Transcendental arguments and the logic of presupposition.

0. Introduction 

In this post I will look at the transcendental methodology employed in philosophy and how far it can be said to be similarly employed in the presuppositional apologetics of Van Til. There is some controversy over the correct logical form of the so-called ‘transcendental argument for God’ (TAG), and I contrast looking at it cashed out using implication, with presupposition, and with ontological dependence. Each has its own difficulties as a rendering of what Van Til says, so in the end I am not sure which way it is supposed to be taken. On the way I discuss how Putnam thought he had refuted the sceptical hypothesis that I could be a brain in a vat, various features of validity in the non-classical logic of presupposition, and end with a discussion about metaphysical dependence.

1 Transcendental arguments.  

Transcendental arguments are somewhat controversial in philosophy. They go back at least to Kant, who used them in his Critique of Pure Reason. There, he was responding to the scepticism of philosophers like Descartes and Hume. It could be that one’s sense data are radically divorced from the external world and it would be impossible to tell, etc. Kant’s strategy is essentially to show that this seemingly neutral starting point between the sceptic and the philosopher, such as the basic fact of one’s own sense-data etc, itself has certain preconditions. These preconditions are things without which the starting point would itself be impossible. Kant wants to drill down into these foundations and show that these often include the very things the sceptic wants to call into question. Thus, when a sceptic calls these certain things into question, she has in fact relied on those things being the case for the question to be meaningful at all. This type of argument is a ‘transcendental argument’.

There is a charming example of such an argument, given in characteristically aphoristic manner by Wittgenstein in On Certainty:

“383. The argument “I may be dreaming” is senseless for this reason: if I am dreaming, this remark is being dreamed as well – and indeed it is also being dreamed that these words have any meaning.” Wittgenstein, On Certainty.

The idea here seems to be that the sceptic is calling into question the existence of the external world, with the suggestion that one may be dreaming. But, says Wittgenstein, in dreams it can seem like a collection of words has meaning, when in actual fact they don’t; one can dream that a word is meaningful, when in fact it isn’t. So the very meaningfulness of each string of words we encounter also becomes one of the things we cannot be certain about, if we entertain the idea that we are dreaming. Thus, the meaningfulness of the sceptical challenge itself is something we must also call into question! This means that in order for one to suspend doubt over the meaningfulness of the sceptical hypothesis (to take it seriously), one must in effect presuppose that they are not dreaming, an act which itself rules out the sceptical hypothesis from consideration.

1.1 Transcendental arguments in analytic philosophy

Apart from their use by Wittgenstein, in the later half of the 20th century this type of argument enjoyed a period of being in vogue in analytic philosophy, primarily due to the work of Peter Strawson, Hillary Putnam and Donald Davidson.

Consider Putnam’s transcendental argument, which is found in chapter 1 of his 1981 book, Reason, Truth and History (read it here). In a sense, he is developing Wittgenstein’s argument from above. Putnam’s argument purports to refute the sceptical hypothesis that we might be brains in vats, merely  being stimulated to have sensations by some evil scientist. Often, this problem is seen primarily in epistemic terms, in the sense that the challenge is how one could know they weren’t brains in vats. Putnam’s approach, in contrast, is not to look primarily into the notion of knowledge per se, but instead to focus on linguistic issues surrounding what would have to be the case for the sentence ‘I am a brain in a vat’ to be true. His claim is that, once these considerations are taken into account, it becomes evident that the sentence ‘I may be a brain in a vat’ is self-refuting:

“A ‘self-refuting supposition’ is one whose truth implies its own falsiry. For example, consider the thesis that all general statements are false. This is a general statement. So if it is true, then it must be false. Hence, it is false. Sometimes a thesis is called ‘self-refuting’ if it is the supposition that the thesis is entertained or enunciated that implies its falsity. For example, ‘I do not exist’ is self-refuting if thought by me (for any ‘me’). So one can be certain that one’s self exists, if one thinks about it (as Descartes argued).

What I shall show is that the supposition that we are brains in a vat has just this property. If we can consider whether it is true or false, then it is not true (I shall show). Hence it is not true.” (Putnam, Reason, Truth and History, 1981 p. 7-8)

The argument is (as stated in the last two sentences):

  1. If ‘I am a brain in a vat’ could be either true or false, then it is false.
  2. ‘I am a brain in a vat’ could be either true or false.
  3. Therefore, ‘I am a brain in a vat’ is false.

