Anaphora and Conditionals
Let me say, before I get started, that I am way out of my league when it comes to issues involving anaphora and conditionals. In spite of that fact, I will be expressing a bit of skepticism about Williamson's treatment of conditionals with anaphora.
Williamson takes the following sentence to express one of the primary propositions in a Gettier style argument against the justified true belief account of knowledge:
(5) If a thinker were Gettier related to a proposition, then he/she would have a justified true belief without knowledge.
Williamson wants to formalize this statement. Surprisingly, Williamson's formalization of (5) is the following (again, I hope everyone can figure out my crude notation system. If not, this is intended to be identical to what Williamson labels (3*) in the book):
(3*) ExEp GC(x, p) []--> AxAp (GC(x, p) --> (JTB(x, p) & ~K(x, p)))
I find this a surprising formalization of (5) because it seems to have a lot more quantifiers and conditionals that (5) does. Put another way, (3*) seems to say that if something were in a Gettier situation, then it would be the case that anything in a Gettier situation has justified true belief without knowledge. But, that doesn't seem to be what (5) says. So, it does not seem like (3*) is a correct formalization of (5).
Williamson happens to have a few argument for the claim that (5) is a correct formalization of (3*). To understand Williamson's argument, we need to first have another candidate formalization in front of us. Here it is:
(10) AxAp (GC(x, p) []--> (JTB(x, P & ~K(x, p))
We also need a couple of candidate formalizations of the following monkey sentence:
(13) if an animal escaped from the zoo, it would be a monkey.
Here are the candidate formalizations:
(14) Ax ((animal(x) & Escaped(x)) []--> Monkey(x)
(15) Ex (animal(x) & escaped(x)) []--> Ax ((Animal(x) & Escaped(x)) --> Monkey(x)
Williamson seems to have two arguments. The first argument says that if (10) is the correct formalization of (5), then (14) is the correct formalization of (13). But, (14) is not the correct formalization of (13). So, (10) is not the correct formalization of (5).
The second argument says that if (15) is the correct formalization of (13), then (3*) is the correct formalization of (5). Moreover, (15) is the correct formalization of (13). So, (3*) is the correct formalization of (5).
Williamson seems to think that the sentences are similar enough that their formalizations will have (roughly) the same logical structure. But, this isn't obviously true to me. For instance, it seems like it follows straightfowardly from (5) that if Neal were in a Gettier situation, then Neal would have justified true belief without knowledge. But, it does not seems to follow straightforwardly from (13) that if George had escaped from the zoo, then George would be a monkey. But, if the sentences have the same logical structure, then the inferences would be both be valid.
I have to admit I've been agonizing over these sentences for a while. In fact, I've been agonizing long enough that I think I should have more to say. Unfortunately, I do not. I am unhappy with both the proposed formalizations and I have not been able to think of any plausible alternatives.
Williamson takes the following sentence to express one of the primary propositions in a Gettier style argument against the justified true belief account of knowledge:
(5) If a thinker were Gettier related to a proposition, then he/she would have a justified true belief without knowledge.
Williamson wants to formalize this statement. Surprisingly, Williamson's formalization of (5) is the following (again, I hope everyone can figure out my crude notation system. If not, this is intended to be identical to what Williamson labels (3*) in the book):
(3*) ExEp GC(x, p) []--> AxAp (GC(x, p) --> (JTB(x, p) & ~K(x, p)))
I find this a surprising formalization of (5) because it seems to have a lot more quantifiers and conditionals that (5) does. Put another way, (3*) seems to say that if something were in a Gettier situation, then it would be the case that anything in a Gettier situation has justified true belief without knowledge. But, that doesn't seem to be what (5) says. So, it does not seem like (3*) is a correct formalization of (5).
Williamson happens to have a few argument for the claim that (5) is a correct formalization of (3*). To understand Williamson's argument, we need to first have another candidate formalization in front of us. Here it is:
(10) AxAp (GC(x, p) []--> (JTB(x, P & ~K(x, p))
We also need a couple of candidate formalizations of the following monkey sentence:
(13) if an animal escaped from the zoo, it would be a monkey.
Here are the candidate formalizations:
(14) Ax ((animal(x) & Escaped(x)) []--> Monkey(x)
(15) Ex (animal(x) & escaped(x)) []--> Ax ((Animal(x) & Escaped(x)) --> Monkey(x)
Williamson seems to have two arguments. The first argument says that if (10) is the correct formalization of (5), then (14) is the correct formalization of (13). But, (14) is not the correct formalization of (13). So, (10) is not the correct formalization of (5).
The second argument says that if (15) is the correct formalization of (13), then (3*) is the correct formalization of (5). Moreover, (15) is the correct formalization of (13). So, (3*) is the correct formalization of (5).
Williamson seems to think that the sentences are similar enough that their formalizations will have (roughly) the same logical structure. But, this isn't obviously true to me. For instance, it seems like it follows straightfowardly from (5) that if Neal were in a Gettier situation, then Neal would have justified true belief without knowledge. But, it does not seems to follow straightforwardly from (13) that if George had escaped from the zoo, then George would be a monkey. But, if the sentences have the same logical structure, then the inferences would be both be valid.
I have to admit I've been agonizing over these sentences for a while. In fact, I've been agonizing long enough that I think I should have more to say. Unfortunately, I do not. I am unhappy with both the proposed formalizations and I have not been able to think of any plausible alternatives.
posted by Joshua at 11:37 AM
1 Comments:
Joshua,
I agree that there is some weirdness going on here.
For one thing, I agree that 3* seems to have way too many quantifiers and conditionals. That was my first reaction, too.
Second, I agree that the monkey sentence doesn't seem to have the same form as the Gettier sentence. In fact, the more I read (13), the more I start to think that it is just straightforwardly false and that (14) is indeed the right formalization of it. But part of the problem here is that (13) is just such a weird thing to say! The problem is similar for so-called "backtracking" counterfactuals, I think...they are all so weird! (For example: "If I were to go to the airport today, then I would have bought plane tickets yesterday.")
Perhaps a better version of the monkey sentence would be: "If an animal escaped from the zoo, then it would be caught within a couple of hours."
Again, it strikes me that the right translation of this is something along these lines:
Ax ((animal(x) & Escaped(x)) []--> Caught(x)
which is just like (14).
So were you thinking that (10) is not a good formalization of (5)? Because it sounds okay to me.
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