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From what I know, the truth of an analytic statement is based solely on its meaning so understanding the meaning of an analytic statement is enough to determine whether it's true. I understand why tautologies such as "either it will or it will not rain" or statements that are true by definition like "all bachelors are unmarried men" would be considered analytic statements, but I still struggle to grasp the concept. Are there less obvious examples of analytic statements?

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    Analytic statements are true via the nature of their meaning. So, really there isn't much, if anything, beyond the idea of "true by definition". The idea of analytic statements is that the predicate in the subject predicate pair is contained within the subject. The concept "Unmarried man" is contained within bachelor. So any sort of proposition that can be analyzed that way would be analytic, but again it mostly just comes down to "true by definition". That's the whole point, separating statements that are true by the nature of meaning, analytic, from ones that aren't, synthetic. – Not_Here Aug 1 '17 at 22:44
  • You may want to read the article on "The Analytic-Synthetic" distinction at Stanford Encyclopedia of Philosophy on the net too. – Gordon Aug 2 '17 at 20:46
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I think a good example of less obvious statements that would be considered analytic are theorems of mathematics - if everything is well-defined, you have a set of axioms, and you follow some given rules of deduction, then the theorems which follow from the axioms are purely analytic.

For example, Euclid's "Elements" is based on some set of axioms and rules of deduction, from which you can analytically derive the Pythagorean theorem - a nontrivial analytic statement.

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    Kant made the argument that mathematics, especially stuff like geometry, were examples of synthetic a priori, not analytic a priori. I.e. "7+5=12" is synthetic because "7" and "5" and "=" are not contained in the definition of "12". Your example of using axioms to derive theorems makes it seem like you're confusing analytic with a priori. – Not_Here Aug 1 '17 at 22:39
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    I agree that if you have a set theoretic definition of ordinal then 7 and 5 are contained within 12, of course, but the addition function is not semantically contained within the definition of 12. – Not_Here Aug 1 '17 at 22:54
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    First, I would like to thank you for pointing out Kant's objection to the conception of mathematics as analytic - I should have mentioned that in the original post, as it is as a famous and important argument. Next, here is the link to the SEP's article on the Analytic/Synthetic distinction and the views of Kant and later logicians on the problem, since it seems particularly relevant: plato.stanford.edu/entries/analytic-synthetic – smb3 Aug 1 '17 at 23:02
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    I think you'd lose Kant and the common meanings of these terms in philosophy when you say "analytically derive". You seem to be using "analytic" as a synonym for "a priori." Analytic is generally opposed to synthetic meaning combining more than one axiom. – virmaior Aug 2 '17 at 6:03
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    But as you go on, you point out that they are extending or restricting Kant's version. Thus, it functions for all of them (and us) as the basis of discourse about analytic-synthetic. If we want to use the term in ways unhinged from Kant (such as generic dictionary definitions), then it's not clear how this is a question for philosophy.SE instead of English.SE – virmaior Aug 2 '17 at 9:24
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I think you've largely got the idea down.

Analytic means something that can be shown true without reference to anything else. This means either pure tautologies or expansions of definitions (if there's some other ingenious category, it does not spring to mind).

The key value of analytic is that it is part of a pair. The other feature in the pair is synthetic where we learn something by adding to two different truth things together, such as combining two axioms. As generally used in philosophy, that's how the pair of terms works.

For most of us, we know the terms through Kant where it occurs in tandem with a two further pairs: necessary vs. contingent and a priori vs. a posteriori. (These, like synthetic / analytic, does not truly originate in Kant but these terms are studied the most in trying to understand Kant). Here, the idea is how do I know something. Something that is a prior analytic can be known just by thinking of the term and working out what it means. That which is a prior synthetic involves the conjunction of multiple a priori truths to reach its conclusion.

A posteriori items work based on evidence. These two can either be things that are identities or things that require working together multiple pieces. If my memory is correct, something that has an a posteriori element would then make the entire argument a posteriori for Kant.

A second big name on this topic is Kripke who is famous for proposing the idea of a posteriori necessity in Naming and Necessity

There's some more examples in a different question: What are examples of analytic a posteriori knowledge?

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