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The Krueger-McHugh Debate: Theism or Atheism


Christopher McHugh

Opening Statement by Christopher McHugh

In this debate, I will present a case for theism, and offer refutations of any atheological arguments that Krueger adduces. In many past debates between theists and atheists, the theist philosopher has used a series of intuitively plausible arguments concerning cosmology, morality and history to construct a cumulative case for theism. I cannot improve upon the work of philosophers such as William Lane Craig in this area, so I will do my best to offer something that is totally new. In this debate, I will use only one argument to ground my case for theism, but this argument is (I believe) a conclusive proof. The reasoning is more technical than what is usually found in a debate of this nature, but I have done my best to make it as accessible as possible within the length limitations. Even though there are brief sections that are written in formal logic, there is no requirement that one understand symbolic logic in order to comprehend the nature of the argument.

It is a foregone conclusion among philosophers of religion that logically valid modal ontological arguments for the existence of God can be formulated. The problem with such arguments, which prevents them from being conclusive proofs, is the challenge of grounding the truth of a few controversial premises. It is my contention that the modal ontological argument can be modified so that each of its premises is necessarily true. I begin with an exposition of some representative attacks on the traditional form of the modal ontological argument. I then show that the controversial premises in the argument can be modified so that they are necessarily true. Finally, I examine some likely objections to the proof.

Consider that there are some things that cannot possibly be unreal. For example, logical laws and mathematical truths do not have the possibility of being mere fictions. In any possible world, there are certain truths like A=A and 2+2=4. Of course, there are some things that do have the possibility of unreality. For example, there are possible worlds in which the Statue of Liberty does not exist. Many philosophers throughout history have argued that one can analyze the idea of God to show that, much like a logical law or mathematical truth, God cannot have the possibility being unreal. Such an argument might go something like this:

1) If God is conceived to be something without deficiency of any kind, then God cannot be conceived to have the possibility of not existing. [This proceeds from the intuition that having the possibility of non-existence is a form of deficiency.]

2) God is conceived to be without deficiency of any kind. [This is from the definition of God as an absolutely perfect being.]

3) God cannot be conceived to have the possibility of not existing. [From 1 and 2]

The above argument may leave one with the suspicion that there is a fallacy involved somewhere. Fortunately, such fears can be allayed by the recognition that this type of argument can be rendered in formal logic so that there is no controversy over its validity.

Charles Hartshorne, in The Logic of Perfection, states the formal version of the modal ontological argument in this way,[1] where “q” = There is a perfect being, “N” = It is logically necessary that, “~” = It is not the case that, “v” = or, and “p -> q” = p strictly implies q:

(1) q -> Nq (Anselm’s principle)

(2) Nq v ~Nq (excluded middle)

(3) ~Nq -> N~Nq (Becker’s postulate)

(4) Nq v N~Nq (from 2 and 3)

(5) N~Nq -> N~q (from 1)

(6) Nq v N~q (from 4 and 5)

(7) ~N~q (intuitive postulate)

(8) Nq (from 6 and 7)

(9) Nq -> q (modal axiom)

(10) q (from 8 and 9)

Premise (1) is St. Anselm’s principle that an absolutely perfect being can only exist if its existence is logically necessary. Premise (7) is the assumption that the existence of a perfect being is not logically impossible. Each of the other premises is either a truth of logic, or is the product of a logically valid deduction.

While the argument is clearly valid, it does have its detractors who contest the truth of the premises. For example, while accepting the logical structure of the proof, atheist philosopher Michael Martin avers that the proponent of this argument must conclusively demonstrate the consistency of the God-concept before one is warranted in accepting premise (7). He writes:

There seems to be little doubt that the argument is valid. The crucial question is whether the premises are true. Clearly the most important ones for our purposes are premises (1) and (7). On Hartshorne’s view, (7) is the hardest to justify. He recommends using one or more of the theistic proofs that he claims demonstrate that perfection must be at least possible. But this seems to have things backward. The theistic proofs presume that the concept of God is coherent; they cannot demonstrate it. Furthermore, … there have been many attempts to show that the concept of God is incoherent. Before one can claim that it is coherent, one at least needs to show that these attempts have failed. Hartshorne has not done this, and consequently premise (7) is unjustified.[2]

Martin continues his attack on the argument by attempting to infirm premise (1). He claims that if Anselm’s principle (that absolute perfection entails necessity) is accepted, then one can use the logic of the ontological argument to prove the existence of all sorts of strange “perfect” entities. Martin writes:

It would seem possible to ‘prove’ the existence of the super absolute evil one [a being like God in all ways except that it is absolutely evil] by a modal argument identical to Hartshorne’s, the basic difference being the interpretation of q. Statement q would now mean ‘A perfectly evil being exists.'[3]

While the above criticisms of Hartshorne’s proof can be called into question on various points, I will accept them (for the sake of argument), and proceed to show how a new version of the modal ontological argument can be formulated so that it is immune to criticisms like Martin’s.

