I argue that meanings are fundamentally fuzzy. In the end, we can interpret things your way, if we think of fuzzy truth-values as sent to “true” or “false” based on an unknown threshhold (which we can have a probability distribution over). However, it is worth noting that the fuzzy truth-values can be logically coherent in a way that the true-false values cannot: the fuzzy truth predicate is just an identity function (so “X is true” has the same fuzzy truth-value as “X”), and this allows us to reason about paradoxical sentences in a completely consistent way (so the self-referential sentence “this sentence is not true” is value 1⁄2). But when we force things to be true or false, the resulting picture must violate some rules of logic (“this sentence is not true” must either be true or false, neither of which is consistent with classical logic’s rules of reasoning).
Digging further into your proposal, what makes a sentence true or false? It seems that you suppose a person always has a precise meaning in mind when they utter a sentence. What sort of object do you think this meaning is? Do you think it is plausible, say, that the human uttering the sentence could always say later whether it was true or false, if given enough information about the world? Or is it possible that even the utterer could be unsure about that, even given all the relevant facts about how the world is? I think even the original utterer can remain unsure—but then, where does the fact-of-the-matter reside?
For example, if I claim something is 500 meters away, I don’t have a specific range in mind. I’ll know I was correct if more accurate measurement establishes that it’s 500.001 meters away. I’ll know I was incorrect if it turns out to be 10 meters away. However, there will be some boundary where I will be inclined to say my intended claim was vague, and I “fundamentally” don’t know—I don’t think there was a fact of the matter about the precise range I intended with my statement, beyond some level of detail.
So, if you propose to model this no-fact-of-the-matter as uncertainty, what do you propose it is uncertain about? Where does this truth reside, and how would it be checked/established?
(I agree with your argument about vagueness, but regarding the first paragraph: I wouldn’t use the 1⁄2 solution to the liar paradox as an argument in favor or fuzzy truth values. This is because even with them we still get a “revenge paradox” when we use a truth predicate that explicitly refers to some specific fuzzy truth value. E.g. “this sentence is exactly false” / “this sentence has truth value exactly 0”. If the sentence has truth value exactly 0, it is exactly true, i.e. has truth value exactly 1. Which is a contradiction. In fact, we get an analogous revenge paradox for all truth predicates “has truth value exactly x” where x is any number between 0 and 1, excluding 1. E.g. x=0.7, and even x=0.5. In a formal system of fuzzy logic we can rule out such truth predicates, but we clearly can’t forbid them in natural language.)
Any solution to the semantic paradoxes must accept something counterintuitive. In the case of using something like fuzzy logic, I must accept the restriction that valid truth-functions must be continuous (or at least Kakutani). I don’t claim this is the final word on the subject (I recognize that fuzzy logic has some severe limitations; I mostly defer to Hartry Field on how to get around these problems). However, I do think it captures a lot of reasonable intuitions. I would challenge you to name a more appealing resolution.
I’m not arguing against fuzzy logic, just that it arguably doesn’t “morally” solve the liar paradox, insofar it yields similar revenge paradoxes. In natural language we arguably can’t just place restrictions, like banning non-continous truth functions such as “is exactly false”. Even if we don’t have a more appealing resolution. We can only pose voluntary restrictions on formal languages. For natural language, the only hope would be to argue that the predicate “is exactly false” doesn’t really make sense, or doesn’t actually yield a contradiction, though that seems difficult. Though I haven’t read Field’s book. Maybe he has some good arguments.
I’m not arguing against fuzzy logic, just that it arguably doesn’t “morally” solve the liar paradox, insofar it yields similar revenge paradoxes.
It has been years since I’ve read the book, so this might be a little bit off, but Field’s response to revenge is basically this:
The semantic values (which are more complex than fuzzy values, but I’ll pretend for simplicity that they’re just fuzzy values) are models of what’s going on, not literal. This idea is intended to respond to complaints like “but we can obviously refer to truth-value-less-than-one, which gives rise to a revenge paradox”. The point of the model is to inform us about what inference systems might be sound and consistent, although we can only ever prove this in a toy setting, thanks to Godel’s theorems.
So, indeed, within this model, “is exactly false” doesn’t make sense. Speaking outside this model, it may seem to make sense, but we can only step outside of it because it is a toy model.
