The Liar Paradox

Seems to me dialetheism describes the contradiction of accepting both evolution and religion. Perhaps ditheism could be applied to someone with a foot in both camps.

jrosenhouse wrote (July 2, 2013):
> The liar paradox is the statement: “This sentence is false.”
> [...] So how do we get out of this mess?

How about interpreting the alpha-numeric string "false" as denoting the negative truth value in all cases/sentences except those that would otherwise appear as self-referential mess?

Similar to resolving some variants of Russell's conundrum by noticing for instance that

- a "barber" can shave all (bearded) men who don't shave themselves, as well as himself; or that

- an "undertaker" can bury all (dead) people who don't bury themselves, except himself.

> this is the sort of thing I’ve been reading about, during all the time I’ve been spending not blogging!

Just the sort of thing I'd be blogging about all the time, if I could stand not to comment it meanwhile.

But you can arrive at the Liar Paradox without being self-referential. Consider the following two statements:

A. Statement B is true.
B. Statement A is false.

You cannot assign a truth value to either A or B without arriving at a contradiction. Kurt Gödel proved that any self-consistent logical system will always have statements that can neither be proven true nor proven false.

Eric, a pair of statements like Statement A and Statement B that lead to a contradiction would not be part of a Gödelian self-consistent logical system. The statements that could not be proven true or false in such a system would still actually BE true or false (without producing any contradictions), but would not be PROVABLE as true or false within the system. If I remember correctly, one of the things you can't prove within such a system is the consistency of the system. Again, if I remember correctly, it's not just any self-consistent logical system, but any self-consistent logical system large enough to encompass the integers - which probably covers just about any logical system that would actually be useful.

I would just say that the concepts of truth and falsehood are inapplicable to such sentences. Calling them "true and false" doesn't achieve anything.

Words get their meaning from how we normally use them, and we shouldn't assume that that they will still make sense when we try to use them in weird cases.

"Philosophers are more interested in using formal languages to capture the way people use natural language."

I think (with Wittgenstein) that philosophers have often been too inclined to treat natural language as if it was a logical calculus (or as an approximation to an underlying logical calculus). I suppose that if you see it as a logical calculus, you may think that weird cases will help you discover the supposed axioms.

Maybe truth is a complex quantity, like sqrt(-1), and we see paradoxes because we are considering only the real component.

While I think this stuff interesting there is always a nagging voice in the back of my head asking: "what exactly does this tell us about the world I live in?"
In other words: I can live with imperfection.

For most mathematicians, logic is just a tool. The principles of logic are the rules of the game. We accept those rules because we find it useful to do so. When it comes to various systems of logic, we don’t think in terms of correct or incorrect. Instead, we think in terms of useful or not useful.

Very similar to the empirical sciences, really. You're not going to find a physicist claiming we ought to stop teaching or using newtonian mechanics just because it breaks down at certain boundary conditions. Its a tool, so it'll continue to be used and taught so long as long as humans find the sorts of problems it solves interesting to solve.

“what exactly does this tell us about the world I live in?”

Don't make perfection the enemy of good, or you may find yourself in a form of philosophical paralysis where no act or belief is justified because your best justification is not perfect justification.
There is a corollary, which Asimov famously pointed out: two tools are not equally wrong merely because they both have boundary conditions beyond which they don't work. "...if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together."

So, I know I'm sort of taking the "don't move your arm like that, then" answer, but I thought Godel should have pretty much put to rest any notion that there could be a better answer. Any symbolic system powerful enough to represent much of anything is going to have these inconsistencies. Philosophers may not like this, but Godel has proved without a doubt that one cannot possibly do any better.

@8 Eric: like I wrote: I can live with imperfection.

It's certainly valid to consider systems of logic with different rules concerning truth and contradiction, and it's not just philosophers who are interested in such concepts. But I'm not sure what the relevance is to natural language. Natural languages have evolved under selection for practicality, and since ambiguous consistency is unlikely to be an issue in most practical situations, they generally make no attempt to enforce some particular notion of consistency,

Read Tarski if you're confused.

Oriental philosophies have no problem dealing with a proposition that might be true and false at the same time. But there is an occidental philosopher, who also knew a lot about physics and math, who came with his concept of the middle third included, which goes against the law of non-contradiction. He was romanian and mainly taught in France. But his work is beginning to be translated. His name is Stephane Lupasco. I think he is one of the greatest mind of the 20th century. Just saying...

Wiki: "In his introduction to Lupasco’s The Principle of Antagonism and the Logic of Energy, Nicolescu (a quantum physics expert) points out that once the reader gets beyond the (relatively few) mathematical formulas in this book, Lupasco’s language is perfectly accessible. “It exemplifies, in a self-referential manner, the ternary aspects of actualization, potentialization and included middle which give it the charm and the privilege of incantations that are at the same time scientific, philosophical and poetic.”
Summarizing briefly, stimulated by Einstein's works and quantum theory, Lupasco founded a new logic, questioning the tertium non datur principle of classical logic. He introduced a third state, going beyond the duality principle, the T-state. The T-state is neither 'actual', nor 'potential' (categories replacing in Lupasco's system the 'true' or 'false' values of standard bivalent logic), but a resolution of the two contradictory elements at a higher level of reality or complexity. Lupasco generalised his logic to physics and epistemology and above all to a new theory of consciousness."

@Robin
You are right, language is an obstacle when it comes to paradoxes states or propositions, even if they exist for real. And since language shapes our way to think, it wouldn't be a surprise that we might miss something about "reality"...

eric --

Very similar to the empirical sciences, really. You’re not going to find a physicist claiming we ought to stop teaching or using newtonian mechanics just because it breaks down at certain boundary conditions. Its a tool, so it’ll continue to be used and taught so long as long as humans find the sorts of problems it solves interesting to solve.

I've seen that exact analogy used on behalf of dialetheism. The basic claim is that classical logic still works fine most of the time, but breaks down in certain extreme cases. Just like Newtonian mechanics is undoubtedly useful in most everyday applications, but fails at the extremes.

Jason, I don't see how that analogy supports dialetheism. I for one don't find it helpful to call Newtonian mechanics "true and false". I would say that Newtonian mechanics is a less accurate model of reality than relativistic mechanics, with the inaccuracy being insignificant under most normal conditions but becoming extreme under other conditions. I don't think the same can be said of the paradoxical sentences we're considering here.

