This is good example of another major problem with logical fallacies. People get into arguments about whether a particular fallacy is being applied correctly as opposed to the actual basis of the argument. — T Clark

And argument requires structure for anything to come from it, and that structure is inescapably part of "the actual basis of an argument", it's how you derive conclusions. This is a complaint that makes no sense. Yes some use these stupidly and don't understand them. I am not one of these people.

Well I didn't misuse it or use it as a short hand for not liking someone's argument. :-)

This is not quite valid since it is missing the premise that winged horses exist. — darthbarracuda

It's invalid in modern logic systems, it was valid in Aristotliean Syllogistic. It doesn't require omniscience, we simply know that syllogism does not lead one from true premises to true conclusions in all models, and hence ought not be valid in a deductive logic.

Well, ↪Posty McPostface just confirmed we are not talking about epistemic content yet, but is this logically fallacious? Is the "existential fallacy" really a logical fallacy? The premises may be false, which makes the syllogism not-sound. But it is still valid. So what does fallacy mean here? — darthbarracuda

"Fallacy" means the conclusion does not follow from the premises. The first two are definitely true, the conclusion is definitely false, and yet Syllogistic (as opposed to classical logic) says the argument form is fine. Yes the Existential fallacy is a fallacy. That argument comes out as invalid in modern systems of logic.

Meinong, unless I'm mistaken, would agree that it's in error. He might think pegasi have being, but not existence in a concrete sense. Then again, I'm not super into Meinongianism so maybe I'm off about it.

Kind of? I'm saying that you could have flawless knowledge of every error in reasoning and still fail to reason perfectly since in practice you will never apply that knowledge infallibly. Like, mathematical logic students and professional logicians aren't immune to common errors in reasoning even if they intellectually know that they're committing a no-no.

It's just a short hand, it makes no difference. The issue is assuming that a class has members without asserting as a premise that the class has members. If someone is confused about what the term referred to, well, we are on the internet.

I don't understand. I don't know if there's a theorem stating that if something in base one logic, applies universally to all other logics? Meaning, if I'm correct in one domain of logic, then by extension it should apply to all other domains of logic. — Posty McPostface

Let me give an example. In the logic Aristotle developed - known as Syllogistic - an argument with Existential Import was valid. So the following argument was valid in Aristotliean logic:

All winged horses are horses
All winged horses have wings
Therefore some horses have wings

(Form: All A's are B's, All A's are C's, therefore some B's are C)

The first two premises seem clearly true, they're analytic statements. But the logic takes us from 2 obviously true premises to an obviously false conclusion: there are, in fact, no horses with wings. Something has gone wrong with the logic itself. That's why the standard logical systems of our day treat arguments like the above as fallacious arguments, they commit the Existential fallacy.

This is just for completeness sake. It's less common to be using the wrong logic in a way that will matter for everyday reasoning, which uses a weak logic to infer things. I believe my other point about imperfectly following rules is more pertinent to your question.

Have I got this about right? A couple of comments. — fishfry

Well summarized, couldn't have done it better myself. My only quibble is that it's "wrecking logic" only insofar as one already has an idea of what the correct logic is beforehand. The Thomists believed those who started using Classic Logic post-Frege were "wrecking logic" by abandoning what Aristotle left for us, but no one ought take that seriously in light of all the standard mathematics can do for us! Granted, paraconsistent mathematics hasn't reached that level (yet, perhaps) so my comparison probably lacks the persuasive force I'd like it to have. :-) Also, I didn't intend to sandbag you, I should have linked it earlier, though I do believe I referenced paraconsistent math.

Why don't you hack logic to allow the existence of a largest prime? Why does one easily proved mathematical fact annoy you so much yet you accept the proof of the infinitude of primes? — fishfry

Because a paradox is not simply a contradiction. The contradiction "It's raining outside and it's not raining outside", as with the supposition there's a largest prime, lacks any persuasive force for it. It doesn't follow from seemingly reasonable principles. Frege believed unrestricted comprehension was "self-evident", though I loathe that term. But of course we know that the very logic he created cannot be paired with that principle of set theory. So we have two choices on offer: remove/rework the principle as best we can or switch logics. At the very least, the Incompleteness Theorems leave the door open about which one you pick, as in either case you will necessarily lose a very desirable trait for a mathematical system (either completeness or consistency).