In order to make the argument work, premise (7) must be taken out of the realm of being an intuitive assumption, and the concept of the essential nature of God must somehow be logically guaranteed to be free from incompatible properties. The argument must also be such that its logic cannot be parodied to yield absurdities.

The task of proving premise (7) will be achieved by the formulation of a consistency-guaranteed concept of what it means to be Godlike. In order to do this, I will limn a class of concepts that can always be consistently conjoined. Richard Gale, an atheist philosopher, has written extensively on the logic of properties, and has concluded that “negative” terms like “non-red”, “non-sour”, “non-colored”, “non-large” etc. are always conceptually compatible with each other. I will directly quote Gale’s well-established position in order to explicate the distinction between negative and positive terms. He writes:

A property P is negative if and only if it specifies no property which is not of the same quality…. Positive properties, in addition to specifying other positive properties, also specify negative properties, i.e., properties of different quality. E.g., blue specifies, in addition to color, non-red, but non-red, as the wide complement to red, specifies no positive property, i.e., property differing in quality from it….[4]

By this criterion, a property such as “being material” is positive because it entails terms of differing quality, like “spatial” and “non-angelic.” By contrast, a negative property, such as “non-colored” entails only properties of the same quality as itself, like “non-red.”

Gale also offers a second criterion based on conceptual compatibility:

Another difference between positive and negative properties in respect to their logical relations, which can also serve as a criterion, is the following: A property P is negative if and only if there is no property of the same quality as it with which it is incompatible. To say that two properties are incompatible means that it is logically impossible for them to be coinstantiated. Red is incompatible with blue, which is of the same quality as it, but there is no property of the same quality as non-red with which it is incompatible.[5]

Gale’s qualifications to his criteria are addressed in the notes.[6], [7] Each of these qualifications will also apply to my use of his criteria.

The following is a list of properties that are logically guaranteed to be conceptually compatible with each other because of their negativity:

(1) Non-contingency [Something possesses the property of “contingency” if and only if its existence and non-existence are both logically possible (e.g., the Statue of Liberty). The existence of something that is non-contingent is either logically impossible (e.g., a “square circle”), or logically necessary (e.g., the Law of Identity).]

(2) Non-dependency [Something that is non-dependent is not caused to exist, and does not rely on the existence of some other reality to sustain it.]

(3) Not being subject to contingent laws [A contingent law is some law that does not obtain in every possible world, like a conservation law in physics.]

(4) Not being natural

(5) Not being a logical law, a number, a mathematical truth, a Platonic form or some other abstraction

(6) Not being spatiotemporal [Something with this property does not change, and is not something that takes up space.]

(7) Non-finitude

(8) Not being deficient in any sense [Note that this negative concept of non-deficiency is more expansive than any rigid positive concept of perfection. It may be the case that a being that is “not deficient in any sense” has a nature that is beyond any positive concept of perfection that one can form.]

Using Gale’s criteria, we can easily demonstrate the negativity of each of these properties by means of their logical relations. For example, it is clear that the property of “contingency” is positive because it is inconsistent with other properties that are of the same quality as itself, viz., being “necessary” or “existing in every possible world.” Non-contingency, however, is negative, for is not inconsistent with any properties of the same quality as itself, such as being non-necessary.[8] The same type of test can be performed on the other properties in the list to prove their negativity.[9]

The property of “being Godlike” will be defined in the following way: Something is Godlike, if and only if it possesses at least the negative properties (1)-(8) essentially. Note that the property of being Godlike may be compatible with some positive properties, but the definition of what it means to be Godlike will consist only of negative properties.[10]

Assuming that Gale’s conclusions on the logic of positive and negative terms are correct, it follows that the property of being Godlike cannot, in itself, entail any conceptually incompatible attributes. With this in mind, one can formulate a sound ontological argument that does not have the same flaws as the argument for a positively conceived notion of a perfect being. Here is a simple statement of such an argument:

1) Either the existence of something Godlike is logically necessary or logically impossible. [The property of being Godlike entails the property of non-contingency by definition, and whatever is non-contingent is either logically necessary or logically impossible.]

