The expression(~p or p )is a Tautology. Consider this statement(p): This statement is false. Now here, Statement p is paradoxical. My question is :- Can we define paradoxes like this as statements which prove Tautologies wrong?
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As an interesting aside, in natural language there are also "anti-paradoxes" like "This statement is true." It turns out that - just as paradoxical statements in natural language wind up yielding important ideas in formal logic - "anti-paradoxes" also reveal rich mathematical structure; see this old math.stackexchange question for some discussion on this (and note that at first glance, the answers there appear to contradict each other!). – Noah Schweber Apr 20 '18 at 16:49
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No. A paradox doesn't just assert something incorrect (e.g. "$0\not=0$") - it asserts something which cannot be consistently assigned a truth value. Just implying the negation of a tautology doesn't mean that a statement is paradoxical: e.g. "$p$ and $\neg p$" is not a paradox, it's just a false statement.
Noah Schweber
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A statement can be provable or not provable, and it can be sound (something we expect to be true, like 1+1=2) or unsound (we expect it to be false, like 1+1=3).
A paradox arises when a statement is unsound but provable. It indicates an error in the logic.
The claim of the speaker in the liar's paradox is clearly unsound: it contradicts itself. But we have no reason to believe it is provable. So it only has 1 of the 2 qualifications needed to be a paradox.
Can we define paradoxes like this as statements which prove Tautologies wrong?
No. A paradox isn't only defined by what it proves (unsound claims) , but by the fact that the statement itself is provable.
DanielV
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