Is God’s Existence Logically Necessary?

0. Introduction

It is a common platitude in philosophy of religion to hear the claim that the existence of God is necessary, or that God exists ‘in all possible worlds’. However, it seems to me that the existence of God can only really be thought of as (at best) metaphysically necessary, but not logically necessary. Strictly speaking, God doesn’t exist in all possible worlds. In this post, I want to explain a bit about what possible worlds are, from a logical point of view, and explain how God’s existence is not logically necessary in classical propositional logic.

  1. Possible worlds

Perhaps the defining idiom of 20th century analytic philosophy is the term ‘possible worlds’. The traditional story is that this terminology dates back to Saul Kripke’s early logical work on the semantics of modal logic (see here, and here). For an excellent summary of the historical lead-up to this development, see this paper by Jack Copeland. As Copeland notes, the roots of the project can be traced back further, to the development of classical propositional logic in its modern form by people like Frege, Russell and Wittgenstein. As Wittgenstein explains in the Tractatus:

“If an elementary proposition is true, the state of affairs exists: if an elementary proposition is false, the state of affairs does not exist. If all true elementary propositions are given, the result is a complete description of the world. The world is completely described by the specification of all elementary propositions plus the specification, which of them are true and which false.” (4.25-4.26)

In the second line of the Tractatus, Wittgenstein defines the world as ‘the totality of facts’. These facts, or states of affairs that exist, are what makes propositions true or false. So if we specify a truth value to each and every proposition, then this would be a way of specifying what the world is like – it specifies whether the states of affairs that correspond to the propositions exist or not. Let’s use as an example the proposition is ‘Donald Trump is the president of the USA’, which we shall refer to as p. If is true, then the state of affairs in which corresponds to p exists; i.e. if p is true, then Donald Trump really is the president of the USA.

Once the truth values of the atomic propositions are given, then this generates the truth values of more complex formulas. So if p is true and q is true, then the formula ‘p & q‘ is true as well, etc. In this way, everything that one can say about the world is said just by giving the truth values of all the atomic propositions.

When we are thinking about this process of describing the world via the truth values of the atomic propositions, we need to specify two conditions:

  • We need to ensure that we give each proposition one truth value or other; we cannot ‘forget’ to specify whether p is true or false.
  • We need to ensure that we do not give the same proposition both truth values; we cannot say that p is both true and false.

Let’s call the first condition maximality, because it is saying that our description of the world is maximal in the sense that no proposition is left out. Let’s call the second condition consistency, because it is saying that our application of truth values to propositions is consistent, in the sense that no proposition is given both truth values. This allows us a nice and precise definition of a ‘world’, as used by Wittgenstein:

A world is a maximal and consistent valuation of atomic propositions

On this understanding, a different world is just a rearrangement of truth values to the basic atomic propositions. So, pretending for a moment that there are only 3 atomic propositions (p, q and r), it might be that in the actual world the following combination of propositions is true: (p, q, r). But at some other world, the following combination is true: (p, ~q, ~r).

The number of worlds is a function of the number of elementary (or atomic) propositions we have. If we have just one proposition p, then there are two worlds, because p could be true, and p could be false. If we have two propositions, p and q, then we have four worlds: one in which both are true, one in which both are false, one in which p is true and is false, and one in which p is false and is true. In this way, we can construct tables which systematically display all the combinations of truth and falsity to the basic propositions. Here is a picture, from the Tractatus, showing such a ‘truth table’ for three propositions, (pq, and r):


The method of truth tables is used by Wittgenstein in the Tractatus as a method of proving whether complex formulas are true or false independently of the values of their atomic elements, and for whether arguments are valid or invalid. In short, it is a proof theory for propositional logic, and it is both sound and complete. It is taught as an introduction to any logic class.

Given that there are two truth values on this picture, if the number of atomic propositions we have is n, then the number of worlds (i.e. maximal and consistent sets of propositions) is 2 raised to the power of n (which WordPress doesn’t seem to have a symbol for); i.e. the number of  worlds doubles for each additional atomic proposition you have.