2) It is not the case that the existence of something Godlike is logically impossible. [The property of being Godlike entails only negative terms, and therefore can contain no conceptually incompatible properties.][11]

3) The existence of something Godlike is logically necessary. [From 1 and 2]

The proof can also be formalized. When this consistency-guaranteed concept is used in the context of Hartshorne’s proof, the two controversial premises (1) and (7) are now necessarily true. Here is the formal proof restated, with “q” understood as “There exists a being that possesses the property of being Godlike:”

(1) q -> Nq [While, in Hartshorne’s proof, this premise was based on the intuition that a perfect being must be non-contingent, it is now necessarily true, for the property of “being Godlike” entails non-contingency by definition.]

(2) Nq v ~Nq (excluded middle)

(3) ~Nq -> N~Nq (Becker’s postulate)

(4) Nq v N~Nq (from 2 and 3)

(5) N~Nq -> N~q (from 1)

(6) Nq v N~q (from 4 and 5)

(7) ~N~q [This premise was previously an intuitive postulate when applied to the positive concept of a perfect being, but now can be shown to be necessarily true. Since no negative properties can be conceptually incompatible, and since the property of being Godlike (according to the definition in use here) entails only negative properties, it follows that this premise is logically necessary.]

(8) Nq (from 6 and 7)

(9) Nq -> q (modal axiom)

(10) q (from 8 and 9)

The argument is valid (even adamant atheist philosophers like Martin agree to this), and each of its premises appears to be necessarily true. The question to be faced now is whether the reasoning can be parodied to argue for strange entities, like necessary islands or perfect devils. It is quite obvious that it cannot, for the “consistency guarantee” will only work for a concept of something the essential nature of which is defined in a way that does not include positive terms at the outset. There is no way to demonstrate the existence of something like a perfect devil or a necessary island by means of this proof, for as soon as one introduces a positive property into the definition of the essence of the hypothetical being to be proven, there is no (known) way to demonstrate the consistency of the concept. Hence, any parodies of this argument that use positive properties are inconclusive, for they can only show that the existence of the parody entity is either logically necessary or logically impossible (Nq v N~q). In the case of the concept of something Godlike, however, we can rule out impossibility because of the necessary coherence of the definition.

Despite its immunity to parodies containing positive properties, it may be argued that the proof can be parodied by adding more negative properties so as to yield conclusions that are unacceptable to theists. Consider the following example:

Definition: Something is “Godlike*” if and only if it possesses negative properties (1)-(8) and is also non-good and non-powerful.

1) Either the existence of something Godlike* is logically necessary or logically impossible.

2) It is not the case that the existence of something Godlike* is logically impossible.

3) The existence of something Godlike* is logically necessary. [From 1 and 2]

Since the terms in the definition are strictly negative, this parody concept succeeds where positive parodies fail, because of its guaranteed conceptual consistency.

It seems that there is no way to escape the conclusion of such a parody, for it functions on exactly the same principles as the proof that attempts to parallel. But, despite its success, the parody fails to pose a problem for the theist because its conclusion is perfectly consonant with the traditional apophatic way of conceiving God (which is the preference of mystics). Many theologians (of the Christian tradition and others) deny literal interpretation of predicates such as “goodness,” “power,” and whatnot of God. They do this not to imply that God is evil or weak, but because the divine essence is believed to transcend all of our positive concepts. The positive terms normally predicated of God, like goodness and power, are considered analogical; they approach the essence of God, but can never reach it adequately. In such a context, the non-deficiency of God is said to surpass all positive interpretations of “perfection.” The denial of positive predicates, like “goodness” is done in order to purify the concept of God from any deficiency that would be entailed by our limited positive understanding of goodness. The non-goodness of God does not imply that God is evil or neutral, but merely that our limited notion of goodness is too frail to be literally predicated of God. This negative understanding of the divine nature seems to present us with a deity that is eminently more worthy of worship than is a being the essence of which can be encapsulated in one of our empirically derived positive concepts.

Despite the fact that this type of parallel argument forces us into a largely negative mode of theology, there is no reason to deny that positive relational properties can be predicated of God, for they do not determine the concept of the essence of God, but only how other beings stand in relation to the deity. So there is no problem posed by saying something like “God is that on which the universe depends,” or “I believe God to be worthy of worship.” Of course, some relational properties may entail positive essential attributes, and must therefore be denied.[12]

Another possible attempt at a negative parody definition is as follows:

Definition: Something is “Godlike**” if and only if it possesses negative properties (1)-(8), and also possesses the negative properties of not being self-identical, not existing, and not being possible.