Premise 2 is no more than the sceptic would concede. The burden is to justify the first premise. This premise is supported by semantic considerations, specifically of the reference for the term ‘a vat’ in the proposition ‘I am a brain in a vat’. Putnam’s argument is that there are three general ways that the phrase ‘a vat’, which is a referring term, could get its reference to the object it refers to. Either a referring term:

  1.  has an intrinsic property of referring to the referent (nomenclaturism),
  2. or it refers to the referent via an internal concept on the part of the speaker/hearer (internalism),
  3. or it refers to its referent due to some external relation the speaker/hearer has to the referent (externalism).

Putnam first goes after the notion that words have intrinsic references. On this view, to produce some words, either by speaking or writing them, is to refer to the things that they name. The refutation of this idea is simple. Take an ant crawling in the sand who happens to write out the name ‘Winston Churchill’. The ant has produced those shapes, but it is obvious that the ant has not referred to Winston Churchill. Thus, signs do not intrinsically refer to things.

The underlying thought here is that if signs are ever used to genuinely refer to things, they need to be supplemented by something. Usually, this something additional which is added to the otherwise non-referential sign is a mental act of intention. The words are internally linked to a concept, and it is because of this internal mental association that they are about something (i.e. genuinely refer to things). This is internalism. However, Putnam also rejects this this thesis, on the grounds that that internal mental images also do not intrinsically refer to things. His counter-example is that of two physically identical depictions of a tree, one on Earth and one on a treeless planet. The one on Earth is formed by the usual photographic process. The one on the treeless planet has been formed by pure chance (say, paint dripping onto the bit of paper at random). The photo of the tree is being looked at by a normal person on Earth, while the picture of the tree is found on the treeless planet by a human who has never seen or heard of a tree. Each person has identical mental sensations upon seeing the photo (because the two pictures are qualitatively identical), but only one of the people thereby refers to a tree.

The reason for the difference in this case, says Putnam, is that there is a causal chain which we could in principle trace back from brain of the thinker of the image on Earth, through the light waves hitting his eyes, back into the photo, which was itself caused to have the arrangement of colours it does because of the light that came from the actual tree. In the treeless planet case, there is no causal link backwards from the event of the light entering the person’s eyes to any actual trees. If reference was fixed in the head, then as the internal situation is the same in both cases, they should both refer to the same object. Yet they don’t. The view that Putnam is advocating here is ‘semantic externalism’. Part of what it means to successfully refer to something is for there to be conditions external to the agent reading, writing, hearing or seeing, etc, the referring term. As he says, when it comes to reference it ain’t all in the head.

When we come to the case of the ‘brain in a vat’ proposition, if we apply semantic externalism to it, then we see that the only way that ‘I am a brain in a vat’ could be true is if ‘a vat’ refers to an actual vat. The reference to (in particular) an actual vat can be secured only if there is a causal chain coming from that vat to the brain. While, in a sense, every sensation that the brains-in-vats have is causally related to the vat they are in (and the electronic current being fed through it), their word “vat” is not semantically linked to it in any particular way (at least, no more than every word they use, and it is not the case that every word a brain uses refers to the vat it is sitting in). Rather, when the brains think propositions like ‘that is a tree’, they refer to the objects they take themselves to be in causal relation to in the virtual world they live in; but they fail to refer to anything in the actual world at all:

“How can the fact that, in the case of the brains in a vat, the language is connected by the program with sensory inputs which do not intrinsically or extrinsically represent trees (or anything external) possibly bring it about that the whole system of representations, the language-in-use, does refer to or represent trees or anything external?”

The answer is that it cannot. The whole system of sense-data, motor signals to the efferent endings, and verbally or conceptually mediated thought connected by ‘language entry rules’ to the sense-data (or whatever) as inputs and by ‘language exit rules’ to the motor signals as outputs, has no more connection to trees than the ant’s curve has to Winston Churchill. (ibid, p.13)

While there is certainly more that can be said about Putnam’s argument, this much is clear. Premise 1 of the argument has been given quite a detailed line of supporting argument, which pits the attractive looking causal theory of reference (semantic externalism) against the other alternatives. Could there be a different theory not considered by Putnam? Sure. Could one of the theories considered by Putnam be rescued against his objections. Sure. The point is just that there is a substantive argument here, and it is clear what Putnam thinks is at stake when he says that the sceptic’s proposal is self-defeating.