Dialetheists hold that classical math/logic fails to account for some pretty crucial data: the truth-value of the Liar sentences, the geometry of Escher spaces, etc. They trivialize one you try to consider such things. Or take unsolved problems that are unprovable (or at least unproven) and we know standard maths leaves one wanting where answers are concerned (Continuum Hypothesis, for example). However, they argue that paraconsistent logics can handle these elegantly and give some real answers, but you can't do it while making an appeal to retain the standard formalism. And so, keeping naive set theory but changing the logic lets you retain a principle that seems very reasonable and you have an explanation for certain data and a reason to give about why some principles and inference rules in standard maths ought to be dropped.

It's a philosophy of science issue, I suppose. Which theory is more theoretically virtuous? That's what they hang their hats on, perhaps.

No. Aside from the fact that no one perfectly adheres to rules even if they believe they ought to l, what counts as a logical fallacy is relative to the logic you're using. Ergo, you may find out you were reasoning according to the wrong logic in some domain and thus have been reasoning fallaciously.

Sure! (This won't hold in standard math obviously)

"It should also be noted that Brady’s construction of naive set theory opens the door to a revival of Frege-Russell logicism, which was widely held, even by Frege himself, to have been badly damaged by the Russell Paradox. If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is deducible from logic via the naive comprehension schema.
(...)
Even more radically, Weber, in related papers (2010), (2012), has taken the inconsistency to be a positive virtue, since it enables us to settle several questions that were left open by Cantor, namely, that the well-ordering theorem and the axiom of choice are provable, and that the Continuum Hypothesis is false (2012, 284)."

https://plato.stanford.edu/entries/mathematics-inconsistent

Would I be right in thinking that one reason to be cool with that approach (the truth learned) is that we don't need unrestricted quantification? — Srap Tasmaner

(I know this wasn't directed at me but I can't resist)

Depends on what you mean by "need". If you aim to prove Logicism then you do need unrestricted comprehension. Without it, we end up with a lot of unprovable hypotheses. For example, it has been demonstrated that naive set theory + a paraconsistent logic lets you prove the Continuum Hypothesis is false. However, in standard maths it's unprovable.

This isn't to say that because one formalism can solve a problem the other can't that we should ditch one for the other. It's just that there are extra-mathematical considerations to what we pick (i.e. assessing theories for their worth/virtue). Most mathematicians prefer working in a consistent system (and a fruitful one at that) so they privilege one which is consistent but lacks a bit over one where some contradictions are provable.

I don't see why. Classical logic goes back to Aristotle. And even math doesn't need set theory. There wasn't any set theory till Cantor and there was plenty of great math getting done before that. Archimedes, Eudoxus, the medieval guys Cardano and so forth, Newton, Gauss, Euler, Cauchy, and all the rest. None of them ever heard of set theory and did fine without it. — fishfry

No, Aristotle created Syllogistic. Classical logic was invented in the 1870s by Frege. These are not the same system, Classical Logic validates a different set of arguments than Syllogistic, it has logical connectives and quantifiers that Syllogistic lacked, and funnily enough, Syllogistic was a type of paraconsistent logic since according to Aristotle you cannot derive anything from a contradiction. Without set theory, we wouldn't understand a lot of maths, it came as part of the program to understand how various kinds of numbers were defined and related to each other. The "classical" in "classical logic" is misleading, if not propagandistic, heh.

Incompleteness is literally a classical result now. Everyone's moved past it. So we can't use the traditional axiomatic method to determine what's true. If anything, that's perfectly sensible. We have to find other paths to truth. That's exciting, not worrisome I think. — fishfry

I didn't say it was worrisome, I was just pointing out a consequence of the theorems. You can use the traditional axiomatic methods, you'll just have an inconsistent theory.

I really don't believe that incompleteness is any kind of nihilistic disaster. Interesting math is being done every day. — fishfry

Now I certainly didn't say it resulted in nihilism, and I don't deny good math is being done. As I said, I don't reject standard math.

