paradoxes; they're an existential threat
— Agent Smith

That's a paradox itself. — jgill

How so?

PPS. I'm not sure, but I think that not just showing that the union is a recursive axiomatization might require simultaneous recursion, I think simultaneous recursion might be required even to define the set that we taken the union of.

Also, what fixed point do you have in mind?

PS. We could take the union of all those theories. (And maybe not hard to show that it is recursively axiomatized? We might need to use simultaneous recursion?) But I don't know whether that would be what you're looking for.

I'd like to add that the books I recommended are beautifully written, precise, and pedagogically excellent. One can see that each of them was written with great care and with both mathematical and pedagocial expertise.

(Though, I forgot to mention that both the Enderton books need to be supplemented by errata sheets, which, hopefully, can be found online if not inserted into the book copies.)

paradoxes; they're an existential threat — Agent Smith

That's true! I cannot understand why more people don't lie awake at night about it! It requires a global response. We need the World Bank, the World Heath Organization, the International Court of Justice, Interpol, the United Nations Security Council, and the entire cast of 'Glee' on this!

Either that, or Agent Smith just doesn't know the meaning of 'existential threat'.

I haven't looked at that one. I did like using the Schaums topology intro, but only as a supplement to other textbooks.

For example, even at page 9, just a few pages in, he writes (Zorns' lemma):


"Say that a collection of sets C is a chain iff for any elements x and y of C, either x is a subset of y or y is a subset of x.

Let A be a nonempty set such that for any chain C subset of A, the set union-of_C is in A. Then there is some element m in A which is maximal in the sense that it is not a subset of any other element of A."


Granted, that doesn't come into play very much in the book (because it's not needed for countable languages), but you see that there's a level of mathematical sophistication taken for granted from the start. On the other, having first studied his set theory book, you'd know Zorn's lemma, its proof, its relationship with the axiom of choice and the well ordering theorem, and its importance generally.

You can study Enderton's mathematical logic book first. However, the book is intended for upper division students who are already studying abstract mathematics such as analysis, abstract algebra, maybe topology. It's not that the material in those subjects is required, but that a level of mathematical sophistication and reasoning is obtained through study of that kind of mathematics.

On the other hand, I didn't study mathematics such as analysis, abstract algebra, or topology prior to studying the book. But I did study the sequence of books I recommended to you. And it worked perfectly for me. I even think that the sequence I recommend is better for leading up to Enderton's book than just studying abstract mathematics.

But if you just start with the Enderton logic book, you'll probably grasp some, maybe a lot, of it. But without the books I recommend first, I bet you'll still be confused about a lot of the Enderton logic book. You surely won't get some of the concepts firmly. Most importantly, I don't think you'll really get what he's driving at without first really knowing how to work in symbolic logic.

When you study the Kalish-Montague-Mar book first, you really understand working in symbolic logic, and that will raise questions in your mind about looking at symbolic logic from above, about not just what it is involved in working in symbolic logic but about questions about certain properties of the logic system itself, i.e. not just the logic but the meta-logic. I.e. learn the logic first in Kalish-Montague-Mar then in Enderton learn the meta-theorems about that logic. The set theory book is also before the Enderton logic book, because with set theory you'll really see that the Enderton logic book is doing things rigorously from the set theoretic axioms.

On the other hand, a lot of people are a lot smarter than I am, and probably they could learn the Enderton book quite well without first reading those other books or first having having studied some upper division math.

Thus a math expression homological to a state-of-affairs, as specified in our example here, expresses contradictory conclusions that are both valid. — ucarr

'valid' has a technical meaning. I wonder whether you are using 'valid' in some other sense.

Anyway, a formula is a contradiction if and only if it is of the form "P & ~P" (or sometimes we say it is a contradiction if and only if it proves a formula of the form "P & ~P").

I'm taking classical logic as the context throughout this discussion unless mentioned otherwise:

A contradiction is not true in any model, so, a fortiori, it is not the case that there is a contradiction that is true in every model (i.e. there is no such thing as a contradiction that is valid).

You're telling me that a math expression that asserts a claim is nonetheless considered theoretical? — ucarr

No. 'Theory' in such contexts is has not the same sense as 'theoretical'. A theory is a set of sentences closed under deduction. An interpretation of the language for a theory may be of any kind of entities or states of affairs we may wish to stipulate, not just "theoretical" ones.

classical logic parameters categorically exclude contradiction — ucarr

'parameters' has technical meanings in mathematical logic and mathematics. I don't know what you mean personally by "logic parameters". And I don't know what you mean by "categorically exclude contradictions". I can tell you that in classical logic:

We can write contradictions. We can put any contradiction as a line in a proof.