2. TAG

It is into this tradition that we find Van Til’s transcendental argument for the existence of God (TAG). Van Til never provided a formal version of his argument, but alluded to it frequently, and we find this reinforced throughout the work of Greg Bahnsen. I have always taken it that the form of the argument is as follows:

  1. If God did not exist, human experience would be unintelligible.
  2. Human experience is intelligible.
  3. Therefore, God exists.

However, I think there is reason to doubt that this could really be the form of the argument, given various considerations I will go through below.

Van Til thought that he was providing more than just another argument for God; not just another argument that sits alongside the ontological argument, or cosmological argument, etc. He thought that he was providing a new and more sophisticated way of defending Christianity. His problem with the traditional arguments is that they seemed to concede something to their opponent which gives the game away already from the outset. This was that it was possible to reason at all independently from God. The idea here is that the approach with the traditional arguments is to see if the existence of God follows from premises which are themselves neutral on the question of whether God exists. These arguments thus start from assumption that there are such premises, ones which are neutral. However, it is precisely this that Van Til found objectionable. In contrast, Van Til wanted to say that there are no such premises; no such neutral ground.

This leads to the curious claim by Van Til that his transcendental argument is neither deductive nor inductive:

“Now the only argument for an absolute God that holds water is a transcendental argument. A deductive argument as such leads only from one spot in the universe to another spot in the universe. So also an inductive argument as such can never lead beyond the universe. In either case there is no more than an infinite regression. In both cases it is possible for the smart little girl to ask, “If God made the universe, who made God?” and no answer is forthcoming. This answer is, for instance, a favorite reply of the atheist debater, Clarence Darrow. But if it be said to such opponents of Christianity that, unless there were an absolute God their own questions and doubts would have no meaning at all, there is no argument in return. There lie the issues. It is the firm conviction of every epistemologically self-conscious Christian that no human being can utter a single syllable, whether in negation or in affirmation, unless it were for God’s existence. Thus the transcendental argument seeks to discover what sort of foundations the house of human knowledge must have, in order to be what it is. It does not seek to find whether the house has a foundation, but it presupposes that it has one.” (Van TIl, Survey of Christian Epistemology, Section 11.)

Van Til’s claim here is strange. The version of TAG above is a deductively valid argument. Let p = ‘God exists’ and q = ‘human experience is intelligible’. Then the form of the argument is:

  1. If not-p, then not-q
  2. q
  3. Therefore, p

If this is correct, then the argument is simply a version of modus tollens, which is a textbook example of a deductively valid argument. It is puzzling why Van Til would think that TAG isn’t deductive.

One option, of course, is that I have given it the wrong logical form. However, I have given it the same form as Kantian transcendental arguments (the same sort of form as that of Wittgenstein and Putnam, etc). The Stanford article on transcendental arguments backs up that my phrasing is correct:

“As standardly conceived, transcendental arguments are taken to be distinctive in involving a certain sort of claim, namely that X is a necessary condition for the possibility of Y—where then, given that Y is the case, it logically follows that X must be the case too.”

So, either the Stanford article and I are wrong about what the form of a transcendental argument is, or Van Til was using the term differently, or he was just wrong about whether it was deductive.

What is the correct logical form of Van Til’s TAG?

2.2. The inadequacy of classical implication

There is another issue with what Van Til said, and it is one that adds weight to the thought that his argument does not have the simple form of a modus tollens. Let’s look again at some particular phrases in the quote from him above:

“…unless there were an absolute God their own questions and doubts would have no meaning at all.”

and,

“…no human being can utter a single syllable, whether in negation or in affirmation, unless it were for God’s existence.”

Van Til is saying more than just that if there were no God then the claims about the existence of logic or the possibility of argument would be false; he is saying that without a God these claims ‘would have no meaning at all‘, and that nothing could be said at all ‘whether in negation or affirmation‘. The logic used in the version of TAG we have been discussing here (the Kantian form) doesn’t capture this feature well at all. Rephrased as a logically equivalent modus ponens, it says:

  1. If logic, then God.
  2. Logic.
  3. Therefore, God.

If the consequent of the first premise (‘God’) is false, then the conditional is only true if the antecedent is also false. This means that, if it is true that God is a necessary condition of logic, and if it is false that God exists, then the claim that logic exists is false. But this is exactly where Van Til’s claims from above seem to go further. He doesn’t say that these claims are false, but, as it were, neither true nor false (‘no meaning at all’, ‘whether in negation or in affirmation’). With the classical logic we are using here, this position is not captured. Thus, we have reason to think that this cannot be what Van Til meant when he used the transcendental argument for the existence of God.

