Monday, May 12, 2008

Modal logic

Just as standard logic is a system for dealing rigorously with the ideas of "true" and "false", modal logic is a system for dealing rigorously with the ideas of "sometimes true" and "always (or never) true".

Modal logic is fine, as far as it goes. You can use it for a lot of different things, so long as your definition of "sometimes" and "always" is consistent. I'm sometimes in my house, but not always; I'm always male. Scientific laws are universals; facts are accidents. It's required that you pay your taxes; giving $3 of those taxes to presidential campaign funding is optional.

But (at least some) philosophers seem to like modal logic because it's easy to create equivocations which are difficult to detect. Plantinga's modal ontological argument is a perfect example of an equivocation fallacy.
  1. It is proposed that a being has maximal excellence in a given possible world W if and only if it is omnipotent, omniscient and wholly good in W; and
  2. It is proposed that a being has maximal greatness if it has maximal excellence in every possible world.
  3. Maximal greatness is possibly exemplified. That is, it is possible that there be a being that has maximal greatness. (Premise)
  4. Therefore, possibly it is necessarily true that an omniscient, omnipotent and perfectly good being exists
  5. Therefore, it is necessarily true that an omniscient, omnipotent and perfectly good being exists. (By S5)
  6. Therefore, an omniscient, omnipotent and perfectly good being exists.

As Graham Oppy observes, "Perhaps somewhat surprisingly, Plantinga himself agrees: the "victorious" modal ontological argument is not a proof of the existence of a being which possesses maximal greatness." When a philosopher denies the conclusion of his own argument, you must suspect he's bullshitting you.

A careful examination of premise 3 — Maximal greatness is possibly exemplified. That is, it is possible that there be a being that has maximal greatness. — shows the problem. But first some background.

One of the uses of modal logic is to examine the concept of logically possible worlds, i.e. those worlds where true statements about that world are logically consistent, but some statements that are true about that world are not true of our world. This is just a rigorous way of talking about subjunctive and counterfactual reasoning, which people routinely employ: e.g. "If I hadn't gone back for my wallet, I would have caught the train," or, "If Ralph Nader hadn't run, then Al Gore would have been inaugurated President in 2001."

This semantic way of employing modal logic, though, assumes that each set of consistent non-modal truths defines a possible world. A non-modal statement (e.g. "Al Gore was inaugurated as US President in 2001") is different from a modal statement (e.g. "There exists a possible world in which Al Gore was inaugurated President in 2001"). The non-modal statement is (sadly) false in this particular world, but it could easily have been true (along with other statements) without any fundamental logical contradiction.

The modal statement, however, is true in all logically possible worlds. Even if Al Gore was or was not inaugurated President in this or any particular possible world, it is true in all possible worlds that some such possible world exists, perhaps elsewhere. A modal statement in possible world semantics does not divide possible worlds into those worlds where it is true and those worlds where it is not. It's either true everywhere or true nowhere.

So on one horn of the dilemma, Plantinga's premise #3 is simply not well-formed, it is not a statement of modal logic.

But perhaps Plantinga does not intend logically possible world semantics. Perhaps, as his comment leads us to believe, he means epistemic possibility: we don't know whether or not God exists; its epistemically possible that God exists. But if so, he seems to use modal logic in a weird way, weird even for a philosopher.

Consider this similar argument:
  1. All true arithmetic statements are true in all possible worlds. (Definition)
  2. If Goldbach's conjecture is true in any possible world, it is true in all possible worlds. (By 1)
  3. It's possible that Goldbach's conjecture is true. (Premise)
  4. Therefore Goldbach's conjecture is true in at least one possible world.
  5. Therefore Goldbach's conjecture is true in all possible worlds. (By 1)
  6. Therefore Goldbach's conjecture is true.
Of course premise 3 is the problem; I was trying to bullshit you; it's the only statement where I don't fully qualify "possible". A more rigorous way of stating premise 3 would be: "It is definitely true in all possible worlds that there exists a possible world in which Goldbach's conjecture is true." Stated so plainly, it should be obvious that I'm simply begging the question.

Plantinga goes on to say, "Take any valid argument: once you see how it works, you may think that asserting or believing the premise is tantamount to asserting or believing the conclusion." His argument can be construed to mean that "either the existence of God is logically impossible or it is logically necessary." I'm sure that those paying Dr. Plantinga's salary are quite pleased that he has gone to elaborate lengths to prove that logical arguments are indeed logical, and modal logic is indeed modal.


  1. I'll see your Plantinga and raise you one Saint Gasoline:

    "My penis is maximally excellent in length and girth. My penis is also maximally great because it is maximally excellent in all possible worlds. A maximally great penis is possible. Therefore a maximally great penis exists necessarily and this we call 'my penis.'"

