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.
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.