I'm still wondering what, if anything, is achieved by calling such sentences "true and false". It seems to me that, without providing some way of understanding how to combine the separate properties of truth and falsehood, this is not telling us anything. It's just providing a label, and a misleading one at that, since it gives the false impression of establishing a substantive fact. If we need a label, perhaps "undecidable" would be better.

I think a couple of the comments are getting to the heart of this issue. This type of sentence doesn't tell us anything about truth, it only tells us something about language. And that is that it is possible to form perfectly grammatically (and other linguistic descriptors) correct, understandable sentences that have no real meaning or relationship to the real world.

Richard Wein:

Jason, I don’t see how that analogy supports dialetheism.

You don't? Seems pretty simple to me: symbolic logic is good at analysing assertions and the relationships between them, but not all of them. Some its just not good at analyzisng. So it has boundary condititions.

I'm actually perfectly okay with dialtheism in principle. Pragmatically, I'd ask whether and how it contributes to solving problems before getting behind it. There's a difference between "I hypothesize that there may be a (different) way to crack this nut" and "here's your cracked nut, are you interested in learning how I cracked it?" If dialtheism is at the former stage, its not all that interesting (at least to me).

Are you inclined to say that M is neither true nor false? Then you have just affirmed that what it asserts is actually the case, which makes it true. . . .

If you say that N is not a proposition at all, then it certainly is not a true proposition. So, by declaring N to be something other than a proposition, then you have just made it true.

I don't find either argument compelling. To say that M and N are not propositions is to say that they do not assert anything—or rather, to be more exact, that nothing is asserted by uttering them. They are merely combinations of words that give the appearance of being the vehicles of assertions. So if I say that M (“This sentence is either false or neither true nor false") is neither true nor false, I am not "affirmed that what it asserts is actually the case," because it does not assert anything. And if I say that N ("This sentence is not a true proposition") is not a proposition, I am not "making it true," because nothing is said by it that could be true.

I concur with Mark P.'s comment: "This type of sentence doesn’t tell us anything about truth, it only tells us something about language."

^Oops! Editorial error: for "I am not 'affirmed,'" etc., please read "I have not 'affirmed,'" etc.

The Liar Paradox and Russell's Paradox both figure heavily in popular expositions of Godel. Among other things Godel proved mathematically that certain of Hilbert's mathematical assumptions were incorrect, or so I thought. I am therefore surprised to see LP and RP dismissed as irrelevant to Math. But then I got a B-- in College Algebra 101, a C-- in College Probability 001, and then fled from the discipline forever.

Here is another view on Lupasco's middle third included logic:
Stéphane Lupasco (1900-1988) has also substantiated the logic of the included middle, showing that it constitutes "a true logic, mathematically formalized, multivalent (with three values: A, non-A, and T) and non-contradictory".[10] Quantum mechanics is said to be an exemplar of this logic, through the superposition of "yes" and "no" quantum states; the included middle is also mentioned as one of the three axioms of transdisciplinarity, without which reality cannot be understood.[11]

I agree with Miles: before we can accept the suggestion that (say) M is a proposition which asserts something about the world, we first need to understand what it is that is supposedly being asserted. A claim relating to the truth value of M can only be agreed to have meaning once we have accepted M as a proposition, and therefore cannot be used to support the claim that it _is_ a proposition.

The given argument is like assuming the existence of a mathematical object en route to proving its existence. E.g. the ontological argument for the existence of God: "God is defined to have the property of existence. Since, by definition, God has the property of existence, God exists."

@16 RW: "a less accurate model of reality"
In some circumstances Newtonian mechanics is flat out false - when something is moving with the speed of light for instance.

konrad wrote (#24, July 3, 2013):
> I agree with Miles [(#19): ...]
> [...] like assuming the existence of a mathematical object en route to proving its existence.

Closely related seems the case, such as Cantor's theorem, of accepting some
("sufficiently innocent looking") string of symbols (such as

$latex B=\left\{\,x\in A : x\not\in f(x)\,\right\}$

as an expression of a mathematical object en route to a proof by contradiction; one consequence of which is in turn that the supposed mathematical object doesn't even exist.

Perhaps, therefore, strings of symbols that are outright acceptable as expression of a mathematical object might be rather:

$latex B=\left\{\,x\in A : x\not\in f(x) \, \text{or} \, \left( x\in B \, \text{and} \, B = f(x) \right) \,\right\}$, or

$latex B=\left\{\,x\in A : x\not\in f(x) \, \text{and} \, B \neq f(x) \,\right\}$.

Oh, well ... perhaps rather:

$latex B=\left\{\,x\in A : x\not\in f(x) \, \text{or} \, $
$latex \left(\exists_{\text unique} S \in {\mathcal P( S )} : S=\left\{\,y\in A : y\not\in f(y) \, \text{or} \, \left( y\in S \, \text{and} \, S = f(y) \right) \,\right\} \, \text{and} $
$latex f(x) = \left\{\,z\in A : z\not\in f(z) \, \text{or} \, \left( z\in f(x) \, \text{and} \, f(x) = f(z) \, \right) \,\right\}$,

or

$latex B=\left\{\,x\in A : x\not\in f(x) \, \text{and} \, $
$latex \left(\exists_{\text unique} S \in {\mathcal P( S )} : S =\left\{\,y\in A : y\not\in f(y) \, \text{and} \, S \neq f(y) \,\right\} \, \text{and} $
$latex f(x) \neq \left\{\,z\in A : z\not\in f(y) \, \text{and} \, f(x) \neq f(z) \,\right\} \, \right)\,\right\}$.

Frank Wappler wrote (#26, July 4, 2013):
> Oh, well ...

… perhaps rather:

$latex B( k ) = \{ \, x \in A : x \not\in f(x) \, \text{or} \, x = k \,\} $, where the variable/index
$latex k \in \{ \, y \in A : f(y) = \{ \, z \in A : z \not\in f(z) \, \text{or} \, ( z \in f(y) \, \text{and} \, z = y \, ) \, \} \, \}$,

or

$latex B = \{\,x\in A : x\not\in f(x) \, \text{and} \, $
$latex f(x) \neq \{\,y\in A : y\not\in f(y) \, \text{and} \, y \neq x \,\} \,\}$.

