Updated: Apr 14, 2020
Immanuel Kant (1724-1804) is responsible for introducing the term “transcendental” to philosophical discussion. All of us, he argued, must concede that knowledge is possible. Else there is no point to any discussion or inquiry. Now, given that knowledge is possible, said Kant, we should ask what the conditions are that make knowledge possible. What must the world be like, and what must the workings of our minds be like, if human knowledge is to be possible? 
Transcendental arguments became a staple of the writings of the idealist school that followed Kant, and from there it made its way into Christian apologetics. The twentieth-century apologist who placed the most weight on the transcendental argument (which he sometimes called “reasoning by presupposition”) was Cornelius Van Til (1895-1987) 
Formulating the Transcendental Argument 
Prove A: The Christian God exists.
Step 1 -A:
(Assume the opposite of what we are trying to prove): The Christian God does not exist.
Step 2 (-A implies B):
If God does not exist, then there is no intelligible experience since God is the precondition of intelligibility
Step 3 (-B):
There is intelligible experience (Contradiction from step 1 and step 2)
Step 4 (- -A):
It is not the case that God does not exist (Modus Tollens on 2 and 3)
Step 5 (A):
God does exist (Law of negation.)
What is Meant by Intelligible Experience?
The requirements for intelligible experience are what is generally called the preconditions of intelligibility. These are things like:
The laws of logic (which are immaterial, unchanging, universally applicable)
Reliability of senses
Reliability of reasoning
Reliability of memory
Uniformity of nature
The argument is quite simple and has very little unbelievers can oppose. The only real opposition can come at step two of TAG. How do we go about proving that if God does not exist, then there is no intelligible experience? How do we go about proving that God is the precondition of intelligibility?
The Word of God
God has revealed in His Word that He is, in fact, the precondition of intelligibility. Hold up. We're trying to prove that God exists remember? We can't appeal to God's Word at this stage, as that assumes the very thing we're trying to prove! The "proof" for God's existence is not the same thing as the premise that God has revealed himself, or that the Scriptures are God's word. Consider an example. You want to prove to someone that your friend exists. One way you can do this is by allowing your friend to talk to the person to prove that he exists. The person then goes: Hold up! You can't allow your friend to talk to me as that assumes that he exists, but that's what you're trying to prove!
As Christians, we accept that Bible as God's self-attesting authoritative Word of God. The Word of God reveals that God is the precondition of intelligibility, and therefore we can know that if we negate the existence of (the Christian) God, we will lose all ground for intelligibility.
This couples in with what Christians call a revelational epistemology. How do we know what we know? Everything that we know must be based directly or in principle on God's revelation, who is Himself the Truth. So how do we know God is the precondition of intelligibility? Revelation. God's revelation is our ultimate standard for truth.
So how do we know the Bible is the Word of God? The Westminster Confession states the following:
The authority of the Holy Scripture, for which it ought to be believed and obeyed, dependeth not upon the testimony of any man, or Church; but wholly upon God (who is truth itself) the author thereof: and therefore it is to be received because it is the Word of God. II Pet. 1:19, 21; II Tim. 3:16; I John 5:9; I Thess. 2:13.
Chapter I, Paragraph IV
We may be moved and induced by the testimony of the Church to a high and reverent esteem of the Holy Scripture. And the heavenliness of the matter, the efficacy of the doctrine, the majesty of the style, the consent of all the parts, the scope of the whole (which is, to give all glory to God), the full discovery it makes of the only way of man’s salvation, the many other incomparable excellencies, and the entire perfection thereof, are arguments whereby it doth abundantly evidence itself to be the Word of God: yet notwithstanding, our full persuasion and assurance of the infallible truth and divine authority thereof, is from the inward work of the Holy Spirit bearing witness by and with the Word in our hearts. I Tim. 3:15; I John 2:20, 27; John 16:13, 14; I Cor. 2:10, 11, 12; Isa. 59:21.
Chapter I, Paragraph V
The next charge comes in the form of a circular reasoning charge. "We believe the Bible is the Word of God because it says it is the Word of God". Check out this article which handles the circular reasoning objection.
