I like the article’s approach, but it’s a bit arbitrary in that “true contradiction” and “false contradiction” are equivalent. But perhaps due to bias towards the positive they get characterized as “true.”
What the Liar’s paradox really demonstrates is that true and false are not general enough to apply to every sentence, and so to deal with such cases satisfactorily we must generalize our logic somehow.
Then the question is—which generalization do we make? Going with the first thing that pops into our heads is probably bad. Well, let’s start with some desiderata:
1) We want it to assign a definite classification to the Liar’s sentence. Fairly straightforward—whether it’s “option 3” or “1/2″ or “0.321374...” we want our system to be able to handle the Liar’s sentence without breaking.
2) It should reduce to classical logic in classical cases.
3) It should not be more complicated than necessary.
4) it should not be obviously vulnerable to a strengthened Liar’s paradox.
5, etc.) Help me out here :P
Desideratum (3) suggests something along the lines of this, but that might fall prey to (4). I think it’s possible that we’ll need to allow a continuous truth value. But for now, sleep!
EDIT: After a little experience with this stuff, I don’t like the article’s approach anymore. “This sentence is not true and is not a ‘true paradox.’”
A little sleep, a little progress. The “fuzzy logic” approach that gives each statement a truth value between 0 and 1 can’t handle the obvious “this sentence is not true,” so it’s out. The other one-parameter approach I can think of is more clever. The thought was that each self-referential statement defines a transformation of it’s own “truth vector” (T, F), so consistency means that the statement should evaluate to eigenvectors of the transformation. Unfortunately, these transformations don’t always commute, so you can get inconsistent answers to “this sentence is not true and is not (1/sqrt(2),1/sqrt(2)).” Still working on that one.
Tordmor’s first sentence below is correct, the system should be boolean arithmetic. (that’s all that’s correct in his post...)
Turing proved that any computational process (if we’re being formalists and saying that our philosophical problems are computations) can be simulated in a universal turing machine, and you can write those in binary; so in some sense you really only have two values to deal with. Given a trinary table of truth values, you can run the same computation in a binary system, and then in that binary system write a liar’s paradox and translate it.
I don’t know what you’d get but it might be something along the lines of “this proposition is (true and false) xor (both)” as a wild guess.
The Liar’s sentence is already uncomputable, so I’ve already abandoned Turning machines by attempting to give it a consistent classification. So his proposed desideratum 5 conflicts with what I consider to be the more important desideratum 1.
The sentence “assign a consistent classification” sounds an awful lot like computing something to me. If you have a different meaning in mind then please elaborate. “Caught by the bug-checker” seems to be what people have settled on elsewhere.
The liar’s sentence isn’t incomputable, it just never returns a value. My point is that you can’t use a third variable to fix everything.
The sentence “assign a consistent classification” sounds an awful lot like computing something to me.
Something does get computed, but not the usual thing. It is possible to write a computer program that can use the symbol “pi.” It is not possible to write computer program to tell you every digit of pi. But on the other hand, if it’s as easy as writing “pi,” there’s not much point to thinking of it as a computer program.
The liar’s sentence isn’t incomputable, it just never returns a value.
If it was computable, it would return a value. If P->Q, then not Q->not P.
My point is that you can’t use a third variable to fix everything.
We agree: in fact, that was a central point—adding more states is still trying to compute the same thing, and so it won’t fix everything for the same reason using boolean arithmetic won’t fix everything. In order to handle the liar’s paradox we need to change the comparison operation (pretty sure, unless we avoid the problem), thus doing away with boolean arithmetic.
When I think “not computable” I think of things which aren’t implementable as computations. For the definition “implementable as a computation of finite length” versus as a program of finite length, pi seems to become incomputable… so that use of incomputable is weird to me.
I do believe that we agree. Creating a different solution to the liar paradox requires us to abandon formalism… but as far as I am aware the whole point of formalism is to give us good criteria for when our answers are satisfying, so I don’t really see how abandoning it helps.
The linked three valued logic failes because it is no boolean arithmetic which is impossible with only three states. You need at least four: true, false, contradictory and ambigous. With these you can not only solve the liar paradox but also the proposition “This proposition is true” which is ambigous. And no, that does not mean it would be false because it states it where true while it actually is ambigous. It is simply ambigous.
As a funny side note, I think that is where Gödel erred. His incompleteness theorem probably rests on a two valued logic. But I’m not a mathematician and can’t proof that.
You won’t create anything worthwhile in math if you don’t study it. To break your current system, consider the proposition “This proposition is either false, contradictory, or ambiguous”.
You are absolutely correct. I haven’t thought this through. Thank you for the lesson.