Or how about:

$latex B( j ) = \{ \, x \in A : x \not\in f(x) \, \text{and} \, x \neq j \,\} $, where the variable/index
$latex j \in \{ \, y \in A : f(y) = \{ \, z \in A : z \not\in f(z) \, \text{and} \, z \neq y \, \} \, \}$.

Miles Rind,

I think you are changing definitions. In logic, a proposition is DEFINED to be any statement that has a truth value. The paradox in Jason's example is evident. Let A be "this sentence is false" and B be "this sentence is neither true nor false". Logically, the sentence is then represented as A or B. The sentence is true if and only if either A is true, B is true, or both are true.

Now, we have three choices for how we can assign a truth value to the sentence, but all are unsatisfactory:

1. Assign a value of true to the sentence. Since we say the sentence is true, A must be false. B must also be false. Therefore, A or B is also false, which contradicts our original assignment.

2. Assign a truth value of false. Therefore A is true. Therefore A or B is true, contradicting our assignment

3. We can assume that the sentence is not a proposition. Therefore it is neither true or false. However, that renders B true, and therefore A or B is true, again contradicting our assignment.

Alternatively, you can just leave out any notion of "proposition" from this and focus on truth values. For any sentence, logically, either it is true, it is false or it is neither. In this case we have a paradox since none of these three assignments makes any sense. That was really the point of this example; there's no way we can conceive of a valid truth value for this sentence.

Sean, I can't follow your objection. I don't see how my comment depends on changing the definition of "proposition." I say of sentence M ("This sentence is either false or neither true nor false") that it is neither true nor false (and I would also say that it is not a proposition) and of sentence N ("This sentence is not a true proposition") that it is not a proposition (and I would also say that it is neither true nor false). I fail to see how this implies anything about the definition of "proposition."

You say:

The paradox in Jason’s example is evident. Let A be “this sentence is false” and B be “this sentence is neither true nor false”. Logically, the sentence is then represented as A or B. The sentence is true if and only if either A is true, B is true, or both are true.

I can't follow this at all. You introduce two sentences, A and B, and then you say "logically, the sentence is then represented as A or B." What sentence is represented as A or B? (I also can't see what work, if any, the word "logically" is doing here, by the way.) Are you talking about the sentence L, "This sentence is false," introduced at the beginning of Jason's post? If so, then what is the sense of saying that L is "represented as A or B"? L is identical with A: you have merely introduced a different letter to designate it; so how can it be "represented as" B, when B is an entirely distinct sentence? I'm sorry, but I just can't follow your reasoning.

My point is a simple one: these supposedly paradox-generating sentences only generate paradoxes if one assumes that they have a truth-value, or, what I take to be equivalent, that they express (or, if you prefer, are) propositions. I see no compelling reason to make that assumption.

Addendum:

3. We can assume that the sentence is not a proposition. Therefore it is neither true or false. However, that renders B true, and therefore A or B is true, again contradicting our assignment.

Wrong. That does not render B true unless you assume that B is a proposition/has a truth-value. I say that nothing is asserted in B: it is not a proposition; it has no truth-value. You simply repeat the assumption that I reject. That is not a counter-argument; that is begging the question.

Having gone back to re-read my original comment (#19), I can now see the point of Sean's criticism (in #29) that I was "changing definitions." That criticism, I take it, is based on my having said this:

To say that M and N are not propositions is to say that they do not assert anything—or rather, to be more exact, that nothing is asserted by uttering them.

Against this, Sean says,

In logic, a proposition is DEFINED to be any statement that has a truth value.

But I still don't think that my argument presumes any change of definition. As I said in my reply (#30), I have no quarrel with defining "proposition" to mean "statement that has a truth value." But in a philosophical context like the present one, it is only fair to ask what that means. What, for instance, does it mean for some string of words to be, or to belong to, a statement? I take it to mean that that the string of words was uttered or written by someone who asserted something to be the case. Not every combination of words that has the grammatical form of a statement is one.

The sentence L could in some instance be (or be the words of) a statement. That is to say, someone could utter or write that sentence and thereby be making a statement. If, for instance, I were looking at a series of statements in writing, I might point to one of them and say, "This statement is false." (I would not myself say "This sentence is false," but I will pass over that for the moment.) But this is plainly not what is going on with the liar sentence. If we consider that string of words as a mere sentence out of context, we have no way of determining what statement, if any, is being made by it. The derivation of logical paradoxes from it presumes that we don't need to do this: it presumes that the sentence simply is in itself a statement and a proposition and has a truth value, regardless of who is saying it, in what context, etc. This seems to me a mistake.

Jason,

One way of dealing with this is to set-up elaborate rules regarding what is, and what is not, a legitimate set. This is the way of axiomatic set theory, and it is a long-standing and honored branch of mathematics. The fact is, however, that most mathematicians pay no attention to it most of the time. We prefer “naive set theory”. Set theory is just a useful formalism. If you get into trouble when you devise super-clever sets then stop being clever! Why should Russell spoil the fun for the rest of us?

I think, here, you understate how things are working wrt this paradox in mathematics and so are either being a bit misleading when it comes to describing philosophy or, at least, are missing an important parallel between the two.

In mathematics, those who are doing axiomatic set theory -- and therefore trying to work out the concepts of sets and all the rules and descriptions of how they work -- care very much about Russell's paradox because it causes them a lot of grief when they try to do that. But most mathematicians are not, in fact, doing that, and so they can get by with what you call "naive set theory" in order to do the work they're trying to do. It's not really a notion, it seems to me, of them "preferring" naive set theory, as they most likely agree with those looking at sets in detail that naive set theory is rough, ready and is almost certainly incomplete. But they don't need the complete story; what we know about sets is good enough for almost anything they do, and if they do get stuck they would certainly turn to axiomatic set theory to see if something they've discovered there will get them out of their impasse.

Interestingly, philosophy works the same way. I've gone though a Master's degree and taken some PhD level courses, and can tell you that this problem is treated in most philosophical fields just like Russell's paradox is treated by mathematicians: as a neat little problem, but something that they, for the most part, don't need to worry about. People working on logic in detail or trying to form a detailed definition of what it means for something to be true probably care, but most of the other fields -- like ethics, epistemology, philosophy of science, philosophy of religion and even, for the most part, philosophy of language -- don't really need to care. Taking epistemology as perhaps the best example, obviously it is really important to understand what it means for a statement to be true in order to claim that you can know it to be true, but they don't have to worry about the truth status of the statement "This sentence is false" to do that. We have more than enough unambiguously true statements to use to define what it means to know them true to worry about that one.