There is no model in which a contradiction is true. So, a contradiction entails every sentence (i.e. the semantic version of the principle of explosion ).

A consistent theory never has a contradiction as a member (i.e. no contradiction is a theorem of a consistent theory).

An inconsistent theory has every sentence as a member (i.e. with an inconsistent theory, every sentence is a theorem of that theory, i.e. the principle of explosion), so, a fortiori, an inconsistent theory has every every contradiction as a member (i.e. with an inconsistent theory, every contradiction is a theorem of that theory).

this is an example of a mathematician modulating axioms to fit a metaphysical principle (LNC) — ucarr

I guess by "moduating axioms" you mean "choosing axioms".

Yes, the axioms of classical logic are chosen so that with them we achieve soundness and completeness:

A sentence P is provable from a set of sentences G if and only if there is no model in which all the members of G are true but P is false.

The logical axioms are true in every model, and there is no model in which a sentence and its negation are both true. So, it is not the case that there is a sentence and its negation that are both provable from the logical axioms alone.

a mathematician can re-jigger axioms to admit contradictions — ucarr

One can depart from classical logic and allow that a system can prove contradictions but without explosion. However, the semantics for such a paraconsistent framework may be more complicated than the 2-value semantics for classical logic (which for the portion that is the sentential calculus alone, are quite simple). So I don't recall for paraconsistent systems, how, if at all, one specifies the difference between a logical axiom and a non-logical axiom or non-logical premise.

Recursive definitions also look circular, but they are not. — Newberry

Correct. But, as far as I can tell, your purported definition is not justified by any instance of a definition by recursion theorem. That is, yours is merely circular and not recursive.

a fixed point of the hierarchy of theories

T' = T + (Ex)Prf_T(x, ⌜F⌝) --> F
T" = T' + (Ex)Prf_T'(x, ⌜F⌝) --> F — Newberry

Then we would have to see a proof that there does exist such a fixed point that has the properties you claim.

even if a proof “exists” it could be infinitely long — Newberry

Though there may be contexts with a notion of infinitely long proofs, I don't know how the notion of infinitely long proofs would enter into this context in which I would take it we are considering the ordinary notion of proof in which proofs are finite.

Absolutely I can recommend the very best textbooks I have found after looking at and reading many of them:

First. Learn symbolic logic. How formulas and formal sentences are made. How many English expressions can be translated into formulas. The basic notion of an interpretation (basically a model). How formal proofs are made in first order predicate logic using a natural deduction system. The book to get (it will empower you with the basic tools for this subject and for critical thinking in general):

Logic: Techniques Of Formal Reasoning - Kalish, Montague and Mar

Next. For the very best explanation of the method of definitions in logic and mathematics, study chapter 8 of:

Introduction To Logic - Suppes

Next. Learn axiomatic set theory. Mathematical logic is itself couched in the terminology and principles of set theory, so learning set theory before mathematical logic is well advised. The book to get:

Elements Of Set Theory - Enderton.

Next. For mathematical logic, the book to get:

A Mathematical Introduction To Logic - Enderton

Next, as a supplement, for possibly the most authoritative overview of the subject of formal logic, read the Introduction chapter of:

Introduction To Mathematical Logic - Church

Let's say for context that we are interested in a particular model M at some point in discussion. So temporarily we'll take 'true' and 'false' to stand for 'true in M' and 'false in M'.

Note:

An argument can be valid even if the conclusion is false. In this case, at least one of the premises would also have to be false, in which case it is not a sound argument though it is a valid argument.

An argument can be valid even if one or more of the premises are false. In this case it is not a sound argument though it is a valid argument.

An argument is not valid only when there is at least one model in which all the premises are true and the conclusion is false.

added steps — ToothyMaw

For this discussion, in order to be as clear as possible, I suggest sticking with my technical distinction between an argument and a proof, even though in ordinary discussions we don't make that technical distinction.

An argument is an ordered pair <G P> where G is a set of sentences and P is a sentence. G is the set of premises and P is the conclusion.

An argument is valid if and only if there is no model in which every member of G is true but P is false.

An argument is sound (per a given model) if and only if the argument is valid and every premise is true (in said model).

