Logic is for tricking people
The following is a theorem of S5:
(L(g>Lg)^Mg)>g.
Now, for some explanation. A note on notation: > is the material conditional, and ^ is the conjunction. L is the necessity operator. It is read as "it is necessary that..." M is the possibility operator. M is read just as you would imagine. g stands for "God exists." So in English, that theorem means something like "If it is the case that God is such that God exists necessarily if God exists at all and it is the case that God possibly exists, then God exists." We now provide an argument for the antecedent. Surely it's analytic that God is a necessary being (if God exists at all). And it's equally obvious that God is possible; after all, God is conceivable. Therefore, God exists.
The proof of the theorem (in a variant of Hughes & Cresswell's system) looks like this:*
* The proof is only 4 lines, but what follows the proof is long and probably boring. I talk about how modal logics are constructed, and present some discussion about choosing a logic. I go on to present two ways to reject the conclusion of the ontological argument. I close with some thoughts on what it means to say that a given statement is a theorem or a logical truth in a particular system and on the relation between possibility and conceivability. Continue if you like.
1. L(g>Lg)>(Mg>MLg)—Theorem K11 + universal substitution
2. (L(g>Lg)^Mg)>MLg—Truth-functional consequence of 1 (exportation)
3. MLg>g—The B axiom + duality
4. (L(g>Lg)^Mg)>g—Consequence of 2,3 (hypothetical syllogism)
You'll have to take my word for it that 1. is a substitution instance of a theorem of K. Notice that this proof this proof suggests that our theorem is actually a theorem in the system KB or any extension thereof. What? You don't know what that means? Well, pull up a chair.
The weakest modal system that's worth a damn is called K, for Saul Kripke. K consists of the axioms of propositional logic and the K axiom. That axiom is L(p>q)>(Lp>Lq). K also includes three inference rules: modus ponens, universal substitution (any substitution instance of a theorem is a theorem), and necessitation (if A is a theorem, LA is a theorem). K is a very weak system. In K, the expression Lp>p is not a theorem. In English, that expression would be read as "If p is necessary, p is true" or something like that. Since that expression isn't a theorem, it means that we're hard-pressed to understand L as necessity in K. To solve this problem, we need a stronger logic. One such logic is T. T is an extension of K; it includes everything in K, plus a new axiom, also called T. The T axiom is Lp>p. There are a number of stronger logics, including S4, S5, B, Triv, and Ver. Each of those systems is an extension of T. S4 adds the 4 axiom, which is Lp>LLp. S5 adds the E axiom, which is Mp>LMp. B adds the B axiom, which is p>LMp. S5 includes both S4 and B. S4 and B are independent; neither contains the other.
We have some ability to mix and match these and other axioms. When I talk about the system KB, I'm talking about the system formed by K + the B axiom. KB is a relatively weak logic. This means that if you accept KB or any stronger modal logic (which just about everyone does), you've got no choice but accept that our ontological proof is a logical truth.
If you don't want to convert (which is eminently reasonable), you've got a couple of options. First, you can reject the argument for the antecedent. The easiest way to do this is probably to reject Mg. That entails claiming that the notion of God is somehow contradictory. It is also possible to reject L(g>Lg). This would mean rejecting the idea that God is a necessary being. Second, you can reject all logics stronger than KB. One of my colleagues takes this approach; he counts the fact that a given logical system can be used to make this type of argument to be a reductio against that logical system. You can still get some strong logics. For example, S4.2 is K + the 4 axiom + the G1 axiom. However, you can't get S5, which is the current front-runner. When I talk about choosing a logic, I don't mean you can just do so arbitrarily. One reason S5 is the most popular logic is that all statements with nested modalities (such as LMLp) reduce to statements with a single modal operator. Consider the English version of LMLp: It is necessarily possible that it is necessary that p. It is unclear what, if anything, that statement means. In S5, you can simply delete all the modal operators except the last one. This means that LMLp is equivalent to Lp. Lp has an easy (and meaningful) translation. In S4.2, all strings of modal operators are equivalent to one of the following strings: L, M, ML, LM. That's about as good as it gets without B.
I take a third option. I see theorems like L(g>Lg)^Mg)>g as artefacts of a formal system. That theorem is interesting in the same way that the Queen's Gambit is interesting. That is, it's an unexpected result of a series of moves in a formal system, and it's got some curious properties—but it doesn't mean anything. In general, I take this position w/r/t the whole of logic and mathematics. In fact, I see the lack of meaning to be the great strength of logic and math. The reason mathematic representations of the world and formal logical representations of thought have been so effective is that we are free to reinterpret the semantic content of the relevant symbols as appropriate for a given context.
There are some other interesting issues raised by this argument. For example, it's not at all clear that all things that are conceivable are possible (or vice versa). I take it that this is one of Putnam's claims with his "Brains in a Vat" article. Time travel might be an example of another such phenomenon which is conceivable (in some sense) but still logically impossible. Similarly, it's plausible that there might be possibilities that outstrip our ability to conceive of them. There's no obvious contradiction in saying there might be a world with twice as many colors in the visible light spectrum, but I certainly can't conceive of what that world might look like. These considerations suggest that God's conceivability may not imply God's possibility.