So, here you seem to imply that philosophers are arguing over these little details and that this is a critically important problem for philosophy in general, while the equivalent in mathematics isn't. However, in general the two are indeed being treated equivalently; mathematics seems to me to take your example a bit more seriously than you imply, while philosophy doesn't take its example quite as seriously as you imply it does.

Miles,

My point is a simple one: these supposedly paradox-generating sentences only generate paradoxes if one assumes that they have a truth-value, or, what I take to be equivalent, that they express (or, if you prefer, are) propositions. I see no compelling reason to make that assumption.

Well, it isn't really an assumption. The question here is that the statement:

This sentence is false.

Really looks like a statement, and the equivalent of saying:

This sentence contains a preposition.

So, we have absolutely no problem saying from the second that:

a) It's a statement.
b) We can assign a proposition to that statement.
c) The proposition is "This sentence contains a preposition".
d) The proposition has a truth value.
e) That truth value is, unless I'm forgetting something about my grammar, is false.

Now ... why can't we do it for the first one? The only reason we're thinking about doing that is NOT because it doesn't look like a statement and not because we can't derive a proposition-looking proposition from it, but because when we try to evaluate it we get a paradox. But that's not a good enough reason to overturn a) it really looking like a statement and b) that we think that all statements state facts and so can be turned into or attached to propositions.

So, for me, the big problem with just saying "Well, then it's not a proposition/statement" is that you can't really give a good reason for doing that. And that's where, I think, the charge of you changing the definition of proposition comes into play.

"This sentence is false" looks like something that should have a truth value. The problem is in figuring out what that truth value is, and that's not a problem that you can use to just declare that it actually doesn't have one.

The problem is the idea that one can arbitrarily assign a value of truth to any of the statements. That assumption is a mistake. The value of science is that what we hold true is not necessarily true. Merely stating that something is "true" does not make it so. And any conclusions from any logical thinking based on such assumptions is presumptuous. So, the logic itself may be paradoxical when the data is bad or misinterpreted.

Well, first, you have to recall that I'm talking about it being a proposition, and therefore being the sort of thing that can take a truth value, which is a far different claim from saying what the truth value of the proposition actually is. And so while originally I thought you were on to something, now I can see that there's a big confusion here when we think about the example.

As you concede, the proposition P = "A sentence S is true" is indeed a valid proposition and clearly asserts something, which is that the sentence -- or, rather, the proposition that aligns with the sentence -- S is true. In both cases, we all agree that you can say that perfectly fine if P and S are not the same. So, what happens when they ARE the same? Well, all that means is that the sentence P is self-referential, but there is again no reason to eliminate self-referential propositions or else the proposition "This sentence contains a proposition" would be problematic as well.

So, we take that example "This sentence is true". We treat it as a proposition, and say that it therefore could indeed be true or false. What it asserts, as we noted above, is that the sentence has a truth value of true. So, if the sentence does indeed have that truth value, then the sentence is true and there is no paradox, and if the sentence is false then the sentence is false and so again there is no paradox. So, there is no issue here. Note that it is at THIS point that the sentence "This sentence is false" runs into problems, because I CAN'T do that: if I assign it a truth value of true, then it asserts that the sentence has a truth value of false, and if I assign it a truth value of false, then it asserts that the sentence has a truth value of true, causing the paradox.

But what it's important to note is that your objection hasn't even come into the picture yet. That comes later, when we try to figure out which of those two truth values the sentence actually has. You are right that in both cases we can't really evaluate what truth value the sentence actually has, because there is no independent proposition or data that can be brought to bear on it. It is indeed purely self-referential and in that case for all practical purposes is meaningless in that sense. So, as I said above, for most practical or philosophical purposes it isn't an interesting example: unless you care about what it means for a sentence to be a proposition or about what it really means to call something "true" or "false". But if you care about that, you stop at my first paragraph, noting that, again, it DOES assert something and therefore looks like it SHOULD have a truth value, but when we try to treat it that way we run into the paradox, and we have no principled way to say that it isn't a proposition while the case where we say "This sentence is true" clearly could be.

In summary, the problem you describe is indeed one that afflicts both sentences, but "This sentence is false" runs into a problem that "This sentence is true" doesn't, and that problem seems to suggest that we should not consider the former a proposition even though we have no issues considering the latter one, and that itself would be a problem.

Replying to Verbose Stoic, #37:

As you concede, the proposition P = “A sentence S is true” is indeed a valid proposition and clearly asserts something, which is that the sentence — or, rather, the proposition that aligns with the sentence — S is true. In both cases, we all agree that you can say that perfectly fine if P and S are not the same.

Actually, what I concede is that such a proposition may be true. If it turns out that S is a sentence that says nothing true or false, or that there is no sentence S, then neither does P say anything true or false--although I suppose that means that the very stipulation that P is a proposition is incoherent: P, because it has no truth value, is then merely a sentence and not a proposition. On the other hand, as long as S is a sentence in which something true or false is stated, then P is all right (though I am uncomfortable with the use of the indefinite article at the beginning of it: taken strictly, P says only that some sentence is true, rather than that a certain specific sentence is true; I take you to have intended the latter rather than the former meaning).

So, we take that example “This sentence is true”. We treat it as a proposition, and say that it therefore could indeed be true or false. What it asserts, as we noted above, is that the sentence has a truth value of true. So, if the sentence does indeed have that truth value, then the sentence is true and there is no paradox, and if the sentence is false then the sentence is false and so again there is no paradox. So, there is no issue here.

