The Liar Paradox and conjunction introduction

Let F be some property had by and only by the sequence of symbols:

1. The sequence of symbols with property F is not a sentence expressing a truth, and 2+2=4.

Then, (1) is nonsense. (For if it makes sense, then it is true if and only if it is not true.) Since sequences of symbols that are nonsense don't express propositions, and hence don't express truths:

2. The sequence of symbols with property F is not a sentence expressing a truth.

And we know:

3. 2+2=4.

Hence, (1), appearances to the contrary notwithstanding, is not a conjunction of (2) and (3). For if it were a conjunction of (2) and (3), it would make sense and be true.

In my previous post, I tried to create space for the idea that in natural language not every pair of sentences can be conjoined. The above argument extends this to sufficiently rich artificial languages, since the above case could be formulated in an artificial language.

We can keep classical logic rules such as:

4. If c is the conjunction of a and b, and a and b are both true, then c is true,

as long as we recognize that in many natural languages and some artificial languages there can be sentences a and b that have no genuine conjunction. They may have a syntactic conjunction—a sequence of symbols formed out of the symbols in a and b and the conjunction symbol—but the preceding post shows that a syntactic conjunction is not the same as a conjunction simpliciter. And (4) must be understood in terms of genuine conjunction, not merely syntactic conjunction. This modification of classical logic does, of course, screw up the meta-theory. It doesn't affect soundness results, but completeness results may be in trouble.

Conjunction and natural language

One of the least controversial rules of logic is conjunction-introduction: from the premises A and B, one infers the conjunction A&B. But now consider the following argument in ordinary English:

1. (Premise) The following claims are all false: 1=2, 2=3, 3=4.
2. (Premise) 5=5.
3. The following claims are all false: 1=2, 2=3, 3=4 and 5=5. (By conjunction-introduction)

Premises 1 and 2 are true, but the conclusion is false.

Of course what we should say is that (3) is not in fact a conjunction of (1) and (2). This shows an interesting thing: it's not that easy to define a conjunction syntactically in natural language. The obvious rule of taking sentences "A" and "B" and forming the sentence "A and B" fails (even if we bracket the fact that in written English we often need to adjust the capitalization of the first word of the second sentence, and adjust punctuation), as (3) shows. In logic/mathematics-influenced written English we can conjoin "A" with "B" by forming the sentence "(A) and (B)". But that's not grammatical ordinary English.

We can, of course, do some paraphrase. Thus, we can say one of the following:

4. 5=5 and the following claims are all false: 1=2, 2=3, 3=4.
5. It is false that 1=2, it is false that 2=3, it is false that 3=4, but 5=5.

But while (4) and (5) express propositions that are obviously logically equivalent to the conjunction of the propositions expressed by (1) and (2), it is not clear that they express the conjunction of the proposition expressed by (1) with the proposition expressed by (2). Moreover, it is not clear that we can in purely syntactic terms specify how to give such a paraphrase in every case.

This is another reason to think that logic within natural language is at least somewhat tricky. The neat distinction in artifical languages between syntactic and semantic properties is much harder to draw. The notion of a conjunction of two sentences may well have no syntactic characterization.

Moreover, there may be sentences that in ordinary English have no conjunction or that have no disjunction. This is because the order of operations in English is foggy. In spoken English, we can do something with tone of voice and emphasis, but it is clear that this cannot be made to work always. In particular, if A,B,C,D,E,F,G,H are ordinary English sentences, I doubt that there is an ordinary English equivalent to "((A or (B and C)) and D) or (E and ((F and G) or H))". Thus, at some point we will have a failure of forming conjunctions or a failure of forming disjunctions.

This is relevant to this post.

A recipe for counterexamples to the Hume-Edwards-Campbell Principle

The Hume-Edwards-Campbell (HEC) principle says that if you have a bunch of items, and each one is explained, then the whole bunch might be explained. In particular, any infinite regress might be a complete explanation. The Hume-Edwards principle replaces the "might" with "is". I've published counterexamples to the HEC before, but here is a cool recipe for generating counterexamples.

Let p₁,p₂,... be an explanatory regress of propositions, so p₂ explains p₁, p₃ explains p₂, and so on. Suppose (as might easily be the case) that there is some proposition q such that (a) q couldn't be self-explanatory, and (b) the p_i are all clearly completely explanatorily irrelevant to q. Now, let q_i=p_i&q. Then q₁,q₂,... are an infinite explanatory regress. But if q couldn't be self-explanatory, this regress can't be completely explanatory as it does nothing to advance the explanation of q.

