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# Does Logic Presuppose the Existence of the Christian God? (2000)

## Michael Martin

The Transcendental Argument for the Existence of God (TAG)--the argument that logic, science, and objective ethical standards presuppose the existence of the Christian God--has been repeatedly used by a small group of Christian apologists operating within the Orthodox Presbyterian tradition. In the recent past I showed that Greg Bahnsen, the leading advocate of TAG, failed to demonstrate that induction presupposes the existence of the Christian God.[1] In the present paper I consider another part of TAG--the claim made by Bahnsen in this debates with Gordon Stein[2] and George Smith[3]--that deductive logic presupposes the existence of the Christian God.[4]

### Deductive Logic and the Claim of TAG

Let us understand deductive logic to be the study of valid deductive arguments; that is, arguments in which the premises necessitate the conclusion. On this common understanding IF the premises of an argument are true, THEN the conclusion must be true. Deductive validity is determined by the form of the argument and not the content of the premises. An example of a valid deductive argument is:

All dogs are brown
Rover is a dog
_______________________
Therefore, Rover is brown.

The argument's validity is not a function of the truth of the premises but its form which is:

All Ds are B,
x is D
______________________
Therefore, x is B

So interpreted, Bahnsen's claim is that the validity of deductive arguments presupposes the existence of God.

What did Bahnsen mean for something A to presume something else B. To put what Bahnsen meant by "presuppose" in terms he often used: To say that A presupposes B is to say that we could not "make sense" of A without assuming B. However, supposing we grant that one must assume B to make sense of A, it does not follow that B is true. For example, if I am trying to communicate to an audience by speaking to them in English, my action makes no sense unless they understand English. But it does not follow that they do. They might only understand Chinese. Scientists listening to radio signals from outer space in order to make contact with extraterrestrial life presuppose that such life is possible. But it does not follow that it is. Similarly, if, as Bahnsen claimed, the Christian worldview is presupposed by the deductive validity, it does not follow that the Christian worldview is true. It might be the case that deductive validity is a myth. TAG would not establish the truth of the Christian worldview but only the inconsistency of atheists who presuppose deductive validity. Moreover, as we shall see, there is no reason to suppose that we cannot make sense of deductive validity without belief in the Christian God.

Although Bahnsen sometimes means by A presuppose B that we could not "make sense" of A without assuming B, this expression may be given another interpretation. A presupposes B often means that A logically implies B. In other words, if A presupposes B, it would be inconsistent to assert A and deny B. For example, being a sister presupposes being a female in the sense that it is inconsistent to claim someone is a sister and not female.[5] Could this be what Bahnsen is getting at? If so, his claim is dubious. There does not seem to be anything inconsistent in asserting that deductive validity exists and that Christianity is false. So interpreted, Bahnsen's claim seems to fail the consistency test. Of course, it is possible that there is a subtle and hidden contradiction that escapes one's notice. But this would have to be shown by arguments that Bahnsen did not provide.

### Atheism, Materialism, and Logic

One of Bahnsen's arguments for maintaining that atheists cannot appeal to logic without presupposing the Christian worldview is that atheism is committed to materialism. Because he maintains that materialism is incompatible with the permanent, abstract, and normative character of logical principles, Bahnsen held that atheists cannot coherently appeal to logic.[6]

There are three problems with this argument. First, Bahnsen gave no argument at all to support his view that materialism is incompatible with logic. Well-known materialists such as D. W. Armstrong and J.J.C. Smart have vigorously defended materialism against attack and Bahnsen did not attempt to deal with their arguments.[7] Second, although some atheists have been materialists there is no reason why they must be, and, indeed, many philosophically trained atheists are not. Many modern atheists assume a pluralistic ontology in which material entities comprise only one kind of entity in their ontology. Bertrand Russell, for example, was an atheist but believed in abstract mathematical entities. Third, even if atheists are committed to materialism and materialism were incompatible with logic, it does not follow that logic presupposes the existence of the Christian God.

