Related: The map is not the territory, Unresolved questions in philosophy part 1: The Liar paradox
A well-known brainteaser asks about the truth of the statement "this statement is false". If the statement is true, then the sentence must be false, but if it false then the sentence must be true. This paradox, far from being just a game, illustrates a question fundamental to understanding the nature of truth itself.
A number of different solutions have been proposed to this paradox (and the closely related Epimenides paradox, Pinocchio paradox). One approach is to reject the principal of bivalence - that every proposition must be true or false - and argue that this statement is neither true nor false. Unfortunately, this approach fails to resolve the truth of "this statement is not true". A second approach called Dialetheism is to argue that it should be both true and false, but this fails on "this statement is only false".
Arthur Prior's resolution it to claim that each statement implicitly asserts its own truth, so that "this statement is false" becomes "this statement is false and this statement is true". This later statement is clearly false. There do appear to be some advantages to constructing a system where each statement asserts its own truth, but the normative claim that truth should always be constructed in this manner seems to be hard to justify.
Another solution (non-cognitivism) is to deny that these statement have any truth content at all, similar to meaningless statements ("Are you a?") or non-propositional statements like commands ("Get me some milk?"). If we take this approach, then a natural question is "Which statements are meaningless?" One answer is to exclude all statements that are self referential. However, there are a few paradoxes that complicate this. One is the Card paradox where the front says that the sentence on the back is true and the back says that the sentence on the front is false. Another is Quine's paradox - ""Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation". One other common example is: "The statement on the blackboard in Carslaw Room 201 is false". The Card paradox and blackboard paradox are interesting in that if we declare the Liar paradox to be meaningless, these paradoxes are meaningless or meaningful depending on the state of the world.
This problem has been previously discussed on Less Wrong, but I think that there is more that is worth being said on this topic. Cousin_it noted that the formalist school of philosophy (in maths) believes that "meaningful questions have to be phrased in terms of finite computational processes". Yvain took a similar approach arguing that "you can't use a truth-function to evaluate the truth of a noun until you unpack the noun into a sentence" and that it would require infinite unpacking to evaluate, while "This sentence is in English" would only require a single unpacking.
I'll take a similar approach, but I'll be exploring the notion of truth as a constructed concept. First I'll note that there are at least two different kinds of truth - truth of statements about the world and truth of mathematical concepts. These two kinds of truth are about completely different kinds of objects. The first are true if part of world is in a particular configuration and satisfy bivalence because the world is either in that configuration or not in that configuration.
The second is a constructed system where certain basic axioms start off in the class of true formulas and we have rules of deduction to allow us to add more formulas into this class or to determine that formulas aren't in the class. One particularly interesting class of axiomatic systems has the following deductive rules:
if x is in the true class, then not x is in the false class
if x is in the false class, then not x is in the true class
if not x is in the true class, then x is in the false class
if not x is in the false class, then x is in the true class
If we start with certain primitive propositions defined as true or false and start adding operations like "AND", "OR", "NOT", ect. then we get propositional logic. If we define variables and predicates (functions from variables to boolean values) and "FOR EACH", "THERE EXISTS", ect, then we get first-order predicate logic and later higher order predicate logics. These logics work with the two given deductive rules and avoid a situation where both x and not x are in the true class which would for any non-trivial classical logic lead to all formulas being in the true class, which would not be a useful system.
The system has a binary notion of truth which satisfies the law of excluded model because it was constructed in this manner. Mathematical truth does not exist in its own right, in only exists within a system of logic. Geometry, arithmetic and set theory can all be modelled within the same set-theoretic logic which has the same rules related to truth. But this doesn't mean that truth is a set-theoretic concept - set-theory is only one possible way of modelling these systems which then lets us combine objects from these different domains into the one proposition. Set-theory simply shows us being within the true or false class has similar effects across multiple systems. This explains why we believe that mathematical truth exists - leaving us with no reason to suppose that this kind of "truth" has an inherent meaning. These aren't models of the truth, "truth" is really just a set of useful models with similar properties.
Once we realise this, these paradoxes completely dissolve. What is the truth value of "This statement is false"? Is it Arthur Prior's solution where he infers that the statement asserts its own truth? Is it invalid because of infinite recursion? Is it both true and false? These questions all miss the point. We define a system that puts statements into the true class, false class or whatever other classes that we want. There is no reason to assume that there is one necessarily best way of determining the truth of the statement. The value of this solution is that this dissolves the paradox without philosophically committing ourselves to formalism or Arthur Prior's notion of truth or Dialetheism or any other such system that would be difficult to justify as being "the true solution". Instead we simply have a choice of which system we wish to construct.
