The Meaning of “It’s Impossible to Prove that God Exists.”

It is not uncommon to hear students and professors, indeed people from all walks of life, asserting with full surety and confidence that it is not possible to prove that God exists. This occurs in various contexts. When the question of the universality and objectivity of moral norms arises, some of these persons link a knowledge and justification of these norms to God in such a way that they can then argue that objective moral principles cannot be known since the existence of God cannot be proven.

Regarding to nature of faith and its relation to reason, people who deny the possibility of proving God’s existence often believe that faith requires that the existence of God be impossible to prove. If the existence of God could be proven, then it would be known and faith would then not be required. In other words, faith in God is necessary precisely because God’s existence cannot be proven. This is a view not confined to the unbeliever.

Some people are so confident that it is impossible to prove God exists that they take their belief as a sufficient refutation of any of St. Thomas’ arguments as well as the ontological argument of St. Anselm. Hence they reason circularly by arguing from their belief that God’s existence cannot be proven to the conclusion that none of these arguments can be a proof that God exists.

Indeed, it should be mentioned that it is not just people who profess either an atheism or an agnosticism who maintain that God’s existence cannot be proven. Catholics and Protestants, Muslims and Jews sometimes affirm that it is impossible to prove that God exists. There is, of course, no logical inconsistency or contradiction in believing that God exists and also believing that human reason is incapable of proving this. Moreover, it must be granted that the justification of a person’s faith does not require that a person be able to construct a proof for God’s existence. Nevertheless, a proof for God’s existence has much value. A person’s faith can be confirmed by such a proof. In addition, while not all revealed truths can be proven through reason, being able to offer a person a proof of God’s existence can open the way for the person’s later acceptance of what is revealed. It is, indeed, unfortunate that the declaration of the First Vatican Council that reason unaided is capable of proving God’s existence is nowadays largely ignored.¹

There are, undoubtedly, a number of responses possible when without hesitation someone affirms the impossibility of proving God’s existence. One might explore the relationship between faith and reason in order to show that faith can be rationally justified. Appealing to differences between ontological and epistemological forms of dependency might persuade someone that, while universal moral norms are grounded in God, the existence of God need not be known first before a moral norm is known.

One might, of course, simply demand that sufficient evidence be provided to justify the belief that it is impossible to prove that God exists. This strategy, however, carries an unwarranted assumption; the assumption that these persons have a clear understanding of what they are asserting when they proclaim, “It’s impossible to prove that God exists.” Indeed, the response which in my experience people find most surprising is my claim that they actually do not understand what their position is and are not exactly clear about what they are asserting. One fundamental confusion concerns what precisely they mean by a proof; that is, what exactly they have in mind by the word “proven” when they proclaim that God’s existence cannot be proven. They are also astonished when I declare that even with a clear and definite notion of what constitutes a proof there are actually six different propositions which they might have in mind when they say, “It’s impossible to prove that God exists.” Of these six different propositions the first proposition is the most logically forceful one since, if it is true, all the other possible propositions are also true. The sixth is the weakest since from its truth nothing follows regarding the truth of the first five.

When the notion of a proof is examined and an explanation of these six propositions is offered along with some consideration of how an acceptance of any of these six propositions could be warranted, I have without exception found people’s confidence in the unprovability of God’s existence to be undermined. Under such circumstances people are often more willing to consider whether certain arguments are proofs of God’s existence than if one simply offers an argument in response to their claim that God’s existence cannot be proven. Generally speaking, people are more willing to consider seriously whether an argument proves God’s existence if they withhold assent on the question of whether God’s existence can be proven than if they believe God’s existence cannot be proven.

Before explaining the six different propositions people might have in mind when they assert that it is impossible to prove that God exists, some preliminary consideration must be given concerning what can be meant by a “proof.” By a proof one might mean simply an argument which logicians call “valid.” A valid argument is one in which the conclusion follows with logical necessity from the premises on account of the form of the argument. In order for an argument to be valid, it is not necessary, however, that all or even some of its premises be true.

