Three types of paradox — EugeneW

Thanks.

The principle of explosion and paraconsistent logic. The solution to all paradoxes. It puts the barber and Russel at ease. All the other paradoxes can be resolved by science or religion. Unless Zen and Nirvana are involved. Is this a paraconsistent dox?

Paradoxes, real ones (p & ~p), are a problem iff disjunction introduction/addition is allowed (ex falso quodlibet). That is to say, classical logic has been superseded by paraconsistent logic, happened a long time ago (since Zeno of Elea revealed his eponymous paradoxes).

We're using, we have to, paraconsistent logic if we're not to end up as confused masses of protoplasm (Zen koans, aporia, mushin no shin). In other words, any philosophical argument that depends on disjunction elimination/addition has to be taken as invalid. Know any?

Mind giving an example of a paradox that exists, and logical people still hold as true?

List of paradoxes

Knock yourself out.

I don't think I was clear. It wasn't a list of paradoxes I was looking for. I was looking for paradoxes that logical people hold as true, or thoughts that logical people still hold despite it falling into a paradox. Most paradoxes are fun accidents that I know of, and no serious logical thinker that I am aware of, entertains a thread of logic that necessarily leads into a paradox.

I was looking for paradoxes that logical people hold as true — Philosophim

The Wikipedia link I provided is a list of paradoxes (which logical people hold as true).

The Wikipedia link I provided is a list of paradoxes (which logical people hold as true). — Agent Smith

You might know they hold them as true, but how do you know they are logical?

The Wikipedia link I provided is a list of paradoxes (which logical people hold as true). — Agent Smith

I'll try one more time in case you aren't understanding my request. I know those are paradoxes that people have come up with. What logical thinker holds onto something that leads into a paradox, agrees that the paradox is sound, but still insists on holding onto logic that leads to that specific paradox?

You're noting that people sweep paradoxes under the bridge to hold certain logical arguments. Which arguments? Which logical argument are people holding onto despite it leading directly into a paradox?

You might know they hold them as true, but how do you know they are logical? — Janus

There are arguments whose conclusions are paradoxes. Check out Wikipedia or the Stanford Encyclopedia of Philosophy, etc.

What logical thinker holds onto something that leads into a paradox, agrees that the paradox is sound, but still insists on holding onto logic that leads to that specific paradox? — Philosophim

That's all of us Philosophim. The Wikipedia list (of paradoxes) I linked to applies to every person, assuming they're genuine paradoxes (no one, as of yet, has attempted to resolve all them; from the multiple disciplines involved, it'll require a team).

Since everyone knows there are paradoxes in classical
logic, we have to stop explosion (ex falso quodlibet) and one way of doing that is doing away with the disjunction introduction/addition rule in natural deduction. That's paraconsitent logic! We're supposed to use it, not the old classical logic systems of Aristotle, Chrysippus, and Frege.

1. P
2. P v Q [disjunction introduction/addition]

Line 2 should not be allowed.

Can anyone link me to a site/book on paraconsistent logic? I want to learn it.

That post and most of what is said in the poster's following posts is terribly ill-premised.

Classical logic itself does not result in contradictions. Rather, adding certain non-logical axioms results in contradictions. For example, classical first order logic is consistent, but if we add an axiom schema of unrestricted comprehension (which is a set of non-logical axioms) then we get the contradiction known as 'Russell's paradox'.

A logical principle is one that is true in all models. A non-logical principle is one that is not true in at least one model.

It is fatal mistake not to understand that difference between logical axioms and rules (the bare bones of deduction) and non-logical axioms or rules.

Classical logic, which includes the explosion principle, is consistent. It is nonsense to claim that classical logic can be consistent only if it eschews the explosion principle.

There are many other terrible misconceptions stated by the poster, but at least it's a start to point out the lack of distinction between the logic itself and what can be derived using the logic from additional non-logical axioms.