I have also seen a few mentions of Tarski's type hierarchies and Kripke's fixed point theory of truth as resolving the paradox. I can't comment too much because I haven't had time to learn these yet. However, the point of this post is to resolve the paradox without committing us to a specific model of truth, as opposed to the general notion of truth as a construct.
Edit: I removed the discussion of "This statement is true" as it was incorrect (thanks to Manfred). The proper example was, "This statement is either true or false". If it is true, then that works. If it is false, then there is a contradiction. So is it true or is it meaningless given that it doesn't seem to refer to anything? This depends on how we define truth. We can either define truth only for statements that can be unpacked or we can define it for statements that have a single stable value allocation. Either version of truth could work.
Some relevant works:
SEP on the Liar
The first major book on the subject.
The book clearly didn't settle the matter, despite its claim to have solved it, because here's another whole book of papers.
So what? “The King of France is bald” is meaningful when France has a king and meaningless the rest of the time. And “The statement on the blackboard in Carslaw Room 201 is false” is meaningless when said blackboard is blank.
I would distinguish between
To me, "the king of France is bald" (when there is no KoF) and "the statement on the blackboard is false" (when there is nothing on the blackboard) are in the third category, which isn't quite the same as either "false" or "meaningless" but is distinctly nearer "false" than "meaningless".
I would say a statement is meaningless, or at least meaningless to me, in so far as I have (or I suspect anyone could have) no clear conception of how its truth value depends on the state of the world. In the case of, e.g., "the king of France is bald" it's pretty straightforward even though for some worlds -- e.g., those with no king of France -- its truth value might be undefined, and for others -- e.g., those where there is a king of France, and he has a little bit of hair left -- it might be unclear. Contrast, say, "the Absolute enters into, but is itself incapable of, evolution and progress" (an example used by A J Ayer) where it's hard to see how to do better than "it's true if the Absolute enters into, etc."; or "The Mungle pilgriffs far awoy / Religeorge too thee worled" (from John Lennon) where it seems unlikely that any proposition with truth values is intended at all.
If you express this claim straightforwardly in first-order predicate logic, it can be either true or false depending on the structure you choose:
If in France it is customary for the king to have his head shaved, then the first formalization is always true, and furthermore the original sentence has an ordinary interpretation which is still true when there is no king (though it is arguably better written as "The Kings of France are bald", to emphasize the scope of the claim, in that case).
The point I intend is that "meaningless the rest of the time" is not fundamental to all reasonable interpretations of the sentence, but a choice you made. (I'd also agree with gjm's comment that "contains a false assumption" is different from "meaningless". (And, yes, first-order predicate logic does not include that distinction.))
Nope. If it is false, then "this statement is true" is false, no flip-flopping. Also note that this is an example that remains undefined under Prior's proposal.
Different assignments of true and false being possible is closely related to model theory, which you might want to look into. (See also some of So8res' posts)
Multiple truth assignments is also symptom of unprovability, which brings us into the realm of the incompleteness theorem. Work in this area demonstrates why type theory is not sufficient to prevent self-reference. Definitely worth learning more about the incompleteness theorem.
Oh and I've updated it to use the statement, "This statement is true or false".
"Nope. If it is false, then "this statement is true" is false, no flip-flopping" - thanks, I've now corrected it.
Model theory sounds interesting. I'll look into it.
Thanks for referring to my comment :-) but now I think it wasn't very good. Today my view of these things is much simpler.
Formal theories like PA support self-reference by the diagonal lemma, a.k.a. quining. You can formulate statements like "this statement is not provable", which will in fact be true and not provable. But you can't formulate "this statement is false" or "this statement is true", due to Tarski's undefinability of truth.
That's pretty much the whole story to me. If the liar statement can't be formulated in any formal theory, then it can't be part of any math I care about. Thinking in terms of formal theories also gives simple answers to many other paradoxes, like Berry's paradox and the Unexpected Hanging paradox.
Every part of math that I care about can perhaps be formulated in some formal theory, but (due to Tarski's undefinability of truth) no single formal theory can express every part of math that I care about. This leads to a problem if I want to write down a UDT utility function that captures everything I care about, since it seems to imply that no formal theory is good enough for this purpose. (I think you read my posts on this. Did it slip your mind when writing the above, or do you no long think it's a serious problem?)