  2. Doesn't Plantinga's premise#3 awaken the possinility of a Modal Problem of Evil,i.e., it is NOT possible that maximal greatness is exemplified because in some possible worlds maximal excellence is contradictory?

  3. There's an issue with premise 5 in the Plantinga argument too - it hinges upon S5 being the 'correct' way to model the world in modal logic (so that 'It is possible that it is necessary that p' entails 'It is necessary that p').

    If you don't accept S5 as 'the correct' modal logic, you aren't going to be convinced by this argument.

  4. Yes, S5 is controversial. On the other hand, Plantinga's argument fails even if one accepts S5, so it's a side issue.

  5. I remember reading Mackie's book 'The miracle of theism' and he debunked a version of Plantinga's modal ontological argument. I think he didn't see the need to get to arguing about the S5 modal system (whatever that is) because the premises were dodgy. I don't recall exactly why though.

  6. I do not think that Plantinga's argument is successful, but I think this is rather in virtue of premise 3 - it being the case that Plantinga has not proven that one being could exemplify maximal greatness. At any rate, let me comment on a few things you stated:

    Plantinga's position, and the position that S5 takes, is that a statement about a possible world ranges across all possible worlds. So your statement about the proposition "There exists a possible world in which Al Gore was inaugurated President in 2001" that Even if Al Gore was or was not inaugurated President in this or any particular possible world, it is true in all possible worlds that some such possible world exists, perhaps elsewhere is simply false. A proposition is possibly true, in any modal metaphysics, iff there is some possible world in which it is true. Now, according to S5, a proposition about some world-indexed possibility, or a proposition about a possible world holds in every possible world - because in S5, what is possible is necessarily possible; ie. no possible world is 'un-accessible' from any other. So Plantinga's statement is well-formed - it is a statement in S5 modal logic concerning the status of a possibility in a particular possible world which, as it were, happens to be a world in which a being which has a necessary trait (one which, according to modal metaphysics, is possessed in all possible worlds) exists.

    On another point, it is well to keep in mind (though Plantinga does not) that the term possible world might refer to what some have termed 'metaphysical possibilities' and not merely 'logical possibilities'; cf. McLeod and Paul. There is an arising group of metaphysicians who would like to separate 'logical' possibility, concerning de dicto statements founded in a-priori considerations, from 'metaphysical' possibility, concerning de re and most usually a-posteriori knowledge.

  7. "There exists a possible world in which Al Gore was inaugurated President in 2001" that Even if Al Gore was or was not inaugurated President in this or any particular possible world, it is true in all possible worlds that some such possible world exists, perhaps elsewhere is simply false.

    I am obviously not being sufficiently clear: I'm trying to make the exact same point you're making.

    I'm attempting to express that same sense by saying that the modal predicate is true even in those particular worlds where the non-modal predicate is false.

    If it is true in one possible world that "Al Gore is president", then it is true in all possible worlds — even those possible worlds where Al Gore is not president — that "There exists a possible world where 'Al Gore is President' is true."

    Plantinga's statement (3) is not well formed at the semantic level, because it is a modal predicate, but it trades on the non-modal predicate schema-definition of "possible world": For every non-modal predicate that is not internally contradictory, there exists a possible world where that predicate is true.

    Since (3) is a modal predicate (it talks about all possible worlds), we cannot apply the schema-definition to infer that there does indeed exist a possible world where a maximally great being exists.

    In other words, we can't even get to S5, because we cannot infer that it is definitely the case that it is possible that a necessary being exists.

    Plantinga wiggles out of this criticism by making his argument vacuous: he concludes that a necessary being exists in all possible worlds or none. But that's just the definition of "necessary"; if this is the best his elaborate argument can prove, it's entirely unnecessary.

  8. We don't agree on much, but on this I'm totally with you; although as a theist I really, really dislike Plantinga's argument, whereas I'd bet that, as an atheist, you must love it when big-name theists produce such rubbish?

    Either (3) to (4) is invalid, or else (3) begs the question. Theists who like this argument also say it is sound, by which they mean that they are theists! I especially hate it when such theists say that the worst this argument can be accused of is begging the question.

    If an argument deduces nothing but the truth of one of its premises then it is not even an argument, in my book. But it is called an ontological argument. So the worst it can be called is a lying piece of rubbish, which insults anyone who agrees with it, because it regards them as foolish enough to be fooled by it. Or rather Plantinga does.

  9. ... I'd bet that, as an atheist, you must love it when big-name theists produce such rubbish?

    No, I don't love it. I hate it. I'm not in it to win, I'm in it to find the truth. *Everyone's* stupidity bugs me.

    1. Presumably that stupidity includes your own?

    2. 11:20am

      You have no credibility for one reason: Edward Feser.