Edit: I did take the lesson that I should think more before making such a claim, however, I wanted to point out that your sentence poses no problem and was not the point.
this p. is false is contradictory
this p. is condradictory/ambigous is false
The conjunction of contradictory and false is contradictory so you have a unique solution.
This is also what intuition tells us since the proposition cannot be true and cannot be false and that would be contradictory.
I don’t understand your solution. If the proposition is contradictory, then it’s true—just look at what it says.
Or maybe I don’t understand how we are supposed to assign truth values to disjunctions (“either/or”) in your system: can a disjunction still be contradictory if one of its clauses is true? And surely if X is contradictory, then the clause “X is contradictory” must be true… or is it?
Any system that does not give this proposition the value of ‘true’ is wrong, for all definitions of true and wrong that are useful, coherent, or reasonable.
Hmm. I was going to say “assign it the value of true, and it returns true. Assign it the value of false, and it returns a contradiction”, but on reflection that’s not the case. If you assign it the value of false, then the claim becomes ¬(A is true), so it returns false.
So I was wrong—the proposition is a null proposition, it simply returns the truth value you assign to it. I don’t know if ambiguous is the best way to describe it, but ‘true’ certainly isn’t.
Tordmor messed up and wrote “This proposition is true” when he probably would have wanted to have referred to “This proposition is false”.
Shokwave correctly notes that “This proposition is true” isn’t ambiguous at all, it essentially returns the value True.
Jonii also correctly observes that the person speaking the claim “This proposition is true” could be lying or mistaken (to the extent that the statement has bearing on facts external to the phrase). Apparent disagreement with Shokwave is likely to be due to ambiguity in the casual English representations of logical dereferencing.
How did you determine that the sentence “This proposition is true” returns the value True?
Again, English is messy. Shokwave was noting (and I was acknowledging) that there is the claim of truth.
To me it doesn’t seem to return any value. Tordmor correctly notes its truth-state is uncertain.
No he doesn’t. He claims it is ambiguous—an entirely different thing. It is an unambiguous claim to be true. Such a claim can itself be false but the meaning is entirely clear. It says it’s true!
Contrast with “This statement is false”.
These distinctions become relevant when Omega throws you box puzzles like this.
I like the article’s approach, but it’s a bit arbitrary in that “true contradiction” and “false contradiction” are equivalent. But perhaps due to bias towards the positive they get characterized as “true.”
What the Liar’s paradox really demonstrates is that true and false are not general enough to apply to every sentence, and so to deal with such cases satisfactorily we must generalize our logic somehow.
Then the question is—which generalization do we make? Going with the first thing that pops into our heads is probably bad. Well, let’s start with some desiderata:
1) We want it to assign a definite classification to the Liar’s sentence. Fairly straightforward—whether it’s “option 3” or “1/2″ or “0.321374...” we want our system to be able to handle the Liar’s sentence without breaking.
2) It should reduce to classical logic in classical cases.
3) It should not be more complicated than necessary.
4) it should not be obviously vulnerable to a strengthened Liar’s paradox.
5, etc.) Help me out here :P
Desideratum (3) suggests something along the lines of this, but that might fall prey to (4). I think it’s possible that we’ll need to allow a continuous truth value. But for now, sleep!
EDIT: After a little experience with this stuff, I don’t like the article’s approach anymore. “This sentence is not true and is not a ‘true paradox.’”
Manfred’s log, stardate 11⁄30
A little sleep, a little progress. The “fuzzy logic” approach that gives each statement a truth value between 0 and 1 can’t handle the obvious “this sentence is not true,” so it’s out. The other one-parameter approach I can think of is more clever. The thought was that each self-referential statement defines a transformation of it’s own “truth vector” (T, F), so consistency means that the statement should evaluate to eigenvectors of the transformation. Unfortunately, these transformations don’t always commute, so you can get inconsistent answers to “this sentence is not true and is not (1/sqrt(2),1/sqrt(2)).” Still working on that one.
Tordmor’s first sentence below is correct, the system should be boolean arithmetic. (that’s all that’s correct in his post...)
Turing proved that any computational process (if we’re being formalists and saying that our philosophical problems are computations) can be simulated in a universal turing machine, and you can write those in binary; so in some sense you really only have two values to deal with. Given a trinary table of truth values, you can run the same computation in a binary system, and then in that binary system write a liar’s paradox and translate it.
I don’t know what you’d get but it might be something along the lines of “this proposition is (true and false) xor (both)” as a wild guess.
The Liar’s sentence is already uncomputable, so I’ve already abandoned Turning machines by attempting to give it a consistent classification. So his proposed desideratum 5 conflicts with what I consider to be the more important desideratum 1.
