An analytic proposition is one in which the conclusion is contained in the premises. We “analyze,” drill down, divide.
A bachelor is an unmarried male…
Smith is bachelor.
Therefore, Smith is unmarried.
Here, the conclusion is less than the sum of the premises. The premises set has more information in it than the conclusion.
A synthetic statement is quite the contrary: the conclusion is “synthesized” from the premises. We put together, harmonize, combine. The conclusion is bigger than any premise.
I’d like to go out to eat tomorrow.
But it’s going to snow.
So, driving may be dangerous.
But we planned this trip a few days ago.
Therefore, we’ll probably go despite the weather.
The conclusion is reached by weighing a bunch of reasons and picking the right decision. It’s almost as if analysis pertains to knowledge, while synthesis, to understanding.
A posteriori refers to sensing; a priori, to reflecting; this is about how we get the premises. Analytic / synthetic is about how we get the conclusions.
“2 + 2= 4″ is an a priori statement, but it is neither analytic nor synthetic but self-evident.
“I’m sitting” is an a posteriori statement and is close to being self-evident, as well. But it can be called analytic. Given the premises “I feel myself sitting” and “My feelings reflect reality,” the conclusion follows.
“Goldbach’s conjecture is true” is an analytic a priori proposition.
“What is colored is extended.” This one is tricky. In order to know colors and extentions, one needs personal experience with these things. On the other hand, one need not experience all colored and extended things in order to be able to say that ∀x [x is colored -> x is extended], even necessarily so. So, this proposition has both a posteriori and a priori aspects. The conclusion is arrived at not from the universal to particular, but the reverse, so this proposition is synthetic. It is synthetic a priori, also like “What is red all over is not green all over”; “All bachelors are unhappy” (not necessarily true); etc.
Update. On second thought, how do we arrive to the conclusion that what is colored is extended? Something like the following:
Every colored thing I’ve seen in my life had a surface. (a posteriori)
I can’t conceive or imagine any colored thing without a concomitant surface. (a priori)
Therefore, every colored thing in the actual universe or even in all possible worlds has a surface. (synthesis)
But a surface by definition is something that is extended in 2- or 3-dimensional space. (a priori)
Therefore, every colored thing is extended. (analytic)
So, this proposition has also both synthetic and analytic aspects.