Saturday, 15 March 2014

Anderson and Welty reply!

Transcendental arguers for the existence of God are notorious for not actually putting forward much of an argument. So I was delighted when a scholarly article by James Anderson and Greg Welty (The Lord of Non-Contradiction: An Argumentfor God from Logic) appeared in the Evangelical Philosophical Society’s journal Philosophia Christi. So excited, in fact, that I submitted a short note for inclusion in the very next issue.
It wasn't accepted.
Oh well.
But now I find that not only was my note featured on the Evangelical Philosophical Society’s website but James and Greg have written a response (to mine and others’), also featured on the EPS website!
The original paper is here, my note is here, and the reply is here.
Right. Time to get a little more formal; switch from "James and Greg" to "Anderson and Welty" and so on:
My note refers to a ‘key lemma’ and Anderson and Welty’s reply focuses on this key lemma together with my arguments surrounding it. As this key lemma seems to be the crux of our disagreement it is worthwhile setting out why I think it key.
“The Lord of Non-Contradiction: An Argument for God from Logic” begins with the, largely, uncontroversial claim that the laws of logic are necessarily true. By the end of the paper we are invited to accept that the laws of logic are also necessarily true thoughts. There is more content in the latter than the former, content that can be argued to be contingent on God. My contention is that is mainly the claim
“(s)ince [the laws of logic] are true in every possible world, they must exist in every possible world.” (Anderson & Welty, 2011)
that allows the introduction of the excess content. Thus, it is key. (But it is not the argument, it is a step on the way, a lemma; if you will a “staging post”.)
Existence can be taken to imply possession of properties, natures, essential natures and the like together with various assumptions about what else is needed for these properties to subsist in. Or "being true" can be taken as a property in itself, entirely sufficient to establish existence of the laws of logic whether or not any other properties are held; in particular whether or not they are thought.
This is the equivocation: before the lemma discussion is limited to being true whether or not thought, after the lemma discussion surrounds a concept of "exists" that requires thought. Thought then entails a thinker, necessary thought a necessary thinker and then to God.  On the “left” we have “thought”, on the right we have “truth and thought”.  My note explored possible ways of making the link (“if thought and truth then thought” is valid, “if truth then truth and thought” is not).  Unfortunately the valid destroy Anderson and Welty’s argument whilst those that support it are invalid. This is as we should expect:
“That the laws of logic are necessarily true entails that they are true whether or not God exists.” (Lloyd, 2012)
Anderson and Welty, in their reply, claim that I offered no argument for that claim. One seems, frankly, entirely redundant. “Necessary” is “not contingent” and “P is not contingent on Q” is equivalent to “there does not exist a Q such that P is contingent on Q”: there is no need to list all the infinite non-existent Qs that P is contingent on or the existent Qs that P is not contingent on.
A reference to “impossible worlds” is then made charging that “necessary” does not entail existence in impossible worlds. I fail to see the point of this objection. Do they mean to suggest that a world without God is an impossible world? If so then (if the laws of logic are necessary) there is no possible world where God does not exist and the laws of logic do not exist.  In a footnote, presumably intended to be explanatory, they invite comparison with “(i)f the proposition God exists is necessarily true then it is true whether or not God exists” (Anderson & Welty, 2013) as if there is something problematical with this statement. What does “P whether or not Q” mean? Its logical meaning is, simply, “P”.  If we wish to emphasise the non dependence on Q: “if Q then P and if not Q then P”. Anderson and Welty’s “problematic” comparator, then, can be written:
□G entails □ (G®G) and □ (¬G®G)
"G ®G” , together with "G", entails "G". “¬G ®G” is just a long-winded way of saying “G”, as the truth table shows:

