ARGUMENTS FOR MONOTHEISM 1752 No necessary propositions are existential – none entails that anything
actually exists.
3 All necessary propositions are tautological – they have a sense property
expressed in an All A is A or All AB is B form.
4 No necessary statements provide genuine information.
We can take these objections in pairs. Objection 1 tells us that all necessary
statements are of the form “if A then B” which neither asserts that A exists
nor that B exists. Objection 2 tells us that no necessary statement asserts that
anything exists. Both 1 and 2 are false, since There are prime numbers larger
than 17 and There is a successor to two obviously are necessary truths but do
assert that something exists. Objection 3 tells us that all necessary
statements are true by virtue of the meanings of the words they contain, like
“All uncles are uncles” and “Any aunt has a niece or nephew.” (It is allowed
that there be a rule to the effect that “A” means “B or C” so that a
proposition expressible in the form All A is A also is expressible in the form
All A is B or C.) Objection 4 tells us that necessary truths provide no genuine
information. But both If proposition P is necessarily true, then it is
necessarily true that P is necessarily true and If proposition P is necessarily
false, then it is necessarily true that P is necessarily false are necessary
truths, and they are not tautological and they do provide genuine
information. So both 3 and 4 are false.^7 None of these common objections
shows that the notion of logically necessary existence is incoherent.
The Ontological Argument
The Ontological Argument is an attempt to state a series of necessarily true
propositions which serve as premises that entail the conclusion God exists. A
successful argument of this sort would prove its conclusion to be necessarily
true – it would show that God has logically necessary existence. Were any of
the four objections just discussed to have succeeded, it would have
undermined the ontological argument.
In constructing perhaps the most interesting version of the ontological
argument,^8 we need some further definitions as follows:
Definition 3: Proposition P entails proposition Q if and only if P, but
not Q is a contradiction.
Definition 4: Proposition P is a maximal proposition if and only if, for
any proposition Q, either P entails Q or P entails not-Q
Definition 5: Each maximal proposition defines an entire possible
world.^9