The following argument is valid: If the majority of people believe that God has a long white beard, then God has a long white beard, and the majority of people believe that God has a long white beard, therefore God has a long white beard. This argument is valid and follows the valid logical pattern of modus ponens; by its form the conclusion follows necessarily from the premises. Yet, each of the three propositions comprising the argument are obviously false. Hence, while the argument has a valid form, it is not what logicians call “sound.” In order for an argument to be sound it must be valid and all of the propositions comprising it must be true.

Now it is fairly clear that proponents of the view that God’s existence cannot be proven do not mean by a proof merely a valid argument. God’s existence is hardly proven if the proposition that God exists is the conclusion of an argument which is valid although one or more of its premises is false. Innumerable arguments can be offered for the existence of God as long as the arguments need only be valid but not sound. Rather, proponents of the view that God’s existence cannot be proven mean by a proof at least a sound argument, which is a valid argument with true premises and therefore a true conclusion. Is this all that they have in mind by a proof? They would probably respond that it is and that by a proof they mean a sound argument. This definition of a proof is sufficient to proceed to an analysis of what people might have in mind when they say, “It’s impossible to prove that God exists.” Further clarifications of the notion of a proof will be made in the course of this analysis.

When people assert, “It’s impossible to prove that God exists,” they might first mean that a sound argument for God’s existence is logically impossible. This proposition shall be labelled “Pl”. P1 posits, then, that no sound argument with the conclusion that God exists is logically possible. In general, a sound argument for a given proposition is logically possible when the given proposition is true and there are other true propositions (premises) which together necessitate the truth of the given proposition. In a similar way, to inquire in geometry whether a proof for squaring a circle is logically possible is to ask whether or not there are true mathematical propositions which together necessitate the true conclusion that in a given instance the area of a square is equal to the area of a circle. Thus, in order for a sound argument for God’s existence to be logically possible, the proposition that God exists must be true and there must be other true propositions which together necessitate the conclusion that God exists. It follows that in order for P’ to be true, that is, in order for no sound argument for God’s existence to be logically possible, the proposition that God exists is false or there are not true propositions which together necessitate the proposition that God exists.²

This understanding of P’ raises a number of issues which center on the relationship between the logical possibility or impossibility of a proof of God’s existence and the ability or inability of any human being to prove that God exists. With respect to the latter it is first necessary to distinguish between the ability of a human being to formulate a proof and his ability to know that the argument he formulates is a proof.

When a person formulates a proof, I do not mean that he is the first person to formulate it. In general, a person formulates an argument when he thinks two or more propositions according to some logical form. Thus, an argument is formulated regardless of whether it is sound or unsound. It is also important to note that in order for an argument to be sound, it is not required that a person know it is sound. More precisely, it is not a condition for an argument being valid or the propositions being true that a person know that the argument is valid or the propositions are true. This explains why it is possible for a person to formulate what is in fact a proof for God’s existence without knowing it is a proof. In other words, the argument he formulates is sound, but he does not know it is sound. Unless this is possible it is difficult to understand how a person can err in formulating an argument he believes is sound when in fact it is not, or believing an argument he formulates is not sound when in fact it is.

Thus, two different interpretations are possible regarding the ability or inability of a human being to prove that God exists. On the one hand, whether or not a human being can prove that God exists might refer to the ability or inability of a human being merely to formulate a proof for God’s existence. On the other hand, it might refer to the ability or inability of a human being to know that an argument is sound. Of course, in order for him to know it is sound, he must have the ability to formulate it, but his ability to formulate it is no guarantee that he is able to know it is sound. In the present article the first interpretation will be used as this accords with the fact that an argument can be sound without any human being knowing it is sound. Therefore, whether or not a human being can prove that God exists is equivalent to whether or not a human being can formulate a sound argument with the conclusion that God exists.³       

On the basis of these remarks it is important to note that P¹refers to what is logically possible, but not to what is humanly possible. It refers to the impossibility of true propositions necessitating the truth of the conclusion that God exists. No reference is made to the inability of any human being to formulate such an argument. In a similar way, a person who asserts that no proof is possible that the area of a square is equal to that of a circle is not referring to the inability of humans to formulate such a proof, but to the logical impossibility of mathematical propositions establishing this.

