Valid but not in Virtue of Form?

Some remarks of the Semantic Terrorist in the post on moral truthmakers got me to thinking. Here is what he said.

consider the following argument which is easily proven to be invalid despite the fact that many analytic philosophers would mistakenly classify it as valid:

1) George Bush is a bachelor.
? George Bush is unmarried.

This argument is in the same logical form as the following:

1) George Bush is a Texan.
? George Bush is unmarried.

As ST points out many analytic philosophers do take that argument to valid. It is standard in logic textbooks to point out that these arguments meet the definition of validity; it is indeed impossible for the premise to be true and the conclusion to be false, but as the point continues, this isn’t because of teh form of the argument. The form, as ST demonstrates, allows for counter-examples, and so the validity must be due to something besides the form of the argument. The reason for the impossibility of the truth of the poremises and the conjunction of the denial of the concusion is said to be due to the definition of ‘bachelor’.

All of this is standard, but why isn’t the argument above seen as having a supressed premise of the form ‘all bachelors are unmarried makes’? Then the argument is formally valid; it is just an instance of a very common categorical syllogism. The same is true of the texan argument, it just happens to have a false suppressed premise ‘all Texans are unmarried males’. I don’t see what the argument against positing the suppressed premise is supposed to be. It is clearly the only way to make the inference legitimate.

Did Quine Change his Mind?

It is well-known that Quine argued that the axioms of logic are revisable. The law of the excluded middle, for instance, while at the center of our ‘web of beliefs’ could, if we had compelling evidence, be revised or even abandoned. But it is commonly thought that Quine changed his mind by the time that he wrote his Philosophy of Logic in 1970. But is this right?

What people seem to have in mind is the passage in chapter 6 on deviant logics where he says, in considering the debate between someone who denies the law of non-contradiction and someone who rejects this denial,

My view of this dialogue us that neither party knows what he is talking about. They think they are talking about negation, ‘~’, ‘not’; but surely the notation ceased to be recognizable as negation when they took to regarding some conjunctions of the form ‘p & ~p’ as true, and stopped regarding such sentences as implying all others. Here, evidently, is the deviant logician’s predicament: when he tries to deny the doctrine he only changes the subject. (p 81)

The idea here is supposed to be that it is impossible to really reject the law of non-contradiction as opposed to simply changing the subject. But this doesn’t mean that the law of non-contradiction isn’t revisable, it simply means that arguments between those who are pro-revision and those who are Conservatives will very often be question begging. Quine goes on to say as much when discussing the law of the excluded middle,

 By the reasoning of a couple of pages back, whoever denies the law of the excluded middle changes the subject. This is not to say that he is wrong in doing so. In repudiating ‘p or ~p’ he is indeed giving up classical negation, or perhaps alteration, or both; and he may have his reasons. (p 83)

He then goes on to canvass the reasons that have been given, which range “from bad to better”. But ultimately Quine rejects them as sufficient to motivate us to abandon classical logic. He appeals to something he calls the ‘maxim of minimal mutilation’, as he says,

The classical logic of truth functions and quantification is free of paradox, and incidentally is a paragon of clarity, elegance, and efficiency. The paradoxes emerge only with set theory and semantics. Let us try to resolve them within set theory and semantics, and not lay fairer fields to waste. (p 85)

He goes on to cite it again in response to the challenge from quantum mechanics,

But in any event let us not underestimate the price of deviant logic. There is a serious loss of simplicity, especially when the new logic is not even a many-valued truth functional logic. And there is a loss, still more serious, on the score of familiarity. Consider again the case, a page or so back, of begging the question in an attempt to defend classical negation. This only begins to illustrate the handicap of having to think within deviant logic. The price is perhaps not prohibitive, but the returns had better be good. (p 86)

It seems clear from this that Quine is not retracting his claim that classical logic is revisable but is instead canvassing the reasons that one may have for such a revision and arguing that we, as of yet, do not have enough reason to abandon classical logic. This is entirely consistent with his views and so we can conclude that he did not change his mind about the revisability of logic.