I am not sure that I follow your reasoning, but I will certainly deny that "there is no issue here," if by that you mean that there is nothing problematic in the reasoning that you offer. The sentence in question--I will call it "R"--says nothing that can be true or false. To be more exact, someone who utters or writes R, intending the subject term "this sentence" to refer to R itself, thereby says nothing true or false. Nothing is asserted in R. So to "treat [R] as a proposition" can make sense only as a counterfactual exercise, just as I can say, e.g., with reference to a couple of items on my desk, "Suppose this pen is a nuclear missile and this book is Washington, DC," in the course of explaining a point about nuclear warfare to someone. But the exercise in both cases remains purely counterfactual: just as, in the latter case, it does not imply that a pen can be a nuclear missile or that a book can be Washington, DC, so in the former it does not imply that R is capable of being true or false. I can certainly say things like "If R is true, then R is true, with no resultant paradox," just as I can say, "If this pen is a nuclear missile and this book is Washington, DC, then once it gets to this point [demonstrating] there is no way to shoot it down." Of course, no sane person would mistake a pen for a nuclear missile or a book for Washington, DC, while sane and very intelligent people do mistake R for a proposition: but, for all that, it isn't one, and the fact that we can, for certain purposes, treat it as one, does not show that it is one or that it can be one (always assuming, of course, that the phrase "this sentence" refers to R itself).

But what it’s important to note is that your objection hasn’t even come into the picture yet. That comes later, when we try to figure out which of those two truth values the sentence actually has.

For the reasons that I have just given, this seems to me false.

Oh, curses: For the first sentence in my just-posted comment:

Actually, what I concede is that such a proposition may be true

please read "what I concede is that such a sentence may be (or express, or whatever) a valid proposition."

I still agree with Miles; here's a different tack:

Sentence L is in the same category as Chomsky's famous sentence "Colourless green ideas sleep furiously". This was proposed as a sentence which has valid syntax but no valid semantics (i.e. no meaning). Before we can agree that Chomsky's sentence has a truth value, we first need to understand what it supposedly says - for this we would need to understand what it means for something to be both colourless and green, what it means for this to be an attribute that could be applied to ideas, what it means for ideas to sleep and what it means for the act of sleeping to be furious. Only once this has been done could the sentence be said to have a truth value, and only after that can we discuss what that truth value might actually be. The point is that, until such time as we have an exact understanding of what a sentence supposedly says (and I claim that for L and the other examples in the original post we do _not_ have such an understanding), we cannot assume that it has a truth value. In particular, we cannot start by assuming all sentences have truth values and then revoke them if they lead to paradox - this would be problematic for Chomsky's sentence, because it is not clear how to construct a paradox out of it and certainly we don't feel that we _have_ to consider it meaningful just because we haven't found a paradox. So the default approach _must_ be to consider sentences meaningless until such time as a clear meaning has been assigned to them.

It may seem that self-reference is important here, but that is a red herring: Chomsky's sentence demonstrates that self-reference is not necessary to render a sentence meaningless, and "This sentence contains a preposition" (which does have a clear meaning) shows that it is not sufficient.

A rule for assigning meaning which (I think) avoids paradox is this: a sentence becomes a proposition only when a (unique) meaning is assigned to it, and we cannot assign a meaning to a sentence if we do not _already_ understand that meaning.

Consider the sentence "Sentence L is false." Before we can assign this a meaning, we must first assign a meaning to sentence L (because claims about the truth value of a sentence are meaningless until we understand the meaning of the sentence about which the truth value claim is being made). But now suppose the sentence "sentence L is false" is itself sentence L. Then a prerequisite for assigning it a meaning is that it must already have one. Hence we can never assign it a meaning.

How about sentence P:"This sentence is not a proposition"? As long as we don't attempt to assign meaning to it, P is not a proposition and it is not meaningful to say "what P says is true", because no meaning has been assigned to the phrase "what P says": P is just a meaningless string of words, and we have no problem.

The problem comes in if we want to interpret P, or the phrase "what P says", as having meaning: we must then attempt to consider P as a proposition by assigning some meaning to it. But the meaning of P, if it were a proposition, would be that it is not the proposition A AND not the proposition B AND not the proposition C, etc for all propositions. One of the claims made by P, if it were a proposition, would be that it is not the proposition P (because if P were a proposition, then this would be one of the propositions we would have to include in the list of propositions P claims not to be). But this means we are again required to assign a meaning to P as a prerequisite for assigning a meaning to P - in other words, we can never assign it a meaning.

Finally, sentence N ("this sentence is not a true proposition"): this is a compound sentence that can be rephrased as "either this sentence is not a proposition or it is not true" - since the first part can never be a proposition, the whole sentence cannot be one either.

Sorry for not replying earlier, but the main issue I have here is that I don't see any reason for saying that a statement like "This sentence is true" has no meaning other than running into issues when you try to figure out what that truth value actually is. So taking that sentence, it is clear that we know what it means for a sentence to have a truth value of true. We can do it easily, as has been pointed out by multiple people, if we refer to a sentence other than itself. So taking this expanded version:

Let S be the sentence "The sky is blue".
The sentence "S" is true.

We don't have any issues at all. Everything works fine. We look at "The sky is blue", determine its truth value, find it in this case to be true, and we're done.

So, what about this case?

Let S be the sentence "The sentence S is true".
The sentence "S" is true (yes, the actual substitution looks weird here, but we have to live with it).

It should work the same way ... except that we can't determine, at least right now, whether S is true, and so can't fill it in in the last sentence. Does this mean that S doesn't have a truth value?

Well, let's take an indexical:

Let S be the sentence "I am laughing", where I is not me.
The sentence "S" is true.

Now we have the same problem. Without knowing who precisely the indexical refers to, we can't determine the truth value of that sentence. But it would be odd, to say the least, to say that indexicals don't have truth values, or only have truth values if we can resolve the indexical. It seems to make much more sense to say that the indexical has a truth value, but until we can resolve the indexical we don't know what it is.

Now, you can argue that we could never have any context that could allow us to resolve "This sentence is true", but this is false. Take this example:

Let S be the sentence "This sentence is true".
Let S be true.

Then I can resolve it.. We axiomatically assign the truth value of "true" to S, which makes it true, and then we check to see what the content of the sentence is in light of that, and it turns out to be true, and we're all happy. And we can do this for false as well:

Let S be the sentence "This sentence is true".
Let S be false.

And it still works out. We say it's false, and lo and behold the sentence being false implies that it saying that this sentence is true is false. All good. Not very interesting, of course -- because we must axiomatically assert its truth value to find out what that value is -- but it works more or less unproblematically.

Now, we try that for "This sentence is false":

Let S be the sentence "This sentence is false".
Let S be false.

And we see that we run right back into the contradiction noted in the original article.

So, again, there's a difference in problem here. But the two statements seem almost identical. So it becomes difficult to just say that the second doesn't have a truth value or isn't really a proposition, without doing just because you see that this is a problem you're having with it.