### Logical Conventionalism

Bahnsen's position is in part based on his rejection of logical conventionalism, the view that logic is a human convention. There are two questions that one can be raised about this rejection:
1.) Has he misunderstood logical conventionalism?
2.) Does the rejection of logical conventionalism entail the truth of Christianity?
In the Stein-Bahnsen debate, Stein assumed that deductive logic is a human convention that has nothing to do with God, Christian or otherwise. Bahnsen in turn criticized conventionalism in logic by arguing that if logic is conventional, it is arbitrary[8] and that this is absurd.

However, sophisticated advocates of logical conventionalism do not agree that logical principles are arbitrary. Conventionalists such as Ernest Nagel have argued that the rules governing deductive validity are not arbitrary since they are adopted for certain purposes that are best achieved when language is employed in a manner that approximates to the norms expressed by these rules.[9] In other words, rules of deductive validity are justified on pragmatic and instrumental grounds.

Stein was in good company in adopting logical conventionalism. However, it is not the only position that is compatible with atheism and it is not clear that it is best position to take. One problem with this approach to logic is that it is unclear how one can interpret the Law of Noncontradiction--not (p and non-p)--in pragmatic terms. This law is presupposed in all valid deductive reasoning and rational thought. However, the negation of this law is not just pragmatically suspect--it is a contradiction. As one commentator has put it, under suitable conventions for the use of "not" and "and" one could not think or speak "in contravention of the principle which under the usual convention is expressed by 'not (p and not-p)'"[10] Surely what is conventional is the language used to express the proposition that it is not the case that (p and not-p) and not the truth of the proposition itself.

### Logic and Metaphysic

However, it is a long way from admitting that the deductive validity, for example, is not conventional to the conclusion Bahnsen needs: namely, that deductive validity presupposes the Christian God. As Bahnsen himself argued, there are many different interpretations of logic.[11] No doubt. But on the vast majority of these interpretations deductive validity is independent of God. What possible arguments could be constructed to arrive at the conclusion that deductive validity presupposes the Christian God? Perhaps this is what Bahnsen had in mind:

(1) Either deductive validity is conventional or it is not.
(2) If it is not conventional, it must have a metaphysical explanation.
(3) If it has a metaphysical explanation, then the best metaphysical explanation is that it is created by the Christian God.
(4) Deductive validity is not conventional.
_________________________________________________
5) Therefore, the best metaphysical explanation of the deductive validity is that it was created by the Christian God.

This argument is formally valid. The crucial question is whether the premises are true. Let us grant premises (1) and (4). However, premises (2) and (3) are problematic.

Consider (2). Just because deductive validity is not conventional it does not follow that they need a metaphysical explanation. On one interpretation deductive validity is explained simply because of the meaning of terms such as "All," "Some," "if ... , then ... ," "not," "and," "or," and so on. In this case, one can see that the premises of valid deductive arguments necessitate the conclusion, and why, simply by understanding the meaning of the terms involved. But then, the explanation of the deductive validity is not metaphysical but semantical, that is, the necessity of deductive validity is a function of the meaning of concepts involved. There is nothing conventional about this, however. The terms used to express these meanings may be conventional, but not deductive validity itself. In short, the explanation of deductive validity is close to the surface, and no deep metaphysical explanation of it is necessary.

Now consider premise (3). There are, of course, other metaphysical interpretations of logic besides a Christian one. For example, Aristotle believed that the laws of thought--the law of identity (if anything is P, then it is P), the law of contradiction (it is not the case that something is P and not P), and the law of excluded middle (either something is P or not P)--(which he believed to be presuppositions of all reasoning) are descriptive of being "as such" and are only secondarily standards of correct thinking. For Leibniz, the laws of thought held in every logical possible world and are "therefore descriptive of facts in such a way that not even God can change them."[12] By implication he held a similar position on deductive validity. Obviously, such interpretations of deductive validity are independent of Christian theology and, indeed, could be embraced by atheists. Moreover, they do not seem any less satisfactory than a Christian interpretation of the laws of thought.

### The Alleged Mystery of the Application of Logic to the World

There is another reason that a Christian apologist might give in support of the view that logic presupposes the existence of the Christian God. Whether or not Bahnsen actually accepted this argument it is certainly in the spirit of his thought. One might argue that only the Christian worldview provides an explanation of why logic applies to the world. Without postulating God one might maintain that the correspondence between formal logic and its application to reality would be a mystery.