It follows very simply from this definition of a world, that a formula of the form ‘p or ~p‘ is going to be true at each and every row of the truth table, which is to say at each world. That’s because each world has the proposition p as either true or false in it due to the maximality constraint. All ‘p or ~p‘ needs to be true is that one or other of its disjuncts is true. So, this formula is considered to be a ‘tautology’ because we do not need to look at the particular variation of truth values in the world we are considering to see if it is true or not. It is true in every world. Similarly, a proposition of the form ‘p & ~p‘ is going to be false regardless of the arrangement of the truth values to the atomic propositions. Because of the consistency condition, each world gives only one truth value to each proposition, and the formula just says that p has both truth values.

Therefore, what we might think of as the ‘law of excluded middle’ is guaranteed by the maximality condition, and the ‘law of non-contradiction’ is guaranteed by the consistency condition. That they are true in every world is a consequence of the definition of a ‘world’.

Also, the notion of validity is just that if the premises of an argument are true, then the conclusion is true. If we plug the argument into a truth table, we can give precise expression to this notion: an argument is valid if there is no row of the truth table on which the premises all come out true but on which the conclusion is also false. This just means that an argument is valid if there is no world in which the premises are true and the conclusion is false.

Although it is often not stated in terms of worlds, the ideas of tautology and validity in classical propositional logic have always made use of the notion of worlds, if construed as maximal consistent sets of propositions.

In essence, all Kripke does to this picture is to add two additional operators to the logic, which say ‘it is possible that…’ (◊p) and ‘it is necessary that…’ (□p). These operators refer, via the semantics (which I will not go into any detail over for ease of reading – though I am more than happy to at a later date) to what is true or false at other worlds, or other ‘possible worlds’. So if ◊is true, then p is true at some other possible world; and if □p is true, then p is true at every possible world. We can afford to leave out most of the details here because we are primarily focussed on logical possibility, which is handy because it gets somewhat technical otherwise.

If we ask a question about whether a given proposition, p, is logically possible, then we can see if there is a possible world where p is true. There is a world where p is true if and only if there is a maximal and complete assignment of truth-values to the atomic propositions where p is true. So is the following proposition possible?:

a) This glass of beer is full, and I am hungry.

We can formalise a) as follows, where p = ‘this glass of beer is full’ and q = ‘I am hungry’:

b) pq

Now, as it happens, this glass of beer is half empty (because I have already been drinking from it), and I am not hungry (because I have just eaten dinner), meaning that p is false and so is q. That means that b) (and thus a)) is false. But that doesn’t tell us whether it is possible or not though. What we have to consider to see whether it is logically possible or not is whether there is a contradiction in supposing that it is true. And there is no contradiction in supposing that the truth-values of the propositions are different. Though p and are both false, they could both be true. As Wittgenstein said when referring to states of affairs,

Any one can either be the case or not be the case, and everything else remain the same. (Wittgenstein, Tractatus, 1.21)

This means that we can vary the truth-values of any of our basic atomic propositions without having to change the others; all the combinations of different truth-values are possible. All that we have to watch out for is that we end up with a proposition to which we have no truth-value, or one that has both truth-values, i.e. as long as we don’t end up with an excluded middle or a contradiction.

Given that there is no contradiction in supposing that ‘This glass of beer is full, and I am hungry’, this means that there is a maximal and consistent set of propositions which contains it, and that means that it is logically possible.

Could the following be true?:

c) This glass of beer is full and the glass of beer is not full.

Now, for c) to be true, both sides of the ‘and’ would have to be true. But these are p and not-p respectively. If both were true, then p would be both true and false. This would be a contradiction, and so (because of the consistency condition) there is no possible world at which this is the case. This means that c) is not logically possible.

Ask yourself this, could the following proposition be true?:

d) This glass of beer is full and God does not exist.

Even if you think that the proposition is in actual fact false (i.e. if you are a theist with a full glass of beer), ask yourself if there is a contradiction which results from supposing that d) is true. It seems that there is not. The logical form of d) seems to just be the following:

e) r & ~g

We can show very easily that this formula is not a contradiction with a truth table:


This shows very explicitly that the formula is false if r and g are both true, and if they are both false, but the formula is true on both middle rows. So it is not always false, i.e. it is not a contradiction. I have just proved that e) is not a contradiction. If someone wants to say that d) is somehow a contradiction, then they need to provide a different logical form for the proposition than e). In propositional logic, this seems to be the only plausible rendering of the form of d), and so it seems that in propositional logic, the formula is not a contradiction. That means that God’s existence is not a logically necessary truth.

