“To us the only thing of great significance in this connection is that it is often found to be more difficult to distinguish our method from the deductive method than from the inductive method. But the favorite charge against us is that we are still bound to the past and are therefore employing the deductive method. Our opponents are thoughtlessly identifying our method with the Greek method of deduction. For this reason it is necessary for us to make the difference between these two methods as clear as we can.” (Van Til, Survey of Christian Epistemology, 9.)
“To put it simply, in the case of ‘direct’ arguments (whether rational or empirical), the negation of one of their premises changes the truth or reliability of their conclusion. But this is not true of transcendental arguments, and that sets them off from the other kinds of proof or analysis. A transcendental argument begins with any item of experience or belief whatsoever and proceeds, by critical analysis, to ask what conditions (or what other beliefs) would need to be true in order for that original experience or belief to make sense, be meaningful, or be intelligible to us. Now then, if we should go back and negate the statement of that original belief (or consider a contrary experience), the transcendental analysis (if originally cogent or sound) would nevertheless reach the very same conclusion.” (Bahnsen, Van Til’s Apologetic, 501-502.)
“Years ago Van Til realized that opponents of presuppositionalism tend to think that there are only two kinds of reasoning: inductive and deductive. Deductive reasoning stands opposed to inductive. However, there is also transcendental reasoning, in which the preconditions for the intelligibility of what is experienced, asserted, or argued are posed or sought. It, too, stands opposed to a purely inductive approach to knowledge. Critics seem to think that, since presuppositionalism does not endorse pure inductivism, it must favor deductivism instead. This logical fallacy is known as false antithesis.” (Bahnsen, Van Til’s Apologetic, 176, n. 55.)
The above three quotes were recently forwarded to me for comment because these sentiments of Van Til and Bahnsen are often (sadly so) misconstrued as affirmation of their denying the place of deduction and induction in the realm of presuppositional apologetics. (Having the books, I have verified the accuracy of the quotes.)
The beauty of the transcendental argument for the existence of God (TAG) as a special kind of deductive argument is not in the reductio but in the transcendental challenge, which demonstrates that to argue against God one must first presuppose that which only Christianity affords. In other words, TAG is certainly a deductive argument, but it's a unique kind of deductive argument, not in its form per se but rather in what it seeks to demonstrate. Transcendental arguments are concerned with the preconditions of any fact of experience – what must be true in order for any fact of experience to be that which it is. Van Til was careful to note that “the Christian method uses neither the inductive nor the deductive method as understood by the opponents of Christianity, but that it has elements of both induction and of deduction in it, if these terms are understood in a Christian sense.” (Van Til, page 10 - emphasis mine.) Why the qualifier "as it is understood by the opponents of Christianity" if Van Til did not believe that TAG incorporated deduction? Why not just say that the Christian method does not use deduction, end of statement? The reason is, TAG has aspects of not just deduction but induction too, as Van Til states with no ambiguity. TAG has a distinctly inductive aspect to it because with TAG the Christian investigates what must be true in order for some experience to be intelligible. Such explorations are inductive in emphasis. Notwithstanding, the manner of the investigation is not open ended because the premises within TAG do not merely support the conclusion, they ensure it. That aspect is unique to deduction. Moreover, the conclusion from TAG is not a mere hypothesis, but rather a sound conclusion derived through a deductive process. Finally, TAG falls short of being fully inductive because there is no asserting the consequent with TAG, as there is with all scientific inference, the playground for induction.
Bahnsen typically employed modus tollens (MT) in his formal argument, yet he distinguished his employment of TAG from garden variety deduction. Mike Butler (at one time Bahnsen’s assistant) to my knowledge, also, has never pitted transcendental arguments against deduction. Butler has written TAG out, which is indeed deductive in form "For x (some aspect of human experience) to be the case, y must also be the case since y is the precondition of x. Since x is the case, y is the case." (Butler, The Transcendental Argument for God’s Existence, 91 The Standard Bearer.)
Van Til and Bahnsen fully appreciated that TAG is a deductive argument strictly speaking (lest they contradicted themselves in practice). However, their focus in this regard was on what distinguishes TAG from the usual kind of deductive (and inductive) arguments. The unique quality of TAG that sets it apart from all other standard deductive arguments is that with the latter we begin with some truths (or inferences) and reason to others - but that to which we reason is not presupposed as a necessary precondition for the intelligible experience of the original fact of experience. In other words, with standard deductive arguments we try to deduce from a fact, or series of facts, other facts; no more, no less. If it's Sunday I'm with a congregation of saints from 9:30-12 in the morning. If I’m not with a congregation of saints at 10:00 a.m., then it’s not Sunday. That it’s not Sunday can be a standard deduction, yet my being with the saints at a certain time does not make Sundays possible. Kant's genius was that TA's are concerned with what must be true in order for something else to be possible. God’s revelation makes intelligible experience possible, whereas my being with the saints at a particular time does not make Sunday between 9:30 and noon possible.
Clearly, Bahnsen applied deduction in his demonstration of TAG. Accordingly, he was either inconsistent with himself, or we should interpret his statements as meaning something other than TAG is not strictly speaking deductive. With little effort we can reconcile Bahnsen's practice of TAG with his description of it. TAG is not like any other deductive argument because it does not reason from fact to fact in the standard Greek sense but rather reasons from fact to the preconditions of fact, which is Kantian, yet when in the hands of a Christian a very powerful tool.