I have started a new chapter and I do not quite understand how you start questions like these? Do i need to just keep doing contradictions to get an answer like this? Can someone point me in the right direction or help me? Premise: B ↔ ¬B Conclusion: J ↔ ¬C

closed as off-topic by Conifold, Jishin Noben, Alexander Gegg, Eliran, YiFan Apr 27 at 11:46

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    Hi, welcome to Philosophy SE. First, it is unclear what "this problem" is. B ↔ ¬B leads to a contradiction from which anything follows by the law of explosion, for example. But second, we do not know what course you are taking and where you are at in it, hence what sorts of arguments/derivations you are supposed to give, and based on what. You have to describe that in your post, along with your attempts to solve it. – Conifold Apr 23 at 23:34
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    What textbook are you using and which problem is that in the textbook? – Frank Hubeny Apr 24 at 0:02

One way to start the problem is to put the premise and conclusion in a proof checker and then use the inference rules programmed into the proof checker to reach the conclusion. For this problem since the premises lead to a contradiction whether you start with "B" or "¬B", you can use the law of the excluded middle to reach any result.

Here is how I did it using the proof checker linked to below with a text book explaining the rules in more detail.

enter image description here

To use the law of the excluded middle I considered both case "B" on lines 2-5 and case "¬B" on lines 6-9. I reached the same conclusion using explosion (X) on lines 5 and 9 and finished by invoking the law of the excluded middle (LEM) on line 10.

This may not match what you need to do but hopefully it provides a hint how you might proceed.

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Winter 2018. http://forallx.openlogicproject.org/

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