I am sorry to see my comments getting ever longer, but, VS, your latest comment multiplies confusions beyond necessity. To begin with:

Let S be the sentence “The sky is blue”.
The sentence “S” is true.

"S" is not a sentence; it is a letter of the alphabet, which you have introduced as a variable. S is, by your stipulation, a sentence, not "S." You commit this confusion repeatedly. It is, however, one that can easily be corrected. So where you write:

The sentence “S” is true.

I will hereafter read:

The sentence S is true.

But you also confuse matters by repeatedly using the single variable "S" to designate now one sentence and now another. I will correct this by adding a numeral to each designation. This will require me to use square brackets to indicate where I have modified quotations from your comment, though these are not intended as material in the symbols themselves. So, to your second use of "S":

So, what about this case?

Let [S1] be the sentence “The sentence [S1] is true”.
The sentence [S1] is true (yes, the actual substitution looks weird here, but we have to live with it).

It should work the same way … except that we can’t determine, at least right now, whether [S1] is true, and so can’t fill it in in the last sentence. Does this mean that [S1] doesn’t have a truth value?

Well, I don't see any basis for saying that this case "should work the same way" as the first one, but, that aside, to say that we merely can't determine right now whether S1 is true seems to me disingenuous. There is simply no sense in saying that S1 is true: it is an empty, though syntactically well-formed, combination of words. Part of the trouble here is that it is not clear whether you are considering the meaning of the term "S1" as being given with the sentence "The sentence S1 is true." In other words, if, say, we had one context in which somebody says, "Let S1 be the sentence 'The sentence S1 is true'; S1 is true"; and another context in which somebody says, "Let S1 be the sentence 'Grass is green'; S1 is true," one might say that the same sentence "The sentence S1 is true" occurs in both instances or one might say that these are two different, though materially indistinguishable, sentences (because the reference of "S1" is different in the two cases). I can't tell which is your standard of "same sentence." My point is that if the meaning of the symbol "S1" is not given with the sentence "The sentence S1 is true," then it should be obvious that that sentence has no truth value, because it makes essential use of a symbol to which no meaning has been given. If, on the other hand, the meaning of "S1" is considered to be given with the sentence, and it is simply a designation of the sentence itself, then it is, I grant, not obvious that the sentence has no truth value. But I still say that the sentence has no truth value rather than that "we can't determine right now" whether it has one. The latter sort of statement would make sense if we were talking about the first sort of case, in which we are simply waiting for a meaning to be assigned to the symbol "S1." It does not make sense in the second sort of case.

Well, let’s take an indexical:

Let [S2] be the sentence “I am laughing”, where I is not me.
The sentence [S2] is true.

Now we have the same problem. Without knowing who precisely the indexical refers to, we can’t determine the truth value of that sentence. But it would be odd, to say the least, to say that indexicals don’t have truth values, or only have truth values if we can resolve the indexical. It seems to make much more sense to say that the indexical has a truth value, but until we can resolve the indexical we don’t know what it is.

Now for one thing, "I is not me" is not intelligible English. You could say "'I' does not refer to me": that would at least be grammatical; but surely what you mean is that S2 is the sentence "I am laughing" as uttered by someone other than you, i.e., other than Verbose Stoic. If that is so then you are considering the sentence to be individuated not merely by the sequence of words that makes it up but by the context of utterance as well.

That much is easily sorted out. But you commit a further confusion. At the beginning, you seem to use the word "indexical" according to its common acceptation as a technical term in the philosophy of language, namely to mean an expression whose reference is determined by the context in which it is used—such as, in this instance, the pronoun "I," or the verb phrase "am laughing." (The latter is indexical because of its tense: it refers to different times according to the context of use.) But then you say, "it would be odd, to say the least, to say that indexicals don’t have truth values." Odd? I should say rather that it is almost trivially true. Indexicals do not even have the right logical or syntactic form to be bearers of truth values. Only statements or sentences or propositions (setting aside the sticky issue of which of these we are talking about) can do that.

The only way that I can make any plausible sense of what you have written is to suppose that you are here using the word "indexical" to signify a sentence containing an indexical. So I take it that what you meant is that it would be "odd" to say that sentences containing indexicals don't have truth values. I would say that this depends entirely on whether by "a sentence containing an indexical" you mean such a sentence in abstraction from the context that is necessary for providing the indexical expressions in it with reference. If it is so taken, then of course it will lack a truth value. If all that is given is a syntactically well-formed string of words, in this case one using the pronoun "I" and a verb phrase in the present progressive tense, without a person saying it or anything else necessary for giving reference to its component expressions, then it is simply senseless to attribute a truth value to it. If, on the other hand, you mean to take the sentence with its context of utterance, then to say that no such sentence has a truth value would be not so much odd as perverse.

Unfortunately, you seem not to mean either of these. You speak of "whether we can resolve the indexical." This seems to me simply to muddy the waters. Are "we" here supposed to be the people who pick the sentences and assign a context of utterance to them? Then to say that we "cannot resolve" the indexical expressions in them is to say only that we have not assigned to the sentences a context that provides the indexicals with a reference; and in that case, once again, it is virtually trivial to say that the sentences lack a truth value. On the other hand, if "we" are not stipulating contexts of utterance for arbitrarily formulated sentences but are talking about actual utterances of human beings, then there may be some cases in which we know enough about the context of an utterance to determine its truth value and others in which we do not know enough to do that. In the latter case, certainly whether the utterance has a truth value is independent of whether we can figure out what the truth value is; but this is not a case in which we can simply stipulate what an expression in a sentence refers to, which seems to be the sort of case that you are talking about.

Now, you can argue that we could never have any context that could allow us to resolve “This sentence is true”, but this is false. Take this example:

Let [S3] be the sentence “This sentence is true”.
Let [S3] be true.
Then I can resolve it.. We axiomatically assign the truth value of “true” to [S3], which makes it true, and then we check to see what the content of the sentence is in light of that, and it turns out to be true, and we’re all happy.

If you think that we can simply "axiomatically assign" truth values to sentences, then you are talking about the sentences of a formal calculus and not about statements made in a natural language. If you want to talk about formal calculi, that is your privilege, but do not then pretend to be making points about sentences and statements in English. If you assign the value "true" to S3 in some formal calculus, this has no bearing at all on whether an English sentence identical with S3 has a truth value. The rest of your comment after this point seems to me likewise vitiated by this equivocation.