If no proof for God’s existence is logically possible, then, of course, no human being can prove that God exists. Moreover, without relying on faith, the only way for a person to know that a proof for God’s existence is possible is by knowing that some argument is a proof; merely formulating it would not be sufficient. It does not follow, however, that if a proof is logically possible, any human being is capable of formulating it. Therefore if someone holds that no human being can prove that God exists, it does not follow that no proof is logically possible. This is why it would be fallacious for people who assert P1 to offer as for evidence for P1 that no human being can formulate a proof.⁴   

In my experience most of the time when people understand the difference between asserting that no proof for God’s existence is logically possible and asserting that no human being can prove that God exists, they admit that they did not state precisely what they meant when they said that it is impossible to prove that God exists. They then maintain that what they originally meant to say was that no human being can prove that God exists. This strikes them as a position easier to establish than Pl; for what possible evidence could establish that no proof is logically possible that God exists? (An answer to this question will be offered later.)

There is one argument which people will sometimes offer for the truth of P1 although the argument is highly problematic. Some people argue that since God’s existence is not empirically verifiable, that is, verifiable through sensory observation, it follows that God’s existence cannot be proven. This conclusion follows from the stated premise, however, only with the additional premise that unless a proposition is verifiable empirically it cannot be proven. In other words, a premise is needed which limits provability to empirical provability. Without this premise, while God’s existence is not provable empirically, His existence might still be provable in some other way. With this needed additional premise the argument would then be: Only propositions which can be verified empirically can be proven; the proposition that God exists cannot be verified empirically; therefore God’s existence cannot be proven.

The easiest way to evaluate critically this argument is to point out that the assertion, “Only propositions which can be verified empirically can be proven,” cannot itself be verified empirically. This does not show that the proposition is false, but it does show that according to the proposition itself people cannot have a sufficient reason to accept it; and therefore people do not have a sufficient reason to accept the conclusion of the argument which includes this premise. In this way, people who limit provability to empirical provability are forced to admit that they lack a sufficient reason for accepting P1.

These remarks reveal a possible confusion people might have concerning the notion of a proof when they assert that it is not possible to prove that God exists. They might tacitly identify provability with empirical verifiability so that their claim regarding the impossibility of proving God’s existence actually refers to the impossibility of empirically proving God’s existence. By doing this their assertion that God’s existence cannot be proven would be more exactly formulated as, “It is impossible to prove empirically that God exists.”

This assertion is to be granted since even arguments for God’s existence which begin with the structure of the world, such as the five arguments of St. Thomas Aquinas, require the introduction of philosophical premises which are not verifiable empirically. The problem then is not with the claim that God’s existence is not empirically verifiable, but with the tacit assumption people might make in this instance, namely, that unless a proposition is empirically verifiable it cannot be verified. This assumption encounters the same problem just mentioned. The assertion that what is provable is limited to what is empirically verifiable is itself not empirically verifiable; no sensory observation can verify it. Therefore according to the very proposition itself there can be no sufficient justification or evidence for accepting it. In this way, it is completely unwarranted to maintain that in order to be a proof, in addition to being sound, an argument must be constituted by premises which are empirically verifiable, for this assertion itself cannot be verified empirically.

As it was mentioned, people generally take the proposition that no human being can prove that God exists as an easier position to establish than P¹. They are sometimes surprised to learn, however, that to assert that no human being can prove that God exists is also not entirely clear for two reasons. First, it must be pointed out that “can” refers to what human beings are capable of, not to what they have actually accomplished. If human beings actually do something, then obviously they can do it; but if they do not actually do something, it does not follow that they are incapable of doing it. Just as there are many things which human beings do not do which, nonetheless, they are able to do, so also it does not follow that human beings are incapableof proving that God exists simply because they do not actually do this. Therefore, the proposition that no human being can prove that God exists must be distinguished from the proposition that no human being actually proves that God exists.

A second ambiguity still remains, however, since the temporal modality of these propositions is unclear. To assert that no human being can prove that God exists might be a proposition about what is presently so, or instead what is so for all of time. The same applies to the proposition about what human beings actually do.