On the other hand, perhaps it's related to postmodernism and the reaction against reason. — fishfry

Eh, I don't think it's Po-Mo at all. It's just that the landscape of possible formal systems worthy of mathematical investigation turned out to not be so limited.

Thanks for the link, looks interesting.

Ok. Just wanted to make sure you accept law of excluded middle and proof by contradiction. — fishfry

Yea, I don't have much issue with Excluded Middle. Then again, I've only passing familiarity with intuitionism. ;)

Hamkins to see what the set theorists are up to. But nobody worries about Russell's paradox because there's nothing to worry about. It just shows that we can't use unrestricted set comprehension. And I still don't know why you think people should be concerned about a run of the mill proof by contradiction. Sure it ruined Frege's day, but it revealed a mathematical truth about the nature of sets. — fishfry

I'm not talking about Russell's Paradox in that bit, I'm talking about the general outlook regarding mathematics post-Incompleteness Theorems. ZFC's development was intentionally practical: we need to get on with the business of doing sensible maths but classical logic cannot function sensibly with an inconsistent set theory. Once it became clear that there was no strict necessity in picking one formalism over another (i.e. no privileged set of indubitable axioms), it seems like mathematicians and logicians became a bit more cavalier about the whole thing. Rightly so, in my view, the interest shifted to the virtues of particular formal systems applied in specific domains, particularly when such systems are fruitful.

Like from the Incompleteness Theorems, we know you can (for systems expressive enough to articulate arithmetic truths) either have an inconsistent but complete mathematics (Paraconsistent mathematics) or you can have a consistent but incomplete maths (Classical math, Intuituonistic math, etc.). Classical logic is so preferred because of its wide usability, but there are known issues and domains where it's questionable (quantum mechanics, representing human reasoning, databases, some evidence paraconsistent logic operations are faster to compute, etc).

So I wonder if this modern openness to more or less any non-trivial logic/math indicates some kind of pluralism. What do you think?

By the way, are you and/or MindForged making some kind of constructivist or intuitionist argument that rejects the law of the excluded middle and/or proof by contradiction? — fishfry

I think I've already articulated my position without recourse to intuitionism. Once you think it over (need not agree obviously) let me know what you think.

I'll admit, I'm something of a logical pluralist so it's not like I'm advocating a wholesale abandonment of standard maths. Honestly, I actually wonder what mathematicians who think about this sort of thing believe (rare-ish to see it done in depth, most don't bother with the foundations of maths these days). Really, it seems like Gödel's Incompleteness Theorems in particular and the death of Logicism (using classical logic) seems to have killed foundationalism in the eyes of mathematicians and logicians, so I wonder if they're pluralists of a sort?

Why is it that in the case of (a) you regard this as a basic mathematical truth; yet in the case of (b) you regard this as a philosophical conundrum perhaps susceptible to attack via paraconsistent logic? — fishfry

Because in the case of A, we have every reason to believe we are in a consistent domain (that of classical mathematics), where proof by contradiction is necessary (on pain of triviality), and we know we can give examples of larger primes . In B, we get a paradox unless we rewrite the rules of naive set theory to get something like ZFC. With A, we have a counter example that let's us dismiss the initial supposition, with B we get a contradiction from what seem like reasonable assumptions on their face. The assumption that there's a largest prime doesn't seem to rest on comparably reasonable principles such as a set being any collection defined by whatever condition you have in mind.

I assume (although you have not confirmed this) that you don't regard the infinitude of primes as being subject to modification or revision based on paraconsistent logic. Why is (b) different? — fishfry

I don't think the infinitude of primes will be much affected by a transition in the logic. Paraconsistent logic dispenses with proof by contradiction and tends to instead rely on proof by non-triviality (these are identical in other logics but not with PLs).

Assume the contrary, derive a contradiction, learn a truth. — fishfry

I suppose the simplest way is to point out there are other concerns that bear on something besides consistency. I can't remember if it was in this thread that I mentioned this, but for example it's just a fact that the early calculus was inconsistent. One had to treat infinitesimals as a non-zero value at one step of proofs and then treat them as having and value of zero at another step of the same proof. This was acknowledged by Newton, Leibniz, criticized by Berkeley, etc., and it remained that way for more than 150 years. Now as far as I can tell, if you really tried to insist on this way of proceeding, you would have been rationally required by your standards to have rejected calculus (and therefore everything learned and built because of it) during that century and a half of it being inconsistent. But that's obviously ridiculous, there are other theoretical virtues besides consistency which made calculus tenable to accept despite the contradictions it required one to adopt.

