One of our members recently posted a link to a great website, The Fallacy Files, on our facebook wall. I did some browsing, and stumbled upon a modal logic fallacy: the modal scope fallacy. I did some more clicking, and ended up reading Professor Norman Swartz’ article on the modal scope fallacy. In it, he claims that the argument against god’s omnipotence based on the immovable stone commits the modal scope fallacy. This is one of my favorite arguments, so I decided to take a closer peek. It turns out that Swartz’s analysis is a bit too coarse-grained, representing “god is omnipotent” as ‘G’, and “god creates an immovable stone” as ‘M’. I thought this might be a nice opportunity to show what modal logicians do. I will give a more precise analysis of the argument, find out that Swartz’s analysis holds, and then show why under the assumption that god is a necessary being, the argument holds, just as Swartz speculates. This will bore and disappoint many of you.
I’ll start with Swartz’s analysis of the argument. He says:
God is omnipotent, i.e. God can do anything which is logically possible. Making a stone which is so heavy that it cannot be moved is logically possible. Therefore God, being omnipotent, can make a stone so heavy that it cannot be moved. But if God makes a stone so heavy that it cannot be moved, then God cannot move it. But if God cannot move that stone, then there is something God cannot do, and hence God is not omnipotent. Thus if God is omnipotent, then God is not omnipotent. But any property which implies its contradictory is self-contradictory. Thus the very notion of God’s (or anyone’s) being omnipotent is logically impossible (self-contradictory).
The argument, as presented just above, is an unholy amalgam of two different arguments, one valid, the other invalid. The valid argument is this (where “G” = “God is omnipotent” and “M” = “God makes an immovable stone”):
Although the immediately preceding argument is valid, its second premise is false. The true premise is used in this next argument, but this next argument is invalid:
To derive ~G from the latter pair of premises, one would have to add the further premise, M. But so long as M is false, the conclusion ~G remains underivable. God, thus, remains omnipotent provided that God does nothing, e.g. making an immovable stone, which destroys His/Her omnipotence.
(Question: What if God is omnipotent – as some have argued – of logical necessity and exists necessarily, i.e. in every possible world? The answer, I’m pretty sure, is that, under these conditions, God’s making a stone so heavy that God cannot move it is a logical impossibility.)
He says the second premise of the valid argument is false. The premise says, “if it is possible for god to create an immovable stone, then god is not omnipotent.” As the English version of the argument indicates (I’ve emboldened the sentence), god’s lack of omnipotence follows from his actually making an immovable stone. The beginning of that sentence is not “If god can make…”, but “If god makes…” This is a big difference, modally. It is like the difference between “If one can murder his father, then one deserves punishment” versus “If one murders his father, then one deserves punishment.” We need god’s omnipotence to be destroyed by the possibility of making an immovable stone, and so far the above argument does not establish that.
Upon reading Swartz’s analysis thus far, I felt the skeptical beast inside me awaken and grumble. I doubted that his analysis was correct, but I recognized that everything else he had said so far in the article seemed correct. I had a good reason to believe what he said. So, I suspended my belief that the argument was valid, and decided to look at it more closely. Two things immediately jumped out at me as requiring closer analysis. First, the notion of an agent’s ability to do things can be analyzed with a system of action logic, called STIT. STIT stands for Sees To It That. It is a logical operator that connects an agent to a sentence, representing the idea that when an agent does something, he makes sure that some sentences are true. For example, when I walk my dogs, I see to it that the sentence, “Sasha and Gypsy are walking outside” is true. We can let the symbol W express that sentence. So, to logically represent my walking the dogs, we can say [seth STIT: W]. Now, STIT logic represents ability like this: [seth STIT: W]; this says ‘seth has the ability to see to it that sasha and gypsy are walking outside.’ With the tools of STIT logic, we can more precisely analyze the parts of the argument that deal with god’s ability to do things. But before we dig in, there is another element to the logic I must explain. That is the second thing that jumped out at me.
To say that god is omnipotent is to say that god can do anything. Some want to qualify this with “… that is logically possible.” I will ignore this qualification. The important thing to recognize is that the claim “god can do anything” requires first order predicate logic, not merely propositional logic. Predicate logic allows us to express things like “All men are mortal,” and “Some people suck” mathematically, and it comes to us compliments of Frege and Russell. Predicate logic lets us generalize over claims about all things and some things. Because “god can do anything” makes a claim that generalizes over all things, we need predicate logic for a precise analysis. The symbols ‘x’ say “for all x’, and the symbols ‘x’ say “there exists an x’ or ‘for some x’. Now we can combine STIT logic and predicate logic to say ‘god is omnipotent’:
1. x[g STIT: x].
In english: For all x, god has the ability to see to it that x. So, x is a variable that can express any sentence. At this point, the “logically possible” qualification might insist that x not be a self-contradiction, but I will just take it for granted that x is self-consistent. To represent ‘god can make an immovable stone,’ I’ll first convert it to the long english version of the logicese: ‘god has the ability to see to it that there exists a stone such that nothing can see to it that the stone moves.’ Let Sy = ‘y is a stone’ and My = ‘y moves’. Then we have the following:
2. [g STIT: y(Sy & ~z([z STIT: My]))].
