A lot of this is just wrong but I'll focus on one thing because @BlueBanana is on the rest.

(3) In other words, if we pair the elements of infinite set A one-to-one with those of infinite set B, we have 10 numbers left over.

This is incorrect. You are confusing lacking particular unique elements (the numbers 1-10) as changing the cardinality. It doesn't. Both sets you specified have the same cardinality so there will be nothing left, they can be put into a one-to-one correspondence with each other.

The set of even numbers lacks all the odd numbers, and yet that set can still be put into a one-to-one correspondence with the set of natural numbers which has both the even and odd numbers. I personally have never met a mathematician who (when speaking to a mathematically literate audience) define infinity as anything other than:

"A set whose members can be put into a one-to-one correspondence with a proper subset of itself."

Obviously that will be not understood by the lay audience. And further, infinity is a concept in common parlance too, it is not just a mathematical concept so the definition won't necessarily match. Perhaps I am misunderstanding you or your point; I am rather low on sleep at the moment, heh.

Yea Explosion is wild. That's why dialetheists adopt a Paraconsistent Logic, most often dropping either Disjunction introduction or disjunctive syllogism. By challenging these, you might also lose something like Excluded Middle or something else as well, but you can then formalize logical theories involving contradictions. Perhaps you want to solve the Liar Paradox or else you want to resurrect Frege's Logicism. After all, Godel's Incompleteness Theorems do suggest that you can build a complete mathematics on an inconsistent foundation, but that's only available when using Paraconsistent Logics.

There are a number of reasons you might want to avoid reasoning explosively besides Dialetheism, these are just some interesting reasons you might consider going to paraconsistency.

Can you show me how that is implied? How do you prove it?

Here's the simplest way. Take 2 arbitrary assertions and assign them to the letters "P" and "Q".

"P" = "It's Monday"
"Q" = "I'm green"

1) P AND ~P
[Assume a Contradiction is the case]

2) P (because if an AND statement is true, each of the components must be true)
[Conjunction Elimination from #1]

3) P OR Q (Only 1 part of a Disjunction needs to be true for the entire statement to be true, and we just proved that P is true)
[Disjunction Introduction from #2]

4) ~P (Remember, if an AND statement is true, both components must be true. In this case, the other component is the negation of P)
[Conjunction Elimination from #1]

5) Q (In an OR statement, if one component is not true, we know the other component must be true. And we just proved above that ~P is true)
[From 3 & 4 via Disjunctive Syllogism]

So if it's both Monday and it's not Monday, I am therefore green.
That's trivial. It let's you prove any and every proposition is true if you assert a contradiction in a logic that is not Paraconsistent.

Once one starts dictating what constitutes logic one is moving away from rather than towards that which constitutes being logical. That which is logical, first to be considered as being logical,, cannot exist as a state of detached thinking in its own right, it has to be shared by a majority of the population. There is no such thing as independent of perception logic. That which is may fall outside of perception, and then it can demand for more than mere logic to take you there. One can argue logic, prove one`s communication skills, and their ability to borrow and steal other`s thoughts,but none of this is philosophy, either one has good instinct for both producing and recognizing that which is or one does not, it`s that simple. Brief is always best in philosophy, not in those other areas, but most definitely in philosophy, it is an instinct. Arguments and counter arguments have no place here because they have no end.. Without this instinct and beyond the mental exercise so called philosophy is a total waste of effort, arguments being infinite. The obstacle is big brains, big egos, and big mortgages, but with no natural instinct, and also of course the whole institution around philosophy.

I swear you sound like a computer generated message.

"God exists" is a claim about our physical world isn't it? Why else would there be so much debate on it?

You're confused. That's an assertion *about* the world. The proposition the assertion expresses is not part of the physical world. It's an abstract object.

Relativity destroys the notion of simultaneity while LNC requires simultaneity.

