Joshua says the GAE is still integral to the Mission

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Modal logic isn’t “correct” or “wrong”. It is a formalism. An extension atop propositional and predicate logic, in order to be able to express so-called modes, chiefly possibility and necessity. There exist other modes, like “sometimes” and “always” (in future and in the past), “permitted” and “obligatory”, etc. Predicate logic already provided “all” and “some”.

The way possibility and necessity are achieved as modes is with an abstract construct called a “possible world”. In simple terms, a possible world is something quite like a “way things could have been” (were they not quite as they happen to be in the actual world). More formally, one could imagine a possible world as a set of all true propositions. Every proposition one can construct that is not in the list, would then be false in that world. So long as a combination of propositions being true (and the entailed falseness of all the other ones) is not internally contradictory, any such combination is definitionally a possible world.

The modes are then defined as such: Let P be a proposition.

  • The statement \Diamond P (read “Possibly P”) means “There exists a possible world where P is true”. That is to say, there exists some combination of truth values for all constructible propositions which is internally consistent and wherein P happens to be assigned the value \texttt{TRUE}.
  • The statement \Box P (read “Necessarily P”) means “For all possible worlds W, P is true in W”. That is to say, there exists no way of combining truth values without either contradicting oneself or assigning \texttt{TRUE} to P.

What one quickly finds with these very basic definitions is difficulty to prove or disprove certain statements that combine multiple instances of a mode. For example, it is intuitive to conjecture that that just because P (i.e. P is true in the actual world) it does not follow that P is true in every possible world. But what if P=\Box Q? In that case Q is true in every possible world, but is it also true in every possible world, that from that world’s perspective Q is true in every possible world? On the one hand, it is still P, so the answer should be no. But if that’s the case, then what on earth does “every” even mean anymore? To put it briefly, naive modal logic is insufficient to decide in the abstract, for all versions of P whether P entails \Box P, and that sort of destroys the entire enterprise of introducing the modes in the first place, and we’re back on square one. If we are to formalize modes at all, what we invent for this purpose must allow us to formalize statements like “It is possible that it is possible that P”. Therefore, we need to decide on just exactly what making a statement about a possible world other than one’s own means. We need to decide which worlds any given world gets to “see”.

This is what’s called the accessibility relation between possible worlds. And there is not one relation that is universally agreed upon, because it all depends on which inferences one wants to make valid or not. The accessibility relation properties are the same as for algebraic relations: Symmetry and antisymmetry, reflexivity, and transitivity, and they mean exactly what one would expect.

It is not the case that modal logic in general claims that a proposition that is possibly necessary is necessarily necessary.

If, for example, accessibility is not transitive and symmetrical, then \Diamond\Box P means that there exists some world W_1, and if one were to look out of that world, every world W_1' one would then see is one where P is true. But without transitivity and symmetry it is not guaranteed that W_1 can see all of the worlds visible out of the actual world W_0, or out of which ever other worlds are hence visible. So it may be the case that there exists a possible world W_2 where \Box P is false, and therefore in W_0 the statement \Box\Box P is false, eventhough \Diamond\Box P is true.

If, on the other hand, accessibility is symmetrical and transitive, then \Box P being true in all worlds visible out of W_1 means \Box P is true in W_0 (symmetry), and \Box P is true in every other possible world, since through a recursive web all of them are visible to W_1 (transitivity). In that case, indeed \Diamond\Box P\therefore\Box\Box P is a valid inference.

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Thanks. Very helpful.

One question: what is modal logic primarily used for? Is there some practical application?

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Quick disclaimer, I am not a philosopher. I am not familiar enough with current or recent philosophical literature and exactly what kinds of arguments most commonly make use of these formal tools. With that in mind, here’s my take:

Natural language can be very sloppy/imprecise. Formalizing arguments can make it much easier to understand why they hold or where they fail. Without modal logic, necessity and possibility would not be attributes one assigns to statements, but rather themselves parts of the statement.

I guess one could liken it to predicates: We would like to say, for example, that “Alice is older than Bob” in natural language entails that “It is not the case that Bob is older than Alice”. However, in pure propositional logic, these are completely unrelated statements and we cannot derive one from the other. We would actually have to interrogate Alice’s and Bob’s ages and individually test both the greater-than and the no-greater-than relations. But that’s silly. So instead we introduce the formalism of predicates, and one such predicate is the “is older than” relation between two persons. Now we can separate that relation from the individuals, swap them around, and derive theorems that naturally feel like they should entail one another.