I might have got the basic idea here from Dan Johnson. I can't remember. The counterexamples depend on the idea that if A explains B and Q is irrelevant, then A&Q explains B&Q. I am a bit less sure of that than when I started writing this post (which was quite a while ago).

Deep Thoughts XXXII

"[T]here is no such thing as being too good."[note 1]

[This is a counterpoint to this. Like it, it rests on ensuring a consistent context of evaluation.]

A curious fact about open future views

Here is a curious fact. Open futurists and fatalists both are committed to the claim that

1. It is not true that tomorrow I will freely choose what to eat for lunch.

But they agree for different reasons. Fatalists think this is so because they think no one will ever freely choose. Open futurists think this is so because whether I will freely choose what to eat for lunch depends on various future contingencies, such as whether I freely choose to stay up until 10 am tomorrow, and then sleep from 10 am to 4 pm tomorrow. It is only the closed futurist non-fatalist who gets to affirm that tomorrow I will freely choose what to eat for lunch.

In fact, just as fatalists are, open futurists may be committed to the claim that

2. It is not true that I will ever make any indeterministic free choices.

For surely whether I will ever make any indeterministic free choices depends on future contingencies, such as whether in the next moment God chooses to take me off to my eternal destiny and remove all indeterminism from my future (or, if one is an atheistic open futurist, whether quantum events rip me to shreds next moment).

Of course, the agreement between open futurists and fatalists only goes so far. Typical open futurists will say that in the past I made some free choices, while fatalists will deny that. Also, fatalists are going to replace the "not true" in (1) and (2) with "false", which some open futurists will resist. But there is still an irony that a view motivated by a desire to maintain one's freedom makes it impossible to say that one will make a free choice.

Closure for knowledge

Here is a closure principle I don't know a counterexample to:

If you know that that s is the conclusion of a sound argument and (non-aberrantly) therefore believe that s, then you know that s.

Propositional logic

As a hobby, from time to time I am thinking about the best way to think about logic, knowing full well that much of what I am thinking about duplicates what other people have done, but since it's just a hobby, that's fine. One of my convictions is that logic should be developed in such a way that it works not only with simple artificial languages like First Order Logic as the target language. Logic concerns truthbearers, and it should be developed in a way that is agnostic as what the truthbearers are—they might be sentences of First Order Logic, sentences of English or propositions.

More generally than truthbearers, logic concerns schemata. Schemata when operated on by sufficiently many quantifiers yield propositions (I think of a name as a kind of quantifier). I don't yet have a clear picture of how I want to characterize schemata.

So what do I have? Well, I have a pretty good picture of how to start thinking about propositional logic—the logic of truthbearers. Let B be the collection of all truthbearers. Let V={T,F} be the truth values. Let V⁰ be any set with only one element. Let Vⁿ be the set of all n-tuples of members of V; let Bⁿ be the set of all n-tuples of members of B. Let O_n be all functions from Vⁿ to V. Let O be the union of all the sets O_n. We call the members of O "truth-tables". Now, for any truth-table f in O_n, there is an (n+1)-ary relation C_f on B, and we read C_f(b₁,...,b_n,b_n+1) "as b_n+1 f-connects b₁,...,b_n". If there exists b₁,...,b_n such that b_n+1 f-connects b₁,...,b_n, then we say that b_n+1's main connective is f. A nice axiom to have, but one that I would ultimately want to do without, is:

1. Unique connective parseability: Each truthbearer has at most one main connective.

A related axiom, also too strong in my opinion, is:

2. Unique component parseability: For any truthbearer b and any truth-table f, there is at most one sequence of truthbearers that b f-connects.

If we're really lucky, we have:

3. Unique parseability: Both unique connective parseability and unique component parseability hold.

Many artificial languages, but certainly not English, satisfy:

4. Unique compositionality: For any truth-table f and any sequence b₁,...,b_n of truthbearers, there is at most one truthbearer that f-connects the sequence.