Top 10 Posts of 2008

OK, so the year isn’t over yet…but these are the most view posts so far…

–Runner up– Reverse Zombies, Dualism, and Reduction

10. Question Begging Thought Experiments

9. Ontological Arguments

8. The Inconceivability of Zombies

7. There’s Something About Jerry 

6. Pain Asymbolia and Higher-Order Theories of consciousness

5.  Philosophical Trends

4. A Short Argument that there is no God

3. Has Idealism Been Refuted?

2. God versus the Delayed Choice Quantuum Eraser

1. A Simple Argument Against Berkeley

Chappell on the A Priori

The a priori seems to be on the rise of late, especially with defenders like Richard Chappell championing the cause. According to Chappell an ideallly rational being would have access to all the metaphysical possibilites. Given that we can ideally (or coherently)  conceive something we can infer that the thing in question is metaphysically possible. This is, of course, the basis for the zombie argument against materialism. Since we can coherently concieve of a zombie world (a world where there are beings like us in every physical way except that they lack conscious experience) that shows that consciousness cannot be a physical property.

The standard (Kripkean) objection to this line of argument is to try to distinguish between metaphysical and epistemic possibility. Some things that are epistemically possible (i.e. seem coherently conceivable) turn out to be impossible (a classic example is to point out that before you learn that the square root of 1,987,690.000 is 1409.855 (rounded up to the nearest thousandth) it is concievable that it be other than 1409.855 but once we find out what it is it is impossible for it to be otherwise. According to the materialist one of these things is the zombie world. While it seems that we can coherently concieve of such a world, we are actually missing some contradiction, or physical difference between our world and the zombie world and so it is not actually (ideally/coherently) concievable. 

Chappell objects to this line of argument for (at least) two reasons. The first has to do with the theoretical extravagance of the materialist’s claim that the identity between (say) H2O and water is necessary. It posits an unexplained strong necessity, wheras the modal rationalist (the one who thinks that it is a metaphysical possibility that water could be other than H2O, not just an epistemic possibility) doesn’t have to posit something like this. All that she needs to posit is a single uniform space of possibilities that we describe in various ways. The materialist has to posit a space of epistemically possible worlds and a seperate space of metaphysically possible worlds. Parsimony and simplicity seem to favore that modal rationalist here.

The second is an attack on the claim that calling something a rigid designator settles the dispute. As Chappell says,

Perhaps our term ‘consciousness’ is, like ‘water’, a rigid designator. But who cares about the words? Twin Earth still contains watery stuff, even if we refuse to call it ‘water’, and the Zombie World still lacks phenomenal stuff (qualia), even if we stipulate that our term ‘consciousness’ refers to some neurophysical property (and so is guaranteed to exist in this physically identical world).

Yes it will, IF we have settled the issue in favor of Chapell’s view and we then think that we are genuinely concieving of a real metaphysical possibility. If there is a question as to whether these kinds of possibility are distinct then Chapell has done nothing more than beg the question.

This is evidenced when he says,

Kripke himself noticed something along these lines. While we can imagine a world where watery stuff isn’t truly water, it’s incoherent to imagine a world where “painy” stuff isn’t truly pain. To feel painful is to be painful.

Pointing out that Kripke begs the same queston as you are beging is not a way to absolve yourself of beging the question. There is a legitimate case to made that being in pain and feeling pain are in fact two seperate things. The evidence for this comes, not from a priori reflection on the nature of pain, but from evidence from cognitive science.

 But suppose that you are not moved by this evidence and you still maintain that a priori analysis reveals that the zombie world is metaphysically (not just epistemically) possible. Is this a coherent position? One objection that immediately pops up is that on this view it seems that we can concieve of various possible worlds that result in contradiction. So, I seem to be able to concieve that God necessarily exists and that God necessarily doesn’t exist (or that numbers do and don’t necessarily exist). Since the claim that conceiveability entails possibility entails that God (or numbers) both necessarily exists and doesn’t exist only one of those possibilities can be a real metaphysical possibility; the other must be an epistemic possibility.

Chappell is of course aware of this objection and tries to deal with it in the post linked to above. Here is what he says,

I agree with Chalmers that the most attractive response for the modal rationalist here is to hold on to their strong position, and instead deny the… conceivability intuitions found, for example,…above. It isn’t at all clear that a necessary being, or a shrunken modal space, is coherently conceivable in the appropriate sense. The modal rationalist will want to hold that their position is not just true, but a priori. They would then expect opposing views to be refutable a priori, and hence not feature in any a priori coherent scenario. Of course, it would beg the question to merely assert: “the thesis is true and hence has no successful counterexamples”. But that is not what’s going on here. Rather, I hope to show that the modal rationalist can explicate their commitments in a way which makes clear exactly why, on their view, the meta-modal cases in question are not taken to be genuinely conceivable. If successful, this should suffice to undermine the charge of internal inconsistency or self-refutation.

