Synthetic proposition

A synthetic proposition is a proposition that is capable of being true or untrue based on facts about the world - in contrast to an analytic proposition which is true by definition.

For example, "Mary had a little lamb" is a synthetic proposition - since its truth depends on whether she in fact had a little lamb.

The truth or falsity of synthetic propositions is contingent - their truth depends on what the universe is like.

Are there synthetic propositions known a priori?
Whether it is possible for a synthetic proposition to be a priori is a matter of considerable controversy in philosophy.

The empiricists hold that there are no synthetic a priori truths.

The rationalists hold a contrary view - that there are synthetic truths that are also a priori.

In rationalist view, certain concepts are derived from experience - but once these concepts have been derived, they are seen as necessarily true.

For example, in this view, we derive the notion that 2 + 2 = 4 from experience - it would not have occurred to us had we not experienced two things, and another two things, becoming four things - but once we do experience it, we recognize it as a necessary truth.

The Kantian view
Immanuel Kant's view was more evolved.

In his view, our knowledge of the world is limited to the phenomenal world - the world as it is known by the mind. The noumenal world - the world as it is when it is not subject to the mind - is fundamentally not knowable. The nature of the phenomenal world is constrained by the nature of the human mind.

Beware. Although often the noumenal world is spoken about as the world "the way it really is", the mere proposition that the noumenal world contains knowledge that somehow isn't accesible to the human mind is false and certainly contrary to the way Kant conceived this distinction. The distinction between phenomenal and noumenal is merely the distinction between "subject to the mind" and "not subject to the mind" ('mind' not to be taken psychologically but epistemologically).

So, certain fundamental truths about the world - for example, that if A precedes B and B precedes C, A must precede C - are truths about the phenomenal world only. We can't prove "A must precede C" in the noumenal world. So such fundamental truths are not universal truths - and thus not analytic. However, they are a priori - in the sense that our minds could not possibly think otherwise.