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cousin_it comments on Unsolved Problems in Philosophy Part 1: The Liar's Paradox - Less Wrong
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This feels like the wrong step in the dance to me. Haven't you just thrown away all of mathematics? What new things do you expect after solving a quadratic equation?
User:Tiiba is correct. Math (after mapping to real-world predicates) allows me to more quickly form reliable expectations about the world. The claim that "this sentence is false" does not. Therefore, I can leave beliefs about the latter unassigned without epistemic or instrumental penalty.
Omega places a series of 50 boxes in front of you, labelled numerically. Two of them contain explosive boobytraps while each of the remaining 48 contain $200,000. The sum of the labels on the trapped boxes is 55 while the product is 714. Which boxes would you not choose to open?
Nice example. To follow up:
Next, Omega places two boxes in front of you. One carries the label "The label on the other box contains a true sentence". The label on the other box reads "The label on the other box contains a false sentence". You are told that the box(es) without false labels contain $1,000,000, whereas the box(es) with false labels are boobytrapped. It is conceivable that the labels are meaningless - therefore not false. It is also conceivable that the labels are both true and false - contradictory, but paraconsistent.
Do you open the boxes?
Quadratic equations are relatively clear-cut.
The issue here is with language and the meaning of the word "false", not with the concept of truth independent of language.
Consider: Omega places a number of colored boxes if front of you. Omega tells you that boxes with an even number of boxes in the same basic color contain $200,000 while those with an odd number contain bombs. The colors are such that speakers of languages with different divisions of the spectrum in basic color words group the boxes differently (Omega repeats the explanation in several different languages in a random order). Does this thought experiment reveal anything interesting about the concept of color?
Omega can only do this if by some coincidence the the even/oddness of each box is the same in every language.
Yes, even so.
The issue where? Are you saying that this thread is about language, rather than truth? Or that my example, as written, is about language rather than (as intended) about truth?
I'm a bit surprised that anyone could conceive of a concept of truth independent of language. I've always considered truth as an attribute of sentences - linguistic objects. Perhaps I am missing your point.
As for your example, yes, I would say that it does point out something interesting, though already known, about the concept of color. That valid classification systems based on this criterion may disagree. This is also true of truth and logic. Some people say that the Liar statement is neither true nor false. Some people say that it is both true and false. Both can be correct, depending on what else they claim.
"Snow is white" is true if, and only if, snow is white - Tarski's material adequacy condition - only makes sense if there is a fact of the matter about whether or not snow is white independent of language.
Tarski left out some of the fine print. That "if and only if" works only under the prior assumption that "snow" designates snow, "white" designates white, and "is" designates the appropriate infix binary relation.
In other words, "Snow is white" is true only if we know that "Snow is white" is a sentence in the English language.
Not really. If "snow" designates grass, and "white" designates green, then "'snow is white' is true if and only if snow is white" is still correct. Same if "snow" designates the sky and "white" designates green.
I'm afraid I don't understand your point.
It should have read: "Same if "snow" designates the sky and "white" designates blue."
It was apparently a nitpick of your first paragraph, ignoring your second paragraph.
If you can't take as a given that statements actually are in the language they appear to be in no statement can have any knowable truth value. If "snow" in the utterance is a word in the same language as the identically spelled word in the statement, and the same for "white" (and "is"), and the rest of the statement means exactly the same as it does in English then the statement is still correct. But if "white" might designate orange "true" might just as well designate bubblegum or "only" designate "to treat like a second cousin".
And further the grammar of English is being assumed... as well as the very concept of languages.
Let me see if I understand you. "Snow is white" is true if and only if "snow" means snow, "is" means is, "white" means white, and snow is white? Because that still only makes sense if there's a fact of the matter about whether or not snow is white. And as ata pointed out, it's also false.
Edit: Maybe Tarski's undefinability theorem applies here. It says that no powerful formal language can define truth in that language. So if, as you say, truth is an attribute of linguistic objects, you have to invoke a metalanguage in which truth is defined. Then you need a meta-meta-language, etc. Of course English is not a formal language, and there is no formal meta-language for English - we talk about the truth of English sentences in English - but that is my point, that it relies for certain things on non-linguistic definitions. When we start discussing sentences like "This sentence is false", there's a tendency to forget that English does not and cannot define the truth of all English sentences.
No, that is not what I said. I said that IF "snow" means snow, "is" means is, and "white" means white, THEN "Snow is white" is true iff snow is white.
I never denied that. But the fact has nothing to do with truth unless you bring language into the discussion. Only linguistic objects (such as sentences) can be true.
