It’s good that you pointed out Kripke’s fixed point theory of truth as a solution to the Liar’s paradox. It seems to be an acceptable solution.
On the other hand, I also agree that “fuzziness as a matter of degree” can be added on top of a binary logic. That would be very useful for dealing with commonsense reasoning—perhaps even indispensable.
What is particularly controversial is whether turth should be regarded as a matter of degree, ie, the development of a fuzzy-valued logic. At this point, I am kinda 50-50 about it. The advantage of doing this is that we can translate commonsense notions easily, and it may be more intuitive to design and implement the AGI. The disadvantage is that we need to deal with a relatively new form of logic (ie, many-valued logic) and its formal semantics, proof theory, model theory, deduction algorithms, etc. With binary logic we may be on firmer ground.
The problem with Kripke’s solution to the paradoxes, and with any solution really, is that it still contains reference holes. If I strictly adhere to Kripke’s system, then I can’t actually explain to you the idea of meaningless sentences, because it’s always either false or meaningless to claim that a sentence is meaningless. (False when we claim it of a meaningful sentence; meaningless when we claim it of a meaningless one.)
With the fuzzy way out, the reference gap is that we can’t have discontinuous functions. This means we can’t actually talk about the fuzzy value of a statement: any claim “This statement has value X” is a discontinuous claim, with value 1 at X and value 0 everywhere else. Instead, all we can do is get arbitrarily close to saying that, by having continuous functions that are 1 at X and fall off sharply around X… this, I admit, is rather nifty, but it is still a reference gap. Warrigal refers to actual values when describing the logic, but the logic itself is incapable of doing that without running into paradox.
About the so-called “discontinuous truth values”, I think the culprit is not that the truth value is discontinuous (it doesn’t make sense to say a point-value is continuous or not), but rather that we have a binary predicate, “less-than”, which is a discontinuous truth functional mapping.
The statement “less-than(tv, 0.5)” seems to be a binary statement. If we make that predicate fuzzy, it becomes “approximately less than 0.5″, which we can visualize as a sigmoidal curve, and this curve intersects with the slope=1 line at 0.5. Thus, the truth value of the fuzzy version of that statement is 0.5, ie, indeterminate.
All in all, this problem seems to stem from the fact that we’ve introduced the binary predicate “less-than”.
If I strictly adhere to Kripke’s system, then I can’t actually explain to you the idea of meaningless sentences, because it’s always either false or meaningless to claim that a sentence is meaningless. (False when we claim it of a meaningful sentence; meaningless when we claim it of a meaningless one.)
I’d like to clear this up for myself. You’re saying that under Kripke’s system we build up a tower of meaningful statements with infinitely many floors, starting from “grounded” statements that don’t mention truth values at all. All statements outside the tower we deem meaningless, but statements of the form “statement X is meaningless” can only become grounded as true after we finish the whole tower, so we aren’t supposed to make them.
But this looks weird. If we can logically see that the statement “this statement is true” is meaningless under Kripke’s system, why can’t we run this logic under that system? Or am I confusing levels?
It’s good that you pointed out Kripke’s fixed point theory of truth as a solution to the Liar’s paradox. It seems to be an acceptable solution.
On the other hand, I also agree that “fuzziness as a matter of degree” can be added on top of a binary logic. That would be very useful for dealing with commonsense reasoning—perhaps even indispensable.
What is particularly controversial is whether turth should be regarded as a matter of degree, ie, the development of a fuzzy-valued logic. At this point, I am kinda 50-50 about it. The advantage of doing this is that we can translate commonsense notions easily, and it may be more intuitive to design and implement the AGI. The disadvantage is that we need to deal with a relatively new form of logic (ie, many-valued logic) and its formal semantics, proof theory, model theory, deduction algorithms, etc. With binary logic we may be on firmer ground.
YKY,
The problem with Kripke’s solution to the paradoxes, and with any solution really, is that it still contains reference holes. If I strictly adhere to Kripke’s system, then I can’t actually explain to you the idea of meaningless sentences, because it’s always either false or meaningless to claim that a sentence is meaningless. (False when we claim it of a meaningful sentence; meaningless when we claim it of a meaningless one.)
With the fuzzy way out, the reference gap is that we can’t have discontinuous functions. This means we can’t actually talk about the fuzzy value of a statement: any claim “This statement has value X” is a discontinuous claim, with value 1 at X and value 0 everywhere else. Instead, all we can do is get arbitrarily close to saying that, by having continuous functions that are 1 at X and fall off sharply around X… this, I admit, is rather nifty, but it is still a reference gap. Warrigal refers to actual values when describing the logic, but the logic itself is incapable of doing that without running into paradox.
About the so-called “discontinuous truth values”, I think the culprit is not that the truth value is discontinuous (it doesn’t make sense to say a point-value is continuous or not), but rather that we have a binary predicate, “less-than”, which is a discontinuous truth functional mapping.
The statement “less-than(tv, 0.5)” seems to be a binary statement. If we make that predicate fuzzy, it becomes “approximately less than 0.5″, which we can visualize as a sigmoidal curve, and this curve intersects with the slope=1 line at 0.5. Thus, the truth value of the fuzzy version of that statement is 0.5, ie, indeterminate.
All in all, this problem seems to stem from the fact that we’ve introduced the binary predicate “less-than”.
I’d like to clear this up for myself. You’re saying that under Kripke’s system we build up a tower of meaningful statements with infinitely many floors, starting from “grounded” statements that don’t mention truth values at all. All statements outside the tower we deem meaningless, but statements of the form “statement X is meaningless” can only become grounded as true after we finish the whole tower, so we aren’t supposed to make them.
But this looks weird. If we can logically see that the statement “this statement is true” is meaningless under Kripke’s system, why can’t we run this logic under that system? Or am I confusing levels?
Call it “expected” truth, analagous to “expected value” in prob and stats. It’s effectively a way to incorporate a risk analysis into your reasoning.
Yes, I have worked out a fuzzy logic with probability distributions over fuzzy values.