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After reading the definitions of both these terms I just confused and I am not able to find good differences between these two.

Wikipedia defines satisfiability as "A formula is satisfiable if it is possible to find an interpretation (model) that makes the formula true."

What I understand is this "a formula P is satisfiable if P is not a contradiction."

Wikipedia defines contingency as "contingency is the status of propositions that are neither true under every possible valuation nor false under every possible valuation."

What I understand is that "a formula P is called contingent if P is neither a tautology nor a contradiction."

After interpreting these two definitions I think the only difference is as follows:

If a formula P is a contingent then P is satisfiable and If Q is satisfiable but not a tautology then Q is contingent. Am I right or misinterpreting it?

Keshav Ranjan
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  • Q can't be contradiction as well. So, it should be Q is satisfiable but not a tautology and contradiction then Q is contingent. So, basically P is contingent iff P is satisfiable. – Ubi.B Jun 26 '20 at 16:27

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Yes, that's right: contingencies are sentences which are satisfiable but which also have satisfiable negations. That is, a sentence is a contingency iff it is neither a tautology nor a contradiction.

Incidentally, tautologies are also often called validities, so we could rephrase this as "contingent = satisfiable but not valid." (It's also worth noting that I've almost never seen "contingent" on the more mathematical side of logic - "satisfiable" is almost certainly the term you want to pay more attention to.)

Noah Schweber
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  • @Learner Max' comments are absolutely correct; I'd ask them to turn them into an answer. The issue is exactly that verifying that a putative counterexample is indeed a counterexample might itself be an undecidable task. – Noah Schweber Sep 07 '19 at 21:16