In a similar manner, modal logic allows us to subdivide statements with necessity and possibility (or structurally similar modes) into those modes and what ever substantive (“atomic”) statement they are put on. An illustrative example from modal logic would be with obligation and permission: Surely, we want to say that for any action X, if “It is not permitted to do X”, then “It is obligatory to not do X”. This inference is not a logical theorem without a way to talk about obligation and permission in the abstract. So to prove it we would actually have to test the pair for all possible actions X individually. Again, this would be kind of silly. With modal logic we achieve a separation of the permission-obligation mode from the action itself, and allow ourselves to make inferences like this, and to construct valid arguments that hinge upon them.

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Interesting. My principal reaction is that while it’s true, as you say, that “natural language can be very sloppy/imprecise,” the impact of formalizing statements in this way is liable in many cases not to be to resolve, but to obscure, that problem. I have very much the sense that when I’ve encountered people who actually think the ontological argument is useful, they LOVE the fact that there is this obscure system of notation, which makes it necessary for the critic to peel apart the actual contents in order to make sense of the thing.

Anyone who’s spent some time trying to draft a statute is familiar with the problem. No matter how precisely and formally one tries to render statements, including careful draftsmanship of definitions, there is always unforeseen difficulty. And the attempt to eradicate ambiguity often only moves the ambiguity from one place to another while making it harder and harder for people to resolve straightforward problems in relation to the statute.

This is, in some sense, the strength of the common law. By attributing legal authority to tradition and usage rather than solely to statute, we import the kind of flexibility which even the best-drafted statutes tend to lack, the lack of which leads often to unforeseen consequences. In other words, rather than attempt to wholly solve the problems inherent in language, the system imports the actual objectives of the whole endeavor – doing justice to individuals – as a kind of failsafe.

The difficulty of formalizing statements, in the law at least, is that language is never actually adequate to describe things. Even a term like “one hundred” can have unexpected meanings that derive from specific contexts, and anything which tries to explain some more complex relation (e.g., “contract” or “agency”) defies any attempt at complete and correct verbal characterization. And when we do try hard to formalize them, we often CAUSE problems rather than solving them. I suspect that modal logic must suffer similarly, all the while burying the problem under obscure notation. Strict logical systems can only be trusted to function fully and correctly in the realm of symbols, not things.

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In my opinion, there is both. The problems of entirely uncontrolled and standard-free natural language are addressed by providing a formal language. That brings its own problems with it, but they are, I would opine, not the same problems. The problem you bring up about the language being constructed for the sake of formality thereby losing what connection it had to “the real world” is fair to some extent. The question then becomes whether pure analytic truth-seeking or a reflection of human experience is the goal.

Mathematics is also an extension of formal logic, and it too has no obligation to map to anything in nature whatsoever. Yet, it feels to me like many more people would acknowledge and maybe even lament its inaccessibility to the untrained, and insist that the point and purpose of mathematics is, in fact, to describe some aspects of common experience. They will often soften their lamenting, too, blaming their lack of access to mathematics on their own inadequacy or lack of training, rather than a careless obfuscationism on the part of the mathematical establishment.

I do not think modal logic is fundamentally different in this regard, except insofar as many attempts are made in philosophy to use it to discuss problems previously treated in natural language.

Of course, things like the modal ontological argument would scarcely exist, were it not for modal logic. It being so formal allows the apologist to sound far ahead of their true target audience, while deserving to be mere laughing stock among their peers in training. It sounds impressive on the surface, but to anyone who can actually correctly parse it, it is obviously either a non-sequitur or a begging of the question. Plantinga’s version specifically, I find, makes the error of treating existence as a predicate, rather than a quantization. There is a problem with that.

