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The following passage that I am quoting at length from Dag Prawitz ("Intuitionistic Logic: A Philosophical Challenge" in Logic and Philosophy edited by G. H. von Wright, Hague, Martinus Nijhoff Publishers, pp. 8-9) may be more illuminating: Intuitionistic philosophers sometimes use true as synonymous with the truth as known, but this is clearly a ...

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Suppose (A <-> ~A) for proof by reductio. Unpacked, this is to assume ((A -> ~A) & (~A -> A)). From this both (A -> ~A) and (~A -> A) follow. Substituting for the conditionals, that implies both (~A v ~A) and (A v A). Suppose A. Then (A & (~A v ~A)) entails ~A. Hence, (A & ~A), i.e. contradiction. Repeat this to show that (A v A)...

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In (X iff ~X) the truth value of X is unstable (if true, then false; if false, then true). So it looks similar to the Liar Paradox. (This statement is false.) As far as I know, there is not yet any satisfactory "solution" to the Liar Paradox. the file I have received to start this problem has a contradiction symbol as step one Usually, ...

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The original post mostly had the right idea. Negation Elimination subproofs are required, we just need to nest them. Assume H and then assume A, aiming to derive contraditions to eliminate the assumptions. 1| H > (A > B) Promise 2| ~K & ~B Promise 3|_ ~A > K Promise 4| |_ H Assumed 5|...

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You can of course first prove Cube(a) v ~Cube(a) using a Proof by contradiction .. the book should have the schema for that But a little more efficient is to use the following set-up:

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Any proof by LEM+disjunction elimination may be rewritten as a proof by Reduction to Absurdity. Take the following structure: | P v ~P TautCon (LEM) | |_ P Assume | | : | | Q derived somehow | + | |_ ~P Assume | | : | | Q derived somehow | Q Disjunction Elimination When Q may be ...

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I figured it out for people that stumble across this again. For the solution I found statement 1 isn't needed.

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The reason you are having trouble doing that, is that that can not be done. It is not a valid derivation. The conjunction of two existences does not entail an existence of a conjunction. Neither do the other two premises enable it to be derived. PS: Existential Elimination requires an existential statement, and the assumption of a witness for that ...

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Short answer: Your confusion stems from not understanding formal systems and their objectivity. Provability in a formal system is almost completely objective, more objective that anything else you can ever hope for. So logical consequence within a formal system is equally objective. Long answer: A mathematically proven statement would be absolutely ...

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