Where?

"This square is not a square" is seen as a self-contradiction on its face, and its truth value is falsehood, and there is no contradiction in saying its truth value is falsehood.

"This sentence is false" also implies a self-contradiction, but it is not so easy to say its truth value is falsehood, since if its truth value is falsehood then its truth value is truth and if its truth value is truth then its truth value is falsehood. — TonesInDeepFreeze

I keep on overlooking the subtleties. You've pointed them out well.

Here's a better justification for why I find the liars paradox to be gibberish:

TMK, given the LNC, a contradiction between X and Y necessitates one of the following three: a) X is valid but Y is invalid, b) Y is valid but X is invalid, or else c) neither X nor Y are valid. But given the LNC, possibility d), that of both X and Y being valid, will be excluded as impossible.

I'm saying "valid" as shorthand for this applying not only to propositions but also to non-propositional criteria, such as percepts and memories. For example, if event E and event F are mutually exclusive, and if one recalls that one did E at time-interval T and also recalls doing F at T, one could then assume a) having a false memory of E at T but not of F, b) having a false memory of F at T but not of E, or else c) having a false memory of both E at T and of F at T. But one does not conclude that both E and F happened at T.

I'm hoping this makes good enough sense without me needing to engage in more in-depth explanations.

If so, applying this type of general rationale to the self-contradiction of the liar paradox, it can be a) true but not false, b) false but not true, or else c) neither true nor false - but, given the LNC, it cannot d) be both true and false at the same time and in the same way.

We know that if we claim (a) it will also be (b) and that if we claim (b) it will also be (a) - with amounts to (d).

This leaves us with possibility (c): neither true nor false.

If so, this amount to the liar's paradox being syntactically coherent gibberish: a statement devoid of any possible truth value.

I might be somehow wrong in this general perspective - and if you have the time to point out how, that would be appreciated - but it's how I've so far appraised the liar paradox: as being syntactically correct gibberish.

given the LNC, a contradiction between X and Y necessitates one of the following three: a) X is valid but Y is invalid, b) Y is valid but X is invalid, or else c) neither X nor Y are valid. But given the LNC, possibility d), that of both X and Y being valid, will be excluded as impossible. — javra

In mathematical logic, 'valid' is used differently from the way you use it. Here's a quick breakdown of the terminology for ordinary first order logic (where 'P' and 'Q' stand for any statements, 'G' for any set of statements, and 'iff' stands for 'if and only if'):


Definitions:

P is true in model M iff [fill in the inductive definition here].

P is false in model M iff it is not the case that P is true in model M.

P is valid iff P is true in every model M.

P is invalid iff P is not valid.

P is contingent iff P is invalid but P is true in at least one model M.

P is a theorem of G iff there is a deduction of P from G.


Meta-theorems:

P is valid iff P is derivable from the logical axioms alone.

P is a theorem of G iff any model in which all the statements in G are true is a model in which P is true.

P is a contradiction iff P is of the form: Q & ~Q. (But sometimes, less formally, we say P is a contradiction iff there is a statement Q & ~Q derivable from {P}).

If P and Q are a contradiction (in the sense that a contradiction is derivable from {P Q}), then there is no model in which both P and Q are true. So, if P and Q are a contradiction, then given any model M, either P is true in M and Q is false in M, or P is false in M and Q is true in M, or P is false in M and Q is false in M.

neither true nor false — javra

Not in classical logic. If M is a model for the language in which P is written, then P is true in M, or false in M, and not both true and false in M, and not neither true nor false in M.

That is, classical logic is 2-valued and the semantics upholds the theorems:

~(P & ~P)

P v ~P

this amount to the liar's paradox being syntactically coherent gibberish — javra

No. First, your argument is semantical not syntactical. You can't make something not syntactical by a semantical argument. Second, you are overlooking another possibility that you are not seeing but that I have mentioned:

The statement cannot be formulated in the theory such that the interpretation of the statement refers to truth values. That is, as I've said, Tarski's theorem is that an arithmetically adequate and consistent formal theory cannot formulate its own truth predicate. If the theory does formulate its own truth predicate then the theory is inconsistent. An arithmetically adequate and consistent theory lacks axioms (assumptions) that would provide expressing a truth predicate for the language in which the theory is written. This is different from the naive notion that "we can't admit the liar sentence as a legitimate sentence", which is not rigorous because it requires ad hoc and post facto fiats about what is a legitimate sentence. Instead, the syntax is objective, decidable, and unchanging, while we do see that among the syntactical sentences there is none that can be interpreted (semantics) as expressing the truth predicate for an arithmetically adequate and consistent theory.


