propositional logic, a valuation of the atomic sentences (or row of a truth table) assigns a truth value (T or F) to each propositional variable $p$. Then the truth values of the complex sentences are calculated with truth tables. In modal semantics, a set $W$ of possible worlds is introduced. ...


Adding this axiom to the axioms for normal modal logic K based on classical propositional logic is equivalent (using propositional logic) to adding the normality axiom (as mentioned before) together with the two axioms $\Box Q \rightarrow \Box (\neg P \vee Q)$ and $\Box P \vee \Box (P \rightarrow Q)$. The first one follows from monotonicity of $\Box$, while ...


This is Gödel's ontological proof, which is fully explained on Wikipedia (see the link). I find this to be an incredible easter egg, since the proof is something which Sheldon says he doesn't understand. And Sheldon has expressed more than once that he doesn't believe in the existence of god. (Originally I thought this would be something pertaining to the ...

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