This distinction generates proposition PI which is that no human being for all of time is capable of proving that God exists; and proposition P³which is that no human being for all of time proves that God exists. (Other temporal modalities will be introduced momentarily.) If P³is true, it does not follow that P2is true; for from the fact that no human being does something it does not follow that no human is able to do that. This suggests how difficult it is to prove P². In other words, if no human being for all of time ever proves that God exists, it does not follow that no human being was capable of proving that God exists.

Moreover,P¹and P³ refer, not just to the past and present, but also to the entire future of mankind. Therefore, even if one were to assume that no one up to the present moment was able to prove God’s existence or had proven God’s existence, what evidence could be offered that no human being in the future would be able to do this or actually do this? Once people realize this they begin to appreciate how difficult it is to prove P²or P³, and usually then maintain that no human being up to this moment has been able to prove that God exists admitting again that they were not precise in stating what they meant when they said, “It’s impossible to prove that God exists.”

Yet, it is still necessary to distinguish, as was done before, between what a human being is capable of and what a human being does. Thus, P⁴, which is that no human being up to the present moment has been able to prove that God exists, is not equivalent to P^5, which is that no human being up to the present moment has proven that God exists. As with the relationship between P²and P³, P⁵does not logically entail P⁴. In other words, it does not follow that no human being up to the present moment has been able to prove that God exists, even if no human being up to the present moment has proven that God exists. Thus, it is difficult to know what evidence could be offered that no human being up to the present moment was able to prove that God exists; for that no one actually did it is not sufficient evidence.

For this reason, many people might find P⁵a reasonable position to hold believing it to be fairly obvious that at least up to the present moment no one has actually proven that God exists. They will, in all likelihood, take the widespread presence of disagreement and disbelief about God’s existence to be undisputed evidence for this.

What must be considered in this context, however, is what conditions are relevant to the truth or falsity of P⁵.⁵ This question invites a further clarification of what people have in mind by the word “prove” with respect to any of the six possible propositions they might have in mind regarding a proof for the existence of God.

It is crucial to distinguish between logical and non-logical factors or conditions. Consider the previous explanation of a sound argument in which it was stated that the following minimum conditions are required for a proof: the argument must have a valid form and the propositions constituting the argument must be true. These are logical requirements for the argument being a proof. As it was previously mentioned, a person need not know that an argument is valid or the propositions constituting it are true in order for an argument to be sound. Neither is it a requirement for the argument to be sound that there be complete or widespread agreement that the argument is sound, or that the argument be convincing to many people, or that the argument be easy to understand. These are non-logical conditions which are not necessary conditions for an argument being sound. To maintain that they are requirements confuses the logical and epistemological realms.

Moreover, a condition for an argument being a proof is obviously not that the argument have been passed down through history.⁶ 

These remarks explain why one has not shown that St. Anselm’s Ontological Argument or the five ways of St. Thomas Aquinas for proving God’s existence are not proofs simplybecause there is disagreement about them or some people are not convinced by them. In order to show that these arguments are not proofs one must show that the arguments have an invalid form or that at least one of the premises is not true.

On the basis of these remarks one can conclude the P⁵(that no human being up to the present moment has proven that God exists) is false if simply a certain set of logical conditions have been realized. If just one person at some time in the past formulated a sound argument for God’s existence, then P⁵ is false. It could be an argument which is not extant and was lost. This is why P5is not proven even if one could show that all of the extant arguments for God’s existence are in fact not proofs; for it is possible that someone in the past formulated a proof which was lost.

These remarks reveal the difficulty in proving that no one up to the present moment has proven that God exists, for this requires a knowledge of the past which is impossible for any human being nowadays to have.

Before turning to the last proposition someone might have in mind when asserting that it is impossible to prove that God exists, we can consider what would be sufficient evidence for propositions Pl through P⁵. Taking into account the fact that one or more of these propositions cover what is logically possible, or what human beings can do or actually do for all of time or for the entire past, it seems that there is only one thing which would be sufficient evidence for these five propositions: a proof that God does not exist. If one could prove that God does not exist, then one will have proven the truth of P¹through P⁵.