Time is not an actual component of the LNC at all. All the LNC says is that the proposition not be true and false in the same sense (i.e. the variable stand for the same thing). Propositional logics and quantified logics don't make use of time at all, it's not part of the formalism. You'd have to go to a temporal logic for that, and even then the LNC would be relativized to particular reference frames.

↪gurugeorge I hold similar views. Logic, at least in its useful form, must conform with experience. Does this "middle-world" you speak of violate the LNC? It does not and how do we actually go about rejecting the LNC? Please read below.

I'm responding to a bit more than what you said to me because I feel more can be said.

Logic need not "conform to experience" in order to be useful. Aside from the fact, that human experience is inherently limited and varied, not all humans interpret or understand their experience in the same way. Like take the so-called "Laws of Thought" (this is a terrible name IMO). People often speak of these are obvious, indisputable certitudes that were never reasonably questioned and the only ones questioning it are modern relativists. But this is demonstrably false. I mean, one need look no further than the "cacuskoti"/tetralemma of Indian philosophers, in which propositions are thought to be capable of being only true, only false, both true and false, or else neither true nor false.

Or if someone wants to just dismiss this as nonsense on the spot because something something Eastern philosophers are wacky frauds, let's take a classical Western Philosopher: Aristotle. Aristotle believed that sometimes the Law of the Excluded Middle was not always true. Specifically, that it failed when making contingent statements about the future. In a sense, this lies beyond experience and yet if Aristotle were right about this, then we would want to have a logic that can handle "violations" of the LEM.

This makes an important point broadly about theories, which is just as applicable to logic. Data in theories are soft, that is, the data itself is also fallible and can be overturned by a good enough theory.

A theory which helps us understand why our data is flawed or incomplete or misunderstood. Logical systems (e.g. Classical Logic (Frege's Logic, different from Aristotle's), Intuinistic logic, etc.) are theories about logical consequence (what follows from what). And in this case, you seem to be saying human experience must be data which logical systems have to conform to. But it can just as easily be the other way around: logical systems can overturn what we believe to be true from human experience, just as Non-Euclidean Geometry upturned assumptions we had about the possibilities of different spaces.

/long-winded, sorry

That's a good point. Rejecting the LNC doesn't require that ALL statements are both true and false. Finding just one statement that is both is a good enough counterexample to the LNC. So, in a sense, rejecting LNC shouldn't be self-referential. Thank you very much.

Well, if you're using an explosive logic (i.e. every type of logic besides Paraconsistent logic), rejecting the LNC does require that all proposition are true and false. It's provably so.

And ironically, according the semantics of standard dialetheic paraconsistent logics, the LNC is a dialetheia, it's both true and false.
— MindForged

Can you explain that a bit. I didn't understand. Thank you.

It takes a bit of delving into the semantics of dialetheic logics, but I'll try to state it better than I did. Essentially, if you believe there are contradictions which are also true (known as Dialetheism), you are committed to believing Dialetheism itself is both true and false. It falls right out of the logic. It seems counter-intuitive (in an already counter-intuitive theory of truth), but it is provably the case.

Man, this thread is hilarious and sad. I just gave up earlier in the page wen some guy basically just substituted "being keen on evidence" for "beliefs" , as if he weren't simply using a different word to mean the same damn thing. This phobia of the word "belief" is annoyingly common among who I assume are atheists.

Kind of a horribly vague question. For one thing, "number" is going to be quite different depending on A) What kind of "number" you're referring to B) What sort of mathematics you're working in (numbers in ZFC + classical logic look quite different than numbers in Paraconsistent Mathematics), etc.

This topic is simply too vast for me and I personally try not to think about it too much, lol.

Vicious circularity is when one is face with some issue, and one's solution to the problem has the same issue and the problem propagates when they try to apply the solution.

Whether or not you admit it, Scientific thinking does not work that way. — uncool

Do you believe that or is it simply false? Seriously, read the research linked. It's about bad belief formation, not that the concept of belief is somehow flawed.