The problem with this line of argument is that it commits the very ‘fallacy’ that Chappell accuses the Kripkeans of making. The strategy that he is here proposing is that of trying to show that there is some possible state of affairs that seems conceivable but which, on reflection, is not in fact metaphysically possible (i.e. that there are possibilities that (seem)concievable but are not metaphysically possible). But if there are possibilities that (seem) concievable but not metaphysically possible then we need an independent argument that the zombie world is not one of these worlds. No such argument has been given. Rather what Chappell does is to assume that it is in fact coherently concievable; but this cannot be assumed if there are any possibilities which (seem) concievable and are not metaphysically possible. Chappell’s own view commits him to there being such possibilites, so by his own view the modal argument against materialism is suspect.

New Classes at LaGuardia

I am lucky enough to come to LaGuardia at a time when they are expanding the philosophy major and we are trying to introduce four new classes to the curriculum. I am responsible for designing two of them; Logic and Philosophy and Medical Ethics (the other two are Aesthetics and Environmental Ethics). I thought I would post the course descriptions and outlines in the hopes of getting some feedback from any LaGuardia students lurking around here on whether or not classes like these sound interesting and would be something you might consider taking if it were offered.

Logic and Philosophy

Course Description: An introduction to modern symbolic logic with a focus on its application to actual philosophical problems. Topics to be discussed include validty, entailment, truth-tables, proofs, translations from English into symbolic form, as well as more philosophical topics like the relation of modern logic to earlier syllogistic logic, the possibility of the use of logic to resolve philosophical problems (e.g. God’s existence or free will), the relation of English to logic, and the possibility of ‘alternative’ logics.

Course Outline

1. Validity & soundness
–Logic and the philosophical method.
–Entailment, inference, and validity.
–Aristotle’s identification of validity with the form of the argument.
–The seperation of validity (formal structure) and soundness (truth of premises).
–The counter-example method of testing validity

2. Syllogistic Logic
–The square of opposition and the cannonical A, E, I, and O sentence forms.
–Categories and Venn diagrams.
–The mood and forms of the valid syllogisms.

3. Philosophical issues in syllogistic logic
–Does ‘all’ imply ‘some’?
–Are some logical truths known by reason alone, independently of experience?
–some arguments cannot be expressed in syllogistic logic.

4. Basic Propositional logic I
–Beginning definitions of formal symbols for ‘and’, ‘or’, ‘not’ and ‘if…then’.
–Simple translations into symbols.
–The truth-table test for validity.

5. Basic propositional logic II
–More advanced translations.
–Introduction of rules for symbol manipulation.

6. Propositional proofs
–Introduction to natural deduction.
–Introduction to truth-trees.
7. Philosophical issues in propositional logic
–paradoxes of material implication.
–why accept valid inferences?

8. Basic quantificational logic
–Introduction of ‘all’ and ‘some’ into the formal language.
–Translations and proofs.

9. Identity and relations
–Introduction of identity into the formal language.
-Introduction of relational predicates (e.g. ‘taller than’).

10. Philosophical issues in quantificational logic.
–Is existence a predicate?
–Do mathematical truths reduce to logical truths?
–Treating names as descriptions.
–Informative identity statements.

11. Basic modal logic
–Introduction of ‘necessary’ and ‘possible’ into the formal language.
–Introduction to possible world semantics.
–translations and proofs.

12. Philosophical issues in modal logic
–The metaphysical status of possible worlds.
–one logic, or many?
–Names and rigid designators.
–different concepts of possibility: Epistemic, metaphysical, and logical.

13 Final Exam

Medical Ethics

Course Description:An introduction to some of the basic issues in medical ethics. The course emphasizes the application of moral theory to the issues that arise in the context of medical research and practice. Topics to be addressed include, among others, the role and responsibility of heathcare givers in death and dying, the use of stem cells and animals in medical research, the use of genetic information to influence the outcome of human pregnancy, cosmetic surgical addiction, and issues involving involuntary psychiatric care.

Course Outline

1. Review of basic ethical theories
–Virtue ethics.

2. Killing those who can’t speak for themselves
–Active vs. passive Euthanasia.
–patients that can’t make their own decisions.
–defective infants.