However, this may be a pseudoproblem for it is difficult to see how logic could not apply to the world. To suppose that God brings about an otherwise inexplicable congruence between logic and the world suggests that we can imagine what it would be like if the world did not correspond to logic. But with respect to deductive validity this is impossible to do. What would the world be like if deductive validity did not apply? That a deductively valid argument with true premises has a false conclusion? However, this makes no sense.

To be sure, it has been claimed that the Law of Noncontradiction does not apply even in our own world. But this claim is based on a deep confusion. One might maintain that the sentence "Michael Martin lived in Cincinnati" was true in 1950 and was not true on 1960. Hence it is true and false and hence the Law of Noncontradiction fails. But there is clearly a misunderstanding here. The complete proposition is either "Michael Martin lived in Cincinnati in 1950" or "Michael Martin lived in Cincinnati in 1960." The one proposition is true and the other is false and there is no need to say that the same proposition is both true and false. To be sure, learning how to apply the Law of Noncontradiction to cases like this takes training and practice, but once this skill is mastered the Law of Noncontradiction remains in harmony with the world, and there is no mystery in this harmony.[13]

### Logic and the Christian God

So far we have not considered what a Christian account of the deductive validity would be. Bahnsen[14] and Van Til, his mentor,[15] provide little help in answering this question, and it is therefore impossible to evaluate the claim of the comparative advantage of the Christian interpretation of the laws of thought.

However, one possibility of what could be meant by the claim comes immediately to mind. To speak of the Christian foundations of deductive validity could mean that the Christian God created the deductive validity, and without God's creative act there would be no deductive validity. However, the problem with this interpretation is that it implies that God could have created the opposite; that is, a deductively valid argument in which true premises necessitate a false conclusion. This is unintelligible, however, and in addition to this problem makes the truth of logical validity dependent on God's whim and arbitrary decision.

If God did not create deductive validity, in what sense is the Christian God the foundation of deductive validity? In this case, the deductive validity would be independent of God, supposing that He did exist. Moreover, deductive validity would remain true if God did not exist.

Something that Van Til once said suggests another interpretation. According to Van Til, non-Christians and Christians alike use the formal law of contradiction. However, he argues that Christian theism holds that:

All predication presupposes the existence of God ... while antitheism holds that predication is possible without any reference of God. This at once gives the terms 'is' and 'is not' quite different connotations. For the antitheist these terms play against the background of bare possibilities. Hence 'is' and 'is not' may very well be reversed. The antitheist has, if effect, denied the very Law of Noncontradiction, inasmuch as the Law of Noncontradiction, to operate at all, must have its foundation in God."[16]

Although what Van Til's argument is, is somewhat obscure, perhaps the following reconstruction of his argument captures his main idea:

(1) If the Christian God did not exist, then predication would operate against a background of bare possibility.
(2) If predication operates against a background of bare possibility, the predication of P to x ( x is P) may be reversed and ~ P might be predicated of x ( x is ~ P)
(3) But if the predication of P to x ( x is P) is reversed and ~ P is be predicated of x ( x is ~ P), then the Law of Noncontradiction must be denied.
________________________________________________
(4) Therefore, If the Christian God did not exist, then the Law of Noncontradiction must be denied.

Every premise of this argument is questionable, however. Consider premise (1). What does operating against "the background of bare possibility" mean? The only meaning that would seem at all plausible is the background of logical possibility. So understood, the Law of Noncontradiction could not be denied. Something by definition is logically possible when the Law of Noncontradiction holds. Consider premise (2). Yes, predication can be reversed in one obvious sense. For example, X can be blue at one time and not blue at some other time; X can be blue at the bottom at time t and not blue at the top at time t, and so on. But X cannot be blue and not blue at the same time in the same respect. Thus, (3) is false if the antecedent of the hypothetical is understood in the proper way. The sense in which predication can be reversed does not affect the law of contradiction; the sense in which it cannot is based on this law. Van Til seems to play on this ambiguity in presenting this argument.