The circular reasoning people believe we are committing boils down to something like this:
Now going back to the formulation of the argument, you'll see that TAG doesn't use the bottom half of the circle. We actually split the circle in two. Consider the following quotation by Van Til:
“The Reformed method of apologetics seeks to escape this nemesis. It begins frankly ‘from above.’ It would ‘presuppose’ God. But in presupposing God it cannot place itself at any point on a neutral basis with the non-Christian. Before seeking to prove that Christianity is in accord with reason and in accord with fact, it would ask what is meant by ‘reason’ and what is meant by ‘fact.’ It would argue that unless reason and fact are themselves interpreted in terms of God they are unintelligible. If God is not presupposed, reason is a pure abstraction that has no contact with fact, and fact is a pure abstraction that has no contact with reason. Reason and fact cannot be brought into fruitful union with one another except upon the presupposition of the existence of God and his control over the universe.
Since on the Reformed basis there is no area of neutrality between the believer and the unbeliever, the argument between them must be indirect. Christians cannot allow the legitimacy of the assumptions that underlie the non-Christian methodology. But they can place themselves upon the position of those whom they are seeking to win to a belief in Christianity for the sake of the argument. And the non-Christian, though not granting the presuppositions from which the Christian works, can nevertheless place himself upon the position of the Christian for the sake of the argument.”
Cornelius Van Til, A Christian Theory of Knowledge p.18
The Christian apologetic, therefore, is a two-part process. We first demonstrate that unless God is presupposed, "reason is a pure abstraction that has no contact with fact, and fact is a pure abstraction that has no contact with reason" (this is in effect doing the internal critique of the unbelieving worldview). Afterwards, we can then invite the unbeliever to place himself upon the position of the Christian for the sake of the argument to see how the Christian position can make sense of reality as we experience it.
The first part, therefore, only reflects the top half of the circle, which in effect says that the preconditions if intelligibility only makes senses when God exists.
The second half of the circle is the apologist inviting the unbeliever to see how the Christian worldview, which uses God revelation as epistemology, can make sense of the preconditions of intelligibility which we all experience. The reason we can split the circle in two is because we are not attempting to prove the existence of the laws of logic or the other elements mentioned, everyone knows they exist and utilises them. But, as Bahnsen said, they are not epistemically self-conscious - which means they haven't thought through the ability of their worldview to be able to justify them assuming the preconditions of intelligibility.
That's the point of the second half. To allow the unbeliever to see how the Christian worldview provides a framework that can justify the preconditions of intelligibility.
It is at this point that unbeliever can challenge the uniqueness of the bottom circle. Why can't we just insert a different God with different revelation in the bottom circle and the argument can be used to prove the existence of any other god?
The Christian can revert to the first circle by doing an internal critique of the proposed worldview to show how this particular proposed god cannot justify the laws of logic, therefore it would not follow that if the laws of logic exist, this particular god (which will be contradictory, arbitrary) exists.
As a last resort, the unbeliever might appeal to a currently unknown worldview that might be able to someday provide suitable justification for the preconditions of intelligibility that is not Christianity.
Bahnsen  answers this objection by stating that if Christianity is able to justify the preconditions of intelligibility, it by necessity follows that it is the only worldview that can justify the preconditions of intelligibility as that is part of it's internal claims (e.g. there is only one God). If there is another worldview that can provide a grounding for the preconditions of intelligibility, then it will follow that Christianity does not, in fact, supply us with the preconditions of intelligibility as it is internally flawed.
If there isn't another one, then Christianity is true. Christianity, due to the nature of the religion cannot be just one of many in a sequence. It's all or nothing. It must be the only one if it is one at all. However, Christianity does provide the preconditions of intelligibility and therefore it must be the only one!
From the own internal character of Christianity, we can eliminate all the other competing worldviews (and hence justify premise 2 of TAG) without having the inductive task of having to refute an endless sequence of possible worldviews. And this brings us back to the Word of God which is foundational to TAG which makes these internal claims.
The Christian Worldview
In what way does the Christian worldview provide for the preconditions of intelligibility? Greg Bahnsen summarised it as follows:
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 , pg. 21.
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Reformed Apologist: The Impropriety of Trying to "Prove" The Absolute Truth Value of a Transcendental Inductively. 2019. [ONLINE] Available at: https://reformedapologist.blogspot.com/2006/03/impropriety-of-trying-to-prove.html. [Accessed 11 November 2019].
The Gospel Coalition Resource Library. 2019. Van Tilian Apologetics, Part 1 of 4 | The Gospel Coalition . [ONLINE] Available at: http://resources.thegospelcoalition.org/library/van-tilian-apologetics-part-1-of-4. [Accessed 11 November 2019].