The symbol string “x exists” is not (or should in my opinion, at any rate, not be treated as) a proposition. Instead a proposition would perhaps be something like “There exists an x such that the predicate p\left(x\right) is true”, i.e. the set of things that have/satisfy property p is non-empty. For instance, when we say “There exist prime numbers smaller than 10”, we are saying not that there are some objects out in the world called numbers. We are not saying that specifically 2, 3, 5, or 7 are objects that exist in nature either, or even that “5 exists” is a true statement in some possible world. We are saying that the way the set of numbers smaller than 10 and the set of prime numbers are defined, they have some overlap. It is a statement about those definitions/constructions, not about existence as a property a thing has. These sets, the way they are constructed, contain elements that can be recognized as identical/equivalent by some agreed upon equivalence relation. That is what existence in formal logic and mathematics is. It is non-contradictory-ness, logically-conceivable-ness.

But suppose what it means if one were to treat existence as a property objects can have. I shall now define a “Shmachelor”: A shmachelor is an unmarried man with the extra property of existing. By definition, something either exists, or it is not a shmachelor. Unicorns, for example, are not shmachelors, for they do not exist (nor are men). A married man is not a shmachelor either, for even if he exists, he is not unmarried. But the bachelor James, who evidently exists, is, by definition, a shmachelor.

But in a world where every man is married, no man is an unmarried man. Therefore, in that world, there are no shmachelors. But a shmachelor has to exist by definition. So a world where all men are married is a world where there does not exist something that exists. This, of course, would be gibberish, and now we have a choice:

  • We bite the bullet, and say that there exists at least one unmarried man in every possible world, even if the world is empty otherwise.
  • Existence is not an actual property that can be assigned willy-nilly by definition the same way unmarried-ness, man-ness, green-ness, warm-ness, or seven-feet-tall-ness can.

The ontological argument invites us to bite the bullet. It treats existence as a property that can be defined onto objects, and a great-making one at that. God is by definition a maximally great being, and that means it is (among what ever other attributes) an existing being, just like a shmachelor. The entire argument then collapses into “God is godly” or “A being that exists exists”.

Also, can I just say, I absolutely love what’s been made out of this thread? Just peak irony.

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Thanks. That’s very well thought-out. I’m not really trying to suggest that a system like this can have no use, but just that I think that we have to take great care in applying “logic,” modal or otherwise, to complex things and phenomena. Most of the questions that are worth asking are best answered empirically, and empirical answers come hedged in with uncertainty: they are best explanations rather than absolute and definitive truths.

I’ve always been uneasy with the notion that somehow “existing” is a property that must be associated with a maximally-what-have-you being, because if one concludes that Sherlock Holmes is the greatest possible detective, one is not warranted in assuming he therefore exists. Maxima can be real, and can be imaginary. But I don’t think I had ever identified that as a core problem with the ontological argument, until reading your thoughts on it.

I tend to take the view that objects do not really have “properties” in the sense of our descriptions of them corresponding neatly to aspects of them. Rather, stuff is stuff, and it does the things stuff does. All description is in some sense abstraction, meant to cause the hearer to imagine the thing or aspect of the thing which is described, and usually to relate those things to practical aspects: will I be able to lift the bag of gravel into my truck, or is it too heavy for me? But the “weight” isn’t a thing, it’s just a way of summarizing how a lot of particles and forces interact, in a way which is meaningful in relation to our experience. It’s a bit like the way all models are wrong, but some models are useful.

Logic applied to abstract theoretical entities is potentially, like mathematics, quite solid and reliable. But so much of what we would actually like to know is better analyzed in terms of the weight of evidence, with logic playing only the role of helping us avoid certain classes of clear error.

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No, that’s not the definition of ‘God’. That’s the definition of some unnamed entity that ‘God’ is then claimed to be, without anyone ever showing that the two are the same.

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Sure, one can disagree with that definition. But for someone, like myself, who agrees with that definition, that solution to the the questions raised by the modal ontological argument is not available,

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Thanks for your very thorough comments, @Gisteron. It will take some time to digest all that.

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While it might not be “arbitrary” (depending on how rigorously you define that), it would seem to be a non-core part of the definition, plus I’m not even sure if I’d accept that “necessary existence” can be an attribute of an as-yet hypothetical being.

This would be a bit like if I was arguing with somebody about the existence of unicorns – I would need to accept that a unicorn is an equine with a single horn (otherwise it is not clear what we’re arguing about), I need not accept only being approachable by a virgin as part of the definition. It is not that the inclusion of this in the definition is arbitrary, it is that it is inessential.

Defining God as necessary is however arbitrary, to the extent that it is a priori, and by assumption, ruling potentially-possible worlds as “impossible” without first demonstrating how they are a logical (or other) impossibility.