3. Conclusion

While the existence of God may be asserted as a metaphysical necessity (however that is cashed out), it cannot be asserted as a logical necessity, if the logic we have in mind is classical propositional logic. I will write a sequel to this paper where we look at the possibilities of cashing out the logical necessity of God’s existence in first order logic, and I will explain how it is not a viable claim there either.

A New Problem for Divine Conceptualism?

0. Introduction

Divine Conceptualism (DC) is an idea about the ontological relationship between God and abstract objects, defended by Greg Welty, in his M.Phil thesis “An Examination of Theistic Conceptual Realism as an Alternative to Theistic Activism“(Welty (2000)), his Philosophia Christi paper “The Lord of Non-Contradiction” (Anderson and Welty (2011)), and his contributions to the book “Beyond the Control of God” edited by Paul Gould (Welty (2016)). Put simply, (DC) identifies abstract objects as something like ideas in the mind of God.

Welty sees his view as being quite close to that of Morris & Menzel‘s (1986) ‘theistic activism’ (TA), according to which:

“…all properties and relations are God’s concepts; the products, or perhaps better, the contents of a divine intellective activity.” (Morris & Menzel (1986), p. 166)

Morris & Menzel’s TA asserts that God created everything which is distinct from God, and that includes the divine concepts themselves. However, as Welty (2000) p.29 observes, TA is vulnerable to ‘bootstrapping’ objections. If God is supposed to be able to create his own properties, then he creates his own omnipotence (because omnipotence is a property); yet it seems that one would already have to have omnipotence in order to be able to create omnipotence. Even more forcefully: God already needs to possess the property of ‘being able to create properties’ in order to create properties. The idea of self-creation is therefore seemingly incoherent.

Welty’s DC can be seen as a modified version of TA; it is TA without the troublesome doctrine of self-creation. On DC:

“…abstract objects … are uncreated ideas in the divine mind; i.e. God’s thoughts.” (Welty, (2000), p. 43

Postulating abstract objects as uncreated divine ideas is designed to avoid the bootstrapping objections from above.

There are of course lots of different types of abstract objects, including propositions, properties, possible worlds, mathematical objects, etc. Here we will only look at propositions. One of the motivations for thinking that propositions in particular are divine thoughts is the argument from intentionality (seen in Anderson and Welty (2011), p 15-18). Propositions are intentional, in that they are about things. So the proposition ‘the cat is on the mat’ is about the cat having a certain relationship to the mat; the proposition is about the cat being in this relation to the mat. In a similar manner, thoughts are also about things. Consciousness is always consciousness of something or other. In Anderson and Welty (2011), it is argued that the laws of logic are propositions, which are necessarily true and really existing things. Given the intrinsic intentionality of propositions, these are argued to be thoughts. However, they cannot be thoughts had by contingently existing entities, like humans, as humans could have failed to exist, whereas laws of logic could not. Thus:

“If the laws of logic are necessarily existent thoughts, they can only be the thoughts of a necessarily existent mind.” (Anderson and Welty, (2011), p.19).

However, I want to point out an objection to this picture, which I have not seen in the literature (a nice summary of existing objections is found here). It is about the definition of the word ‘thought’. (It may be that this problem has been adequately documented in the literature somewhere that I have not seen. Maybe someone can let me know in the comments section.)

  1. Thought

It seems to have gone unnoticed that Welty in particular oscillates between DC being construed in two different and incompatible ways. It has to do with the word ‘thought’. There is no completely standardised usage of this term in the philosophy literature. And it is a term which needs careful definition in a philosophical argument because in natural language the word ‘thought’ is sometimes used to refer to the thinking and sometimes the thought-of; it is either the token of a type of mental activity called ‘thinking’, or it is the content, or object, of the thinking. For example, we may have the intuition that my thought is private, and that it is metaphysically impossible for you to have my thought (which makes thoughts similar to perceptions in this respect). But we may also have the intuition that we can ‘put our thoughts on paper’ or ‘share our thoughts’ with other people. It seems to me that this ambiguity infects Welty’s version of DC due to his not clearly and carefully defining what he means by ‘thought’ so as to disambiguate the term between thinking and thought-of. Welty (2000), for example, doesn’t actually contain a definition of a ‘thought’ anywhere in it, even though it mentions ‘thought’ 135 times in 85 pages.