### Conclusion

This part of TAG fails. Bahnsen gave no good reason why logic presupposes the Christian God. There is nothing inconsistent in asserting that deductive validity is possible and that Christianity is false. He seems to have misunderstood both materialism, and logical conventionalism. The argument that without the Christian God it would be a mystery of how logic applies to the world has been shown to be without merit. Logic may not need a metaphysical foundation as Bahnsen assumes and, even if it does, it need not be a Christian one. Moreover, the exact notion of a Christian account of logic is unclear. On one interpretation, it is incoherent and on another it is problematic.

### Notes

[1] See Michael Martin, "Does Induction Presuppose the Existence of the Christian God?," Oct. 11, 1997 <http://www.infidels.org/library/modern/michael_martin/induction.html>. This paper first appeared in Skeptic, 5, 1997, pp. 71-75.

[2] A tape of this debate is available through Still Water Revival Books, 4710-37A Ave, Edmonton, Alberta, Canada, T61-3T5.

[3] This debate, entitled "A Case For/Against God: Debate with author George Smith," is available on tape through the Covenant Tape Ministry, 22005 N. Vendo Dr. Sun City West, AZ. 85375.

[4] A similar argument was used by Phil Fernandes in a debate against Jeff Lowder on Sept. 26, 1999 at the Friday Center, University of North Carolina, Chapel Hill, NC.

[5] According to Strawson presuppositions are distinguished from implications. If p implies g, then if g is false, p is false. But when p presupposes q, so does not-p. If q is false, p is neither true or false. This obviously cannot be what Bahnsen had in mind. For example, he did not think that the invalidity of induction presupposes the Christian worldview and he does not suppose that if the Christian worldview is false, then it would neither be true or false that induction was invalid. I infer from this that for Bahnsen implication and presupposition was identical. See A.R. Lacey, A Dictionary of Philosophy (New York: Charles Scriber's Sons, 1976), p. 91.

[6] See his taped lecture "How to Argue With Atheists?" A tape of this lecture is available through Still Water Revival Books, 4710 -37A Ave, Edmonton, Alberta, Canada, T61-3T5.

[7] For a survey of arguments against materialism and their refutation as well as a bibliography on contemporary materialism see Keith Campbell, "Materialism," Paul Edwards, The Encyclopedia of Philosophy, Vol. 5 ( New York: Macmillan Publishing Co., Inc. & The Free Press, 1967), pp. 179-188.

[8] Bahnsen used the same sort of argument against John Dewey. See Bahnsen, "Pragmatism, Prejudice, and Presuppositionalism," Foundations of Christian Scholarship, ed. G. North, (Vallicito, Calif.: Ross House Books, 1976), pp. 256-257.

[9] Ernest Nagel, "Logic Without Ontology," Paul Benacerraf and Hilary Putnam (ed.), Philosophy of Mathematics (Englewood Cliffs, N.J.: Prentice Hall, Inc. 1964), pp. 302-321.

[10] S. Korner, "Laws of Thought," Paul Edwards (ed.), The Encyclopedia of Philosophy, Vol. 4 (New York: Macmillan Publishing Co., Inc. & The Free Press, 1967), p. 415.

[11] See Greg Bahnsen, "Science, Subjectivity and Scripture (Is Biblical Interpretation Scientific?) " This paper is available through the Covenant Tape Ministry, 22005 N. Vendo Dr. Sun City West, AZ. 85375.

[12] Ibid., p. 416.

[13] It is beyond the scope of this paper to consider whether the law of excluded middle always holds. But there have been various reasons offered why this law should be rejected in some contexts. On this point see Susan Haak, Philosophy of Logic, (Cambridge University Press, 1978), pp. 208-213.

[14] For example, although Bahnsen in his debate with Stein alluded again and again to the Christian basis of logic, he did not indicate what that would be.

[15] See Cornelius Van Til, The Defense of the Faith (Philadelphia, PA.: The Presbyterian and Reformed Publishing Co, 1955), pp. 295-300.

[16] Ibid., pp. 296-297.