Yeah your statement above doesn't remove that unless evidence is prioritized, no amount of belief or passion delivers results. — uncool

Belief is not passion and if you cannot even understand the role of evidence in motivating believing or disbelieving some proposition, you are beyond my help. No one has said to prioritize blind belief and the fact that you think evidence is somehow immune to bias is stunning.

I'm done, I didn't think this thread could be as bad as I had assumed.

So you continue to ignore evidence.

The evidence simply states that belief does not largely facilitate that evidence is considered.

How in Bill Gates' name does something that mostly permits evidence ignorance (i.e. belief) become compatible with something that generally facilitates that evidence is generally considered (i.e. Science)? — uncool

How am I ignoring evidence? Beliefs *don't* mostly permit ignorance, that's a vapid assertion. The whole point of evidence is that it motivates *believing* some proposition to be true or false. Evidence is *for* something, it is not an end in itself. If you can't understand that then this is a waste of time.

The obvious response is to ask "Do you believe what you're asserting?" If your answer is "no", then you're simply saying that what you're asserting is false and we can all get on with our day.

That scientists believe on non-evidence, does not suddenly remove that science generally facilitates that evidence is considered.

In fact, you've demonstrated that when scientists fail to prioritize evidence, they fail to make progress in a regime where evidence generally facilitates progress. — uncool

You are unreal. It's not that they believed on "non-evidence", it's that only a fool immediately changes everything they believe at the first inkling of doubt. It was reasonable to be skeptical about new developments in physics overturning centuries of beliefs about the world that Newtonian physics gave us. It was precisely new evidence and new models which motivated *believing* that the newer theories were correct (or at least covered more cases correctly) than the old models.

Looking at your earlier quote below, that is precisely what belief generally promotes: — uncool

You do realize that I was using your proposal and you just agreed (umwittingly) that it was indistinguishable from belief, don't you? Your proposal is paper thin fear of religion or some other silly thing.

That statement of yours is demonstrably false, because scientific thinking has been less susceptible to evidence ignorance and evidence distortion, than what belief generally facilitates.(Scientific thinking has brought technological/scientific progress, and promoted that old mistakes were and are repaired rather than being maintained regardless of contrasting evidence) — uncool

What a whopping non sequitur. That science has improved over time doesn't show that it is above exactly the bias you complained about beliefs being susceptible to. See the move from Newtonian dynamics to 20th century developments, especially qusntum mechanics.

No one is saying false beliefs should be kept, yet simultaneously, it is the height of foolishness to always abandon a belief the moment it isn't 100% certain. There a balance here you don't seem to recognize exists. Beliefs ought to be formed rationally. That's why science is rather successful, not because scientists abandoned the concept of belief.

If you go back to the OP, or my exchanges with you, you may quickly notice what I underlined all along; scientific thinking generally permits evidence consideration, while belief generally permits evidence ignorance.

I don't know how many times I can literally just substitute your phrase "keen on evidence" to show its being used exactly the same as belief is until you get it. Beliefs are part of science. Ask a scientist if they believe quantum mechanics is the best theory we have of the quantum world and they will say yes. Do you know why? Because a belief is just what you hold to be true or false. The fact that it can be susceptible to bias (like literally everything else humans do) is the stupidest reason to discharge a concept that is used in every field of science, mathematics, etc.

If you constantly observe evidence, rather than only looking for data that agrees with your prior stance, you're already doing what is contrary to the concept of belief. — uncool

How does that change a thing? My point is people's biases will do the exact same thing if they are "constantly observing evidence" instead of believing. The evidence motivates believing some set of conclusions, that's the whole point of bringing up evidence: it provides reason to accept (provisionally) some proposition as being true or false. It is no less immune to the manipulation of personal bias and to try and pass it off as such is so naive as to be scary.