This is ‘god can make an immovable stone’ in predicate STIT logic. Suppose statement (1) is true, that god is omnipotent. It follows, then, that (2) is true, that god can make an immovable stone. In order for (2) to be true, there must be some possible world in which god does create an immovable stone. That’s what it means for something to be possible. We’ll call that possible world ‘U’. At possible world U, god creates an immovable stone. Now, U may not be the actual world, in which case god does not actually create an immovable stone. This is why Swartz’s analysis makes the argument invalid. However, at world U, god does create an immovable stone, meaning at U the sentence [g STIT: y(Sy & ~z[z STIT: My])] is true. So, y(Sy & ~z[z STIT: My]) is also true at U, meaning that at U, there exists an immovable stone, we’ll call it ‘s’. So, at U, ~z[z STIT: Ms] is true, meaning ‘nothing can see to it that the stone moves.’ Logically equivalent to this, we can say z~[z STIT: Ms]. For all z, z cannot see to it that the stone moves. But, we can let z = g, because z is just a universally generalized variable. So, ~[g STIT: Ms]. At U, it is not possible that god sees to it that the stone moves.
To say that something is not possible is to say that it is necessarily false. So, ~[g STIT: Ms]; necessarily, god doesn’t see to it that the stone moves. This means, in all possible worlds, including U, god doesn’t see to it that the stone moves: ~[g STIT: Ms]. In Swartz’s analysis, god remains omnipotent in the actual world as long as he doesn’t create the immovable stone. We see now that if he creates the immovable stone, then he can’t move it, but his inability to move it does not follow from his ability to create it. So, in no possible world does he move the immovable stone, but so long as he doesn’t create the immovable stone in the actual world, it does not threaten his omnipotence in the actual world. At U, however, god is not omnipotent.
In order to get a contradiction with (1), we need ~[g STIT: Ms] true in the actual world. So far, we haven’t established that. All we have at the actual world is that ~[g STIT: Ms]; god doesn’t move the immovable stone (because there isn’t one to move). So far, the argument is still invalid for the same reasons Swartz says! Imagine my disappointment! However, this defense rests on a crucial suppression: (1) is not necessarily true. This means, that there are some possible worlds, like U, at which god is not omnipotent. So, in order for the argument against god’s omnipotence to be invalid, one must admit that god is not necessarily omnipotent! This is a surprising result, which most theists would want to deny. Because, they want the following to be true: x[g STIT: x]. Necessarily, god is omnipotent.
1*. x[g STIT: x].
If we suppose (1*) is true, then by the exact same reasoning as before, we will wind up with ~[g STIT: Ms] at world U. But now, because (1*) is a necessary truth, we also have [g STIT: Ms] at U. [g STIT: Ms] and ~[g STIT: Ms] contradict each other. So, (1*) is false. Now, most theists think god is a necessary being that exists in all possible worlds, and that by definition god is omnipotent. However, we just showed that god is not necessarily omnipotent. So, if god is omnipotent by definition, then god does not necessarily exist, that is, god possibly doesn’t exist. I am content with this result, because it undermines a crucial claim the theist wants to make, but I would rather make the move from possible non-existence to actual non-existence. Unfortunately, the argument against god’s omnipotence can show only that s/he’s not necessarily omnipotent. So, if god is omnipotent by definition, then god is not a necessary being, which means god possibly doesn’t exist. That’s pretty neat.
Seth Kurtenbach here. The above link is a claim that can be formalized with modal logic. Modal logic is a branch of logic that deals with necessity and possibility qualifiers on truth. Depending on how one specifies the type of necessity and possibility, one can formalize arguments from many domains that cannot otherwise be precisely represented with regular old classical logic. Classical logic is strictly extensional, because it was developed to explore the foundations of mathematics. It is complicated why this is so, but trust me, it is. An intensional logic is one that does not guarantee the substitution of identities. For example, suppose the number of planets = 8. Furthermore, suppose, reasonably, that 8 = 8. In regular extensional logic, one can substitute these equivalencies, due to transitivity. However, suppose one wants to say that necessarily, 8 = 8. Now, one cannot validly make the substitution. It is not necessary that the number of planets = 8. It is possible that the number of planets = 9. This is an example of intensionality disrupting the logic. In the last half of the 20th century much work has gone into exploring the various types of modal logic and their rules of inference.
During the recent XKCD hubbub about all Wikipedia roads leading to philosophy, I encountered a Wikipedia page about the formal sciences. The page indicates that these sciences are different from most other sciences insofar as the formal sciences are a priori, while the other sciences are a posteriori. That is, the formal sciences are not empirical, but instead amass knowledge based on definitions and axioms.
Among the so-called formal sciences is logic and its various subfields. I do work in modal logic, and so it would seem that if Wikipedia is correct, then I am a sort of scientist. But, I have never thought of my work as being a type of science. I usually consider logic to be a subfield of analytic philosophy, and I consider analytic philosophy to be distinct from science. In that same vein, I am not sure any of the so-called formal sciences are actually sciences, because my impression is that all science is primarily empirical.
I would like to know what others think about these so-called formal sciences, specifically about logic. Am I a formal scientist, or is that a misnomer?
This week’s posts
- RT @AmericanAtheist: .@todayshow @alroker @NMoralesNBC Atheists are citizens too. Leave your god out of journalism. Remember we don't all b… 1 week ago
- Buying CAFO products is bad, mmm-kay, Part 1: youtu.be/s8C9ajXYZOI?a via @YouTube 2 weeks ago
- Buying CAFO products is bad, mmm-kay, Part 2: youtu.be/NGrCwXW084U?a via @YouTube 3 weeks ago
- About to get started at Speaker's Circle. Come out and help us #DefendDissent 4 weeks ago
- Come to Speaker's Circle today at noon to #DefendDisssent facebook.com/events/1828297… 4 weeks ago