3. Physician-assisted suicide

4. Ethical issues in reproductive science
–surrogate motherhood.
–fertility treatments.
–Over population.

5. The use of human embryos in scientific research

6. Elective cosmetic surgery and surgical addiction
7. The use of animals in scientific research

8. Issues involving justice and the allocation of medical resources
–transplants and alchoholics.
–Transplants and the black market.
–Expensive treatments.

9. Involuntary psychiatric care

10. Issues in genomics (genetic counseling/genetic engeneering)

11. Universal heathcare

12. Issues involving HIV/AIDS

13 Final Exam

Where Am I?

I’m back!!

 The plane ride there was long and super bumpy (and I hate flying!!) and then I got strep throat and the plane ride back was a red eye that got into JFK at six a.m. (and I REALLY hate flying!!!!)…but other than that California was fantastic! 🙂

The APA was fun, though I got there on the last day of the conference and since I wasn’t feeling well (I was chaining Sucrets one after the other) I left after my talk. But I did see the session before mine, by Hanna Kim, on a proposed compositional semantics for metaphors which was interesting. She sketched an account that borrowed Jason Stanely’s idea of a hidden unarticulated variable that was context sensitive to metaphorical meaning. This would allow one to get the meaning of the metaphor in a way that was completely determined by the meaning of the parts (including the hidden, context sensitive variable). Marga Reimer responded with a couple of objections. One of which was the Gricean kind of objection one would expect. She invoked Grice’s modified Occam’s razor and asked why we need a semantic account of metaphor’s when we have a perfectly good account from Grice that appeals to speaker’s intentions and doesn’t posit all of these weird hidden variables? (Here! Here!) Kim’s answer, in part, was to point out that Grice’s account cannot take care of ‘impossible metaphors”.  The basic idea behind impossible metaphors is that there are semantic and syntactic constraints on what kinds of sentences we can make metaphors from. I don’t recall any of her examples and I can’t find the handout…but still, I wonder about this kind of strategy. Why is an objection to Gricean theories to point out that sentence construction is constrained by syntax? A speaker is constrained by what she can reasonably assume will alert a hearer to her communicative intention and thereby fulfil that very intention. The syntax of a language is definitely one thing that would suggest itself as something which would constrain which utterances a speaker can reasonably expect a hearer to successfully infer what one is communicating. No problem.

My talk went well, I think. We had some interesting discussion. The commentator (Imogen Dickie) posed a dilemma for me. If we can have rigid designation in thought then either the problem of necessary existence reoccurs at that level and we haven’t solved the problem or we can have rigid designation without the problem of necessary existence (in thought) and so we shouldn’t be worried about it in language. This is especially pressing when we think that S5 is attractive because it is supposed to be a logic for thought.     I responded that the problem of necessary existence is only a problem when we try to regiment our thoughts into a formal language. There is no problem with having a singular thought about Socrates, the problem is trying to formalize a sentence representing that thought. This is the evidence that we have that we need an separate account of the semantics of language. But S5 is still a logic of modal thought because we can formulate descriptions in it that ‘single out’ the object of thought without rigid designators. The absence of singular terms in our logic is nothing more than an inconvenience. She also mentioned, in passing, that Williamson thinks that necessary existence is not as terrible as one might think. One might argue that I exist in all possible worlds but in some worlds I exist without any properties. This was quite shocking to me, as I can’t really fathom what that would mean. Really, what does that mean? Anyone know?

From the audience I was asked several good questions. One was from Tim Lewis on how I felt about the fact that names on my account would fail the Church translation test. That is, we expect that ‘Richard’ and ‘Ricardo’ to be synonyms but if the really stand for ‘the bearer of “Richard”‘ and ‘The bearer of “Ricardo”‘ then pretty clearly they aren’t synonyms since they each have a separate quoted name in them. I thought that was a pretty nice objection. At the time I said that I would argue that names are not part of a language. So, in a complete dictionary of English there would be no ‘Richard’ or ‘Doug’ (forget about the dictionaries around now, they are half encyclopedia, I am talking about just a list of the words of a language and their conventional meanings, pronunciation guide, and syntactical/grammatical categories. That seems right to me, but then on the plane home, in a half trance, I started to think that maybe we could use Seller’s notion of ‘dot quotes’ to solve the problem if people don’t like the position on names. So instead of ‘the bearer of “Richard”‘ we could have ‘The bearer of *Richard*’ where ‘*P*’ is ‘dot-quote P’ and basically serves to single out all of the functional types that play the role that ‘Richard’ does in English. This would allow one to preserve the intuition that other language cognates of English names are synonyms. Or so it seemed on the plane…and besides I like the bit about names not being part of the language…

