Tordmor comments on Unsolved Problems in Philosophy Part 1: The Liar's Paradox - Less Wrong

4 Post author: Kevin 30 November 2010 08:56AM

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Comment author: Manfred 30 November 2010 10:08:43AM * 3 points [-]

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.'"

Comment author: [deleted] comment score below threshold [+] (14 children)

Comment author: cousin_it 30 November 2010 03:49:13PM 4 points [-]

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".

Comment author: [deleted] 02 December 2010 06:41:46AM * 0 points [-]

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.

Comment author: cousin_it 02 December 2010 06:58:57AM * 0 points [-]

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?

Comment author: [deleted] 02 December 2010 09:16:43AM 1 point [-]

Ok, I get it now. So, I was wrong on that too. Thank you.

Comment author: Oscar_Cunningham 30 November 2010 07:34:04PM 0 points [-]

What do you do with "This sentence is contradictory"?

Comment author: [deleted] 02 December 2010 06:57:29AM 0 points [-]

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.

Comment author: wedrifid 02 December 2010 07:34:20AM 0 points [-]

What do you do with "This sentence is contradictory"?

false.

This makes no sense.

Comment author: shokwave 30 November 2010 04:53:33PM -1 points [-]

"This proposition is true" which is ambiguous

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.

Comment author: Jonii 30 November 2010 05:28:02PM 1 point [-]

Mind explaining why? I don't see any reason it's any more true than it is false.

Comment author: shokwave 01 December 2010 01:46:49AM * 0 points [-]

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.

Comment author: wedrifid 30 November 2010 06:18:42PM * 0 points [-]

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.

Comment author: ArisKatsaris 30 November 2010 07:34:26PM * 0 points [-]

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.

Comment author: wedrifid 01 December 2010 04:21:01AM * 0 points [-]

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.

Tordmor comments on Unsolved Problems in Philosophy Part 1: The Liar's Paradox - Less Wrong

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