An Objection Against Theism
In the moral argument and the Leibnizian cosmological argument I’ve argued for the sort of God whose existence is a necessary truth, where a necessary truth is a truth that can’t be or couldn’t have been otherwise. For example, “There is no married bachelor” is a necessary truth. The idea that God has necessary existence can be motivated further by the idea that God is the greatest conceivable being, and as the greatest conceivable being God would have the greatest possible form of existence (necessary existence).
One objection an atheist could make against God’s necessary existence is that the only way something can be necessarily true is if its denial is logically contradictory. For example, There is no married bachelor being false would mean that there is a married bachelor, which is selfcontradictory because bachelors are by definition unmarried. However, there’s nothing selfcontradictory about God not existing, so God’s existence isn’t a necessary truth.
While I agree that God’s nonexistence isn’t selfcontradictory, the idea that the only necessary truths are those whose denials are selfcontradictory (and by extension, the idea that the only necessary falsehoods are those that are selfcontradictory) has a fatal problem that I’ll talk about this blog entry.
Logic and Lingo
Before going on I’ll introduce some logic and lingo. In a branch of logic called modal logic, “necessary” and “possible” are often thought of in possible world semantics, where a possible world is a complete description of the way the world is or could have been like. A necessary truth is said to be true in all possible worlds, and a proposition is said to be possibly true if it is true in at least one possible world. Where p is a placeholder for some proposition, □p is shorthand for “p is true in all possible worlds.”
One interesting thing about possible world semantics: if □p is true in one possible world, then p is true in all possible worlds.[1] To illustrate, suppose □p is true in one possible world that we’ll call Alice. Then p is true in all possible worlds, because if p is false in some possible world (call that world Bob) then it wouldn’t be true in Alice that p is true in all possible worlds (since Bob is a possible world where p is false).
In philosophy, an analytic statement is a statement that is true by virtue of what it means such that a selfcontradiction is present in the meaning of its denial, e.g. It is not the case that Sam is a married bachelor is an analytic truth because the meaning of its denial (Sam is a married bachelor) contains a selfcontradiction (bachelors are by definition unmarried). A synthetic statement is a statement that is not analytic, i.e. its denial isn’t selfcontradictory. “Abraham Lincoln had a beard” is an example of a synthetic statement. One point that will be important later on: even if it is true that no synthetic statement holds in all possible worlds, this is not part of the definition of a synthetic statement.
Since an analytic truth is one whose denial is selfcontradictory, another way of saying “The only necessary truths are those whose denials are selfcontradictory” is “All necessary truths are analytic.” A truth is necessary only if its denial is necessarily false. Thus if all necessary falsehoods are selfcontradictory, then the only necessary truths are those whose (necessarily false) denials are selfcontradictory, which would mean that all necessary truths are analytic. The upshot is that if all necessary falsehoods are selfcontradictory, then all necessary truths are analytic. However, the proponent of “All necessary falsehoods are selfcontradictory” or “All necessary truths are analytic” has a fatal problem.
The Fatal Problem
To see the fatal problem let’s take this step by step. Notice that statement (1) below:
(1)  All necessary truths are analytic truths 
entails this:
(2)  There is no nonanalytic truth that is a necessary truth. 
Nonanalytic truths are synthetic truths, so (2) entails:
(3)  There is no synthetic truth that holds in all possible worlds. 
The denial of (3) is this:
(4)  There is a synthetic truth that holds in all possible worlds. 
By virtue of what (1) means, (1) and (3) are logically equivalent. Here’s the problem: is (3) a necessary truth? If the answer is “No” then there is some possible world where (3) is false, and thus there is some possible world where a synthetic truth is a necessary truth—some possible world where □p is true for some synthetic proposition p. But recall that if □p is true in some possible world, then p is true in all possible worlds. So denying that (3) is a necessary truth would mean that (3) is false, for there would have to be some synthetic statement that is true in all possible worlds.
The proponent of (1) could say that (3) is a necessary truth, but according to the proponent, only analytic statements can be necessary truths, and so for (3) to be an analytic truth its denial (4) must be selfcontradictory. But there doesn’t appear to be anything about the meaning of (4) that contradicts itself like “Sam is married and not married” and “Sam is a married bachelor,” since it just isn’t true by definition that a synthetic statement doesn’t hold in all possible worlds. Thus, it is not true that all necessary truths are analytic.
Conclusion
While “all necessary truths are analytic” and “necessary falsehood requires selfcontradiction” may seem like a reasonable claims on the surface, it ultimately doesn’t work. This is interesting because it leaves the door open (at least for the agnostic and theist) for a God who exists necessarily even though God’s nonexistence isn’t selfcontradictory.
While I found it somewhat surprising the way in which “all necessary truths are analytic” can be disproved, the idea that not all necessary truths are analytic has for me been rather unsurprising. Consider for example the proposition It is morally wrong to torture infants just for fun. There doesn’t appear to be anything selfcontradictory about that claim, yet there is no possible world where someone torturing infants just for fun isn’t doing something morally wrong. Incidentally, the necessity of at least some moral truths also helps make for an interesting moral argument for a necessary being grounding morality.
[1] For philosophy nerds: I’m basically claiming that ◊□p → □p, which is a theorem in S5modal logic. The proof:
 ◊□p conditional proof assumption
 ¬□p indirect proof assumption
 ¬¬◊¬p 2, □equivalence
 ◊¬p 3, double negation
 ◊¬p 4, S5reiteration
 □◊¬p 5, necessity introduction
 ¬◊¬◊¬p 6, □equivalence
 ¬◊□p 7, □equivalence
 ◊□p ∧ ¬◊□p 1, 8, conjunction
 □p 29, indirect proof
□ 

 ◊□p → □p 110, conditional proof