The other question that I remember was from Adam Sennet (there were a couple of others that I am forgetting). He echoed Williamson’s point that since we know quite well what a rigid designator is and how one would introduce them into a formal langauage it is then quite odd to say that there aren’t any. I responded that we know what it would be like for there to be all kinds of things that don’t exist. I know what it be like for there to be square circles (it would be for there to be one object that is both square anc circular at the same time), but that doesn’t mean that there are any. This is exactly what one would expect. We know what it would be like for there to be flogisten or tachyons or any other theoretical posit we come up with. It would be like finding the thing that we posited, but someimes we find out that they don’t exist. Interpreting that syntactical category proper noun as a rigid designator is a natural attempt at capturing what it is that we do when we think about some particular thing but when we do model that category that way we get the problems with necessary existence, which means that it is a mistake to model it in that way. I compared it to what happens when we try to mix quantuum theory with relativity theory. When we try to calculate the probabilities for things which we have well worked out answers for we get crazy results (like the probability of some event occuring being infinite). This let’s us know that there is a problem and then you get all of the different answers to solve the problem. Our finding the proofs for necessary existence in S5 are like the infinite probabilities in physics; it is an indicator that something needs to be done.

This is, by the way, why I disagree with Chappell’s charge that logic is over rated and that, in particular, my

employed logical apparatus merely serves to build in misunderstandings. The formal steps of the argument may be flawless, but that’s all for naught if the entire argument is based on a mistake — due to failing to understand precisely what all those formalisms really mean.

I understand what the formalisms mean and I am using them to apply pressure to a person who holds a certain kind of view. The proofs count as evidence that some assumptions don’t work. This is exactly what formal logic is good for…though I do agree that one needs to also make the argument in prose as well as symbols.

OK, well that’s enough for now, I gotta get to work on my Tucson presentation and grade some exams!!!!!!!

Language, Thought, Logic, and Existence

Well, I’m off to go present my paper at the APA! I’ll be back on Monday. I guess I have Philosophy Sucks! to thank, since I was noticing that the paper grew out of some interesting discussion I had here last year. Thanks to everyone who participated!!

You can enjoy the virtual version here (and on the sidebar with the other virtual presentations), which is a recording of a rehersal I did today (It may take a second to open since I recorded it in stereo, which I haven’t before).

More on the Ontological Argument

The traditional version of the ontological argument is usually criticized for treating existence as a predicate. If existence is a predicate, then it is a predicate that always applies, and as Russel quipped that is the sign of a mistake. A predicate must be fail to apply to some objects in order to count as expressing a genuine property.  But Kripke has shown that it is easy to introduce an existence predicate into our formal language that avoides this and related difficulties. We do so as (1),

(1) E(y) (y=x)

Where ‘E(y)’ is the standard existential quantifier and identity is understood normally.(1) is an existence predicate because it is an open sentence that can be satisfied by the values of the free variable x. Intuitively (1) says that x has the property of being identical to some thing (y) and this captures what we mean when we say that existence is a property.

(1) can fail to apply, the model is very easy to give. Imagine a universe of two distinct objects A and B. Now, say that there are two distinct worlds in this universe one containing only A and the other containing only B. E(y) (y=B) will be false at the world where only A  exists.

As Kripke points out the problem only arises when one mistakenly thinks that the claim that existence is a predicate is the claim that (2) makes

(2) (x) (E(y) (y=x))

(2) says that every thing exists and this cannot fail to apply and is a necessary truth, but it is also not a predicate (it is not an open sentence, it is closed by the universal quantifier ‘(x)’).

So if existence is a property then it makes sense to think it is the kind of property that God must have, since He is a being who would be without equal and if He lacked the property of existence then I or you would be His equal and better. So, He must exist. To concieve of God not existing is to concieve of an object that has everything and yet lacks something which is just as contradictory as concieving of an object that is triangular yet lacking three sides.

A Puzzle About Reductios

I finally got my internet connection back up at home now. Turns out I had a bad cable out on the side of the appartment building. Now all I have to do is get to the backlog of super interesting comments! I hope I’ll have some time to do that this weekend.

 In the meantime here is something I was puzzeling about today. Consider the following argument

If P then P
Not P
So, not P

Is this a valid argument? That would depend on whether it is an instance of modus tollens or denying the antecedent; but how can we tell which one it is an instance of? We have the same problem for modus ponens and asserting the consequent.

