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.