For instance, in English, there is more than one way of expressing a conjunction, and hence unique compositionality fails. Say that b* is a direct subtruthbearer of b provided that b f-connects some sequence containing b*. Say that b* is a subtruthbearer of b if and only if there is a finite chain of direct subtruthbearer relations from b* to b. Many nice collections of truthbearers will satisfy:

5. Well-foundedness: The subtruthbearer relation is a partial well-ordering (i.e., there are no infinite decreasing chains of subtruthbearers).

We can say that a truthbearer is atomic provided it has no subtruthbearers.

If f is a truth-table in O_n, say that the system is f-compositional provided that for any sequence b₁,...,b_n there is a truthbearer b_n+1 that f-connects the sequence. For instance, English is negation, finite-conjunction and finite-disjunction compositional, as can be seen by the fact that for any sentence, we can form a negation of it by prefixing with "It is not the case that", for any finite collection of sentences we can form a conjunction by prefixing with "All of the following are the case" and then stringing the sentences with appropriate punctuation, and a disjunction by prefixing with "At least one of the following is the case".

For a proof theory, we specify bunches of rules, such as standard introduction and elimination rules. We specify them using f-connectedness. For instance, if f is binary conjunction (i.e., the function that takes (T,T) to T and all other pairs in V² to F), then a reasonable rule says that at if at some point in a proof you have b₁ and b₂ accessible, then you may write down anything that f-connects them.

That's the start, on the side of syntax. The start on the side of semantics is straightforward. A truth assignment is a function v from truthbearers to V such that whenever b_n+1 f-connects b₁,...,b_n, then v(b_n+1)=f(v(b₁),...,v(b_n)) (if n=0 then v(b₁) is equal to the value of v at the unique point of V⁰).

One can now easily prove this: If well-foundedness and unique parseability hold, then any function from atomic truthbearers to V can be uniquely extended to a truth assignment.

But in the end I'd like to work with logics that aren't uniquely parseable. For instance, English probably isn't uniquely parseable. "The sky is blue and the sea is blue and snow is white" can be parsed as a ternary conjunction, but it may also be parsed as a binary conjunction of "The sky is blue and the sea is blue" and "Snow is white". A nice replacement semantic property is:

6. Weak (strong) truth-definability: Any function from atomic truthbearers to V can be (uniquely) extended to a truth assignment.

There are all sorts of other cool properties one can define. The payoff of all of that is that one will get to do logic with natural languages or directly at the level of propositions.

Like I said, I know a lot of this stuff has been worked out by real logicians (I think someone once even told me what real logicians call what I called "unique parseability"). But just as it's fun to build a telescope focuser yourself rather than buying one online, so too it's fun to develop logic rather than getting one from the library, as long as one is only doing this as a hobby.

Sceptical non-theism

Sceptical theists respond to the problem of evil by, very roughly, telling us that we can't say what kinds of worlds God is more likely to create. This is a very rough formulation, and perhaps not entirely fair, but I think it is defensible. After all, if we have no reason to think that the values we know are representative of the larger realm of value, then we are unable to say what kinds of worlds a being whose actions are entirely guided by correct value considerations is likely to create.

I defined sceptical theism in such a way that it does not entail theism. Thus, an atheist could be a sceptical theist, and in fact it's fair to say that Anthony Flew, before he became a theist and while he was pressing the claim that the God hypothesis has no empirical consequences, was committed to something like sceptical theism.

It seems likely—though this can be questioned—that the conjunction of theism with sceptical theism commits one to wide-spread scepticism. If we don't know what sorts of worlds God is more likely to create, and we think this world is one that God created, we really shouldn't trust induction, etc. This is good reason for theists not to opt for sceptical theism.

But there is another position to be considered: sceptical non-theism. Sceptical non-theism tells us that we can't say what kinds of worlds are more likely to exist if God doesn't exist.

Just as sceptical theism does not imply theism, so too sceptical non-theism does not imply non-theism. If one is both a non-theist and a sceptical non-theist, then one is pushed towards more general scepticism. But if one is a theist and a sceptical non-theist, then one can resist scepticism, it seems. So sceptical non-theism doesn't carry the danger to the theist that sceptical theism does. Moreover, sceptical non-theism answers the problem of evil just as well as sceptical theism does. The theist who is a sceptical non-theist can say: "Granted, it sure looks like these horrendous evils are very unlikely given theism. I grant you that! But I have no reason to think that they are any more likely given non-theism. And in order for these evils to be an argument against theism, they would have to be more likely given non-theism."