3. The logic of  presupposition.

In 1905, Russell published a paper called ‘On Denoting‘. In that paper, he advocated a semantics for descriptions, i.e. phrases like ‘the third planet from the sun’, ‘your favourite ice cream flavour’, and ‘the present king of France’. In particular, he was interested in the latter type of example, as these cases (where there is apparent reference to things that do not exist) had posed problems for previous theories, such as Frege’s. His solution was essentially to say that ‘the present King of France is bald’ has a logical form which is more complex than it appears on the surface; it is in fact a conjunction of two claims:

  1. There is exactly one thing which is the king of France, and
  2. That thing is bald.

Because the first conjunct is false (because there is no king of France), the whole conjunction is false as well. It remains false for the same reason if we change the second conjunct to ‘that thing is not bald’. Thus, ‘the present King of France is bald’ and ‘the present King of France is not bald’ are both false.

In 1953, Peter Strawson proposed an alternative theory to Russell’s. According to Strawson, the sentence ‘the present King of France is bald’ should be considered to be neither true nor false. The reason for this is that it presupposes that there is a king of France. Unlike Russell, who claimed that the sentence implicitly implied there is a king of France, Strawson said it has this as a presupposition.

Presupposition, in Strawson’s sense, differs from implication precisely on the issue of the consequent being possibly neither true nor false. This idea is cashed out by Van Fraassen here. In that we find the standard Strawsonian definition of presupposition:

Presupposition)      (A presupposes B) iff (if A is either true or false, then B is true)

This says that when A presupposes B, A has a truth-value only if B is true; if B is false, then A is neither true nor false.

3.1 TAG with Presupposition instead of Implication

This definition of presupposition does considerably better at capturing the spirit of Van Til’s claims from above. He wanted to ‘up the ante’ by saying that its not just that if what the atheist says is false then God exists, but that if what the atheist says is meaningful at all, then God exists. This is captured by saying:

  1. Whatever an atheist says presupposes that God exists.
  2. Therefore, for whatever an atheist says, if it is either true or false, then God exists.

We are not talking about the specific truth value of what the atheist says, but into the conditions which make it such that it has either truth value.

This also seems to do justice to the following remarks of Van Til:

“Thus the transcendental argument seeks to discover what sort of foundations the house of human knowledge must have, in order to be what it is.”

Thus, we have some reason for thinking that the logical form of Van Til’s argument involves presupposition in this sense. This is the view of the presuppositionalist Don Collett (see this).

3.2 Presuppositional validity

The logic of presupposition, a hot topic in philosophy of language today, has some interesting features. One thing that is particularly relevant here is how far this notion of presupposition differs from classical implication.

The first thing to notice about it is that it is a non-classical logic. This is because there can be formulas which lack a truth value altogether. It is standard to think of the semantics for this sort of logic as the strong Kleene tables.

The fact that some propositions can lack a truth-value makes the notion of validity for presupposition different to that of implication. For instance, while modus ponens is valid for presupposition, modus tollens is not. This means that the following is valid:

  1. A presupposes B
  2. A
  3. Therefore, B.

But the following is not:

  1. A presupposes B
  2. not-B
  3. Therefore, not-A

This is because if A presupposes B, and B is not true, then A is neither true nor false. And in the strong Kleene semantics, if A is neither true nor false, then so is not-A.

It also follows from this that in the logic of presupposition the following form, which is invalid in classical logic, is valid:

  1. A presupposes B
  2. Not-A
  3. Therefore, B

Call this argument form ‘modus presuppans‘. If A presupposes B, then even if not-A is true, B is true. Even the falsity of A entails B, if A presupposes B.