But that has not been demonstrated for God. That 2 + 2 = 4 is a demonstrable logical certainty.

You have still failed to explicate metaphysical possibility. At most you have ruled out logical impossibilities as metaphysical possibilities. You have not however demonstrated that one of ‘God exists’ or ‘God does not exist’ is a logical impossibility.

Of course we are. “Necessary existence” is a subset of existence. If we accept that God necessarily exists by defintion, then we have already accepted his existence. Q.E.D.

Hence my prior comment about defining God into existence.

Including necessary existence as part of the definition of God, in an argument for God’s existence, is blatantly Begging the Question.

You first have to demonstrate that something exists, in order to demonstrate that it necessarily exists.

It is blatantly illogical to allow anyone to a priori posit “necessary existence” as an “attribute” of the very being whose existence you are arguing about. To my mind, this point of logic trumps any purported definition.

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I think part of the problem is that this argument is intermingling definitions and assertions/propositions.

The Eiffel Tower is a tower built by the company of Gustave Eiffel in Paris in the late 19th century.

… would appear to be a valid definition of what we mean by “Eiffel Tower”.

The Eiffel Tower is a tower built by the company of Gustave Eiffel in Paris in the late 19th century. It exists.

… would appear to be problematical, as if the Eiffel Tower were destroyed tomorrow, it would still be what we meant by the Eiffel Tower.

Likewise, “God” is the same hypothetical being, whether he exists or not, or even if he necessarilly exists. Therefore including (the assertion of) his (necessary) existence in his definition would appear to serve no legitimate logical purpose, and in fact (as @Gisteron alluded to above) is deeply problematical.

The issue is also of concern in the opposite direction.

The original P2 and P3 do not appear to be independent propositions, but rather definitions, that cannot be evaluated independently, but rather modify our understanding of P1.

Allowing definitions to be conflated with propositions allows for such trickery as the following ‘propositions’:

  1. The Eiffel Tower exists.

This would seem to be uncontroversially true.

But then:

  1. The Eiffel Tower was a library in Alexandria that was likely destroyed in the Third Century CE.

This definition of course completely changes our understanding of the first proposition.

Whilst the Modal Ontological Argument is not as capriciously dishonest as the above, and whilst Modal Logic may well have non-bullshit uses, it would appear that the only reason Modal Logic has been introduced into the Ontological Argument is to allow a similar bait-and-switch and thus it is engineered to deceive rather than enlighten.

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Exactly. It’s simply an illegal move to just assert X is necessary, just by definition. Necessity is a contingent property. Something is only necessary if it follows necessarily from something else. To assign it as a brute property of some entity is simply a question begging move. Necessity must be demonstrated logically.

To show what sorts of absurdities follow if brute necessity can just be declared in this way, we can disprove anything by just defining some entity to have it.
The psaklyton is a necessarily existing entity that only has two properties, that it necessarily exists, and that it cancels God’s necessity. We now have a contradiction and one of them has to give. If the psaklyton exists, God can’t be necessary. And obviously the psaklyton must exist, because it’s necessary. I have now disproven God.

It’s obvious how idiotic this “define it to be necessary” move becomes once you just stick it on to something.

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Necessity – just like existence – is not a property of individuals at all. At best it can be a property of propositions. God, or the psaklyton, or the shmachelor are not propositions. They are not \texttt{TRUE} or \texttt{FALSE}, and the symbol string “Necessarily God” is meaningless. A proposition can be necessarily true (or false), in that it is true (or false) in all possible worlds, but an individual cannot be necessary in this modal logical sense, because modal logic as a language simply does not provide the grammatical structure to express a sentence like that.

What the modal ontological argument tries to do is to say not that “God is in all possible worlds”, but rather that “The statement ‘God exists’ is true in all possible worlds”. And that only works in one of two ways:

  • “God exists” is an atomic proposition. It is an item that has a truth-value, but cannot be subdivided into smaller parts. It is not a ‘statement about God’, because ‘God’ is not a part of that proposition. The proposition being atomic means it has no parts. Obviously this is not useful to the apologist, since the entire point and purpose of their constructing the argument is to conclude a meaningful statement about an individual, namely a proposition that says that God does something: existing.
  • “God exists” is comprised of the invidual ‘God’ and the predicate ‘exists’.