However, we do get the ability to state ever-stronger Liar sentences with a “definitely” operator (“definitely x” is intuitively twice as strong a truth-claim compared to “x”). So the theory deals with revenge problems in that sense by formulating an infinite hierarchy of Strengthened Liars, none of which cause a problem. IIRC Hartry’s final theory even handles iteration of the “definitely” operator infinitely many times (unlike fuzzy logic).
In natural language we arguably can’t just place restrictions, like banning non-continous truth functions such as “is exactly false”. Even if we don’t have a more appealing resolution. We can only pose voluntary restrictions on formal languages. For natural language, the only hope would be to argue that the predicate “is exactly false” doesn’t really make sense, or doesn’t actually yield a contradiction, though that seems difficult.
Of course in some sense natural language is an amorphous blob which we can only formally model as an action-space which is instrumentally useful. The question, for me, is about normative reasoning—how can we model as many of the strengths of natural language as possible, while also keeping as many of the strengths of formal logic as possible?
So I do think fuzzy logic makes some positive progress on the Liar and on revenge problems, and Hartry’s proposal makes more positive progress.
That seems fair enough. Do you know what Field had to say about the “truth teller” (“This sentence is true”)? While the liar sentence can (classically) be neither true nor false, the problem with the truth teller is that it can be either true or false, with no fact of the matter deciding which. This does seem to be a closely related problem, even it it isn’t always considered a serious paradox. I’m not aware fuzzy truth values can help here. This is on contrast to Kripke’s proposed solution to the liar paradox: On his account, both the liar and the truth teller are “ungrounded” rather than true or false, because they use the truth predicate in a way that can’t be eliminated. Though I think one can construct some revenge paradoxes with his solution as well.
Anyway, I still think the main argument for fuzzy logic (or at least fuzzy truth values, without considering how logical connectives should behave) is still that concepts seem to be inherently vague. E.g. when I believe that Bob is bald, I don’t expect him to have an exact degree of baldness. So the extension of the concept expressed by the predicate “is bald” must be a fuzzy set. So Bob is partially contained in that set, and the degree to which he is, is the fuzzy truth value of the proposition that Bob is bald. This is independent of how paradoxes are handled.
(And of course the next problem is then how fuzzy truth values could be combined with probability theory, since the classical axiomatization of probability theory assumes that truth is binary. Are beliefs perhaps about an “expected” degree of truth? How would that be formalized? I don’t know.)
I argue that meanings are fundamentally fuzzy. In the end, we can interpret things your way, if we think of fuzzy truth-values as sent to “true” or “false” based on an unknown threshhold (which we can have a probability distribution over). However, it is worth noting that the fuzzy truth-values can be logically coherent in a way that the true-false values cannot: the fuzzy truth predicate is just an identity function (so “X is true” has the same fuzzy truth-value as “X”), and this allows us to reason about paradoxical sentences in a completely consistent way (so the self-referential sentence “this sentence is not true” is value 1⁄2). But when we force things to be true or false, the resulting picture must violate some rules of logic (“this sentence is not true” must either be true or false, neither of which is consistent with classical logic’s rules of reasoning).
Digging further into your proposal, what makes a sentence true or false? It seems that you suppose a person always has a precise meaning in mind when they utter a sentence. What sort of object do you think this meaning is? Do you think it is plausible, say, that the human uttering the sentence could always say later whether it was true or false, if given enough information about the world? Or is it possible that even the utterer could be unsure about that, even given all the relevant facts about how the world is? I think even the original utterer can remain unsure—but then, where does the fact-of-the-matter reside?
For example, if I claim something is 500 meters away, I don’t have a specific range in mind. I’ll know I was correct if more accurate measurement establishes that it’s 500.001 meters away. I’ll know I was incorrect if it turns out to be 10 meters away. However, there will be some boundary where I will be inclined to say my intended claim was vague, and I “fundamentally” don’t know—I don’t think there was a fact of the matter about the precise range I intended with my statement, beyond some level of detail.
So, if you propose to model this no-fact-of-the-matter as uncertainty, what do you propose it is uncertain about? Where does this truth reside, and how would it be checked/established?