Gah! I always find mistakes after I post a comment! Where I wrote "the single variable 'S'" please read "the single constant 'S.'"

Miles,

I think the biggest part of your confusion here is that you are nitpicking over specific style and format, rather than simply trying to understand what I am saying based on the context. Surely the "S" vs S and using the same variable in completely different arguments -- ie contexts -- would not confuse you unduly. The only thing that I might need to clarify is that when I used the term "indexical" I meant it as "indexical SENTENCE", which is a perfectly valid way to use the term and it seems that even in that case you figured it out ... mostly.

Anyway, what I was saying there is that if we -- ie the person reading the sentence -- does not have the context in order to determine if the indexical sentence is true or false, it seems ridiculous to suggest that it therefore does not, in fact, have a truth value at all. The indexical sentence clearly DOES have a truth value, or at least is the right sort of sentence that can be associated with a proposition that can have a truth value, but that the truth value is indeterminate does not in any way mean that you should just declare it to not have one. Again, the indexical sentence I gave in most cases WILL have a truth value even when we do not have the context to evaluate it.

That carries on to "This sentence is true". Again, the sentence clearly seems to be the sort of sentence that we can associate a proposition to that would have a truth value, but in that case in most cases we don't have any context that can provide it -- except axiomatic assertion. Which can assign a truth value to that sort of sentence.

If you think that we can simply “axiomatically assign” truth values to sentences, then you are talking about the sentences of a formal calculus and not about statements made in a natural language. If you want to talk about formal calculi, that is your privilege, but do not then pretend to be making points about sentences and statements in English. If you assign the value “true” to S3 in some formal calculus, this has no bearing at all on whether an English sentence identical with S3 has a truth value. The rest of your comment after this point seems to me likewise vitiated by this equivocation.

Unless you are talking about propositions -- ie not natural language sentences -- you can't assign truth values to a sentence. If we can assing truth values to any of the sentences we've talked about, I can do it by providing a sentence that tells me precisely what that truth value is. So either you are assuming that this is about natural language when it absolutely cannot be, or else you are now quibbling over assigning truth values to natural language sentences despite all of the previous discussions assuming that we can do it just like we would in formal calculus. Whichever way you are going here, you are makign an invalid counter, and I am not equivocating in the slightest.

Anyway, what I was saying there is that if we — ie the person reading the sentence — does not have the context in order to determine if the indexical sentence is true or false, it seems ridiculous to suggest that it therefore does not, in fact, have a truth value at all. The indexical sentence clearly DOES have a truth value, or at least is the right sort of sentence that can be associated with a proposition that can have a truth value, but that the truth value is indeterminate does not in any way mean that you should just declare it to not have one. Again, the indexical sentence I gave in most cases WILL have a truth value even when we do not have the context to evaluate it.

I find this completely unpersuasive, and indeed false. Consider the sentence:

It weighs more than 5 grams.

Your position appears to be that this sentence has a truth value, only we don't know it. I find such a position patently absurd. It is just a sentence: it is not something said by someone in a context in which the word "it" has a reference. You seem to think that any time words are put together in the grammatical form of a declarative sentence, they become a truth-valued assertion. I cannot understand why you believe that, and it seems to me patently false. To suppose that every sentence containing an indexical and having the grammatical form of a statement has a truth value is analogous believing that every pronoun—not, mind you, every pronoun as used by someone on some specific context, but simply any such word as "he," "it," "they," etc.—refers to someone, only we don't know whom. I can see no more reason for the first view, which seems to be yours, than I can see for the second, which surely even you will grant to be absurd.

I do not understand your last paragraph at all.

Okay, I have reread the last paragraph of your comment (#44), and it looks to me now as though we have misunderstood each other. The passage that I was criticizing, in your previous comment (#41) was this:

Take this example:

Let [S3] be the sentence “This sentence is true”.
Let [S3] be true.

Then I can resolve it.. We axiomatically assign the truth value of “true” to [S3], which makes it true, and then we check to see what the content of the sentence is in light of that, and it turns out to be true, and we’re all happy.

The dispute between us is whether sentences like "This sentence is true" have a truth value. My criticism is that you cannot simply stipulate that such a sentence has a particular truth value, if it is supposed to be a sentence in a natural language. The business about sentence vs. proposition seems to me irrelevant to this particular point, but I can also put my point in those terms, by saying that it is begging the question to assume that S3 expresses a proposition. In the absence of a context, S3 has no truth value. As I have said, once can imagine contexts in which it would have a truth value: e.g., if someone looks at some sentences in which statements are made and says, with reference to one of them, "This sentence is true." But in the absence of any such context, S3 has no truth value; and if the words "this sentence" in it are supposed to refer to the very sentence in which they occur, then the sentence cannot have a truth value.

My criticism is that you cannot simply stipulate that such a sentence has a particular truth value, if it is supposed to be a sentence in a natural language.

I have two comments here, presuming that you mean stipulating that such a sentence has a particular truth value to be something like adding another sentence that says "The sentence is true" (note, not "THIS sentence is true"):

1) Why not? Why can't I add that, noting that I already said that it would be, for the most part, pragmatically useless?

2) Even if you can do that, why do you think that the Liar Paradox refers to natural language sentences as opposed to propositions?

The business about sentence vs. proposition seems to me irrelevant to this particular point, but I can also put my point in those terms, by saying that it is begging the question to assume that S3 expresses a proposition.

But I don't assume it. I point out that structurally it really looks like a proposition. For "This sentence is true", I can say:

a) It is a statement rather than a command or question, and so statements can generally take or be translated to propositions.

b) It has the same format as similar statements about sentences, with the only difference being that it is self-referential (and note that you yourself declared the self-reference as being irrelevant).

c) I can assign truth values to the sentence without causing any interpretive problems or in any way impacting the perceived meaning or purpose. In short, I can do a truth table for it and it works and makes sense.

d) The only reasons I've seen for NOT treating it like a proposition have to do with there being a missing context that would allow us to determine what specific truth value it has, but any sentence or proposition that depends on context for its truth value will have the same issue, and no one -- except perhaps you -- is willing to claim that they can't be propositions if that's the case.

For "This sentence is false", c) fails. But even that doesn't seem good enough cause to claim that it isn't really a proposition.