One reason for thinking that this is a more faithful way of rendering Van Til’s idea is how well it fits with other claims he made. In one of his more memorable illustrations, Van Til said that the unbeliever is like a child who can only slap her father in the face because he his supporting her on his knee. The point is supposed to be that even the claim that Christianity is false presupposes that God exists. This result seems to be obtained if we grant that Christianity presupposes that God exists. It is in fact just the argument form from above:

  1. Christianity presupposes God.
  2. Christianity is false.
  3. Therefore, God.

This argument form is valid given Strawson’s logic of presupposition. It seems then that we have a form of TAG that fits well with Van Til’s aims.

4. Problems

The notion of validity for presupposition outlined here might be considered to capture some of the intuitions and ideas of Van Til. However, it also faces some serious problems.

  1. Firstly, it might be completely arbitrary, or even actually inconsistent.
  2. Secondly, there is a disanalogy between the most natural renderings of the first premise of TAG and textbook cases of Strawsonian presupposition, and this suggests that it is a different relation altogether.

4.1 Arbitrariness, or Inconsistency?

It seems quite clear that the central existential claim in Christianity could be cashed out in the following biconditional:

‘Christianity is true if and only if God exists’.

Assume we mean by ‘God’ the Christian God, i.e. the triune God referred to in the Bible, etc. Then this looks fairly watertight. Could Christianity be true if this God does not exist? Could (the Christian) God exist and Christianity not be true? It seems quite clear (to me) that the answer to both questions is ‘no’.

The main claim of the presuppositionalist argument, when cashed out using presupposition rather than implication is that Christianity presupposes that God exists, because every fact is supposed to presuppose that God exists. But this causes a problem with the existential biconditional above. They can’t both be true, or we get a contradiction.

The following argument (the ‘from truth to existence’ argument) is valid:

  1. Christianity is true if and only if God exists
  2. Christianity is true.
  3. Therefore, God exists.

We can also reason the other way (the ‘from falsity to non-existence’ argument):

  1. Christianity is true if and only if God exists
  2. Christianity is false.
  3. Therefore, God does not exist.

But if we also add in that Christianity presupposes that God exists, then ‘from falsity to non-existence’ becomes invalid:

  1. The truth of Christianity presupposes the existence of God.
  2. Christianity is false.
  3. Therefore, God exists.

This is just a version of modus presuppans, and is valid on the Strawson/Kleene semantics. It means that if Christianity presupposes the existence of God, then the falsity of Christianity is compatible with the Christian God existing. And we can also reason the other way as well:

  1. The falsity of Christianity presupposes the non-existence of God.
  2. Christianity is true.
  3. Therefore, God does not exist.

Thus we have an inconsistent set of propositions. If the existential biconditional is true, then the truth of Christianity is incompatible with the non-existence of God. If it is true that the truth of Christianity presupposes that God exists, then it is compatible with the non-existence of God. They are either compatible or incompatible, which means either the existential biconditional has to go or the claim that Christianity presupposes that God exists has to go. I find the biconditional much more obviously fundamental to Christianity, and I find it hard to make sense out of the result that Christianity is true and God does not exist. For me, that is pretty strong evidence that the biconditional is to be kept at the expense of the presuppositional claim.

I want to point to another problem before suggesting why this problem is happening.

4.2 The Disanalogy

We can begin to see a disanalogy between the usual first premise of TAG and standard examples of Strawsonian presupposition. Here are some examples of Strawsonian presupposition:

  1. ‘The King of France is bald’ presupposes that ‘there exists a King of France’.
  2. ‘I have stopped beating my wife’ presupposes that ‘I have a wife’.
  3. ‘Julius is a bachelor’ presupposes that ‘Julius is an unmarried male’.
  4. ‘He set me free’ presupposes that ‘somebody set me free’, etc.

In most of these cases, the relationship between the antecedent and consequent of the presupposition is very obvious:

  • 3 seems to be merely a case of definition (which is linguistic),
  • 4 is just existential generalisation (which is linguistic),
  • and arguably so is 1 (so it is also linguistic),
  • 2 is an example of a leading question (which is linguistic).

On the other hand, it is not so obvious that the existence of logical laws (etc) presupposes that God exists. Part of the reason for this difference is because 1-4 above are all obviously linguistic phenomena; the relationship being brought out in the examples is between elements of language. In contrast, when Van Til states his first premise as “unless there were an absolute God their own questions and doubts would have no meaning at all” and (as I discuss below) this seems more naturally considered to be a metaphysical claim; i.e. not it is not a relation between elements of language, but a relation between things that actually exist.

