Man is a metaphysical animal. We philosophers ought to encourage this tendency in our fellow mortals. This morning's mail brings me a long disquisition by a Spanish lawyer (abogado), Daniel Vincente Carillo, entitled "A New Argument on the Existence of God." It consists of numerous definitions, axioms, and theorems. I don't have time to comment on the whole thing, which can be found here, but I will remark critically, and I hope helpfully, on his modal axioms.
Axiom 1
An opposite of what is impossible is either possible and not necessary, impossible or necessary (what is impossible is opposed to everything).
Better: What is not impossible is either possible but not necessary, or possible and necessary.
Example: I am not impossible because I am possible but not necessary. God and the number 7 are possible and necessary.
Axiom 2
An opposite of what is possible and not necessary is either possible and not necessary or impossible (what is possible is opposed to everything, except to what is necessary).
Incoherent as it stands.
The possible is the opposite of the impossible. That is, x is possible iff x is not impossible. The possible divides into the necessary and the contingent. The contingent divides into the actual but possibly nonexistent and the nonactual but possibly existent.
For example, I am contingent and so is a talking donkey. The difference is that I am actual but possibly nonexistent while the talking donkey is not actual but possibly existent.
Of course, the modality in play here is broadly logical or metaphysical.
Axiom 3
An opposite of what is necessary is impossible (what is necessary is concomitant with what is possible and what is necessary, and only is opposed to what is impossible).
Incorrect. What is not necessary is either impossible or contingent. Possibility and impossibility are opposites. Necessity and impossibility are not opposites. Why not? Because the contingent is neither necessary nor impossible.
In the patois of 'possible worlds': the contingent is that which exists (or is true in the case of propositions) in some but not all possible worlds, whereas the necessary exists or else is true in all worlds and the impossible in none.
This patois is a very useful façon de parler for rendering modal relationships graphic.
Axiom 4
What is not necessary is contingent.
Incorrect. What is not necessary is either contingent or impossible. For example, I am not necessary because I am contingent. A round square is not necessary because it is impossible.
Axiom 5
What is contingent is either possible or impossible.
Incorrect. What is contingent is possible but not necessary. The impossible exists or else is true in no possible world whereas the contingent exists or else is true in some but not all possible worlds.
Or think of it this way. The contingent is either actual or unactual, and if actual, then possibly nonexistent, and if unactual, then possibly existent. For example, your humble correspondent is contingent and actual, hence possibly nonexistent, while a flying armadillo is contingent and unactual, hence possibly existent.
Axiom 6
What is not impossible is possible.
Correct!
Axiom 7
Everything that exists does so by itself or by another being.
Correct!
Axiom 8
God, the universe, and nothingness, if possible, are the only possible beings.
Incorrect or at least highly controversial. Many philosophers maintain, with good reason, that there are so-called 'abstract objects,' all or most of which are necessary beings. (Candidates: properties, propositions, numbers, sets.) Now everything necessary is possible. So these abstract objects are possible beings. Therefore, there are possible beings that are distinct from God and the universe.
Second, nothing is in no sense a being. Hence it cannot be a possible being. It is, however, at least a question whether there could have been nothing at all. I examine this question in the entry referenced below.
UPDATE 2/26: David Gordon writes,
You are of course right that nothing is both contingent and impossible; but this does not show that Axiom 5 is incorrect. "What is contingent is possible" is true; and from this "What is contingent is either possible or impossible" follows. If, as I gather from your account, Carillo denies that the class of impossible contingent things is empty, he is mistaken, but his axiom isn't. One can object to it that it isn't an axiom, as its truth depends on the truth of "What is contingent is possible." This is probably too trivial a point to have written to you about, but I pass it along anyway.
The problem is the ambiguity of Carillo's formulation. I took him to be saying that some contingent beings are possible while other are impossible -- which is surely false.
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