That's what I'm arguing, sort of. Sure, Russell's paradox is a paradox. That was never the dispute. The issue was always that the principles that gave rise to the paradox in naive set theory seem pretty damn reasonable. So the way out of it was to come up with ad hoc restrictions on what constituted a set. There were extra-mathematical considerations which led to that response, not simply a proof by contradiction because that argument itself relies on already dismissing the possibility of paradoxes, which is the very thing under dispute of you accept Russell's Paradox. There has to be a reason (besides arguing against the conclusion) for why you reject the principles that give rise to the paradox, otherwise it seems like the objection is circular. One can get around it the way ZFC does, but the question is if that is more rational or if it results in a more theoretically virtuous theory. Perhaps it does, but it's certainly not answered by pointing out there's a contradiction.

MindForged, you are completely misunderstanding the difference between a veridical paradox and a plain old proof by contraction. Moreover, Russell's paradox has absolutely nothing to do with Gödelean incompleteness. Simply nothing. — fishfry

I didn't mistake anything. The paradox resulted from rules that Frege thought were indubitable. And you completely misrepresented what I said. I didn't say RP had anything whatsoever to with the Incompleteness Theorems. I said the regimentation of set theory was motivated by a desire to.prove mathematics was consistent and complete, but Gödel showed you could only have one or the other, not both (if your system is capable of articulating arithmetic truths).

I know how a proof by contradiction works, pleasedon't patronize me. What you don't seem to get is that avoiding a paradox is not the same as proving your solution ought to be adopted. I can avoid the Liar paradox by disallowing all self reference. But then all hell breaks lose because perfectly sensible sentences like "This is an English sentence" get the axe. The solution has to be justified. Modern.maths avoided the paradox to get on with the business of mathematics, no one disputes that the resolution was ad hoc. They simply take that theoretical black mark as preferable to an inconsistent theory. If the principles which cause the paradox are more theoretically virtuous than the consequences of evading the paradox, then one ought to accept the paradoxical conclusion. I mean, to just throw up "It's a proof by contradiction" is easily abusable:

Early calculus was indisputably contradictory. One had to treat infinitesimals as a non-zero value at one step of the proof and then as having a value of zero at another step. Ergo, we should have dropped calculus as a viable mathematical branch immediately instead of it knowingly being left to languish as an inconsistent theory for 150+ years.

Long story short, it's a lot more complicated than you seem to think. Paired with a Paraconsistent Logic, one can work with Naive Set Theory (and thus with Russell's Paradox) and draw interesting, non-trivial conclusions. Proof by contradiction is only going to work this way if you're already assuming that contradictions are off the table, meaning you're already rejecting Russell's Paradox before you even do the proof.

I didn't say the barber shaves two distinct groups P & Q. I said the barber "shaves all and only those who do not shave themselves". That is a condition defining one set.

Dude, I "changed it" A) By mistake, since I gave the classic wording of the paradox and B) the words I changed didn't alter the paradox at all. "All and only" is the same thing as "every".

This is ridiculous.

??? Bro, I said it was a paradox on the previous page. I accept Russell's Paradox but I'm an outlier. The Barber Paradox is simple: no such barber exists, such a barber is fictional.

Well said, you articulated it better than me.

That's where the paradox comes from. He ends up having to shave himself and shave those who cannot shave themselves. He can't do both.

The barber paradox splits a town into two distinct groups: those who shaves themselves and those who don't. The barber is proposed to be the one who shaves those in the latter group. The paradox comes from the fact that the barber must end up being part of both groups. There's no paradox without 2 groups.

There is no semantic difference between what I said and what you said. "All and only" will capture the same set of objects as "every".

I'm sorry, but this is like saying it's bad to call murderers evil because you're being intolerant of them. Like, really?

