What is the Nature of Counterexamples to Analytic Truths?

Sometimes when someone, in order to show that there are analytic truths, proposes some sentence as an example of analytic truth, other people give counterexamples to it to show how the sentence can’t be analytic truth.

For example if I say that ‘if a male is unmarried then he is a bachelor’, one can give such counterexample by saying ‘A unmarried homosexual man who lives with his partner in a long-term relationship is unmarried male, but he is not a bachelor’.

But if this is to count as a counterexample, it needs to be true. But what kind of truth is it? It doesn’t seem that it is an empirical truth. Is it analytic? Does this mean that the counterexamples to examples of analytic truths are not of any value in the arguing against possibility of analytic truths?

But then, how does one respond to those examples, if not by giving counterexamples?

Say that someone gives example ‘if a male is unmarried then he is a bachelor’ as an example of analytic sentence. And we don’t give counterexamples, but just say that in light of future experiences, we might encounter a case of an unmarried male which we wouldn’t classify as a bachelor.

But then can’t we ask… in virtue of what we wouldn’t classify that person as a bachelor? It surely can’t be just in virtue of simple ‘thisness’ (to use Hegelian phrase). There must be a reason why we won’t classify him as a bachelor. But then, if we put those reasons in a sentence, wouldn’t this sentence fall again under same case as previous counterexamples?

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