Furthermore, sceptical non-theism is at least as well motivated as sceptical theism. Consider the fact that non-theism is a disjunction of a large number of very different views, including: polytheism, kakotheism, pantheism and naturalism (I am tempted to include "open theism" here, too). Naturalism in turn is a disjunction of a large number of very different views, since any logically coherent all-encompassing physical theory gives rise to a version of naturalism. On any one of the disjuncts, it's hard to figure out what sort of world would likely exist. And it's hard to figure out which disjunct is more likely than which. Hence, figuring out what world would be more likely to exist if non-theism held seems to be an insuperable task. This remains true even if we replace "non-theism" with "naturalism".

I do not endorse sceptical non-theism. But I do endorse the thesis that it is a better move for theists than sceptical theism.

The first rule for the professor of higher faculties

From the Jesuit Ratio Studiorum of 1599:

It should be the set purpose of the teacher, both in his lectures as opportunity offers and on other occasions, to inspire his students to the love and service of God and to the practice of the virtues which He expects of them, for this is the sole purpose of all their activities.

Deep Thoughts XXXI

The adequate is good enough.

[If I wrote in a letter of recommendation that someone was an adequate candidate that would probably the kiss of death. But why? Putative explanation 1: It is a tautology that the adequate is good enough. However, it is only a tautology if the same context is held fixed throughout the sentence. Thus, someone might be adequate as a student, but not good enough for admission to a highly competitive program. Putative explanation 2: Letters of recommendation are overblown.]

No one but you yourself can reliably make you be evil

This argument for incompatibilism is inspired by an excellent paper by Patrick Todd that I just heard. I don't know if Patrick would endorse the argument I give, though.
Start with this principle:

1. No one but you yourself can reliably make you be evil.

Explanation is needed. Evil isn't just a bad here—it is a particularly serious kind of bad. Nor is being evil just a matter of having an evil character. For I could, through no fault of my own, be brainwashed into having an evil character, but that wouldn't make me be evil. If I were brainwashed into having a seriously bad character, my actions and my character would be worthy of condemnation, but I would be deserving of pity, and not the kind of condemnation that evil people deserve.
Now, people can cause you to be evil. For instance, they can present you with the temptation to do an evil, and if you fall prey to it your character becomes distorted and they have successfully caused you to be evil. But this temptation was either one that you were very likely to fall prey to or it is not on that you were very likely to fall prey to.
If it was a temptation that you were very likely to fall prey to, then you already had an evil character. For a character that makes one very likely to fall prey to a temptation to do an evil seems to be an evil character. So, the tempter did not make you have an evil character. This may seem to show that the tempter did not make you be evil, but that's not right—it leaves out one possibility, which is that previously you had an evil character but were not evil, because you were not sufficiently culpable for the evil character, but now that you've falled into this temptation, that made you be evil. But I think this can't be so. For if an evil character that you were not sufficiently culpable for made the evil action very likely, then you were not very culpable for the evil action—you were only somewhat culpable for it—and that isn't enough to make you be evil. So if the temptation was one that you were very likely to fall prey to, then either you were already evil, or else you lacked sufficient culpability, and you still lack sufficient culpability.
Oone might worry about boundary cases. Let's say you're pretty bad, but not quite evil. You're just one micromoriarty short of being evil. In that case, maybe, the tempter can produce a temptation such that very likely you'll take it, and then it'll push you over the edge, giving you that micromoriarty, and make you be evil. But I think it's not correct to say then that the tempter made you evil. The tempter only helped you a little bit—you already were almost all the way there.
On the other hand, suppose that the temptation was one you were not very likely to fall prey to. Then quite possibly you did become evil, but because you were not very likely to fall prey to the temptation, the tempter's method by which he made you evil wasn't a reliable method, and so we still do not have a counterexample.
One might have another worry about the argument. There may be some temptations that only a moral hero would have much chance at withstanding. Let's say that only a moral hero would have much chance at failing to betray her friend given some particularly nasty torture T. But we can imagine that if she betrays, she will acquire an evil character. So if you're not a moral hero, it will be very likely that the torture will make you betray, and hence will become evil. But I deny that in this case you will become evil—only your character will. The reason is that temptation that only a moral hero would have much chance at withstanding makes you only slightly culpable, and that isn't enough to make you be evil, unless you were already just a shade short of being evil.
What I said above about single temptations generalizes to long-term plans of temptation. If the plan is pretty sure to work, you already have an evil character, or else you don't become culpable when you fall prey to the plan.
Suppose that (1) is correct. Then we can have the following argument against compatibilism:

2. If compatibilism is true—i.e., if freedom is compatible with all one's mental life being determined—then by setting up appropriate environmental and genetic conditions, one can reliably make someone be evil.
3. Therefore, compatibilism is false. (By 1 and 2)

Epistemic self-sacrifice and prisoner's dilemma

In the fall, I attended a really neat talk by Patrick Grim which reported on several computer simulation experiments by Grim. Suppose you have a bunch of investigators who are each trying to find the maximum ("the solution to the problem") of some function. They search, but they also talk to one other. When someone they are in communication with finds a better option than their own, they have a certain probability of switching to that. The question is: How much communication should there be between investigators if we want the community as a whole to do well vis-a-vis the maximization problem?

Consider two models. On the Local Model (my terminology), the investigators are arranged around the circumference of a circle, and each talks only to her immediate neighbors. On the Internet Model (also my tendentious terminology), every investigator is in communication with every investigator. So, here's what you get. On both models, the investigators eventually communally converge on a solution. On the Internet Model, community opinion converges much faster than on the Local Model. But on the Internet Model the solution converged on is much more likely to be wrong (to be a local maximum rather than the global maximum).

So, here is a conclusion one might draw (which may not be the same as Grim's conclusion): If the task is satisficing or time is of the essence, the Internet Model may be better—we may need to get a decent working answer quickly for practical purposes, even if it's not the true one. But if the task is getting the true solution, it seems the Local Model is a better model for the community to adopt.

Suppose we're dealing with a problem where we really want the true solution, not solutions that are "good enough". This is more likely in more theoretical intellectual enterprises. Then the Local Model is epistemically better for the community. But what is epistemically better for the individual investigator?

Suppose that we have a certain hybrid of the Internet and Local Models. As in the Local Model, the investigators are arranged on a circle. Each investigator knows what every other investigator is up to. But the investigator has a bias in favor of her two neighbors over other investigators. Thus, she is more likely to switch her opinion to match that of her neighbors than to match that of the distant ones. There are two limiting cases: in one limiting case, the bias goes to zero, and we have the Internet Model. In the other limiting case, although she knows of the opinions of investigators who aren't her neighbors, she ignores it, and will never switch to it. This is the Parochial Model. The Parochial Model gives exactly the same predictions as the Local Model.

Thus, investigators' having an epistemic bias in favor of their neighbors can be good for the community. But such a bias can be bad for the individual investigator. Jane would be better off epistemically if she adopted the best solution currently available in the community. But if everybody always did that, then the community would be worse off epistemically with respect to eventually getting at the truth, since then we would have the Internet Model.

This suggests that we might well have the structure of a Prisoner's Dilemma. Everybody is better off epistemically if everybody has biases in favor of the local (and it need not be spatially local), but any individual would be better off defecting in favor of the best solution currently available. This suggests that epistemic self-sacrifice is called for by communal investigation: people ought not all adopt the best available solution—we need eccentrics investigating odd corners of the solution space, because the true solution may be there.

Of course, one could solve the problem like this. One keeps track of two solutions. One solution is the one that one comes to using the biased method and the other is the best one the community has so far. The one that one comes to using the biased method is the one that one's publications are based on. The best one the community has so far is the one that one's own personal opinion is tied to. The problem with this is that this kind of "double think" may be psychologically unworkable. It may be that investigation only works well when one is committed to one's solution.

If this double think doesn't work, this suggests that in some cases individual and group rationality could come apart. It is individually irrational to be intellectually eccentric, but good for the community that there be intellectual eccentrics.

My own pull is different in this case than in the classic non-epistemic Prisoner's Dilemma. In this case, I think one should individually go for individual rationality. One should not sacrifice oneself epistemically here by adopting biases. But in the classic Prisoner's Dilemma, one has very good reason to sacrifice oneself.

A dignity argument against most abortions

1. (Premise) If x has dignity, it is wrong to intentionally kill x primarily for the sake of a benefit to someone other than x.
2. (Premise) Elderly people whose minds are functioning very poorly have dignity.
3. (Premise) If elderly people whose minds are functioning very poorly have dignity, fetal humans have dignity.
4. Therefore, it is wrong to kill a fetal human primarily for the sake of a benefit to someone other than the fetus.