The point is that beyond the philosophical definitions which you quoted above (on which there is no concensus) there exists experimental data on the other hand, showing that belief generally occurs such that evidence is ignored. — uncool

I already answered this and showed the shell game. Let's try again:

You are quoting it referring to people who form beliefs in an irrational way. Great argument, I never realized that people could use a tool incorrectly or have biases. Let's try this:

I will now be "keen on evidence". Oh what's that, you have some evidence which contradicts what evidence I am currently "keen on"? Well that can't be correct, I will ignore your evidence and only pay attention to the evidence which supports the evidence I am "keen on".

See what i mean? This is the stupidest semantic deception I've seen in a good while. It's literally just changing the label of what we refer to as a "belief" — MindForged

People will do exactly the same thing if they are "keen on evidence" (e.g. believe) as suggested by you. This is nothing but a smokescreen and ignores that the research you quoted was about people forming beliefs irrationally (overly driven by bia.

Don't forget this line:

"There is, for example, no philosophical consensus on what belief is." — uncool

Your point? Are you serious? Yes people disagree on what exactly a belief is. What does that have to do with anything? There is no philosophical consensus on what "evidence" is either, and even about how evidence "supports" the truth of some proposition. This is more absurd by the minute.

You are quoting it referring to people who form beliefs in an irrational way. Great argument, I never realized that people could use a tool incorrectly or have biases. Let's try this:

I will now be "keen on evidence". Oh what's that, you have some evidence which contradicts what evidence I am currently "keen on"? Well that can't be correct, I will ignore your evidence and only pay attention to the evidence which supports the evidence I am "keen on".

See what i mean? This is the stupidest semantic deception I've seen in a good while. It's literally just changing the label of what we refer to as a "belief".

You are reaching. Again, this is neither state in the research and is even contradicted by it.
:

"Belief can be defined as the mental acceptance or conviction in the truth or actuality of some idea (Schwitzgebel, 2010). According to many analytic philosophers, a belief is a “propositional attitude”: as a proposition, it has a specific meaning that can be expressed in the form of a sentence; as an attitude, it involves a mental stance on the validity of the proposition (Schwitzgebel, 2010). Beliefs thus involve at least two properties: (i) representational content and (ii) assumed veracity
(...)
Beliefs, or perhaps more realistically belief systems, provide the ‘mental scaffolding’ for appraising the environment, explaining new observations, and constructing a shared meaning of the world (Halligan, 2007)."

I mean if you literally get rid of one of the most basic concepts in human experience I guess being ridiculous isn't a problem for you.

As I said, you are literally just taking time concept of "belief" (holding some proposition to be true or false) and renaming it "being keen on evidence".

And try reading that research in the OP. It is primarily about delusions (false beliefs, as defined in the research) and belief formation. You are misguided and misrepresenting research because of some silly fear of religion.

Remember, that evidence doesn't depend on beliefs; scientific equations don't suddenly work because scientists chose to believe in them.

It doesn't work until you follow evidence, and no amount of belief or passion poured into work, affords that said thing works. — uncool

That is completely irrelevant. Scientists *believe* (meaning they hold it to be true) that some such equation is veridical. No one is proposing some causal relation between "X believes Y" and "Therefore Y is true" in virtue of being believed. This should be extremely simple.

Belief no more "permits one to ignore evidence" anymore than your weasel phrase "being keen on evidence" (which is just another way of saying "belief").

We can trivially avoid belief by generally being keen on evidence.(which is contrary to the concept of belief which generally permits the opposite, evidence ignorance) — uncool

Man are you serious? Being "keen on evidence" literally just means believing what the current evidence seems to indicate. This is hilarious. Evidence is a nebulous term, which people can just as easily be fallible or even stubborn about. Not everyone agrees on what some set evidence suggests, nor even on what counts as appropriate evidence for some proposition or view.