But if existence is a predicate, that means we can put it into the definition of things, just like redness, younger-than-Alice-ness, or seven-feet-tall-ness. For every object we can imagine, we can now easily construct a counterpart defined in all of the same ways we’d normally define, except with the additional property of ‘existence’. Doing so means we bar ourselves from ever imagining a world in which these objects do not exist, since, by definition, they exist, or else we are talking about something else. Unicorns may or may not exist in some possible world, but shmunicorns, which are unicorns that exist, wouldn’t be shmunicorns unless they existed in that world, so they do. Shmunicorns exist in all possible worlds.

This is, of course absurd, and the only ways to get rid of such nonsense is to say that existence is either not a predicate at all, or that it is a special predicate somehow, that one cannot just willy-nilly assign to what ever one makes up. In the former case, the apologist loses, because their argument is just grammatically nonsensical. In the latter case, the apologist needs to start back at square one, as now they have to justify why we are allowed to assign existence to God specifically, which was the entire point of their argument in the first place.

The way they justify that assignment, then, is by saying that existence is (for some reason) ‘great-making’, and God being maximally great by definition must have it. But if a maximally great being is a being that exists, then “God exists because it is maximally great” translates to “God exists because it exists”.

Needless to say, the modal wrapping doesn’t do anything to fix this fundamental flaw, and to those who can parse it, it does precious little to obscure it either…

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I would say that existing is a prerequisite for actually having properties. If I said “Santa has a white beard” I would not be claiming that there is an actual white beard that belongs to Santa - the beard is as imaginary as Santa himself.

With this understanding assigning existence as a property becomes redundant at best.

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Yes, I appreciate this perspective. And you say “at best”. I think a more forceful rejection of existence is indeed necessary (no pun intended).

If existing is a prerequisite to having properties, then “Santa has a white beard” is actually false, unless Santa really exists. “Santa does not have a white beard”, however, is also false, for the same reason. All statements about non-existent things are false, because the moment anything is said about a thing “the thing exists, and…” is implicitly prepended, rendering the conjunction false, no matter what else is said of a non-existent thing. Treating existence as a base property like that required before any other properties can be analyzed completely bars us from considering hypotheticals. Unicorns are not horses with a horn on their forehead, because in order to be a horse or to have a horn a thing must first exist.

Talk of fiction might be destroyed too: We cannot say that Gandalf is powerful, or that Samwise is caring, because neither character is real out in the actual world. One could try and fix this, by separating ‘existence’ into existence within fiction and existence within the actual world, but that, too, only gets us so far, and at quite a cost at that.

More problems still remain: We cannot say that every daughter of a colour blind man or a haemophiliac will be born a carrier of their respective father’s defective gene, until we can know with certainty that either man will, as a matter of fact, beget a daughter. For if the girls do not exist at any point in the future, then they cannot have the other properties of inheriting their fathers’ X chromosomes.

Yes, on the face of it, it feels like there is some redundancy in adding “… and also exists” to descriptions of real things. As I hope I could express in this message and the last few, I believe there are good reasons to not only dismiss it as redundant, but as outright absurd, and to reject treating it as an attribute at all for this reason. No matter how we slice it, attempts to do so introduce more and far, far greater problems than we ever felt like we had before the attempt.

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If you interpret “Santa has a white beard” as meaning that there is an actual white beard belonging to Santa then of course it is false.

But I would interpret it as meaning that the concept of Santa includes “having a white beard” which is true.

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I don’t think it is being used in a manner inconsistent with ordinary usage. When one says it is possibly necessary that 2+2=4, this is trivially true. That’s because we know, beyond any doubt that it is necessarly true. To say it is possibly true, then is redundant. It is just true, period. (This assumes agreement that mathematical truths are necessary truths i.e. that they hold in every possible world.)

Now, suppose I say it is possibly necessary that 2+2=5. This statement is false. Not only is it not true that 2+2=5. It is not possibly true. That is to say, there is no possible world in which 2+2=5.

Another way of saying it: Since there is no situation in which something necessarily true in one world, and not another, there is no situation in which adding the modifier “possibly” to the term “necessarily true” provides any additional information or changes the conditions in any way under which the proposition in question could be deemed true or false.

That’s how it seems to me, anyway.

