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.
- 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 p 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 q is false, and one in which p is false and q 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, (p, q, 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 ◊p 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) p & q
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 q 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.
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.