This is simply absurd. Beliefs *are* part of science, no matter what asinine claim your favorite science communicator tells you. Scientists, within their work, believe and assert things as true. Yet, those claims are rarely taken to be claims of infallibility. Even things that we know to be true are beliefs, that's literally part of the classical definition of knowledge. Things I know to be true are also thing since believe to be true. This holds in mathematics, formal logic, science, you name it.

I cannot believe how jumpy people have let their fear of the influence of religion make them become. Utter foolishness.

This is just a stupid thread. A belief is simply a propositional attitude. I.e., what you hold the be true or false. The idea that belief is somehow a religious concept or that it ought to be abolished is mind-numbingly ridiculous.

"Beliefs ought to be abolished" do you hold that to be a true statement? If you do, that means to you believe that statement.

This is a stretch but thoughts, propositions included, are, so far as we know, matter-based. Is it too much, then, to say that the ToR applies to propositions that aren't about our physical world? — TheMadFool

That's absurd. How on earth are propositions (an abstract object) "matter-based"? Show exactly where a proposition is in the physical world.

Relativity does not violate non-contradiction. Propositional truth values would be relativized to particular reference frames, so no contradiction ever actually arises.

People seem to think "logic" means Aristotle's logic from 2400 years ago. Nothing could be further from the truth. Today logicians are perfectly comfortable embracing and formalizing contradictions. — fishfry

While you are correct that there is a whole panoply of logics, some formalizing non-explosive inferences from contradictions, and even the truth of contradictions (dialetheism), you misunderstood the person. What they said was "contradictions in reality", not in a formal system (e.g. in a dialetheic paraconsistent logic). Logics and reality aren't the same thing and if the 2 are even properly related to each other is unclear (it's hard to even state how this would be so) I myself endorse a paraconsistent logic, it doesn't commit me to something in reality (I hold dialetheia to be purely semantic or linguistic in nature).

Yes and no. SQL's logic is many-valued, but that's not what causes it to violate Excluded Middle. Multiple truth-values causes a violation of the Principles of Bivalence. Lots of Many-valued logcs still validate Excluded Middle, since the LEM isn't about the semantic values relating to propositions. For example, Intuitionistic Logic has only has 2 truth-values, yet it still violates LEM.

SQL's logic violates LEM and Bivalence, as well as the Law of Identity when dealing with the NULL object (NULL = NULL does not return true, it returns the 3rd truth-value "UNKNOWN").

Sorry about being a pedantic ass, lol.

Not really. I mean, one can develop a Paraconsistent Logic (including a Paraconsistent meta theory) which has truth tables and which is dialetheic, meaning it does not treat the Law of Non-contradiction as always true (in fact, it is both true and false in dialetheic paraconsistent semantics).

No. There are scores of deductive logics, many of which violate *any* axiom you can think of.

Logics where the Law of Identity is either not present or is restricted? Check out Non-reflexive logics and quasi-set theory (Newton da Costa).

Logics where the Law of Non-contradiction fails to be strictly true? Check out paraconsistent logic, more specifically, the theory known as Dialetheism (namely, from logician Graham Priest).

Logics where the Law of the Excluded Middle fails to be a tautology? Check out Intuitionistic Logic, the most studied type of paracomplete logic.

There is more in logical heaven than one might assume.

Now if the plain arithmetic cannot be stated to be consistent then what can? nothing on earth. This is exactly a fatal blow to Hilbert as pioneer supporter of maths.
Finally if nothing is consistent then where should you place your trust on? — Dzung

Yes it broke Hilbert's program, but it didn't prove that nothing is consistent in maths. For example, classical propositional logic is provably consistent (though we obviously want to use stronger logics than a propositional theory).

That said, you can work without consistency. After all, in response to Godel's Incompleteness Theorems, one can choose to go the inconsistent route and adopt a Paraconsistent Logic. This allows one to develop a mathematical theory on inconsistent foundations, yet because Paraconsistent logics lack the Law of explosion, theory is non-trivial.