A proof is a sequence of formulas such that every line in the sequence is either an axiom or a premise or follows by the deduction rules from previous lines in the sequence. (Note that, contrary to ordinary usage, there is no such thing as an "incorrect" proof. Something is either a proof or not. If a sequence has a line that is not an axiom, premise or follows by the deduction rules from previous lines in the sequence, then that sequence is not an "incorrect" proof but rather it simply is not a proof.)*

* Note that I am simplifying by not mentioning that there are other proof forms such as sequents (not to be confused with sequences), trees, natural deduction proofs (which can be rendered as sequences of lines that each have an index number, a formula, and the set of index numbers that are the undischarged assumptions for that line), and probably others.

/

an argument, A, whose premises include the entire set of the correct premises of sound argument B — ToothyMaw

I don't know what you mean by "correct" premise. So I'll take the above as just saying:

A and B are arguments with the same set of premises.

and has the same conclusion as B — ToothyMaw

Then A and B are the same argument, since they have the same premises as each other and the same conclusion as each other.

but the conclusion is unsound for A — ToothyMaw

I wouldn't use the terminology "the conclusion is unsound for the argument". Maybe you mean that the conclusion is not entailed by the premises, i.e. the argument is not valid.

because of added steps or premises — ToothyMaw

Steps pertain to proofs not to arguments. Proofs have steps; arguments don't have steps.

So let's look at proofs. If you have a proof and add steps that are not needed, it's still a proof. There is no requirement that every step in a proof must be a needed step.

Now, every proof is relative to its axioms and premises. So let H be the set of axioms and premises that are mentioned in the proof and let T be the last line in the proof, then we say the proof is a proof of T from H.

Now, back to arguments, a very very important thing to keep in mind about classical logic is that it is monotonic. This means that if an argument <G P> is valid than any argument <H P> is valid too where G is a subset of H. In other words, adding premises to a valid argument still results in a valid argument. That holds without exception in classical logic.*

* As an aside, this should be recognized to hold even informally in such cases where there is a valid argument but an ad hominem is gratuitously added . For example, suppose the argument is:

Premises = {"All fish are creatures", "All trout are fish", "My debate opponent is stupid"}.
Conclusion = "All trout are creatures".

That is a valid argument, despite that there is an ad hominem among the premises. That's because the logic is monotonic. The argument without the ad hominem is valid, so adding another premise doesn't result in invalidity.

Doesn't that give a model in which every member of G could be true but the conclusion, P, be false? — ToothyMaw

No.

Also, better not to say "could be true", which complicates with subjunctive modality. Just say "is true".

recursive step in a valid model — ToothyMaw

I don't know what you mean by a "step in a model" and even more I don't know what you mean by a "recursive step in a model".

an instance in which correct premises are applied to correct premises — ToothyMaw

Again, I don't know what you mean by "correct premise".

Per a given model, a sentences is either true or false (and not both true and false).

So sentences come in these varieties:

logically true (i.e. valid*, i.e. true in every model)

not logically true (i.e. not valid, i.e. false in at least one model).

logically false (i.e. false in very model)

And a sentence that is not logically true can be either logically false (false in every model) or contingent (true in at least one model but also false in at least one model).

* Notice that the word 'valid' has two different contexts: (1) A valid argument, as defined earlier, and (2) a valid sentence, as defined just above . (Also, there is the notion of valid formulas, but that requires explaining the technical difference between a formula and a sentence, and the difference between satisfaction of a formula per a model and an assignment for the variables and truth of a sentence per a model.)

in such a way that the conclusion P of argument G becomes false even though all the premises stay true? — ToothyMaw

Sentences don't change truth value per arguments. Rather, contingent sentences have different truth values per models.

Would that not be a model that would defy the formal definition? — ToothyMaw

I don't know in what sense something could be a model though "defying the formal definition".

Models, arguments, and proofs are different things, though related.

Regarding the mathematical handling of Russell's paradox as opposed to dealing with the problem informally, Russell gives a non-mathematical analogue:

Suppose there is a person who shaves all and only those who do not shave themselves.

There 'shaves' is in place of 'is a member of'. And a solution works analogous to the mathematical approach:

There is no relation 'x shaves y' such that for all x there is a y such that y is shaved by x if and only if y does not shave y (there is no person such that that person shaves all and only those who do not shave themselves). Just as, in mathematics, there is no membership relation such that for all x there is a y such that y is a member of x if and only if y is not a member of y.

For "This sentence is false" we have something similar: In a consistent theory, there is no sentence that says of itself that it is false.

Also, don't forget that a paraconsistent logic, depending on its formulation, can have both LNC and non-explosiveness.

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.

AP — Tate

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

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.

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 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.

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 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.

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.

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.

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.

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.

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.

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.

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.