The sentence “assign a consistent classification” sounds an awful lot like computing something to me. If you have a different meaning in mind then please elaborate. “Caught by the bug-checker” seems to be what people have settled on elsewhere.
The liar’s sentence isn’t incomputable, it just never returns a value. My point is that you can’t use a third variable to fix everything.
Something does get computed, but not the usual thing. It is possible to write a computer program that can use the symbol “pi.” It is not possible to write computer program to tell you every digit of pi. But on the other hand, if it’s as easy as writing “pi,” there’s not much point to thinking of it as a computer program.
If it was computable, it would return a value. If P->Q, then not Q->not P.
We agree: in fact, that was a central point—adding more states is still trying to compute the same thing, and so it won’t fix everything for the same reason using boolean arithmetic won’t fix everything. In order to handle the liar’s paradox we need to change the comparison operation (pretty sure, unless we avoid the problem), thus doing away with boolean arithmetic.
When I think “not computable” I think of things which aren’t implementable as computations. For the definition “implementable as a computation of finite length” versus as a program of finite length, pi seems to become incomputable… so that use of incomputable is weird to me.
I do believe that we agree. Creating a different solution to the liar paradox requires us to abandon formalism… but as far as I am aware the whole point of formalism is to give us good criteria for when our answers are satisfying, so I don’t really see how abandoning it helps.
5) It should be a boolean arithmetic
The linked three valued logic failes because it is no boolean arithmetic which is impossible with only three states. You need at least four: true, false, contradictory and ambigous. With these you can not only solve the liar paradox but also the proposition “This proposition is true” which is ambigous. And no, that does not mean it would be false because it states it where true while it actually is ambigous. It is simply ambigous.
As a funny side note, I think that is where Gödel erred. His incompleteness theorem probably rests on a two valued logic. But I’m not a mathematician and can’t proof that.
You won’t create anything worthwhile in math if you don’t study it. To break your current system, consider the proposition “This proposition is either false, contradictory, or ambiguous”.
You are absolutely correct. I haven’t thought this through. Thank you for the lesson.
Edit: I did take the lesson that I should think more before making such a claim, however, I wanted to point out that your sentence poses no problem and was not the point.
this p. is false is contradictory this p. is condradictory/ambigous is false The conjunction of contradictory and false is contradictory so you have a unique solution. This is also what intuition tells us since the proposition cannot be true and cannot be false and that would be contradictory.
I don’t understand your solution. If the proposition is contradictory, then it’s true—just look at what it says.
Or maybe I don’t understand how we are supposed to assign truth values to disjunctions (“either/or”) in your system: can a disjunction still be contradictory if one of its clauses is true? And surely if X is contradictory, then the clause “X is contradictory” must be true… or is it?
Ok, I get it now. So, I was wrong on that too. Thank you.
What do you do with “This sentence is contradictory”?
false.
The method would be to ask: Can it be true? Can it be false?
If yes to both it is ambigous, if no to both it is contradictory.
This makes no sense.
I’m neither a mathematician nor a linguist but I think you mean ‘prove’.
Any system that does not give this proposition the value of ‘true’ is wrong, for all definitions of true and wrong that are useful, coherent, or reasonable.
Mind explaining why? I don’t see any reason it’s any more true than it is false.
Hmm. I was going to say “assign it the value of true, and it returns true. Assign it the value of false, and it returns a contradiction”, but on reflection that’s not the case. If you assign it the value of false, then the claim becomes ¬(A is true), so it returns false.
So I was wrong—the proposition is a null proposition, it simply returns the truth value you assign to it. I don’t know if ambiguous is the best way to describe it, but ‘true’ certainly isn’t.
edit: perhaps cata’s ‘trivial’ is a good word for it.
Interesting. If I infer correctly...
Tordmor messed up and wrote “This proposition is true” when he probably would have wanted to have referred to “This proposition is false”.
Shokwave correctly notes that “This proposition is true” isn’t ambiguous at all, it essentially returns the value True.
Jonii also correctly observes that the person speaking the claim “This proposition is true” could be lying or mistaken (to the extent that the statement has bearing on facts external to the phrase). Apparent disagreement with Shokwave is likely to be due to ambiguity in the casual English representations of logical dereferencing.
How did you determine that the sentence “This proposition is true” returns the value True?
To me it doesn’t seem to return any value. Tordmor correctly notes its truth-state is uncertain.
Again, English is messy. Shokwave was noting (and I was acknowledging) that there is the claim of truth.
No he doesn’t. He claims it is ambiguous—an entirely different thing. It is an unambiguous claim to be true. Such a claim can itself be false but the meaning is entirely clear. It says it’s true!
Contrast with “This statement is false”.
These distinctions become relevant when Omega throws you box puzzles like this.