Hume's Fork is an ontological principle which states that all statements fall into one of two discrete categories. Statements can either be analytic or synthetic. Note that this says nothing about the truth value of the statements, although bivalence is assumed.
Analytic statements are based on a priori knowledge, are necessarily and definitively true or false, and true or false by definition. Their truth or falsehood is often, but not always, obvious. All tautologous ('all hamburgers are hamburgers'), axiomatic ('this triangle has 6 sides'), mathematical ('2+2=5'), or predicate-contained ('all bachelors are unmarried') statements are analytic. Analytic reasoning uses deductive logic.
Synthetic statements are based on a posteriori knowledge in the physical world, are likely to be true or false based on the strength of their supporting arguments. Scientific ('at sea level and under normal atmospheric pressure, pure water boils at 100 degrees centigrade'), physical ('the sun will rise tomorrow'), existensial ('God exists'), and most day-to-day ('my car is red') statements are synthetic. Synthetic reasoning uses inductive logic.
Many arguments for the existence of God and arguments against the existence of God are based on synthetic statements and inductive reasoning. A notable exception to this is the ontological argument, which is wholly analytic.