Can you explain in more detail why no single formal theory can express every part of math that you care about?
I just wrote a post on a related but simpler question, about math beliefs rather than math values. It might apply to your question as well.
I assume he's claiming to care about a great deal of math, including at each stage the equivalent of Morse-Kelley as a whole rather than just the statements it makes about sets alone.
But I don't know what post Wei Dai referred to, and I doubt I read it. Quick search finds this comment, which seems grossly misleading to me - we could easily program an AI to reason in an inconsistent system and print out absurdities like those we encounter from humans - but may have something to it.
Why not? Or rather, why is this a normative claim rather than either an observation, or a decision about a formal system of statements?
This is the neatest resolution to the liar's paradox that I've ever seen, and in retrospect it seems obvious. It's extremely non-intrusive, only coming into play to render self-negating statements false.
"It's extremely non-intrusive, only coming into play to render self-negating statements false." That's a good point, it is an extremely neat solution. I'm simply arguing that this should be accepted as a useful model, rather than the model of reality. What about the model where we declare "This statement is false" to be meaningless? If we want to accept Prior's resolution as the model of reality we need to show that meaningless is an invalid model. I can't see how you would be able to do this. Therefore, it seems more elegant to just accept Prior's model as a useful model.
I just wanted to make one thing absolutely clear - when I talk about "truth as a construct" I don't mean it in the pseduo-philosophical manner ("but you aren't objective") that it is used in post-modern philosophy.
Thanks to those who explained why they disagreed with this article!
This post gives me the standard impression of "a little learning is a dangerous thing". You learned enough to feel that you can contribute, not realizing that you are nowhere near the necessary level.
Why do you say that? And what knowledge would one need to have to be at the necessary level?
Downvoted with extreme prejudice for not explaining what the specific problem with the post is.
That'll teach me to reply to "please explain your downvotes" (now edited out) comments.
Sorry, I removed it after I saw Manfred's and Azathoth's replies (and before I saw your reply) because people were explaining why they disagreed with the post or where I made mistakes, so there didn't seem to be any need for it.
What I meant was "give better explanations" rather than "give up on giving explanations", but it's entirely possible I was too harsh.
The sentence’s structure is true. The meaning behind the words have no value.
The sentence/statement, "This sentence is false." has no content, no "aboutness", and thus can't participate in the true-false game. It's like a bishop on a chessboard. Why is this explanation wrong?
One can easily add some content and get something easily deducible to be true, yet which has clearly absurd implications. For example, "Either this sentence is false or you should give me all your money."
In more formal terms, if for every sentence Q there a sentence P such that P is equivalent to "either P is false or Q is true", then every Q is provable and the system is inconsistent.
So, what's the "content" in your example? I don't see that the example sentence has any content and so I don't see how it's relevant. If one were to say, "It is false." the natural response would be, "Huh?" or "What's the 'it'." There's nothing there that can be false. it's the same with the sentence, "This sentence is false." (Or, for that matter, "This sentence is true.") In order for something to be true or false, there need be something referred to.
I understand the stakes here and the ultimate conclusions that Godel came to with a related inquiry, but I can't get past the fact that there needs to be some content for the sentence to be admitted to the true or false game.
The part that says "you should give me all your money". This is a clearly meaningful, contentful sentence. (Unfortunately for me, a false one.) Embedded in the reflexive sentence, it gives a reflexive sentence containing content. However, such sentences render the system inconsistent, so excluding empty circularities like "this sentence is false" is insufficient to resolve the problem of deciding what circularities can be admitted.
That sounds very similar to Yvain's unpacking approach. This is a valid approach, but what if the question said: "This sentence is true or false"? Assigning this sentence a truth value of "true" would be valid too.
Using the non - cognitive approach you could dismiss statements in symbolic logic that do not refer to constructs or events that could come into being. I am referring to constructs in Deutsch's theory
Pace your later comments, this is a wonderfully pithy solution and I look forward to pulling it out at cocktail parties.
Here's my solution. A statement is a grouping of symbols(which is what words are) and can, itself, be treated as a symbol. Symbols are used to refer to something other than themselves, but when you have the statement refer to itself, you get a paradox. When it refers to itself it becomes infinitely interpretable causing "This statement is false." to cycle between True and False for as long as you try to interpret it.