    3. Anonymous, if you want to engage, pick a handle and make a substantive argument. Who are you? Why do you think Edward Feser's work is so important? I haven't read him, but the reviews are not particularly promising.

  10. This is an off topic comment,sorry. I was just wondering if you had seen or what you thought of fatfists web page. Yours is one of the best pages I've seen so I though I would see if you had an impression of him. I think he makes very good points and very bad points regarding this topic. Wow, I'm random. Anyway, I don't know if anyone is responding on this page anymore considering how old the last comment was.

  11. I was just wondering if you had seen or what you thought of fatfists web page.

    I haven't seen it. Link?

  12. This is a little late but here is the fatfist page.

    I think he's mostly just rejecting the use of the S5 axiom. This is a off-topic, but I like your's and Paul Almonds pages on this (and other topics) much more then some of the other pages I've seen. This also plays into my criticism: I was looking through fatfists pages and I know that i should judge an argument on its own merit on not on the who is making the argument but fatfists other pages kind of make it hard for me to take him as seriously.

  13. This comment will probably violates the comment policy. I guess its related to Enigman's comment about not agreeing with another theist. I guess I wouldn't be steamed if I hadn't heard what I'm going to talk about coming from atheists other then this one guy (Fatfist). He argues for classical physics over modern physics mostly on the basis of not being able to imagine something not explicitly related to space and time as we are familiar with them on a personal level. I don't see any reason as to why the universe *must* follow our systems of logic or agree with our sensibilities in regards to space-time, other dimensions, other universes, a finite/eternal universe or anything else. Things that directly affect us very well could exist in other dimensions. Space and time could be (and is) different on a deeper level from what we experience them on a personal level. It bothers me when people argue that something is true on the basis that it is intuitive. Many of the things that we have discovered through empiricism (I mean through science specifically) have been counterintuitive yet we know that our current understanding of physics comprise of the best models we have come up with to date because of highly specific predictions that can be made using them. One thing I often hear is that one must "see it to believe it." However, this isn't the case. Radio waves for example can't be (to my knowledge) detected by our five senses. We can however design tests and evaluate whether highly specific predictions formed by modeling radio waves existence are met. Atheists (fatfist) who argue from personal incredulity (what is intuitive) bother me just as much as theists who argue from personal inrdulity. Atheist who are absolutely certain that no gods exists (I'm not saying your doing that) or argue that gods are impossible (epistemically?) irritate me as well. I only claim practical knowledge as in I know no leprechauns exist (I'm gnostic atheist in this sense). Anyway I guess my views come as a result from trusting the most refined (in my opinion) inductive methodology and process over what seems intuitive or logical. I feel that (if needed to) I can present a pretty good case for why scientific methods and why scientific community should be trusted as having the best models of reality. I think systems of logic and philosophy certainly have their place but my opinion is that they are often both deductive reasoning based on premises formed out of inferior (to scientific method) inductive premises. Sorry for my ranting, that was certainly off topic. I really like your site, I guess that doesn't give me the right to rant on it. Note I wasn't saying the scientific community had anything to say on the existence of gods. Anyway, I don't know why I feel the need to rant to insightful people I come across. Sorry.

  14. Anyway, I'd like to clarify, my off topic rant was directed at people who try to discredit modern physics (I'm not saying our understanding is 100%, it never is) by applying systems of logic developed often over a thousands years ago at a time when we had relatively (compared to now) poor understanding of the universe. I do think different systems of logic and philosophical thought have their place in providing support for an argument though. If something is proven within a system of logic i don't think that means it is "proven" in reality but I do think that if the system of logic is reasonable then it can be offered as support for an argument. Gee I wish I had taken philosophy or logic in college. Anyway, I don't that Plantinga's argument is sound. I don't think Plantinga provides any support for his position, and I like your (and Paul's) critique although there are parts I'm not sure I completely understand in the critiques. I mostly don't think that having "it is possibly necessary that x is true" is a good premise. If the premise was "it is possible that x is true" and he didn't make necessity part of the definition of x I think that would be fine. I don't really care what x is. I understand if these comments don't get through moderation by the way. Thank you for your time and consideration.

  15. Whoops. Think the 2011 Anonymous post was mine. How embarrassing. My writing was idiotic and childish. Apologies. Wish I could delete them. Will be "Moniker" from now on. Haven't posted much that I remember.

    1. I dislike deleting comments; all of us have stupid things, but they're still part of us. And although there's much room to improve your organization, your comments are not terrible.

  16. I'm not the 2012 or 2014 anonymous's by the way. Everyone really should have a consistent (preferably unique) name.

    1. Hence my exhortation. Sadly, Blogger has nothing between allowing anonymous comments and requiring registration; I find the latter too restrictive.


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