= 27d567bc2d48d9f40e9ccc20ea370b15)
Manfred comments on Unsolved Problems in Philosophy Part 1: The Liar's Paradox - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (130)
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.