That’s what I conclude. But going thru this exercise is one way to demonstrate this.

But maybe that’s not a bug, but a feature. By seeing how far we can go using abstractions and logic alone, it helps us avoid errors when applying logic to concrete things in the real world.

I don’t see how.

Whereas I think the modal ontological argument provides warrant to conclude that such being is not merely unlikely, but impossible.

I think that’s a reasonable point of disagreement. But, personally, I do consider necessary existence a core attribute. I would be less inclined to call a being “God” if he was omniscient and omnibenevolent but unnecessary, than I would be one whose existence was necessary, even if he was kinda stupid and a bit of a jerk.

No, it has not been. However, if 2+2=4 is possibly true in one world, and it is, then it is necessarily true in all worlds. Same with a necessary being, except we cannot say “and it is.” The point being we can only say God (so defined) does not exist in our world by saying he cannot possibly exist in any world. I’m perfectly fine with that, but others might not be, and then they’ll have problems dealing with this argument.

No, because we still have to show his existence is possible.

Again: “2+2=5” is not merely untrue. It is not possibly true. By saying it is not true in just a single possible world, we are saying it is not true in every possible world.

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Yes, because necessary existence entails existence – so we would have already accepted existence.

Necessary existence means existence in all possible worlds, this world is a possible world, therefore by agreeing to God’s necessary existence you have agreed to his existence in this world.

Yes, but you have not already agreed to the necessary truth of “2+2=5” as you have to God’s necessary existence. Therefore you are not precluded from arguing against the former, only the latter.

I don’t actually see this as really being ‘about’ God at all. I see it as simply an arbitrary attempt to restrict what possible worlds I am allowed to conceive of.

Why?

Because Classical Theism says so, apparently.

Then Classical Theism can go screw itself, in my not-so-humble opinion.

I see no reason not to conceive of any “possible world” I can think of, independently of each other – as long as each one cannot be shown to be logically impossible. This might include a possible world where Classical Theism is true, it might include possible worlds where Classical Theism is not true. I see no a priori reason to let the viewpoint of the former possible worlds (and their rules) intrude into the latter.

Things are only “necessary” to the extent that they can be shown to be a logical certainty.

And I don’t think anybody has shown God to be one, as yet.

I also don’t see how God being the foundation of all existence in one possible world, leads to him being a necessary being who exists in all possible worlds. Each of these worlds is a separate ‘existence’ and there would seem to be no logical requirement for them to all have the same foundation of existence or even a foundation of existence.

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Well there’s no stipulation and it’s hardly controversial that 2+2=4 in our universe. I don’t think we can be certain that even a maximally excellent being is possible.

That’s a step further. Once you have concluded that it is more than reasonable to regard the premise as false - rather than true as the proponents claim - the rest follows. The warrant given must be at least as strong as that claimed by the proponents - arguably more so since the reasons for rejecting the premise are stronger than those for accepting it.

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But if God cannot possibly exist, then he does not exist in our world, or any other world.

IOW, there is no possible world in which he exists, just as there is no possible world in which 2+2=5.

No, I have only agreed that, if they are possible, then they are necessarily true.

If it was possible that 2+2=5, then it would be true in every possible world. That is because any mathematical proposition that is true (and therefore possibly true) is necessarily true (i.e.true in every possible world). But since it is not possibly true, it is not true in any possible world.

Similarly, if God could possibly exist, he would do so in every possible world. But if he cannot possibly exist, then he does not exist in any possible world.

The attempt here, as I understand it, is to demonstrate that if a necessary being possibly exists, then it does exist. As far as I can tell (the caveats @Gisteron has raised regarding the soundness of modal logic itself not withstanding), it succeeeds in doing so.

As I have already said, the question then remains as to whether such a necessary being possibly exists. And while we are not able to say it cannot possibly exist, as we can say that 2+2=5 is not possibly true, we are also in no position to say it CAN possibly exist. On that basis alone, any attempt to use this argument to convince an atheist to convert the theism ought not succeed, so long as the atheist is willing to reject the position that God can possibly exist. Unless he falls for the rhetorical trick of confusing epistemic possibility with metaphysical possibility, there is no reason he should not be so willing.

Yes, Again, it seems to me all that is needed to defeat this argument is accept that the existence of such a being might be impossible. You just can’t waffle and say it is possible.