So the statement simplifies to "□G".
A further claim is that I “presuppose(d) that the laws of logic are not ontologically dependent on God”. There was, of course, no “presupposition”:“that P entails Q” is a statement of logical and linguistic analysis that takes no position, presupposed or otherwise, on the truth either of P or of Q.
(A reader interested in a proof of "that P is necessary entails P whether or not Q" is invited to consult the box at the foot of this post.)
At the end of my note I sketched out the idea that all arguments to God from logic are liable to fail. Naturally, anyone arguing that logic depends on God needs to argue that logic is contingent on God. There are plenty of people who are quite happy to accept logic's contingency, but on something much more mundane than God. Perhaps Paula thinks that the laws of logic are thought.  She is quite happy that they exist because there are beings, at least in this world, where there are minds, like Paula’s, that think them. Paula’s position is reasonable and the God-from-logic proponent requires necessity (having admitted contingency) in order to discomfit Paula’s position.  The God-from-logic proponent needs, so to speak, to place the laws logic in a world where Paula is not there to think them.  The laws of logic must be there, because they are necessary, but can’t (according to Paula) because they are contingent on her. The assumption of necessity per se though is not enough to save the argument.  Necessity removes the contingency and without contingency the laws of logic cannot be contingent on God.  So the laws of logic must be held both necessary and contingent.
Now P can be both necessary and contingent if the necessity is qualified. A proper subset of possible worlds is those possible worlds where the physical laws of our universe hold. The laws of physics could have been different, so there are possible worlds were they are different.  But we may limit our considerations to those possible worlds where the laws of physics do hold.  Something, such as the speed of light in a vacuum, which holds in all of these physically possible worlds, is physically necessary.  As the speed of light could have been different there is at least one ‘metaphysical’ world where it is.  Thus the speed of light in a vacuum is physically necessary and metaphysically contingent.
I do not think this helps the God-from-logic proponent. It rescues him from contradiction, but also rescues Paula.  She, herself, may adopt the position that the necessity of the laws of logic she has been presented with is a qualified necessity.  She may admit that the laws of logic are X-ly necessary and still maintain that they are Paula’s- mind-ly contingent.
There is much to do, should anyone wish to pursue it, in analysing the interrelations of qualified necessity.  On first sight it would seem that metaphysical contingencies can be physically necessary but metaphysical necessities cannot be physical contingencies.  What of other ways of qualifying necessity; ontological, logical, epistemological and the like?
I suspect that no argument for God from logic will succeed mostly because I suspect that no combination of qualifications will place Paula in a bind and not give her the very tools to free herself. 
There is also the feeling that logical necessity is the nearest a qualified necessity gets to necessity simpliciter.  And logical necessity appears to take a special role in the argument.  Take Anderson and Welty’s own summary of the qualification of the contingencies in their argument:
"The laws of logic are “contingent on God” only in the sense that they are metaphysically dependent on God’s existence, in precisely the way that God’s thoughts are metaphysically dependent on God’s existence. This doesn’t entail that the laws of logic exist contingently or are true contingently (where contingently is a modal operator equivalent to not necessarily)." (Anderson & Welty, 2013)

Note, in particular the status of truth.  The metaphysical contingency of the laws of logic does not mean that they are not necessarily true.  That does not sit well with:

“(O)ne can logically argue against God only if God exists” (Anderson & Welty, 2011)

Just what are Anderson and Welty to say to someone who argues against God using logic? “Yes, yes, it may be true that God does not exist but part of the logic you use to conclude that is ontologically suspect”?

Proof that □P entails □(Q®P) & □ (¬Q®P)
◊¬ (Q®P) v ◊¬ (¬Q®P)
Negation of the conclusion: “possibly not (Q®P) or possibly not (¬Q®P)”
To test the first option we need to open up a new possible world. As (by hypothesis) “possibly not (Q ® P)” there must be a world that has ¬(Q ® P)
¬(Q ® P)
And here it is.
5 is true (and thus (Q ® P) false) just when Q is true and...
P is false
But from our premise, P is in this world
Lines 7 and 8 contradict
Open up another world. As “¬(¬Q ® P)” is (by hypothesis) possible there must be a world that has ¬(¬Q ® P).This need not be the same world as before, so a new one is needed.
¬(¬Q ® P)
10 is true (and thus (¬Q ® P) false) just when ¬Q is true and...
P is false
But from our premise, P is in this world
Which, again, is a contradiction


Anderson, J. N. & Welty, G., 2011. The Lord of Non-Contradiction: An Argument for God from Logic. Philosophia Christi, 13(2).
Anderson, J. N. & Welty, G., 2013. In Defense of the Argument for God from Logic. [Online]
Available at:
[Accessed 15 3 2014].
Lloyd, T., 2012. An Equivocation in Anderson and Welty’s “Argument for God from Logic”. [Online]
Available at:
[Accessed 15 3 2014].