Such a parody concept does not really pose any problem for the argument for a Godlike being because the qualifications to Gale’s criteria that I mention in notes 6 and 7 explain that terms with a universal or null extension are not to be counted as properties. As such, they are outside of the positive and negative distinction. Since some of the “properties” in this definition have a null extension, and are not really negative properties at all, it follows that this parody fails to adequately parallel the argument for a Godlike being (which uses only “authentic” negative properties).

I will now examine some objections to the proof in order to forestall some of the more common rebuttals.

One may attempt to counter the argument with the contention that it is not legitimate to conclude the existence of something solely through analysis of its definition. It may be argued that affirmations of existence are always extrinsic to definitions because they are judgments about what has been defined, and can never be part of the descriptions themselves. The famous Kantian criticism of the ontological argument, for example, argues that the term “exists” can never be part of a definition:

By whatever and by however many predicates we may think of a thing–even if we completely determine it–we do not make the least addition to the [description of the] thing when we further declare that this thing is. Otherwise, it would not be the same thing that exists, but something more than we had thought in the concept; and we could not, therefore, say that the exact object of my conception exists.[13]

Upon careful reflection, it becomes clear that the above objection does not militate against the argument that has been presented here. This is because the definition of what it means to be Godlike does not, in and of itself, imply any ontological commitment. The non-contingency of a Godlike being merely entails that it either necessarily exists or necessarily does not exist. Since its ontological status remains indeterminate even though contingency has been denied, it follows that one cannot rightly say that the definition contains a judgment of existence within itself. One achieves a proof of the existence of something defined as non-contingent only when one combines the definitional denial of contingency with the additional judgment of the logical possibility of what has been defined. Of course, this judgment about possibility is not part of the definition itself, and is reached through independent considerations. Hence, it follows that the Kantian criticism has no power against this version of the ontological argument.

Another objection is that one can imagine a possible world in which a Godlike being does not exist, and therefore it must be the case that the existence of such an entity is not logically necessary. But even though it is prima facie imaginable that such a being does not exist, it is not really a coherent thought, for any possible world is subject to all conceptual laws and logical truths. As such, the concept of what it means to be Godlike would be a consistent concept (since it entails only negative properties), and would still imply the denial of contingency as a matter of definitional truth. From these facts, the existence of something Godlike can be deduced using nothing but truths of logic. When one makes the objection that they can conceive of a world without something Godlike, they are ignoring the implications of abstract reasoning, and are not recognizing that the conclusions derived as a matter of conceptual necessity are binding in all possible worlds.

There is also the question of why the conceptual coherence of a description should be thought to guarantee its metaphysical possibility. Metaphysical possibility is a species of real possibility, and it seems highly questionable that what is really possible be decidable on the basis of what we can coherently conceive to be the case in some possible world. For example, one can conceive of a “female president of the United States that held office in 1982” in some possible world, but it doesn’t follow that this is a real possibility in the actual world. This question of metaphysical possibility can be easily answered when one considers that a conclusion that is logically necessary (true in all possible worlds) must be true (and hence metaphysically possible) in the actual world too. The modal ontological argument allows us to go from the recognition that, if a Godlike being exists in some possible world, then it exists in every possible world (including the actual world). The logical necessity of this conclusion entails its metaphysical possibility.


It has been shown that the modal ontological argument can be modified to prove the existence of a Godlike being. Unless a sound rebuttal is forthcoming, one must conclude that atheism is false.


[1] Charles Hartshorne, The Logic of Perfection, (La Salle, Ill.: Open Court, 1962), p.51.

[2] Michael Martin, Atheism: A Philosophical Justification, (Philadelphia: Temple University Press, 1990), pp. 89.

[3] Ibid., p. 91

[4] Richard M. Gale, “Negative Statements,” American Philosophical Quarterly, Volume 7, Number 3, 1970, p.215.

[5] Ibid.

[6] Gale writes: “[This criterion] must be restricted to qualitatively homogenous properties; for non-red color, an obviously negative property, is incompatible with some other property of the same quality–non-colored.” (Ibid.)