Someone could instead try to prove that human reason is limited or defective and for this reason is incapable of proving God’s existence. There are two problems with this position, however. First, such an argument would not disprove P¹since a proof for God’s existence might still be possible even if no human being is able to discover it. Second, it is hard to understand how a limitation or defect in human reason makes it incapable of proving God’s existence, but not incapable of proving that human reason is limited. If human reason is limited or defective how is it able to prove that human reason is unable to prove God’s existence? Why would a limitation in human reason extend to man’s capacity to prove that God exists, but would not extend to man’s capacity to prove that human reason is limited? It seems quite arbitrary to claim that human reason is limited when it comes to proving that God exists, but not limited when it comes to proving that human reason is limited and is unable to prove God’s existence. Indeed, there is no guarantee that a proof regarding the capacities and limitations of human reason would not be as complex or difficult as any proof for the existence of God.

If it is true then that only a proof that God does not exist would be sufficient evidence for propositions P^lthrough P⁵, at least two interesting consequences follow. First, people who hold any of these five propositions assume a burden of proof to demonstrate that God does not exist. This is a burden many of them will probably be surprised to find they have; for while propositions P¹through P⁵do not logically entail that God does not exist, the evidence which would warrant a belief in any of the five propositions commits one to the view that God does not exist. Thus, people might be less willing to accept any of these five propositions if they realize that this forces them to argue that God does not exist.

A second consequence is that this shows that there is less difference between agnosticism and atheism than people might think. If agnosticism denies that man’s reason is capable of proving God’s existence, but an acceptance of agnosticism is only warranted by a proof that God does not exist, then an agnostic must either move to the position of atheism, continue his belief in agnosticism while admitting that it is unwarranted, or withhold assent on the question of whether God’s existence can be proven.⁷

There still remains one last proposition which people might have in mind when they assert, “It’s impossible to prove that God exists.” This is P⁶which is that they themselves are not aware of any argument which proves that God exists. This, of course, may indeed be true, although justifying this position can be more complex than it at first appears. It seems that this position can be warranted by people simply internally examining what they know and what they do not know. If one is asked what the first five Chinese dynasties were, it is rather simple for one to confess ignorance. One is hardly mistaken about whether or not one knows this. The same seems to be the case regarding one’s knowledge of a proof for God’s existence.

Two different circumstances need to be distinguished, however. These two circumstances differ on the matter of whether someone is aware of any arguments for the existence of God. If people who are unaware of any argument for God’s existence assert P⁶, that is, that they are unaware of any argument which proves that God exists, then their belief in this is certainly correct and warranted; for they know what they are or are not aware of. One could say that they are unaware of any arguments which prove that God exists because they are not aware of any arguments. The situation is different for people who are aware of one or more arguments for the existence of God. If they assert P⁶, much more is required for their belief to be warranted. Unlike the first situation, they are aware of at least one argument for the existence of God. Therefore, in order for them to be justified in believing that they are unaware of any arguments which prove that God exists, they must provide evidence for why the arguments they are aware of are not proofs. Thus, unlike the people in the first situation, the people in the second situation have a more complex burden of proof for their belief; one which in many instances they will probably be surprised to find they have, and which is more difficult to achieve than simply reporting their ignorance of any arguments for God’s existence.

The preceding analyses of what people can have in mind when they assert that God’s existence cannot be proven is logically quite limited. No proof for the existence of God has been offered. Indeed, these analyses have not shown that a proof of God’s existence is possible, or even that the various propositions distinguished are false. Yet, these analyses do form a useful pedagogical foundation for a serious and more intelligible exploration of the arguments for the existence of God. They can lead people to a greater willingness to examine the arguments for God’s existence, more so than if their denial of the possibility of proving God’s existence is met simply with an argument for His existence. These analyses make the focus of the discussion not an argument itself, but a critical examination of the person’s own commitment. This explains the advantage of such an approach in possibly awakening people to the importance of the Socratic dictum that “An unexamined life is not worth living.”