And that is not the paradox in the OP. Which is my point. — Jeremiah

Are you serious? Your OP:

The town barber, who is a man, shaves exactly every man in the town who does not shave himself. — Jeremiah

I have to assume there's some communication issue here. I accept Russell's Paradox, but the Barber doesn't seem difficult: no such barber exists, problem solved. If the barber is tangential to your question, fine. But you did in fact bring it up.

Because people don't always behave rationally even if on other things they are reasonable. Racism doesn't just drop out of moral character or beneficial evolutionary strategies. That sounds like the unprovable just-so stories that get tossed around evo-psych from time to time.

It does not actually say he shaves all. It is every man who does not shave himself, — Jeremiah

That's is literally what I said. My post:

I'm saying that even as metaphysical dialetheist I do not believe a "barber who shaves all and only those who do not shave themselves" cant exist. — Me

I already said I accept the paradox as a valid argument, but unlike most that's because I accept naive set theory. In ZFC, the set is not a valid one, sets cannot contain themselves in ZFC.

I'm saying that even as metaphysical dialetheist I do not believe a "barber who shaves all and only those who do not shave themselves" cant exist. Obviously vanilla barbers can exist.

Racism arises from a misunderstanding, not evil intent. Perhaps the most abysmal aspect of racism is that it's nothing personal. The target race is just "vermin" or what have you. It's no more evil than identifying rats as pests. — frank

Do you hear yourself? "Nothing personal blacky, I just believe you're a pest because of your skin color". Who cares if it's personal, people can be maniacally bad to people they don't know, of course it doesn't have to be personal, but that doesn't mean it has no bearing on character. I personally think believing someone is vermin because of their race or ethnicity is evil, it is a sign of poor character. It shows your willing to make blanket assumptions about a group with no reason as to why (other than the one's you make up so you don't eel bad about it) and go on to use these to justify whatever you want done to them. That's a sign of poor character.

Pretending that these will simply exist in a vacuum and not have an affect on one's behavior is ridiculous. Most people want to get rid of vermin. Pair that with a belief that "race/ethnicity X is vermin" and you get "We should get rid of X".

But the barber is psychically cable of shaving himself or not, our problem is with the group. Does he fit the group, or not? — Jeremiah

He fits in the group and does not fit in the group, but only in a fiction can it obtain, even for metaphysical dialetheists. To motivate anything more you'd have to show such a barber does exist. So even if you're like me and think certain aspects of.reality might be inconsistent, the Barber Paradox doesn't show there is or can be such a barber.

Good God Almighty. Russell's paradox was resolved in 1922 by the axiom schema of specification. — fishfry

As I say, that's a somewhat naive view. The specification scheme allows one to avoid the paradox, but it doesn't necessarily solve the paradox. The whole point of regimenting set theory this way was to make make math consistent (or at least not provably inconsistent). But it comes with well known issues, like a number of unsolved questions that have known answers in other systems (e.g. Continuum Hypothesis).

It's kind of like saying you solve the Liar paradox by simply disallowing self-reference in your language. It's true in a sense (you can no longer articulate the paradox) but the debate is in the merits and justification in doing so. It's simply incorrect to say the see foundational issues were solved; no one studying modern fundamental mathematics would say it with such certainty, in any case.

I see you made other threads with titles like this, but I'm too lazy to check them.

Anyway, in modern mathematics Russell's Paradox isn't so much "solved" as it is avoided. The Unrestricted Comprehension Scheme allows for this paradox , despite Frege thinking it to have been "self-evident". Russell tried to get away from it but the whole motivation behind this, Logicism, died with Gödel's Incompleteness Theorems.

But really, it's an open question. The Incompleteness Theorems didn't so much destroy Logicism as much as it made the lay of the landscape clear. If you want to endorse Logicism, your mathematical system must be inconsistent. If you want to avoid triviality (e.g. proving all sentences are true) then you have to adopt a Paraconsistent logic to go with your inconsistent set theory. But if you want to keep consistency, you cannot accept Logicism. Of course for those like the Intuitionists this isn't an issue for their view. They never accepted Logicism.

Probably why foundationalism died a horrible death for mathematicians and logicians. There's a whole panoply of options for constructing mathematical and logical systems which are equally open to interesting and valid mathematical investigation.