[7] Gale writes: “It might be contended that there are properties not of the same quality as non-red which it specifies, viz., being an entity, being something, being self-identical with itself, etc. These ‘properties’ will not be counted as properties, since they logically must have a universal extension. We shall also not countenance as properties any property that must have a null extension. (Ibid.)”

[8] In personal correspondence with Gale, he confirmed that the property of non-contingency is, in fact, negative according to his criteria.

[9] The property of dependency is positive because it entails terms of differing quality, like “caused” and “non-necessary.” Its complement is negative because it only entails properties of the same quality, such as “non-caused.” The property of “being subject to physical laws” is positive because it entails terms of differing quality such as “limited” and “non-omnipotent.” Its negation, however, entails only terms of homogenous quality. The property of “being natural” is positive because it is incompatible with properties of the same quality as itself, such as “being an angel.” “Natural” also entails terms of differing quality, like “being subject to physical law,” and “not being an angel.” “Non-natural,” since it is a negative term, is compatible with all properties of the same quality as itself, such as “non-red,” and “non-physical.” The ascriptions of being a Platonic form, a logical law, a number or a mathematical truth are clearly positive because they are incompatible with other properties of the same quality, like being “bright” or “solid.” These positive terms also entail properties of differing quality such as “being abstract” and “non-material.” Their negations do not entail properties of differing quality, and are therefore negative. “Finitude” is positive because it entails the positive property or “being measurable in theory” and the negative property of “not being the set of all numbers.” Its complement “non-finitude” is negative because it only entails terms of the same quality as itself. The notion of “being deficient” is positive because it entails the positive term “inferior to something conceivable” and the negative term “non-perfect.” Its complement “not being deficient in any sense” is negative because it entails no positive terms.

[10] One can deduce at least one positive property of something Godlike by combining the negative definition of what it means to be Godlike with additional premises. For example, the property of being Godlike entails the negative property of non-contingency, but when one combines this non-contingency with the positive fact that the existence of something Godlike is logically possible, one can deduce that a Godlike being can possess the positive property of “being necessary.” It may be possible to deduce other positive properties in a similar way, through the use of additional premises.

[11] In personal correspondence with Graham Oppy, he pointed out that there are other factors that can prevent a property from being exemplified in any possible world even though that property may be logically consistent. Consider, for example, the property of “being the actual female prime minister of Australia in 1998”. There is no logical inconsistency in this property (one can’t tell a priori that there is nothing in the actual world that exemplifies it): nonetheless, since the prime minister of Australia in 1998 was male, there is no possible world in which anything falls under this (rigidified) property. It is important to realize that this issue that Oppy raises only occurs with very specific types of modal vocabulary that prevent something from existing in more than one possible world (such as when something is defined as “actual,” or as “non-contingent and non-necessary”). Therefore, this phenomenon is not really relevant to the soundness of my argument because the property of being Godlike does not entail any of these very specific modal properties that prevent it from being exemplified in more than one possible world.

[12] When discussing this proof, Oppy raised the point that if one starts with a two place predicate, Rxy, one can always define related one-place predicates R’x and R”x. Sometimes, these related predicates will have implications for the nature of that which falls under them, and sometimes not. So, for example, from the sentence “God is the creator of the actual universe”, we get–by immediate analysis–Created (God, the actual universe), and then–by the kind of process hinted at above–Creator of the actual universe (God). What does it take to be a creator of something? It seems that, plausibly, being a creator requires mind, intention, intelligence–etc. But, if that’s right, then we now have positive attributes of God: God has a mind; God is intelligent; God has intentions, etc. Oppy argued that if this is denied, then it is difficult to see how any sense can be attributed to the claim that God is the creator of the actual universe. Despite Oppy’s observations, though, it may be the case that the creator has a positively incomprehensible super-essence that is something like a mind (at least in the sense that it is able to create), but is beyond the limitations that are concomitant with our concept of mind (like propositional thought, or deliberation). There is no reason to believe that being the creator entails having “mind” as we positively understand the term. Oppy also raised the same point about the claim that God is worship-worthy. He argued that if it is literally true that God is worship-worthy, then it must also be literally true that God is good. Contra Oppy, someone defending a strictly negative view could say that God is worship-worthy because God is the source of all that we call “good,” but that the super-essence of God is simply beyond our limited notion of good, not less than it.

[13] Immanuel Kant, The Critique of Pure Reason, trans. Norman Kemp Smith, (London: Macmillan, 1929), p.505, reprinted in Martin, Atheism: A Philosophical Justification, p. 81.

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