James N. Anderson and Greg Welty have published a paper entitled The Lord of Non-Contradiction: An Argument for God from Logic. Having worked out similar arguments in unpublished manuscripts, I am very sympathetic to the project of arguing from the existence of necessary truths to the necessary existence of divine mind.
Here is a quick sketch of the Anderson-Welty argument as I construe it:
1. There are laws of logic, e.g., the law of non-contradiction.
2. The laws of logic are truths.
3. The laws of logic are necessary truths.
4. A truth is a true proposition, where propositions are the primary truth-bearers or primary vehicles of the truth values.
5. Propositions exist. Argument: there are truths (from 1, 2); a truth is a true proposition (3); if an item has a property such as the property of being true, then it exists. Ergo, propositions exist.
6. Necessarily true propositions necessarily exist. For if a proposition has the property of being true in every possible world, then it exists in every possible world. Remark: in play here are 'Fregean' as opposed to 'Russellian' propositions. See here for an explanation of the distinction as I see it. If the proposition expressed by 'Socrates is Socrates' is Russellian, then it has Socrates himself, warts and all, as a constituent. But then, though the proposition is in some sense necessarily true, being a truth of logic, it is surely not necessarily existent.
7. Propositions are not physical entities. This is because no physical entity such as a string of marks on paper could be a primary truth-bearer. A string of marks, if true, is true only derivatively or secondarily, only insofar as as it expresses a proposition.
8. Propositions are intrinsically intentional. (This is explained in the post which is the warm-up to the present one.)
9. The laws of logic are necessarily existent, nonphysical, intrinsically intentional entities.
10. Thoughts are intrinsically intentional.
The argument now takes a very interesting turn. If propositions are intrinsically intentional, and thoughts are as well, might it be that propositions are thoughts?
The following invalid syllogism must be avoided: "Every proposition is intrinsically intentional; every thought is intrinsically intentional; ergo, every proposition is a thought." This argument is an instance of the fallacy of undistributed middle, and of course the authors argue in no such way. They instead raise the question whether it is parsimonious to admit into our ontology two distinct categories of intrinsically intentional item, one mental, the other non-mental. Their claim is that the principle of parsimony "demands" that propositions be constued as mental items, as thoughts. Therefore
11. Propositions are thoughts.
12. Some propositions (the law of logic among them) are necessarily existent thoughts. (From 8, 9, 10, 11)
13. Necessarily, thoughts are thoughts of a thinker.
14. The laws of logic are the thoughts of a necessarily existent thinker, and "this all men call God." (Aquinas)
A Stab at Critique
Line (11) is the crucial sub-conclusion. The whole argument hinges on it. Changing the metaphor, here is where I insert my critical blade, and take my stab. I count three views.
A. There are propositions and there are thoughts and both are intrinsically intentional.
B. Propositions reduce to thoughts.
C. Thoughts reduce to propositions.
Now do considerations of parsimony speak against (A)? We are enjoined not to multiply entities (or rather types of entity) praeter necessitatem. That is, we ought not posit more types of entity than we need for explanatory purposes. This is not the same as saying that we ought to prefer ontologies with fewer categories. Suppose we are comparing an n category ontology with an n + 1 category ontology. Parsimony does not instruct us to take the n category ontology. It instructs us to take the n category ontology only if it is explanatorily adequate, only if it explains all the relevant data but without the additional posit. Well, do we need propositions in addition to thoughts for explanatory purposes? It is plausible to say yes because there are (infinitely) many propositions that no one has ever thought of or about. Arithmetic alone supplies plenty of examples. Of course, if God exists, then there are no unthought propositions. But the existence of God is precisely what is at issue. So we cannot assume it. But if we don't assume it, then we have a pretty good reason to distinguish propositions and thoughts as two different sorts of intrinsically intentional entity given that we already have reason to posit thoughts and propositions.
So my first critical point is that the principle of parsimony is too frail a reed with which to support the reduction of propositions to thoughts. Parsimony needs to be beefed-up with other considerations, e.g., an argument to show why an abstract object could not be intrinsically intentional.
My second critical point is this. Why not countenance (C), the reduction of thoughts to propositions? It could be like this. There are all the (Fregean) propostions there might have been, hanging out in Frege's Third Reich (Popper's world 3). The thought that 7 + 5 = 12 is not a state of an individul thinker; there are no individual thinkers, no selves, no egos. The thought is just the Fregean proposition's temporary and contingent exemplification of the monadic property, Pre-Personal Awareness or Bewusst-sein. Now I don't have time to develop this suggestion which has elements of Natorp and Butchvarov, and in any case it is not my view.
All I am saying is that (C) needs excluding. Otherwise we don't have a good reason to plump for (B).
My conclusion? The Anderson-Welty argument, though fascinating and competently articulated, is not rationally compelling. Rationally acceptable, but not rationally compelling. Acceptable, because the premises are plausible and the reasoning is correct. Not compelling, because one could resist it without quitting the precincts of reasonableness.
To theists, I say: go on being theists. You are better off being a theist than not being one. Your position is rationally defensible and the alternatives are rationally rejectable. But don't fancy that you can prove the existence of God or the opposite. In the end you must decide how you will live and what you will believe.