.

A more general subject: entailment. It occurs to me that perhaps a good way to express the advantage of the rigorous model theoretic notion of entailment over the naive notion is this:

The naive notion is that a set of statements G entails a statement P iff it is impossible that all the members of G are true but P is false.

The rigorous model theoretic notion is that a set of statements G entails a statement P iff there does not exist a model in which all the members of G are true and P is false.

So, instead of the modal notion 'possible', we have the more fundamental notion 'exists'.

the sense you mention a 'truth predicate', we actually say a 'truth function'. On the other hand, as to truth predicates, (Tarksi) for an adequately arithmetic theory, there is no truth predicate definable in the theory.

For a language, per a model for that language, in a meta-theory (not in any object theory in the language) a function is induced that maps sentences to truth values. It's a function, so it maps a statement to only one truth value, and the domain of the function is the set of sentences, so any sentence is mapped to a truth value. — TonesInDeepFreeze

In the real world we don't use sentences as truth bearers. I don't think we need to break from ordinary language use in assessing Russell's paradox. I'm just pointing out that the solution you've been talking about is artificial.

In the real world we don't use sentences as truth bearers. — Tate

Sure we do.

"Provo is in Utah" bears truth.

"Provo is not in Utah" bears falsehood.

See the Introduction in Alonzo Church's 'An Introduction To Mathematical Logic', which is wonderfully cogent, beautifully written, and arguably the very best overview of the subject.

I don't think we need to break from ordinary language use in assessing Russell's paradox. — Tate

Russell's paradox was first presented in context of formal theories. And, at least usually, the interest in Russell's paradox centers around mathematics.

I'm just pointing out that the solution you've been talking about is artificial. — Tate

I don't know how you evaluate for "artificiality". However, of course, since the subject of mathematical logic is conveyed courtesy of human intellect, I guess it's "artificial" in the same sense that just about any other area of study presented by humans is "artificial". Moreover, even if arguably mathematical logic is especially artificial in some sense, it is an excellent artifact - the product of the great intellect and sagacity of many rigorously critical scholars - that makes rigorous sense of many notions that otherwise would suffer from the vagaries of amphiboly and subjectivity. Wouldn't it be better to learn about the subject rather than glibly dismissing it out of hand as "artificial" before familiarization with even its basics?

Anyway, it's not clear to me that you understand the solution per mathematical logic.

Sure we do.

"Provo is in Utah" bears truth.

"Provo is not in Utah" bears falsehood. — TonesInDeepFreeze

No, we don't. A sentence has to be contextualized by some form of utterance to qualify as a truthbearer.

Russell's paradox was first presented in context of formal theories. And, at least usually, the interest in Russell's paradox centers around mathematics. — TonesInDeepFreeze

I think you'll need more weight than this offers to show that we can't evaluate Russell's paradox using ordinary English rules.

don't know how you evaluate for "artificiality". However, of course, since the subject of mathematical logic is conveyed courtesy of human intellect, I guess it's "artificial" in the same sense that just about any other area of study presented by humans is "artificial". — TonesInDeepFreeze

It's artifical in the sense that we could change it if we wanted to, at least we can imagine doing so.

Anyway, it's not clear to me that you understand the solution per mathematical logic. — TonesInDeepFreeze

I do.

A sentence has to be contextualized by some form of utterance to qualify as a truthbearer. — Tate

Oh come on, of course we admit that natural language utterances don't have a single definitive unequivocal context. But given some reasonable understanding of given contexts, we do view sufficiently clear sentences as being true or false. When I say "Provo is in Utah" as we both reasonably understand the ordinary context, we agree that that sentence bears truth.

But, indeed, it is mathematical logic itself that rigorously explicates the notion of context by the method of models.

you'll need more weight than this offers to show that we can't evaluate Russell's paradox using ordinary English rules. — Tate

I said that we can evaluate it by formal methods. I didn't say that we must evaluate it only by formal methods.

But given some reasonable understanding of given contexts, we do view sufficiently clear sentences as being true or false — TonesInDeepFreeze

A sentence is no more than a string of words that conforms to some linguistic rules. Once you add context you have more than just the sentence. You have a statement. The statement can have the property of truth. The string of words can't, not in ordinary language use.

This is pretty standard stuff.