According to Anderson and Welty (2011), they seem to indicate that a thought is not the content of thinking, but the token of the act of thinking. In a footnote on page 20, they say:

We could not have had your thoughts (except in the weaker sense that we could have thoughts with the same content as your thoughts, which presupposes a distinction between human thoughts and the content of those thoughts, e.g., propositions).”

The distinction that is being made here is between thoughts, which are individualised occurrences not shareable by multiple thinkers, and the contents of those thoughts, which are generalised and shareable by multiple thinkers. I can have a thought with the same content as you, even though we cannot have the same thought. In Fregean terms, a ‘thought’ (as Anderson and Welty use the term above) is an ‘apprehension’. When one thinks about the Pythagorean theorem, one is apprehending the proposition. In order to be explicit about what I mean, I will disambiguate the term ‘thought’ by referring to the token act of thinking as an ‘apprehension’, and the content of the thought as the ‘proposition’.

2. Blurred Lines

However, in Welty (2000), this distinction is repeatedly blurred. One of the main thrusts of the position defended there is that God’s thoughts function as abstract objects:

“God and I can have the same thought, ‘2+2=4’, in terms of content. But my thought doesn’t function in the same way that God’s thought does. My thought doesn’t determine or delimit anything about the actual world, or about any possible world. But God’s thought does. Thus, it plays a completely different role in the scheme of things, even though God and I have the same thought in terms of content. Thus, God’s thought uniquely functions as an abstract object, because of his role as creator of any possible world. I am not the creator of the actual world (much less, any possible world), and thus my thoughts, though they are in many cases the same thoughts as God’s, don’t function as abstract objects in any relevant sense.” (Welty, (2000), p. 51)

Welty says that God and I can have ‘the same thought in terms of content’, which blatantly smudges the sharp distinction between the apprehension and proposition. We can each apprehend the same proposition. But can I share in God’s apprehension of the proposition? It seems that the answer would have to be: no. God’s apprehension of a proposition is surely private to God, just as my apprehension of a proposition is private to me.

Then Welty ends the passage with “my thoughts, though they are in many cases the same thoughts as God’s, don’t function as abstract objects in any relevant sense”. The only sense in which my thoughts are “the same thoughts as God’s” is in terms of the propositions that I think about being the same as the ones that God thinks about. In that sense they do function as abstract objects, precisely because they are abstract objects, namely propositions! The sense in which ‘my thoughts’ don’t function as abstract objects is in terms of the token act of thinking (the apprehension). That doesn’t function as an abstract object, but then that is not something I share with God. So Welty cannot have it that there is something, x, which is both something I share with God and which doesn’t function as an abstract object. The only reason it seems like this is possible is because of a failure to distinguish clearly between thought as apprehension, and thought as propositional content.

This confusion pops up again and again. Take the argument from intentionality, found in all three Welty publications referenced in this post. Part of the motivation for DC is that propositions are (supposedly) thoughts (because they are intentional) but that they cannot be human thoughts; a non-divine conceptualism, the doctrine that abstract objects like propositions are human thoughts, cannot do the job here. The reason for thinking that they cannot be human thoughts is as follows:

“There aren’t enough human thoughts to go around…, human thoughts don’t necessarily exist, and whose thoughts will serve as the intersubjectively available and mind-independent referents of propositional attitudes (referents that are also named by that-clauses)?”

There are three reasons given against human thoughts being able to play the role of propositions: a) there aren’t enough of them, b) their existence isn’t necessary, c) they aren’t intersubjectively available.

While these considerations look somewhat compelling when trying to think of a human conceptualism without the benefit of the distinction between apprehension and proposition, it quickly loses its force when we apply the distinction. The problem is the combination of two types of properties that propositions need. One type of property is associated with divine apprehensions, and the other type of property is associated with divinely apprehended propositions. Being of sufficient plentitude to play the role of propositions (a), and having necessary existence (b), are of one type, and being ‘intersubjectively available’ (c) is of the other. As I shall show, you cannot have both of these types at the same time, without smudging the distinction between apprehensions and propositions.

Firstly, let’s consider non-divine conceptualism, where thoughts are construed as apprehensions.