So what I deny is that "This sentence is true" cannot have a truth value. I say that we can specify it axiomatically. You say that we can't do that for natural language sentences, but I disagree, at least to the extent that we can assign truth values to natural language sentences at all, since I say that we can do that for natural language sentences that can, shall we say, be expressed as propositions, and if the sentence can be expressed as a proposition then we can declare the proposition true and therefore the sentence true. Not pragmatically usefully of course, but that's not the real point here.

1) Why not? Why can’t I add that, noting that I already said that it would be, for the most part, pragmatically useless?

2) Even if you can do that, why do you think that the Liar Paradox refers to natural language sentences as opposed to propositions?

To (1): You can add all the sentences you want. The question is whether you are achieving anything by doing so. My point, as I have said before, is that when what is at issue is whether a certain sentence is capable of having a truth value, then to purport to assign it a truth value is to beg the question.

To (2): Your remark seems to me disingenuous. I remind you that almost every version of the liar paradox discussed here, including the versions that you have used, have been expressed in terms of "this sentence," "sentence S," and so on.

But I don’t assume it. I point out that structurally it really looks like a proposition. For “This sentence is true”, I can say:

a) It is a statement rather than a command or question, and so statements can generally take or be translated to propositions.

b) It has the same format as similar statements about sentences, with the only difference being that it is self-referential (and note that you yourself declared the self-reference as being irrelevant).

c) I can assign truth values to the sentence without causing any interpretive problems or in any way impacting the perceived meaning or purpose. In short, I can do a truth table for it and it works and makes sense.

d) The only reasons I’ve seen for NOT treating it like a proposition have to do with there being a missing context that would allow us to determine what specific truth value it has, but any sentence or proposition that depends on context for its truth value will have the same issue, and no one — except perhaps you — is willing to claim that they can’t be propositions if that’s the case.

The fact that something "looks like" a proposition shows only that it is a grammatically well formed sentence, not that it has a truth value. Every one of (a) through (c) applies equally well to pseudo-statements like "Colorless green ideas sleep furiously." They therefore do nothing to show that "This sentence is true" is a genuine statement. As to (d), my point about context is (1) that the mere form of words, "This sentence is true" could be used in a context of a certain sort to say something true or false (namely, a context in which the speaker is referring to some sentence other than the one he is uttering), (2) that such contexts have been excluded ex hypothesi and are not what you want S3 to be, and (3) that no other context has been provided or suggested in which S3 could be an intelligible statement. I will go further and venture to say that no such context can be provided: I have no proof of this, but, since you are the one maintaining that S3 has a truth value, I believe that the burden of proof is on you to provide such a context.

Yes, S3 looks like a statement, verbally and grammatically; and one can run a certain empty logical exercise with it of treating it as if it were a statement. But it isn't, because it says nothing that can be true or false: it is just a logical run-around, like a pair of signs that I recently saw at the two entrances to a building, each of which directed visitors to the other. To say "Sentence S is true" passes the burden of making an assertion on to S: if S contains no intelligible assertion then neither does the sentence just quoted. If it turns out that S is the very sentence just quoted, then nothing has been asserted at all.

Minor correction: for "grammatically well formed sentence" please read "grammatically well formed declarative sentence."

Happy now?

I think Mark P. @ 17 said it well--though I didn't locate the posts to which he refers.

"This sentence is not a true proposition." Which sentence?

"This one." Which one is 'this one" ?

“This sentence is false.” Which sentence?

Could we assert, "I am a sentence and I am false." and mean anything by that ?

If Kurt Gödel didn't prove that a sentence cannot meaingfully refer to "itself", surely he intended to do that, if not newtn, then "one of these days.". He was very busy, though--and believed (I am not making this up!) in sprites, fairies--yes, the supernatural sort--which are agents of evil and whose work is to sow mischief and trip us up in our thinking.

To (1): You can add all the sentences you want. The question is whether you are achieving anything by doing so. My point, as I have said before, is that when what is at issue is whether a certain sentence is capable of having a truth value, then to purport to assign it a truth value is to beg the question.

I'll repeat my c) from above:

c) I can assign truth values to the sentence without causing any interpretive problems or in any way impacting the perceived meaning or purpose. In short, I can do a truth table for it and it works and makes sense.

If I can build a truth table for it, then it is capable of having truth values. The only open questions would be "Does it have a truth value?" and "What truth value does it have?" In order to determine that, you are correct that we would need to know what the sentence means, but I argue that we DO know what the sentence "This sentence is true" means, and from that we know that it can't take one pragmatically, but only axiomatically. But that is not enough to claim that it doesn't have a truth value.

Note that you cannot build a truth table for "This sentence is false". Which is the paradox: why can't I? The only difference between it and "This sentence is true" is that the former claims that it itself is false, and that's not enough from a logical/propositional standpoint to say that it isn't a proposition or that it can't take truth values.

In short, if I can do a truth table for something, then it can take truth values, and if I can't then that is not sufficient to say that it can't take truth values.

If I can build a truth table for it, then it is capable of having truth values.

This seems to me a patently silly argument. Consider the string of words "Even numbers are greasier than odd numbers": call it S4. Here is a truth table for S4:

S4
T
F

Is this supposed to show that S4 is capable of having a truth value? Presumably not; it is a completely empty exercise in table-construction. I see no reason why the exercise of constructing a truth table for S3.

Of course, any sentence, indeed any string of symbols, is in some sense "capable of having a truth value." If, for instance, I assign a special interpretation to S3, stipulating that by "Even numbers are greasier than odd numbers" I mean "Even numbers are divisible by 2 without remainder," then S4 becomes a sentence not merely capable of bearing a truth value but actually bearing one, namely the value "true." But just as it stands, S4 is, at best, a poetical fancy whose meaning is too unclear for it to be held to be true or false.

As I have said, S3, "This sentence is true," is capable of bearing a truth value in an appropriate context, namely one in which somebody is talking about some sentence other than S3 itself. But, of course, that sort of contextual interpretation of S3 has been ruled out ex hypothesi. In our discussion, the "this" of S3 has to refer to S3 itself. Under that restriction, S3, though a syntactically well-formed declarative sentence, is not one in which anything is asserted that can be true or false. The possibility of constructing a truth table shows nothing about it.

Correction again: for "why the exercise of constructing a truth table for S3" please read "why the exercise of constructing a truth table for S3 should be essentially different."