Just a minor point:
"when you have the statement refer to itself, you get a paradox" is not necessarily true. For example, the statement "this statement has five words" is self-referential and true. No paradox. Even a self-referential statement that includes its own truth value can be non-paradoxical: "This statement is true and has two words" is merely false.
By the way, this leads me to consider Prior's resolution as somewhat problematic:
"This statement is true and has eight words" "This statement has eight words"
The first statement is true and the second false, hence they cannot be equivalent. Nevertheless, adding "This statement is true and " to any statement should not change the statement's truth value if we accept that every statement implicitly states its own truth.
That's not really a problem with Prior's resolution. Rather, it's a different problem with self-reference, which appears whether we adopt Prior's resolution or not.
Compare: "P" and "P and P" are usually equivalent. But
"This sentence has five words." and "This sentence has five words and this sentence has five words."
don't have the same truth value. The problem seems to be that the meaning of "this sentence" isn't the same in the two ostensibly equivalent sentences. Whatever your favorite solution of this problem is, it seems that Prior could just graft that solution onto his own.
Prior's solution to the liar paradox needn't solve all paradoxes of self-references. As long as his solution is compatible with other solutions to other paradoxes, Prior has still contributed something of value.
Can we really say that world-states and mathematical formalism are totally independent? I would argue otherwise.
In particular, I would call your attention to the halting problem. Much like Godel's incompleteness theorem, the proof of the unsolvability of the halting problem calls on recursive self-negation. In other words, our computational abilities are limited in this way precisely because of the liar's paradox.
Now, let's take this a step further. Our computers are physical devices- when we write programs, and when we write meta-programs that analyze other programs, we're arranging physical and electromagnetic components in certain ways. Ergo, that formal proof describes the inherent limitations of physical objects, not just abstract formalisms.
Which is to say, that the liar's paradox cannot be dissolved, because it has material consequences for atoms and energy states. Mathematical and empirical systems are not cleanly separable, at least not here.
"Can we really say that world-states and mathematical formalism are totally independent? I would argue otherwise." I understand that we can formulate propositions that involve both real world states and mathematical formulations, but there generally needs to be some kind of bridge, ie. "let's assume that the moon is size X with mass Y around the earth with size A and mass B in a perfect circle".
The notions of truth involved in basic real world statements and mathematical expressions appear to be quite different. The truth of real world statements seems to be defined in a kind of objective manner in that the world corresponds to this state or it doesn't (even if we don't know what the state is). Mathematical statements seem to be constructed as they can only ever said to be true within a particular system or not.
"In other words, our computational abilities are limited in this way precisely because of the liar's paradox" - the Liar paradox and the proof of the halting problem both use self-reference, but that really isn't grounds to say that the halting problem is due to the Liar paradox.
To paraphrase the proof of the halting problem in dangerously abbreviated common English: We can program a machine such that 'Machine X halts iff Machine X never halts.' Does the liar's paradox contain any properties of interest that are not also contained in this proof? A reasonably complex formal language should be able to (and indeed, does) capture this paradox just as easily as a translation to Spanish or Mandarin.
The sequence, How to Convince Me That 2+2=3 may be germane to this discussion as well, for its discussion of belief-in-arithmetic as an empirical process. It is true enough that you can create mathematical models that do not map easily on to our material experiences, i.e. certain non-Euclidian geometries. But at the same time, it's trivially obvious that our experiences are structured in predictable ways. When we prove things about the limits of what structure is, in otherwise abstract mathematical ways, we have also produced useful predictions about the behavior of the material world.
(Incidentally, quotes are a bit more readable if you use the ">" notation, demonstrated under the 'show help' menu.)
"We can program a machine such that 'Machine X halts iff Machine X never halts.'" - good point. I can see that now.
So why did you remove the post modern philosophy part of the discussion and focused only on the math portion of this examination?
Is there good enough categorization of paradoxes that you can say "this statement is false" is a certain kind of paradox?
Actually, there is: http://en.wikipedia.org/wiki/Insolubilia
The problem with that approach is that you still need a meta-language and a notion of "meta-truth" to talk about these models, and then you're right back where you started.
That's a good point, but I don't think that it invalidates the whole approach. Non-classical logic is normally formulated within classical logic. I believe that other formulations of set theory are usually analysed from within standard set theory (can someone else confirm?).
The liars paradox is a paradox in "meta-logic". Standard set theory already has ways of dealing with it (by disallowing use of the word "truth").
My point was that just as some notion of set theory is necessary to talk about the different kinds of set theory, some notion of truth is needed to talk about the different notions of truth.