Twentieth-century developments in the Reformed tradition of theology have led a number of apologists to assert that there exist arguments which they call transcendental, which are neither identical with nor reducible to arguments of deductive or inductive form, and that only these arguments are faithful to God's plan for ministering to the unregenerate, that is, to non-Christians. However, this position leads us to certain problems---namely, if transcendental arguments are not deductive or inductive, then in what sense ought we regard their conclusions as justified? Collett's paper appears framed at least in part as an attempt to help answer this question by appealing to the logical system induced by van Fraassen's presuppositional semantics, since he evidently believes that this system is something altogether different from either induction or deduction. However, we shall see that Collett does not, in fact, accomplish this task, nor does he suggest any strategies for doing so, for the very simple reason that, contrary to Collett's analysis, van Fraassen's system is indeed one of deduction.
While Collett never unequivocally states that he believes transcendental arguments to be distinct from deductive arguments, the implication remains strong throughout his paper. For example, he writes (emphasis original):
Thus in contrast to both deductive and inductive forms of argument, a transcendental argument allows the concept of God to function as a logically primitive rather than logically derivative proposition, thereby bearing witness to the non-derivative character of God's existence on an argumentative level. To state matters another way, in [Cornelius] Van Til's Christian-theistic construction of transcendental argument, the truth of God's existence is not a deductive consequence of the premises of the argument, but rather the ontological and logical ground for the very possibility of the premises themselves. This is undoubtedly one of the reasons, if not the chief reason, why he believed that transcendental arguments were uniquely suited for the task of placing into sharp relief the non-deductive character of the truth of God's existence.
This is not an isolated passage; he peppers similar suggestions throughout his paper to the effect that, according to Van Til's conception of transcendental arguments, they are "non-deductive." In this way, a natural reading of Collett gives us the clear impression that he shares Van Til's view in this respect; if this is so, however, then he requires a further defense of it. Now, this is not to say that he intended such a purpose for his paper; on the contrary, he seems to have voluntarily restricted himself there to attacking one particular mode of reduction, namely that once argued by apologist John Frame, instead of generally condemning all possibility of reduction. Nevertheless, Collett's choice to describe transcendental arguments in terms of van Fraassen's presuppositional semantics, which induce an unmistakably deductive system, leads him ultimately to a view of transcendental argumentation entirely incompatible with the notion that transcendental inferences are always distinct from deduction. In other words, however Collett thinks his transcendental arguments cash out, they must do so deductively if we are to frame them using van Fraassen's semantics. With this in mind, we may examine his deductive method.
Following Peter Strawson, van Fraassen characterizes the two-place presupposition relation in the following way: If A and B are sentences in a presuppositional language, then A presupposes B if and only if the following hold:
(a) T(A) only if T(B); and
(b) T(¬A) only if T(B),
where we read 'T(A)' as 'A is true.' So the following is deductively valid in van Fraassen's system:
(1) P presupposes Q;
(3) therefore Q.
This is because we have by the presupposition relation that T(P) only if T(Q), while taking P as a premise gives us T(P); and from this it follows by modus ponens that T(Q), which is to say that we conclude Q is true.
Notice that even though we applied modus ponens, we can still construct (albeit not in van Fraassen's system) deductively valid arguments which do not make use of it. This is because deduction is not limited to a single kind of logic, e.g. classical logic. In fact, in this case we are not even using classical logic! for van Fraassen's system does not always obey the principle of bivalence; that is, we permit some sentences in his language to be neither true nor false. Instead, all we need in order to make deductive inferences is a language, semantics, and a set of rules of inference. Strawson, for instance, describes a deductive system thusly:
A small number of logicians' statements or rules (e.g., to the effect that certain formulae are analytic) are taken as premises, and further rules derived from them by the use of one or two higher-order rules of inference.
It is under the umbrella of such rules that we may understand arguments as valid or invalid. So, we might wonder, then, what does Collett have in mind when he asserts on one hand that his transcendental argument is valid, yet implies on the other that it is non-deductive? Ironically enough, the only argument Collett explicitly identifies as "transcendental" is not actually valid at all under the logical system in which he frames it. For this is what he provides (numbering added):
(4) Causality presupposes God;
(5) Not God;
(6) Therefore neither causality or not causality [sic].