(I agree with your argument about vagueness, but regarding the first paragraph: I wouldn’t use the 1⁄2 solution to the liar paradox as an argument in favor or fuzzy truth values. This is because even with them we still get a “revenge paradox” when we use a truth predicate that explicitly refers to some specific fuzzy truth value. E.g. “this sentence is exactly false” / “this sentence has truth value exactly 0”. If the sentence has truth value exactly 0, it is exactly true, i.e. has truth value exactly 1. Which is a contradiction. In fact, we get an analogous revenge paradox for all truth predicates “has truth value exactly x” where x is any number between 0 and 1, excluding 1. E.g. x=0.7, and even x=0.5. In a formal system of fuzzy logic we can rule out such truth predicates, but we clearly can’t forbid them in natural language.)
Any solution to the semantic paradoxes must accept something counterintuitive. In the case of using something like fuzzy logic, I must accept the restriction that valid truth-functions must be continuous (or at least Kakutani). I don’t claim this is the final word on the subject (I recognize that fuzzy logic has some severe limitations; I mostly defer to Hartry Field on how to get around these problems). However, I do think it captures a lot of reasonable intuitions. I would challenge you to name a more appealing resolution.
I’m not arguing against fuzzy logic, just that it arguably doesn’t “morally” solve the liar paradox, insofar it yields similar revenge paradoxes. In natural language we arguably can’t just place restrictions, like banning non-continous truth functions such as “is exactly false”. Even if we don’t have a more appealing resolution. We can only pose voluntary restrictions on formal languages. For natural language, the only hope would be to argue that the predicate “is exactly false” doesn’t really make sense, or doesn’t actually yield a contradiction, though that seems difficult. Though I haven’t read Field’s book. Maybe he has some good arguments.
It has been years since I’ve read the book, so this might be a little bit off, but Field’s response to revenge is basically this:
The semantic values (which are more complex than fuzzy values, but I’ll pretend for simplicity that they’re just fuzzy values) are models of what’s going on, not literal. This idea is intended to respond to complaints like “but we can obviously refer to truth-value-less-than-one, which gives rise to a revenge paradox”. The point of the model is to inform us about what inference systems might be sound and consistent, although we can only ever prove this in a toy setting, thanks to Godel’s theorems.
So, indeed, within this model, “is exactly false” doesn’t make sense. Speaking outside this model, it may seem to make sense, but we can only step outside of it because it is a toy model.
However, we do get the ability to state ever-stronger Liar sentences with a “definitely” operator (“definitely x” is intuitively twice as strong a truth-claim compared to “x”). So the theory deals with revenge problems in that sense by formulating an infinite hierarchy of Strengthened Liars, none of which cause a problem. IIRC Hartry’s final theory even handles iteration of the “definitely” operator infinitely many times (unlike fuzzy logic).
Of course in some sense natural language is an amorphous blob which we can only formally model as an action-space which is instrumentally useful. The question, for me, is about normative reasoning—how can we model as many of the strengths of natural language as possible, while also keeping as many of the strengths of formal logic as possible?
So I do think fuzzy logic makes some positive progress on the Liar and on revenge problems, and Hartry’s proposal makes more positive progress.
That seems fair enough. Do you know what Field had to say about the “truth teller” (“This sentence is true”)? While the liar sentence can (classically) be neither true nor false, the problem with the truth teller is that it can be either true or false, with no fact of the matter deciding which. This does seem to be a closely related problem, even it it isn’t always considered a serious paradox. I’m not aware fuzzy truth values can help here. This is on contrast to Kripke’s proposed solution to the liar paradox: On his account, both the liar and the truth teller are “ungrounded” rather than true or false, because they use the truth predicate in a way that can’t be eliminated. Though I think one can construct some revenge paradoxes with his solution as well.
Anyway, I still think the main argument for fuzzy logic (or at least fuzzy truth values, without considering how logical connectives should behave) is still that concepts seem to be inherently vague. E.g. when I believe that Bob is bald, I don’t expect him to have an exact degree of baldness. So the extension of the concept expressed by the predicate “is bald” must be a fuzzy set. So Bob is partially contained in that set, and the degree to which he is, is the fuzzy truth value of the proposition that Bob is bald. This is independent of how paradoxes are handled.
(And of course the next problem is then how fuzzy truth values could be combined with probability theory, since the classical axiomatization of probability theory assumes that truth is binary. Are beliefs perhaps about an “expected” degree of truth? How would that be formalized? I don’t know.)