This can result in an interesting mathematical theory which proves true (and simultaneously false) some of Frege's Logicism, as logicism becomes provable here and one can prove that the Continuum Hypothesis is false (in this formalism).

That said, when we investigate logic itself, we find we're immediately involved in a circular argument. Simply put, we need proof that logic is the best mode of thinking but thinking this way presupposes that logic is the best mode of thinking. Note that we're looking for a deductive proof that logic is the best mode of thinking. — TheMadFool

Why would we need to presuppose that? For one, logic is *not* a mode of thinking, that's silly. Logic refers to a theory of logical consequence, that is, a theory of what follows from what. In other words, when some things are true, it appears other truths can be derived from it. E.g. If it's raining, then there are clouds in the sky. It's raining. Therefore there are clouds in the sky. No thinking is needed here, as computers (which are not sentient) operate according to the rules of (essentially) classical propositional logic. And really, if logic were simply a mode of thinking, what logic do we think in accordance with? If anything, it's an extremely weak logic that can hardly derive very much unless we force ourselves to adhere to and specific set of inference rules (because we often have inconsistencies in what we think, meaning we cannot reason via classical logic because of the law of explosion).

So, how does one get out of this predicament, The Fallacy of Logic?

Logic is not innate to the mind. We have to learn it. From where? From the external world. We learn the rules of logic by observing the world. Deductive logic works fine at the macroscopic level. In our everyday lives we never see violations of logical principles and deductive and inductive logic work well.

I've heard that this isn't the case at the quantum level. I believe there are many situations where contradictions (a no-no in logic) arise. For such experiences we need a different kind of logic - something that accomodates the ''strange'' behavior of quantum objects.

This is just wrong and I wish people would stop saying it. Quantum mechanics is a consistent theory, it doesn't portray an inconsistent quantum realm (unless you're Newton da Costa). Quantum physicists use the standard mathematical formalism (ZFC + classical logic) so it's necessarily consistent since otherwise trivialism would follow. Quantum logic is a separate logical system, but that logic only discharges the Distribution Axiom, not the Law of Non-contradiction.


This isn't a logic specific problem at all, it's just a broad epistemological issue. One simply picks their logic in accordance with a model of theory choice (logical systems are competing theories about logical consequence after all) and proceeds to do their work from there. Not everything needs to be justified so this is not an interesting problem to me.

I think you're barking up the wrong tree. Nothing in the microscopic world has even been suggested to have an inconsistent nature. Quantum mechanics uses the standard mathematical formalism (Zermelo-Frankel set theory + classical logic). There is such a thing as quantum logic, but that formal system does not violate the LNC, it gets rid of the Law of Distribution. The only respected defender of a truly inconsistent quantum theory I know of is Newton da Costa, but even he knows he is almost alone in his view regarding the nature of superpositions.

The claim is circular and kind of ignorant, quite honestly. Dialetheism has been around as a legitimate theory for decades now, as has paraconsistent logic. The fact that one may deny the LNC doesn't entail one accepts the LNC. It simply means you believe there is at least one proposition which is true and has a true negation. And ironically, according the semantics of standard dialetheic paraconsistent logics, the LNC is a dialetheia, it's both true and false.

Seriously, Graham Priest (and others) has a thorough defense of this view. Work has even been done to develop purely paraconsistent meta theory (see Patrick Girard's work). The book you referred to is simply incorrect.

It sounds like you think there is a representation of the sentence in a formal logical language. I would be very surprised to see such a thing, and suspect that it either lacks in formality or doesn't sufficiently represent what people feel the natural language version says.

Post the representation, with details of the formal language being used, and we can discuss it.

I'm about to leave for a New Years party, so I don't have the time to get into the nitty-gritty. However, Graham Priest gives a fuller, comprehensive look at the paradox (including formal representations of it) in his book "In Contradiction" (and probably elsewhere too).