Here is a way of thinking about it which makes it easier to see why Van Til’s statement seems to be metaphysical and not linguistic. Once we rearrange Van Til’s statement into modus ponens form, we see what the antecedent is, and we can state one of its presuppositions:

1a. The atheist’s own questions and doubts have meaning.

And a presupposition of 1a is claimed to be this:

1b. God exists.

Now compare someone saying 1a with someone saying 2a, along with one of its presuppositions:

2a. I have stopped beating my wife.

2b. I have a wife.

If the Strawsonian account of presupposition, which applies to 2a, is supposed to apply to 1a, then we should expect the way these sentences are related to their respective presuppositions would be quite similar, i.e. the way 1a is related to 1b and the way 2a is related to 2b should be quite similar. But it seems clear to me that the reason that 2b is presupposed by 2a is primarily a linguistic reason. It is a product of the meaning of the words, as used in normal contexts. Most people have the linguistic intuition that 2b is a presupposition of 2a, and this means that we are happy to grant it as true if used as a premise in an argument. There are tricky cases of presupposition, for sure, but 2a-2b isn’t one of those cases. We could even disagree with Strawson, and perhaps agree with Russell, on the details of the semantic relation between 2a and 2b, but it is not seriously disputed that they have some linguistic/semantic relation or other that preserves the rational inference from 2a to 2b.

The relation between 1a and 1b doesn’t seem to be linguistic like that. It doesn’t seem to be part of the meaning of the words “The atheist’s own questions and doubts have meaning” that “God exists”. At the very least, it isn’t a commonplace statement of linguistic meaning, like 2a and 2b. This is why people (other than presuppositionalists) are not happy to concede it as a premise in an argument. It isn’t obvious at all, unlike with 2a and 2b. This utter lack of semantic intuition here is evidence that the claim that ‘“The atheist’s own questions and doubts have meaning” semantically presupposes “God exists”‘ is just false.

4.3 Metaphysical dependence

I would go further and claim that this is intentional. Why is it that 1a implies 1b, on the Van Tilian picture? The answer is essentially that all truths are metaphysically grounded in God, on this view. Van Til often says things which make it clear he has this sort of metaphysical idea in view:

“Man’s ethical alienation plays upon the background of his metaphysical dependence.” (Van Til, Survey of Christian Epistemology, chapter 14, emphasis mine).

It is the fact that man (and everything there is at all) is metaphysically dependent on God that is motivating Van Til. His point is that whatever an atheist might appeal to, anything that exists in any sense, it will end up being something which is metaphysically dependent for its existence on God. This metaphysical dependence is what seems to be driving the idea of presupposition here, and it is not a linguistic phenomenon. The claim isn’t that 1a presupposes 1b in the linguistic Strawsonian sense, but in a stronger metaphysical, we might say ‘Van Tilian’, sense. If this is right, we should really drop the talk of presupposition, and talk explicitly of metaphysical grounding, or metaphysical dependence.

But if we go down this road, we seem to have ended at a destination that is quite far from a transcendental argument, for now the argument is something like this:

  1. For everything there is, if it exists, then God exists (metaphysical dependence claim)
  2. If an atheist questions whether God exists, then the atheist exists (assumption)
  3. If an atheist questions whether God exists, then God exists (1, 2, modus ponens)
  4. An atheist is questioning whether God exists (assumption)
  5. Therefore, God exists.

This argument is valid, and premises 2 & 4 are very likely to be granted by an atheist, and 3 follows from 1 & 2, so all that is required to be supported is 1, which is itself the Van Tilian metaphysical dependence claim. All the Van Tilian needs to do is justify the first premise (their main claim) and they will be able to prove that God exists merely from the presence of an atheist questioning whether God exists. This seems to capture rather well the Van Tilian idea of the child slapping their father in the face.

So it seems that premise 1 is what needs to be justified. But there already is an argument which attempts to get us to this destination, which is the argument from contingency. In fact, the metaphysical dependence argument above is just a special instance of the argument from contingency; we could call it the argument from dependency. If this is correct, then there is no special transcendental method in TAG, and it is just another classical argument for God, alongside the other well-known deductive arguments.

5. Conclusion. 

In conclusion then, the precise form of TAG remains illusive. It seems very hard to square everything that Van Til said into one logical system that doesn’t also give up something seemingly important to how he described it.

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