It's artifical in the sense that we could change it if we wanted to, at least we can imagine doing so. — Tate

Yes, mathematical logic offers the freedom for anyone to present alternative formulations, definitions, methods, and paradigms. That's a good thing.

In any case, ordinary language and ordinary naive approaches not only can be imagined to change but we know that they do change.

said that we can evaluate it by formal methods. I didn't say that we must evaluate it only by formal methods. — TonesInDeepFreeze

And this makes a world of difference. If we can evaluate it by ordinary standards, the paradox stands.

Yes, mathematical logic offers the freedom for anyone to present alternative formulations, definitions, methods, and paradigms. That's a good thing. — TonesInDeepFreeze

I'm sure it is good for many purposes. But a solution that's subject to revision is not a strong solution.

In any case, ordinary language and ordinary naive approaches not can be imagined to change but we know that they do change. — TonesInDeepFreeze

I don't propose that they change anything.

Once you add context you have more than just the sentence. You have a statement. The statement can have the property of truth. The string of words can't, not in ordinary language use. — Tate

Again, you're not seeing the point among your unnecessarily split hairs.

Sometimes informally we use 'sentence' and 'statement' synonymously. Whether or not to do that is a matter of choice in definition. We don't need to get bogged down in disputes about such choices. Meanwhile, the distinction you mention is usually made in logic as the difference between a sentence and a proposition. And there it becomes a matter of the particular development of the subject whether we say that sentences bear truth values or whether only propositions bear truth values.

The way mathematical logic does it is this: A sentence is a syntactical object. It has no truth value as merely a syntactical object (except valid (i.e. logically true) sentences that are true in every model). However, given a model for the language of the sentence, there is a truth value for the sentence per that model (a model being an "interpretation of the language", i.e. the meanings of the words and then meanings of sentences as they are built from the meanings of the words). So, when I informally say that sentences bear truth, of course, more formally I mean they bear truth per a given model. In the case of "Provo is in Utah" I mean the ordinary interpretation we share of the city we know of and its location in the state we know of.

If we can evaluate it by ordinary standards, the paradox stands. — Tate

I have never proposed any argument that it is not paradoxical in ordinary language.

But a solution that's subject to revision is not a strong solution. — Tate

It's subject to revision in the sense that anyone can propose different approaches. Meanwhile, in terms of its ordinary mathematical context, it has proven to be pretty strong as it permits a (presumably*) consistent axiomatization of the mathematics that the system found to suffer from the contradiction also intended to axiomatize.

* Presumably only, because, one can always doubt the axioms or principles used for any consistency proof of even PA.

Again, you're not seeing the point among your unnecessarily split hairs.

Sometimes informally we use 'sentence' and 'statement' synonymously. Whether or not to do that is a matter of choice in definition. We don't need to get bogged down in disputes about such choices. Meanwhile, the distinction you mention is usually made in logic as the difference between a sentence and a proposition. And there it becomes a matter of the particular development of the subject whether we say that sentences bear truth values or whether only propositions bear truth values. — TonesInDeepFreeze

It's not me splitting the hairs. AP gets very specific about what a sentence is when comparing Tarski's project to ordinary language use. If you wander through the SEP articles touching on the issue you'll get up to speed pretty quickly.

In the case of "Provo is in Utah" I mean the ordinary interpretation we share of the city we know of and its location in the state we know of. — TonesInDeepFreeze

You're providing a context for the sentence, so it's more than just the string of words. It's a statement.

I don't propose any argument that it is not paradoxical in ordinary language. — TonesInDeepFreeze

I think we're broadly in agreement.

I don't propose that they change anything. — Tate

And I don't propose any specific changes to the explication of the paradox per mathematical logic. On the other hand, no matter what you propose or do not propose, natural language changes drastically, so if change is your determinant of 'artificiality' then natural language is quite artificial too.

And I don't propose any specific changes to the explication of the paradox per mathematical logic. On the other hand, no matter what you propose or do not propose, natural language changes drastically, so if change is your determinant of 'artificiality' then natural language is quite artificial too. — TonesInDeepFreeze

I'm open to being corrected, but I don't think we can imagine changing the rules of natural language the way we can imagine changing a formal system.

If you wander through the SEP articles touching on the issue you'll get up to speed pretty quickly. — Tate

Your condescension is belied by comparing our familiarity with the subject.

And you just skipped what I wrote about this. It depends on the author whether 'sentence' and 'statement' are taken as synonymous or whether 'statement' is taken only as 'proposition' or a similar rubric.