I'm not really sure what you're talking about. Analysis of the Liar sentences, even by strictly Classical Logicians (like Timothy Williamson) use logical formalisms to provide a rigorous assessment of the Liars. The usual response by these people is not that there is a problem with the reasoning involved. After all, in standard logic, Falsity just the truth of negation (and vice-versa).
So yes, if the Liar is false, it is also true. So a better solution would be to question if it's a proper proposition at all.

The evidence is the straightforward proof by contradiction. That the Liar sentence is not truth-apt is a readily established fact. Now, you may wonder what makes it so, but that's a different — SophistiCat

You didn't provide evidence. Very few people deny that the Liar Paradoxes are contradictions. The point is that, prima facie, they appear to dialetheia - a proposition which is both true & false. You haven't established that the Liars aren't truth-apt. The fact that the Liar Paradox is an unsolved problem with no standard solution speaks to the fact that you are getting way ahead of yourself.

What does the proof by contradiction have to do with anything here? If your argument is that the truth of the Liars would result in trivialism, I agree. But that makes the assumption that the Principle of Explosion should be accepted as a valid argument (Paraconsistent Logics give us another route). So this is far from a simple matter, which is the reason why it's unsolved amongst logicians.

Yes, see the first reply to this discussion. ;)

Lol, not sure how I missed that. But that aside, as I said, even Kripke said that his proposed solution was probably suspectible to a Revenge Paradox, such as the ones I listed previously.

Yes, it's true if it's false. But what does it mean for it to be true? Are you saying that it being true means that it's false (and vice versa)? So in the context of the liar paradox, "true" and "false" mean the same thing? If so then a) there is no contradiction and b) the terms "true" and "false" in the context of the liar paradox mean something other than what they mean in ordinary usage. And then you still need to explain what it actually means for the sentence to be true/false.

To say that it's true merely means that the proposition is related to (in the mathematical sense) the value "true", and to say it's false simply means it is related to the value "false". If you're asking for a theory of truth, that is a discussion independent of logical formalisms. Formal logic is, generally speaking, neutral as to the meaning of those predicates (that's why there are a number of theories of truth). Or you could just check out Tarski's work on the matter.

No, in the paradox, "truth" and "falsity" do not mean the same thing (otherwise the formalism would simply be a trivial system). As in the standard usage, truth and falsity are duals. A proposition is deemed "true" when it's "not false", and vice-versa. If we're working in, say, Classical Logic, all propositions must have a truth-value they relate to, and that value must be either "truth" or "falsity". So unless you can show specifically how the LPs aren't propositions, there's really nowhere to go. You either have to reject Classical Logic or just accept the paradox.

Its syntax is what misleads people into believing it's truth-apt.

But *how so* is my question. If it has the same structure as other truth-apt sentences, clearly it's not the syntax which is the issue.

The Liar Paradox is a natural language sentence, not a sentence made in some formal system.

There has been no better luck in solving the LPs in formal languages either; the Liars can & have been articulated in formal systems too, otherwise we would reject it as a pseudo-problem. An LP is a sentence L which is true iff it isn't true (L <=> ~L). There are even purely syntactic versions, such as Russell's Paradox in naive set theory (R ∈ R <=> R ∉ R). And anyway, logicians and mathematicians (especially) almost always reason in vernacular anyway. The problem exists for both kinds of languages, which (as an aside) was the reason Tarski's suggested jettisoning natural language in favor of an artificial language to solve the LP.

Sorry, I wasn't clear. I meant that it being true or false doesn't mean anything given that there's no evaluable fact in virtue of which it is either true or false.

It doesn't mean anything empirical, but that doesn't mean it lacks a meaning. It's a sentence which is true when it is false, and vice-versa. I think you're treading down the path that Kripke went down. And speaking of Kripke, you should see some of his work on the Liar Paradoxes. There are versions of the LPs which are actually tied up in empirical facts. They are called the Contingent Liar Paradoxes.