First we need to be very clear in our terminology:

Formally, an argument is merely an ordered pair <G P> where G is a set of statements and P is a statement. G is the set of premises and P is the conclusion.

An argument is valid if and only if there is no model in which every member of G is true but P is false. So 'validity of an argument' is a semantical notion.

This formal sense of 'argument' is different from the informal sense of an argument being a presentation of reasoning in order to demonstrate a proposition.

On the other hand, formally, a proof is a sequence of statements such that every statement in the sequence is either an axiom (or an axiom or premise) or follows by the deduction rules from previous statements in the sequence. So this 'proof' is a syntactical notion.

However, informally, mathematicians commonly use the rubric 'proof' to mean something along the lines of 'a correct and convincing argument', in the informal sense of 'argument'. Arguably, in almost any case, such an informal proof is one that can be translated into a formal proof.

we know the premises, but don't know the format of the argument — ToothyMaw

Suppose we do know the "format", i.e. we are given the premises and the formal sequence of steps toward the conclusion, then there is an algorithm that will determine whether the steps are correct so that we know that indeed the sequence is a proof of the conclusion from the premises.

However, even if we are not given the sequence of steps, in some cases we can still demonstrate that the conclusion is entailed by the premises, either by filling in steps ourselves or by making a semantical argument that the premises entail the conclusion.

In some other case though, we might not happen to know how to see whether the premises entail the conclusion, and there can be no general algorithm that determines whether any given statement is entailed by a given set of premises (this is undecidability, which stems from Godel's incompleteness, et. al).

In any case, notice that whether the premises are true or not is a separate from the questions of validity and proof. Validity of an argument is only that any model in which the premises are true is a model in which the conclusion is true.

And a conclusion may be proven from a set of premises, irrespective of whether or not the axioms or premises are true in a given model; this is the "relative" nature of proof - our mechanics of proof only provide validity and don't ensure truth.

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'.

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 sub-branch of philosophy — ToothyMaw

In mathematics (and often crossing into philosophy) there is the subject of mathematical logic in which the notions of truth and provability are given rigorous explication.

I am not familiar with Wittgenstein's views on Moore's paradox.

"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.

gibberish — javra

But it's not gibberish. It's syntactical and it talks about the property of truth as pertaining or not to a given sentence, which is a well understood notion. We don't throw out expressions from the language merely because they present logical problems. The expression is well formed; it is only upon further analysis that we find it is problematic. It would be poor analysis to throw out sentences ad hoc only on the basis that de facto they are problematic.

A square is a circle — javra

That's not paradoxical. Rather, with definitions of 'square' and 'circle' and some theorems of mathematics, it simply, without any paradoxical aspects, implies a contradiction.

Yes, I didn't write that correctly. What I meant:

Classical logic with added mathematical axioms works just fine for a vast amount of the mathematics for the sciences.

Rather than get bogged down in whatever vagaries there might be in the Epimenides paradox, I would suggest the clearer, simpler, mathematically "translatable" simpler and more starkly problematic "This sentence is false".

“I am a member of humankind, and all humans I know of (including myself) are liars (on account of having lied at some point in their lives)” to hold a truth value.

Again, as someone ignorant in the formalities of the matter, cannot this latter sentence as expressed be mapped to a truth value in formal logics? — javra

No arithmetically adequate and consistent theory can define a truth predicate by which to then formulate a predicate 'is a liar'.

Keep in mind that Tarski's theorem is a claim only about certain kinds of theories (arithmetically adequate and consistent) formulated in classical logic.

However, given predicates Hx for 'x is a human', and Sxy for 'x states y', and Ly for 'y is a lie', and m for you, then we can write:

Ax(Hx -> Ex(Sxy & Ly)) ... "All humans lie sometimes"
Ey(Smy & Ly) ... you lie sometimes.

And those would have truth values.

But so what? It's not in question that we can formalize a lot of non-problematic things. That we can formalize a lot of non-problematic things doesn't refute that "This statement is false" is problematic.

Of course, if we wish to have theorems that are contradictions but without explosion, then classical logic doesn't work, and if one wishes to have contradictions without explosion, then one may say that classical logic is thereby wrong.

But the particular argument given by Agent Smith does not vitiate classical logic itself. Classical logic works just fine for a vast amount of the logic for the sciences. You and I are now communicating with computers built by principles of classical logic. And that does not overlook that for arguably paradoxical statements in a wider scientific context, classical logic may be inadequate.

In particular, Russell's paradox does not ruin classical logic, which gives us an immediate and wonderfully simple solution with the theorem that there does not exist such a contradictory relation.

Yes, so?