I think something broader than (1) is true—it's also wrong to kill an innocent for the sake of a benefit to x. But (1) will be less controversial. I think (3) is probably the most controversial premise. One argument for (3) is this:

5. (Premise) Dignity is either had by all humans or only by those who satisfy achievement-type conditions for personhood, such as being able to solve relatively sophisticated problems or communicate on a large variety of topics.
6. (Premise) Fetal humans are humans.
7. (Premise) If dignity is only had by those who satisfy achievement-type conditions for personhood, then elderly people whose minds are functioning very poorly do not have dignity.
8. Therefore, if elderly people whose minds are functioning very poorly have dignity, then dignity is had by all humans, and hence by fetal humans as well.

Alas, premise (2) may be somewhat controversial. Here is an argument for it.

9. (Premise) x can only suffer an indignity if x has dignity.
10. (Premise) Elderly people whose minds are functioning very poorly can suffer indignities (and often do).
11. Therefore, elderly people whose minds are functioning very poorly have dignity.

Horrendous evil and indignity

Many horrendous evils are horrendous largely because of horrendous indignity to the sufferer. Such "horrendous indignities" may seem to provide evidence against the existence of God. But on reflection, I think, they do not. For only a being with a dignity can suffer an indignity. It is no indignity for a rock to have mud poured over it. Making fun of a monkey does not harm the monkey. Moreover, only a being with great dignity can suffer a great indignity. Thus, that some beings suffer horrendous indignities entails that these beings have great dignity.

The evidence that many suffer horrendous indignities thus tells us that:

1. There are many finite beings with great dignity,
2. who suffer great indignities.

Now, is this better predicted by theism or by naturalism? Theism predicts (1), though it may not predict (2). Naturalism does not predict (1)—it could even be that naturalism is incompatible with (1) since perhaps great dignity requires being in the image of God; I do not know what it says about (2) (even conditionally on (1)). In any case, the existence of horrendous indignities is also a problem for naturalism, because the existence of beings of great dignity is a problem for naturalism.

Let us step back and ask if (2) is usually such a great problem given theism. We need distinguish "o-dignity", the innate "ontological" dignity of a being, from the "m-dignity" which is a manifestation of o-dignity. In (1) we are talking of o-dignity. Indignity, however, is not the opposite of o-dignity, but of m-dignity. The person who suffers an indignity still has o-dignity—if she didn't have it any more, she wouldn't be suffering an indignity, just as a rock cannot suffer an indignity because it lacks o-dignity. (One's pride can only be hurt when one has pride; otherwise, at worst one's former pride is hurt, and that's not a present hurt.) Thus manifest indignity as such highlights the o-dignity of the being suffering from the indignity. Only the evidently holy can be manifestly blasphemed.

In other words, manifest indignity is a kind of m-dignity. Manifest indignities are self-defeating—they highlight the dignity of that which they demean. (This may remind one of Hegel's master-slave dialectic and of the mockeries Christ suffered.) But horrendous indignities tend to be manifest. As such they paradoxically conduce to the manifestation of o-dignity, and hence there is reason for God to allow them to occur.

True kingship is most manifest when stripped, on the cross and with the side pierced.

Compatibilism and classical theism

Standard definitions of compatibilism say something like:

1. Compatibilism holds if and only if there is a world where there is determinism and at least one exercise of free will.

Standard definitions of determinism say something like:

2. Determinism holds at w if and only if for any time t in the history of w, and any world w* such that (a) the laws of w hold at w* and (b) w* exactly matches w at t, the worlds w and w* exactly match in the future of t.

If we define compatibilism and determinism as above, then classical theism entails compatibilism. According to classical theism, God is outside time, free and omnipotent. God could, then, create a world at which determinism holds, since determinism only concerns the beings that are in time, and hence the determinism would not apply to him. (A tricky issue is: Couldn't God always produce miracles, thereby making the right hand side of (2) be unsatisfied? Well, we could imagine that in the deterministic world, at every time, there is heard a divine promise to do no miracles.) And if he did this, then that would be a world where there is determinism and at least one exercise of free will—namely, God's exercise of free will.
I think this only shows that the standard definitions of (in)compatibilism are wrong. Instead of saying that determinism is incompatible with freedom, they should say that determinism is incompatible with beings like us having freedom.