There are, of course, only finitely many human apprehensions of propositions; there are only finitely many times people have apprehended propositions. Also, human apprehensions of propositions are contingently existing things, because human minds are themselves only contingently existing things. Human apprehensions are also inherently private, and thus not intersubjectively available. So apprehensions cannot be thought of as ‘doing the job’ of abstract objects for these reasons. That much is quite clear.

On the other hand, there may be infinitely many divine apprehensions, so there would be ‘enough to go round’, and perhaps they each exists necessarily. In this sense, they seem suited to play the role of propositions. However, as apprehensions, they would not be ‘intersubjectively available’. Can I actually share in God’s apprehension of a proposition? Unless I can, they cannot play the role of an abstract object.

Thus, when considering apprehensions, although non-divine conceptualism is not suited to play the job, neither is divine conceptualism. The problem is just that apprehensions are private. So let’s compare non-divine and divine conceptualism, where we construe ‘thought’ as the contents of thoughts.

Right away it is obvious that there is no reason to think that the content of human apprehensions are limited in the same way as their apprehensions were. The contents of human apprehensions just are propositions, so of course they can play the role of propositions!

Equally, if divine thoughts are construed as divinely apprehended propositions, then there will be enough to go round, they will exist necessarily, and they will be intersubjectively available. But in both cases, this is just because propositions themselves are sufficiently plentiful, necessary and intersubjective to play the role of propositions. Obviously, propositions can play the role of propositions. Being apprehended by God, rather than humans, is not what bestows the required properties on them.

3. Begging the question?

But perhaps I have begged the question somehow. Maybe the defender of DC can stipulate that, although my apprehensions are private, God’s apprehensions are somehow intersubjectively available. Call this theory ‘divine accessibility’ (DA). So on DA, propositions are divine apprehensions (which are plentiful, and necessary existing) and crucially also intersubjectively available to humans; they can be the content of humans’ apprehensions.

So, let’s say that I am thinking about the Pythagorean theorem. Let’s say that my apprehension is A. According to DA, the content of my apprehension, what A is about, is a divinely accessible apprehension, D. But the question is, what is the content of the divine apprehension, D? What is it that God is thinking about when he has the thought which is the Pythagorean theorem? There seem to be only a few options:

Either God’s apprehension, D, has content, or it does not. If it has no content, then what is it about D which links it to the Pythagorean theorem, rather than to some other theorem, or to nothing at all? It would be empty and featureless without content.

But, if it does have content, then either the content is that ‘the square of the hypotenuse is equal to the sum of the squares of the other two sides’, or it is something else.

If it does have this as its content, then it seems like the content of D is doing all the work. It seems like the only reason God’s apprehension is linked in any way to the Pythagorean theorem is that it has the theorem as its content. If that is right, then we need to have the proposition itself in the picture for God’s apprehension to be in any way relevant.

Consider what would be the case if the content of God’s apprehension was of something else entirely, like the fact that it all bachelors are unmarried men or something. In that situation, there  would be no reason to say that this apprehension was the Pythagorean theorem. The only divine apprehension that could, even plausibly, look like it is playing the role of the proposition is one which has the proposition as its content.

And if we ask what role God’s apprehension plays here it seems that the answer is that it is just a middle man in between my apprehension and the theorem. It seems to be doing nothing. When I think of the theorem, I have an apprehension, A, and all this is about is one of God’s apprehensions, D, which is itself about the theorem. If p is the Pythagorean therem, and x ⇒ y means ‘x is about y’, then we have:

A ⇒ D ⇒ p

God’s apprehension is just an idle cog which does nothing. Why not just have:

A ⇒ p

Why not just say that I have the theorem as the content of my thought? It would be a much simpler suggestion. Given that for God’s apprehension to be in any way relevant to the proposition in question it has to have the proposition as its content, we seem to require the proposition in the picture anyway. Ockham’s razor should suggest shaving off the unnecessary extra entity in the picture, which is the divine apprehension.

4. Conclusion

Thus, there are really two problems with DC. If construed as the contents of God’s thoughts, divine ‘thoughts’ just are propositions. So for DC to be in any way different from the traditional Fregean picture (where propositions are abstract objects), we have no other option but to construe divine thoughts as divine apprehensions. However, it seems that apprehensions are inherently private, and so they are unsuited to play the role of propositions. Even if we postulate that somehow divine apprehensions are accessible to everyone, they seem to become idle cogs doing nothing.