(L(g>Lg)^Mg)>g.
Now, for some explanation. A note on notation: > is the material conditional, and ^ is the conjunction. L is the necessity operator. It is read as "it is necessary that..." M is the possibility operator. M is read just as you would imagine. g stands for "God exists." So in English, that theorem means something like "If it is the case that God is such that God exists necessarily if God exists at all and it is the case that God possibly exists, then God exists." We now provide an argument for the antecedent. Surely it's analytic that God is a necessary being (if God exists at all). And it's equally obvious that God is possible; after all, God is conceivable. Therefore, God exists.
The proof of the theorem (in a variant of Hughes & Cresswell's system) looks like this:*
* The proof is only 4 lines, but what follows the proof is long and probably boring. I talk about how modal logics are constructed, and present some discussion about choosing a logic. I go on to present two ways to reject the conclusion of the ontological argument. I close with some thoughts on what it means to say that a given statement is a theorem or a logical truth in a particular system and on the relation between possibility and conceivability. Continue if you like.
1. L(g>Lg)>(Mg>MLg)—Theorem K11 + universal substitution
2. (L(g>Lg)^Mg)>MLg—Truth-functional consequence of 1 (exportation)
3. MLg>g—The B axiom + duality
4. (L(g>Lg)^Mg)>g—Consequence of 2,3 (hypothetical syllogism)
You'll have to take my word for it that 1. is a substitution instance of a theorem of K. Notice that this proof this proof suggests that our theorem is actually a theorem in the system KB or any extension thereof. What? You don't know what that means? Well, pull up a chair.
The weakest modal system that's worth a damn is called K, for Saul Kripke. K consists of the axioms of propositional logic and the K axiom. That axiom is L(p>q)>(Lp>Lq). K also includes three inference rules: modus ponens, universal substitution (any substitution instance of a theorem is a theorem), and necessitation (if A is a theorem, LA is a theorem). K is a very weak system. In K, the expression Lp>p is not a theorem. In English, that expression would be read as "If p is necessary, p is true" or something like that. Since that expression isn't a theorem, it means that we're hard-pressed to understand L as necessity in K. To solve this problem, we need a stronger logic. One such logic is T. T is an extension of K; it includes everything in K, plus a new axiom, also called T. The T axiom is Lp>p. There are a number of stronger logics, including S4, S5, B, Triv, and Ver. Each of those systems is an extension of T. S4 adds the 4 axiom, which is Lp>LLp. S5 adds the E axiom, which is Mp>LMp. B adds the B axiom, which is p>LMp. S5 includes both S4 and B. S4 and B are independent; neither contains the other.
We have some ability to mix and match these and other axioms. When I talk about the system KB, I'm talking about the system formed by K + the B axiom. KB is a relatively weak logic. This means that if you accept KB or any stronger modal logic (which just about everyone does), you've got no choice but accept that our ontological proof is a logical truth.
If you don't want to convert (which is eminently reasonable), you've got a couple of options. First, you can reject the argument for the antecedent. The easiest way to do this is probably to reject Mg. That entails claiming that the notion of God is somehow contradictory. It is also possible to reject L(g>Lg). This would mean rejecting the idea that God is a necessary being. Second, you can reject all logics stronger than KB. One of my colleagues takes this approach; he counts the fact that a given logical system can be used to make this type of argument to be a reductio against that logical system. You can still get some strong logics. For example, S4.2 is K + the 4 axiom + the G1 axiom. However, you can't get S5, which is the current front-runner. When I talk about choosing a logic, I don't mean you can just do so arbitrarily. One reason S5 is the most popular logic is that all statements with nested modalities (such as LMLp) reduce to statements with a single modal operator. Consider the English version of LMLp: It is necessarily possible that it is necessary that p. It is unclear what, if anything, that statement means. In S5, you can simply delete all the modal operators except the last one. This means that LMLp is equivalent to Lp. Lp has an easy (and meaningful) translation. In S4.2, all strings of modal operators are equivalent to one of the following strings: L, M, ML, LM. That's about as good as it gets without B.
I take a third option. I see theorems like L(g>Lg)^Mg)>g as artefacts of a formal system. That theorem is interesting in the same way that the Queen's Gambit is interesting. That is, it's an unexpected result of a series of moves in a formal system, and it's got some curious properties—but it doesn't mean anything. In general, I take this position w/r/t the whole of logic and mathematics. In fact, I see the lack of meaning to be the great strength of logic and math. The reason mathematic representations of the world and formal logical representations of thought have been so effective is that we are free to reinterpret the semantic content of the relevant symbols as appropriate for a given context.
There are some other interesting issues raised by this argument. For example, it's not at all clear that all things that are conceivable are possible (or vice versa). I take it that this is one of Putnam's claims with his "Brains in a Vat" article. Time travel might be an example of another such phenomenon which is conceivable (in some sense) but still logically impossible. Similarly, it's plausible that there might be possibilities that outstrip our ability to conceive of them. There's no obvious contradiction in saying there might be a world with twice as many colors in the visible light spectrum, but I certainly can't conceive of what that world might look like. These considerations suggest that God's conceivability may not imply God's possibility.
posted by Dan @ 9:55 AM
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