Presuppositional apologetics is a form of Christian apologetics, largely Calvinist that asserts that the acceptance of the proposition "God exists" and the truth of the Christian Bible is necessary for making the world intelligible. Presuppositional apologetics usually hinges on the rejection of Thomist apologetics, which attempt to establish logical proofs for the existence of god.
The Transcendental Argument for the existence of God is often considered a claim of presuppositional apologetics.
Critique from Modal Logic
One critique of presuppositional apologetics is that it makes an existential claim (i.e. 'God exist') primitive. It is generally accepted, in logic, that only universal claims (like tautologies) can be primitive.
In modal logic, existential claims about logical possibilities (like 'God exists') are generally believed to be true in at least one possible world and false in some other possible world. Because they are generally accepted as being false in some possible world, they are not considered necessary truths. Only necessary truths can be primitive.
List of Presuppositional Theologians
Cornelius Van Til - Gordon Clark - Greg Bahnsen - John Frame - Rousas John Rushdoony - Francis Schaeffer - Douglas Wilson