If P then P
Therefore, P

So what are we supposed to say about this? I suppose one could deny ‘p –> p’ since it is equivelent to the law of the excluded middle (~p v p) and there are those who would deny that it is true but that doesn’t seem to solve the problem. We still won’t be able to tell what argument form this is an instance of and so can’t know if it is valid or not. But if that is the case then we may be commiting a fallacy when we infer that a sentence must be true because its negation can’t be true and that would mean that reductio arguments have a deep problem.

So, any thgoughts on whether these are valid arguments or not?

Free Will and Omniscience, again

A while ago I was obsessed with trying to show that God’s foreknowledge of our actions was incompatible with Human free will. I have had some time to reflect on the issue and I want to take another stab at it.

So, let ‘K’ be ‘knows that’ and ‘G’ stand for God, and ‘R’ for Richard Brown (me). Then (1) says that if God knows that I will do some action then it is necessary that I do that action.

(1) (x)(K(G,R,x) –> [](D(R,x))

(1) captures the intuition that God’s knowledge necessitates our actions. I think that this is true, so to prove it I tried to show that denying it leads to a contradiction and, since it can’t be false it must be true. Here is the proof.

1. ~(x)(K(G,R,x) –> []D(R,x)) assume

2. (Ex)~(K(G,R,x) –> []D(R,x)) 1, by definition

3. (Ex)~~(K(G,R,x) & ~[]D(R,x)) 2, by def

4. (Ex) (K(G,R,x) & ~[]D(R,x)) 3, by def

5. K(G,R,a) & ~[]D(R,a) 4, EI

6. K(G,R,a) 5, CE

7. []K(G,R,a) 6, necessitation

8. ~[]D(R,a) 5, CE

9. (x)[] (K(G,R,x) –> D(R,x)) assumption (2′)

10. [](K(G,R,a) –> D(R,a)) 9, UI.

11. []K(G,R,a) –> []D(R,a) 10, distribution

12. ~[]D(R,a) –> ~[]K(G,R,a) 11, contraposition

13. ~[]K(G,R,a) 8,11 MP

14. []K(G,R,a) & ~[]K(G,R,a) 7,13 CI

15. (x)(K(G,R,x) –> [](D,R,x)) 1-14 reductio

The main objection centered on step (lucky number) 7 and my use of the rule of necessitation. 7 says that it is necessay that God knows that I perform action a. That means that it would have to be true in every possible world that God (in that world) knows that you perform action a. This may seem unreasonable if one thinks that there is a possible world where you do not perform action a. But if actions are events that can be named then it is easy to show that they must necessarily exist, in which case I would have to perform that action in every world where I exist, and snce it is just as easy to show that I must necessarily exist it follows that God would indeed know that I perform action a in every possible world and so 7 comes out true. So if one accepts S5 then one should not have a problem with 7.

But suppose that one rejects, or modifys S5 to avoid the embaressment of necessary existence? Then 7 starts to look fishy again. But is it? Say that there is some world where I do in fact perform a and some other world where I do not. Call them ‘A’ and ‘~A’. The in A God knows that I perform a but in ~A He doesn’t know that I perform a because it is false that I perform a and God does not know falsehoods. But is it really true that in ~A God does not know that I perform a? He knows everything, so He knows what is possible and so He knows that there is a possible world where I do perform a. Yes, but that just means that He knows “possibly Richard performs a’ not ‘Richard performs a'”, or in symbols; he knows <>D(R,a) not D(R,a). This I admit, and so it seems that there is a conception of God’s foreknowledge that is compatible with Human free will. But there does seem to be a sense in which He still knows that I do a; He knows in which possible worlds I do it and in which I don’t. But maybe that isn’t enough to justify 7 and so enough to avoid the issue.

But notice that it is a conception of God as confined to particular possible worlds where he knows all and only the truths in that world that is the actual world. The possible worlds are not real worlds but formal descriptions or specifications of how the actual world could have been and God has maximal knowledge of that. If one were a modal realist and thought that the possible worlds were real worlds that exist then there would be a problem here. In each world God would know either that you perform action a in that world or that you perform it in world-x. In both cases He knows that you perform action a and so it will true in all worlds that He knows that you do a. So 7 would be true again.

So I conclude that there are some interpretations where 7 comes out true; in which case there are some metaphysical systems in which God’s omniscience is incompatible with Human free will. Or He’s a dialetheist…