Anyway, I simply accept the paradox. Nothing has gone wrong with it, it's a veridical argument.

It fails where you say possible truths are a subset of the contingent ones. Possibility encapsulates contingency.

A is not equal to A, then we would know A could not exist. People overlook the fact that a mathematical object can only exist if its existence is consistent with logic. It works both ways.
9d — LD Saunders

This is presumptuous. There are developments in quantum mechanics - extending to its ontology and thus the formal logic used - where quantum objects lack identity. They are not self-identical, sometimes terms "non-individual objects". Also, there are non-classical logics (and thus corresponding non-classical mathematics) where contradictions can be proved without trivialism, thus allowing one to prove the existence of inconsistent mathematical objects (e.g. the Russell Set).

Well, my initial thought is that when art breaks the 4th wall, it sort of ends up incorporating the real world into itself, making it inconsistent since things that are true "there" aren't necessarily so "here" and vice versa. But without such breaks, that world is incomplete since the details of the world could never be fully fleshed out (and besides which, we aren't usually interested in every detail!)

This was more of a random thought. Probably just me being silly, I grant.

I was trying to grasp why we assume that a theory of everything is possible, and what it might mean to accept that no such theory can ever exist. — frank

Well, depending on what you mean by "everything", many do not accept that a theory for everything is possible. If the theory is supposed to explain everything in every field of learning, then I'd suggest that's just a pipe dream. If it's just supposed to explain fundamental physics, that's probably possible.

Could you explain what expressiveness is to a lay-person?

I'll give it a try! So what mathematicians discovered back in the 30s was this. If your formal (logical or mathematical) theory or language contain certain features, then that theory will let you validly construct a paradoxical sentence, one that ends in a contradiction. The classic Liar Paradox, essentially. So if you're using a language or a logic, even natural languages, which can create self-referenial sentences and which contain the concept of truth, you can make:

This sentence is false.

Which when explores ends up being both true and false at the same time; a contradiction that falls right out of basic logical principles. So what Kurt Godel (and also technically Tarski) proved was that if your language/logic contained these 2 features (self-reference and truth), it will be contradictory. The only way to avoid this is to limit the expressive power of the language, meaning you restrict how you can apply truth to sentences or you discharge or otherwise modify self-reference. This inherently limits what can be proved in mathematics if you want to stay consistent. If you're consistent, many things in math will be unprovable, but if you want complete provability then you'll end up proving the truth of some contradictions.

As it turned out, the entire thing is a mine field, and it's arguably defensible to embrace the contradiction to see the interesting theories that can result from accepting certain contradictions as true (Paraconsistent mathematics/paraconsistent logic)! Or not. It's fun anyway. Somewhat similar to your 4th wall break, actually.

I think it's sorting itself out though.

I think that's naive. It's certainly getting better, but not by itself. That's what people think after the fact, but it ignores that people are actively working for it to bring that change about against a tendency to keep a status quo.

To what extent is this same aesthetic in force in regard to what we think of as the real world?

I don't know of any examples of incompleteness in reality (by which I assume you mean the physical world). In mathematics sure, but the reason for Incompleteness there has to do with how expressive the theory is. Don't know what the physical analogue for that is.

As Malcolm X said, the house negro has played the victim card for decades. For some women, being thought of as a victim paradoxically grants them a feeling of being powerful, seeing as they are able to influence others to think a certain way about them.

Bringing up Malcolm X here makes no sense. His point there was that the house negroes were traitors to their people, that he identified himself with his master as much as he could.

The house Negro usually lived close to his master. He dressed like his master. He wore his master's second-hand clothes. He ate food that his master left on the table. And he lived in his master's house--probably in the basement or the attic--but he still lived in the master's house.

So whenever that house Negro identified himself, he always identified himself in the same sense that his master identified himself. — Malcolm X: "The Race Problem"

Whatever your views about what @Janus is saying, Malcolm X isn't likely to come down on your side of it since it isn't what they're arguing about oppressed people's and the methods of expression that is acceptable for each group to use.

The point of the OP is the purported inconsistency of how the rules are carried out. Also, I'd argue that philosophical inquiry is hardly ever useful, it's more intellectually entertaining that anything else.