I happened to be using 'statement' for 'sentence'. That is not essential. I could just as easily say that, since that usage conflicts with other usage you have come across, then I could confine to 'sentence' and 'proposition' or whatever stipulated uses we choose to agree upon. When reasonable people find an innocent and understandable terminological clash, they may accept from one another that they just happened to have different meanings in mind and then agree to a shared meaning going forward.

You're providing a context for the sentence, so it's more than just the string of words. It's a statement. — Tate

Please, you purely disregarded what I said about that. I already agreed that sentences do not have truth values without an interpretation. Do you ordinarily go around disputing people when they say things like "He spoke spoke a true sentence when he said 'Provo is in Utah', and you would continue to dispute them even when they granted that some people don't take 'sentence' in the same sense as 'statement' so to take their remark, mutatis mutandis, per whatever agreed upon stipulation so that, of course, one means an utterance or expression in combination with some interpretation of the meaning of the words.

I don't think we can imagine changing the rules of natural language the way we can imagine changing a formal system. — Tate

Ordinary language changes in the course of millions of individual choices toward variation but also sometimes in decisive strokes. If you wish to argue that that mitigates that even ordinary language is artificial, then okay I suppose. But then I don't see much persuasiveness in the argument that mathematical (especially mathematical logic) has its explanatory potency diminished by the fact that it always can be augmented in clear, unambiguous, and rigorous ways. Moreover, empirical sciences are always subject to emendation, so we would take them too as "artificial" and dispute their explanatory value while giving a fair amount of weight, at least as far as the criterion of artificiality, to more naive explanations, even superstitions, that have been more stable even if ignorant.

You don't know really anything about the subject of mathematical logic, yet you are persistent to somehow fault it in a quite flimsy way. I wonder why.

You'rr correct! The properties of the whole tend to be inexplicable from a parts point of view (holism: the whole is not just the sum of its parts). The fact that a Unified Theory of Everything has eluded us till now is evidence of that I suppose (the world of the small doesn't quite jibe with the world of the large).

But then I don't see much persuasiveness in the argument that mathematical (especially mathematical logic) has its explanatory potency diminished by the fact that it always can be augmented in clear, unambiguous, and rigorous ways. — TonesInDeepFreeze

I didn't say anything about its explanatory potency.

You don't know really anything about the subject of mathematical logic, yet you are persistent to somehow fault it in a quite flimsy way. I wonder why. — TonesInDeepFreeze

You don't appear to know the basics of the philosophy of truth, so we're even. :razz:

I don't propose any argument that it is not paradoxical in ordinary language.
— TonesInDeepFreeze

I think we're broadly in agreement. — Tate

I don't know how you ever came up with the strawman that I don't take the statement as paradoxical in its everyday language context. Indeed, very much to the contrary, I have argued that it is a formalized version that provides not paradox but merely that for any relation R, ~AxEy(Ryx <-> ~Ryy).

I didn't say anything about its explanatory potency. — Tate

You did in so many words. If you object to the paraphrase, then substitute the actual words you used.

You don't appear to know the basics of the philosophy of truth — Tate

Appears to you, whose perception is poor.

Moreover, even as I grant that I am not expert in philosophy, I do know some basics, and I point out that I haven't made very much, if anything, in the way of philosophical claims. Mainly I addressed the technical matters that are behind certain claims about logic and mathematics made by another poster. And even though I am not a true expert in those technical matters, I know enough to see outline the nature of his mistakes.

so we're even — Tate

No, because I do know at least something about philosophical notions of truth, while you know virtually nothing about the context of mathematical logic that was the context of my remarks regarding the poster's claims about the logic used for mathematics and about paraconsistent logics.

AP — Tate

What does 'AP' stand for? And what article online (if it's online) do you refer to in relation to Tarski?

AP: Analytical philosophy.

A fair chunk of AP as it relates to truth revolves around Tarski. I didn't refer to any specific article.

@javra

To my understanding neither multivalued logic nor fuzzy logic deny the LNC. They seem to be about truth value, how many of them are there or necessary to make sense of reality to be precise - they both reject the principle of bivalence though.

A system for use with a multi-valued semantics can be paraconsistent or not.

However, as far as I know, a paraconsistent system can't have a classical 2-value semantics.

And a while back I corrected your misconception about LNC and paraconsistent logic. A paraconsistent logic, depending on its formulation, can have both LNC and non-explosiveness.