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Logic

Why is the Law of Contradiction so important to Philosophers? Can its truth be proven? What are the consequences for philosophy of an answer to the question, what's at stake? Thanks so much for your help. Krystle

Here's what I said to an earlier question asking why contradictions are bad (November 23). As you'll see, my answer implies that it's not just philosophers for whom the avoidance of contradiction is important. It's important for anybody who is aiming at the truth. "A simple answer is that sentences of the form 'A and not-A' cannot be true. So if you're aiming at truth, such sentences should not be endorsed. (And if other claims led you 'A and not-A' by valid reasoning (reasoning that never goes from truths to falsehood) they cannot all be true either, and so similarly shouldn't all be endorsed.) Why can't sentences of the form 'A and not-A' be true? Because of the meanings of 'not' and 'and'. The classical semantic analysis of 'not' is that prefixing it to a sentence ('A') gives you a new sentence ('not-A') that is true in just those circumstances where A is not true. The analogous analysis of 'and' is that placing it between two sentences gives you a new sentence that is true just in...
Logic

What is the principle underpinning logic's rules aimed at avoiding contradiction? We know that contradiction is "bad", e.g. if a line of argument can be reduced to the statement A & ~A, (or if such an assertion can be extracted from an argument) the argument may be invalid. Where does this principle that contradiction leads to invalidity come from? Is it a "just because" axiom? Is it from overwhelming empirical observation? (I've certainly never seen something that both is and isn't at the same time.) More broadly (if this isn't too much) what is the relationship between basic concepts in epistemology (non-contradiction and cause/effect) and axioms of logic? What metaphysical connections or bindings exist between axioms of logic, epistemology, and objective reality? Thanks

A simple answer is that sentences of the form 'A and not-A' cannot be true. So if you're aiming at truth, such sentences should not be endorsed. (And if other claims led you 'A and not-A' by valid reasoning (reasoning that never goes from truths to falsehood) they cannot all be true either, and so similarly shouldn't all be endorsed.) Why can't sentences of the form 'A and not-A' be true? Because of the meanings of 'not' and 'and'. The classical semantic analysis of 'not' is that prefixing it to a sentence ('A') gives you a new sentence ('not-A') that is true in just those circumstances where A is not true. The analogous analysis of 'and' is that putting it between two sentences gives you a new sentence that is true just in case the original two sentences are both true. Putting these together, it follows that 'A and not-A' cannot be true. (There are other (non-classical) ways of analysing 'not and 'and'; most (though not all) will similarly explain why 'A and not-A' cannot be true.)