Somehow, I feel that we are talking past each other.
ETA:
And now I know we are talking past each other.
That makes a lot more sense, thanks.
I think we're getting somewhere. I thought that you were saying that whether or not a statement is true is a property of language. Tarski's saying that whether or not a sentence is true is determined by whether it corresponds to reality. You're saying that whether or not it corresponds to reality is determined by the meaning the language assigns to it.
I'm still not convinced that truth is to do with language, though. Consider a squirrel trying to get nuts out of a bird-feeder, say. The squirrel believes that the feeder contains nuts, that there's a small hole in the feeder, and that it can eat the nuts by suspending itself upside down from a branch to access the hole. The squirrel does actually possess those beliefs, in the sense that it has a state of mind which enables it to anticipate the given outcome from the given conditions. The beliefs are true, but I'm certain that the squirrel is not using a language to formulate those beliefs in.
Can you clarify more exactly what you mean by "valid?" Because my initial reaction is that of course, you can come up with many classification systems for any set of things. It's not yet clear to me what interesting thing we can take away about how people are using the classifications of "true" and "false", other than the fact that they don't work very well for classifying certain unusual statements.
It seems to me that we have seen people in this thread advocate two value logics, three value logics, and four value logics. You can have workable systems of logic with and without the law of the excluded middle, and with and without a law of contradiction. There are intuitionistic logics, relevance logics, classically consistent and paraconsistent logics. To say nothing of linear logic, modal logics, and ludics.
Follow the links to the SEP articles on dialethi and paraconsistency. And then follow the citations from there to learn that logic is pretty big and flexible field.
The latter, though the former might also be the case.
"Independent of language" as in independent of the conventions of English, or Chinese, or Python, or street signs, or dolphin calls or whatever, not removed everything that could bear it.
Well, arguing about words is not very interesting to me, nor is the insight that words are just conventions and to a large degree arbitrary.
A good follow up. My response is no Omega didn't. The very nature of Omega prohibits writing such things. If someone gave you that problem it was someone other than Omega.
OH NO HE DI'INT
Yes. Quantum immortality.
<commits quantum suicide>
I will expect new things about where the zeros are. That means I can expect new things about my graphing calculator.
What about the sentence "This English sentence has six words?" It's self-referential, but it's certainly not meaningless, is it? And yet if you believe it, it doesn't tell you anything except something about itself.
(EDIT: Maybe my claim is conflating the proposition TESHSW with the written representation of the proposition, which it describes and does tell you something about. Perhaps simply "This sentence is either true or false" is a better example -- I don't think that is meaningless at all, either, just trivial.)
Hmm. If you visualize meaning as a mapping between representation space and some subset of expectation space, "this English sentence has six words" forms a tight little loop disconnected from the rest of the universe. That seems to me like as good an indication as any that the statement has no useful consequences.
The distinction between "meaningless" and "trivial" seems pretty semantic to me.
In my mind, I have the category "meaningless" as statements which can't be assigned a truth value without breaking the consistency of our system, and "trivial" as statements which can be assigned a truth value, but don't pay any rent at all.
Try this way: Working in boolean logic, "This sentence is either true or false" can be true, and it can't be false, right? If we can make these definite remarks about its properties within our system, can we still call it meaningless? Even though it doesn't have useful consequences. (A formal way of saying it doesn't have useful consequences, I guess, is to say that for our useless statement B and for all A, P(A) = P(A|B) -- it isn't any evidence for anything at all.)
Given your definitions, that makes sense. One of the points I was trying to make, though, is that "meaningless" is one of those words with several related but slightly different interpretations, and that a lot of the trouble in this thread seems to have come from conflicts between those interpretations. In particular, a lot of the people here seem to be using it to mean "lacks evidential value" without making a distinction between the cases you do.
As to which definition to use: I'd say it depends on what we're looking at. If we're trying to figure out the internal properties of the logical system we're working with, it's quite important to make a distinction between cata!trivial and cata!meaningless statements; the latter give us information about the system that the former don't. If we're looking at the external consequences of the system, though, the two seem pretty much equivalent to me -- in both cases we can't productively take truth or falsity into account..
To echo Tiiba but more formally: given a specific physical circumstance (transistors designed to do a computation) you can predict the result of the computation exactly and arbitrarily quickly (or as fast as you can look it up), because you have already done that computation.
For more abstract theorems, proving two things are equivalent leads me to expect one in the presence of the other.
For example, before proving Fermat's Last Theorem, I might expect there to be a right triangle whose three sides were squares of integers. Now I expect not to.