First of all, none of these strings are sentences in the English language, unless we read 'causality' as an abbreviation for the sentence 'there is such a thing as causality,' or 'causality exists,' or some such; and similarly with 'God.' If we assume that this is his intent, which seems natural enough, then we may further abbreviate his 'causality' and 'God' sentences by 'C' and 'G,' respectively. Then his argument consists of:
(4’) C presupposes G;
(6’) therefore ¬(C ∨ ¬C).
Notice that there are no rules in van Fraassen's system which allow us to deduce (6’) from (4’) and (5’). Moreover, no argument with (6’) as its conclusion could possibly be valid in van Fraassen's system, because (6’) is always false by the law of excluded middle. However, we can construct the following valid argument, which perhaps Collett had intended instead to communicate:
(7) C presupposes G;
(9) therefore ¬(T(C) ∨ T(¬C)).
For since we take (7) as true, we have T(C) only if T(G), and T(¬C) only if T(G), by the properties of the presupposition relation. By contrapositive, ¬T(G) only if ¬T(C), and ¬T(G) only if ¬T(¬C). Meanwhile, we have that (8) is true if and only if G is false by the valuation rules for statements. Then T(G) is false, which is to say, ¬T(G). It follows by modus ponens that ¬T(C) and ¬T(¬C), which gives us (9).
On this more charitable interpretation of Collett, he still commits himself to the position that at least some transcendental arguments are deductive, which seems unsatisfactory given his presumed intent to follow Van Til; for he believes Van Til held that "the truth of God’s existence is not a deductive consequence of the premises of the argument," which is clearly inconsistent with argumentation under van Fraassen's semantics. Perhaps more importantly, if there really is a class of arguments which are neither inductive nor deductive, but deserve the unique label 'transcendental,' then what are they? No mere appeal to van Fraassen's presuppositional semantics, which afford us only rules for deduction, will answer this question.
What, therefore, is the solution to the Reformed apologist's unusual conundrum? How can we compel a uniquely transcendental inference, whatever that might mean? If an answer is possible, then regardless of its content, we have sufficient information even now to know that any argument distinct from deduction and induction, whether we call it transcendental or not, will have to satisfy at least two obvious criteria: First, it mustn't consist of recognizing some kind of order or constancy of the universe, our cognition or language, nor any other aspect of reality; indeed, such abstracting of regularities falls squarely within the scope of inductive inference. Second, it cannot have us follow a system of carefully-constructed logical rules in order to draw a conclusion from a body of existing knowledge or belief, since this describes a deductive procedure. Unfortunately, once we have set all those aside, it seems difficult to imagine that any other kind of inference should remain; so I do not envy the apologist's task in this matter.
To sum, we have seen that Collett's solitary example of a transcendental argument given in his paper is invalid by the very system he borrows in order to construct it. Further, his characterization of transcendental arguments, even charitably interpreted, requires that we understand them as plainly deductive. In this way, van Fraassen's system, the basis of Collett's analysis, seems fundamentally unfit to serve Reformed apologists in their efforts to distinguish between deductive and transcendental inferences, on account of its own unavoidably deductive nature.
 Collett, Don. "Van Til and Transcendental Argument Revisited," pp6-7.
 Here we refer to Frame's Cornelius Van Til: An Analysis of His Thought (1995), p315, as cited by Collett, p9. Frame has since retracted some of his views, attributing his shifts to Collett, in the online essay "Reply to Don Collett on Transcendental Argument" [sic] (2003).
 van Fraassen, Bas. "Presupposition, Implication, and Self-Reference," The Journal of Philosophy, Vol. 65, No. 5 (Mar. 7, 1968), pp. 136-152.
 Strawson, Peter. Introduction to Logical Theory (1952), p175.
 van Fraassen, pp137,43.
 Strawson, p58.
 Collett, p34.
 Collett, p33.
 van Fraassen, p142.
 Namely, property (3) from van Fraassen's paper.
 This is by rule (8b) from van Fraassen.
 By rule (18a).
 Collett, p7.
 Here I class, not uncontroversially, abductive arguments among inductive.