Transcendental (n.) - A condition of experience. Something transcendental is the condition for some kind of experience.
Form: For p to be the case, q would have to be the case because q is a precondition of p. p is the case, therefore q is the case.
Example 1: For there to be carpet laid (p) there must be a floor (q) to have it. There is a carpet laid (p), therefore, there is a floor (q). (Narrow, ontological argument-type).
For us to have a have a conception of creation (p) there has to be a concept of causation (q) because the concept of causation is a necessary precondition of creation. Because there is creation (p), we must have causation (q). (General, conceptual argument-type).
A transcendental is an ultimate criterion for p to be the case. How do we show that q is necessary to explain p?
Indirect Method: (-q) implies absurdity or renders the operational feature impossible: not believing a floor (-q) is necessary for a carpet to be laid (p), not believing causation (-q) is a necessary precondition for creation (p).
Dilemma Method: Take p, show how it implies q, and show that the denial of p implies q as well: Making the statement that one believes in causality (p) implies causality (q). Making the statement that one rejects causality (-p) still implies causality (q).
Form: p implies q and -p implies q. But it must be true that p or -p. Therefore, q is true regardless.
The Christian Worldview - Dr Greg Bahnsen
The Bible sets forth a definite metaphysical scheme. It begins with God who is personal, infinitely perfect, pure spirit (Ex. 15:11; Mal. 2:10; John 4:24). The triune God (2 Cor. 13:14) is unique in His nature and works (Ps. 86:9), self-existent (Ex. 3:14; John 5:26; Gal. 4:8-9), eternal (Ps. 90:2), immutable (Mal. 3:6), and omnipresent (Ps. 139:7-10). Everything else that exists has been created out of nothing (Col. 1:16-17; Heb. 11:3), whether the material world (Gen. 1:1; Ex. 20:11), the realm of spirits (Ps. 148:2,5), or man. Man was created as the image of God (Gen. 1:27), a being who exhibits both material and immaterial character (Matt. 10:28), surviving bodily death (Eccl. 12:7; Rom. 2:7), with personal awareness of God (2 Cor. 5:8), and awaiting bodily resurrection (1 Cor. 6:14; 15:42-44).
In creation God made all things according to his unsearchable wisdom (Ps. 104:24; Isa. 40:28), assigning all things their definite characters (Isa. 40:26; 46:9-10). God also determines all things by His wisdom (Eph. 1:11), preserving (Neh. 9:6), governing (Ps. 103:19), and predetermining the nature and course of all things, thus being able to work miracles (Ps. 72:18). The decree by which God providentially ordains historical events is eternal, effectual, unconditional, unchangeable, and comprehensive (e.g., Isa. 46:10; Acts 2:23; Eph. 3:9-11).
These truths are paradigmatic for the believer; they are ultimate principles of objective reality, to be distinguished from the delusions set forth in contrary views of the world.
Greg L. Bahnsen, Always Ready (Nacogdoches, TX: Covenant Media Press, 11th printing, Sept. 2011), 21.
Rejecting the Christian's metaphysical scheme (-q), man inescapably becomes the final authority concerning all matters. He must then come up with a cogent explanation for truth, human dignity, uniformity (both present and throughout time), logic, knowledge, absolute regularities, prescriptions and descriptions, a priori and a posteriori knowledge, cognitive reliability, causation - the preconditions for intelligibility (p).
How do we show that the Christian worldview (q) is a necessary precondition, a transcendental, to the preconditions required for intelligibility (p)? Standing on the unbeliever's worldview (-q) renders all of the above operational features (p) impossible. (There are only 2 worldviews, q & -q). Because man is helplessly the final reference point, he can only appeal to subjectivity. And subjectivism, by definition, can never conjure up absolute, objective justification for the "ultimate principles of objective reality". They can never appeal to anything operating as a transcendental because any appeal they make is subjective. Even if they argue for the anonymity of r (principles of logic) being a transcendental (this in itself just runs into more problems), knowledge of that alleged anonymity is only a product his subjective, finite, fallible cognition. Anything that proceeds from the unbeliever's mouth is only the result of this fallibility. Man either appeals to the absolute, objective God of Scripture, or he appeals to himself. Therefore, -q renders p absurd and impossible.
Given p is the acknowledgment of the truth of the Christian worldview, and q that the Christian's metaphysical scheme (above) justifies the prerequisites necessary for intelligibility, p obviously implies q. In order for p to be the case q must be the case. Making an acknowledgement in affirmation of anything, let alone q, assumes absolute truth, logic, uniformity, causality, rationality, etc. So p implies q. However, denying the truth of Christianity (-p) still implies q because in order to reject anything you still utilize absolute truth, logic, uniformity, causality, rationality, etc. But it must be true that p or not -p. Therefore, the Christian worldview, q, is true regardless.
One True Worldview
By definition, there can only be one transcendental and therefore only one true worldview (which supplies the transcendental). The Christian, therefore, does not need to set out and refute every worldview in existence, either actual or hypothetical. There is either the Christian worldview which appeals to the objective, absolute God as the final reference point (and transcendental), or autonomous worldviews which appeal only to the subjectivity of man and lead to absurdity. Because the Christian worldview is shown to be internally consistent and make sense of the objective operations of reality, by default every other worldview, hypothetical or not, is false. There cannot be two true worldviews, or even the possibility of another true worldview 1. Because Christianity claims exclusivity, and 2., as Bahnsen rightfully stated:
...there can only be one transcendental. There cannot be two “ultimate authorities”. Why? Because if there are two systems of truth you’ve then lost unity, coherence, continuity, and therefore intelligibility and truth. If you had two transcendentals you’d have to ask what the relationship is between the two systems. What’s further is that in order to make sense of the claim that there are two systems by which facts can be made intelligible requires another system in terms of which you’re saying that about the other two. But you see if these two transcendental systems are by hypothesis, we can’t even get behind them to have the one that unites the two… To even talk about there being two there needs to be one perspective in which you talk about the two. But these two that you’re talking about are by definition ultimate! And so there can’t be a one that unites the two, even to talk about their relationship.
Bahnsen, "Transcendental Arguments" course
The proof of the Christian worldview, the proof that God exists is from that without God as the ultimate transcendental there is no basis or possibility for intelligibility. Standing on an autonomous worldview renders such preconditions impossible (indirect method). -q renders p impossible. If the unbeliever objects to this, he just runs into the dilemma method - his objection (-p) still implies the Christian worldview (q) is true, so he's lost either way.