You need to first learn basic symbolic logic. I recommend 'Logic: Techniques Of Formal Reasoning' by Kalish, Montague and Mar. Then basic set theory in which the systems of logic (including paraconsistent logic) are formalized. I recommend 'Elements Of Set Theory' by Enderton. Then mathematical logic, which is a deeper study of logic. I recommend 'A Mathematical Introduction To Logic' by Enderton, supplemented with 'Introduction To Logic' by Suppes for his discussion of mathematical definition, which is the very best discussion of that subject I've seen.

That will give you a clear and rigorous understanding as opposed to flitting about among various bits from improperly edited Wikipedia pages, taken out of context or even grossly misunderstood and then irresponsibly misrepresented by you on this Internet forum.

Paradoxes break (classical) logic. — Agent Smith

This is ill-phrased. (P & ~P) does not "break" logic because it is not a theorem. That is, it cannot be deduced in classical logic.

A situation in which (P & ~P) would be one in which "classical" logic was inapplicable. There are, as you note, other logics.

Again, the appropriate way to think of this is as choosing what language - which logic - is applicable to a given circumstance.

Hence it is

really just playing with language — T Clark

but that is not to trivialise logic; playing with language is what we do.


Ah, I see @TonesInDeepFreeze has made much the same point.

Well, paradoxes break math (Bertrand Russell and his set of all sets that don't contain itself) for the reason, I was told, that ex falso quodlibet renders such mathematical systems trivial (all well-formed formulae are true). :grin:

Why I said what I said was because the LNC (the law of noncontradiction) is an axiom (vide the 3 laws of logic); paradoxes, if real, imply that the LNC can no longer be an axiom, it has to go. That's what I meant.

On the larger point...

If true paradoxes exist (re heterological paradox), we're either

1. Using paraconsistent logic, a version of it

or

2. We're ignoring them but not in a systematic manner (picture a bomb disposal unit at work)

Bear with me...this is mainly guesswork.

paradoxes break math — Agent Smith

No, it doesn't. 2+2 still makes 4, regardless of Russell.

Bear with me...this is mainly guesswork. — Agent Smith

Indeed. Take @TonesInDeepFreeze' advice and study up on logic some more. There is a lot to be said for doing a formal course, since only in that close interaction will someone be able to pick apart the multiple, small errors in your comments.

No, it doesn't. 2+2 still makes 4, regardless of Russell. — Banno

:snicker: 2 + 2 = 5 or chimpanzee too if Russell's set is allowed in math based on set theory. I recall reading how Russell attempted damage control by, in a sense, making his set of all sets that don't contain itself illegal in a manner of speaking.

What sayest thou, kind sir?

My apologies for sounding a bit curt in that last post. it's just that it seems to me that you have not actually set out the issue clearly, but nevertheless you are drawing conclusions, and as a result your conclusions are muddled.

Nothing Russell might say, and certainly no paradox, can lead us to conclude that 2+2=5. You have the tail wagging the dog. If one's argument concludes that 2+2=5, one has made an error.

(p & ~p) is false in both propositional logic and first order predicate calculus. Hence, it is difficult to see what you mean by

What do we mean a logical system is trivial? Simply this: every proposition is true — Agent Smith

humans are clever animals — Tom Storm

We are animals? Sometimes, I say this to my wife. Sometimes she agrees...

paradoxes break math
— Agent Smith

No, it doesn't. 2+2 still makes 4, regardless of Russell. — Banno

Yes they do.

The paradox is that approximations in math are the exact solution. An apparent contradiction. A oaradix, eeehhh... paradox.

I don't see why you should object to what is the official position on Russell's paradox. Russell's set (of all sets that don't contain itself), in colloquial language, breaks math; to be precise, it, via ex falso quodlibet, means every mathematical statement is true. In my world math is broken when that (trivialization of math) happens; it happens because the set theory based axioms of mathematics allows Russell's set to, well, exist; what happens next is common knowledge.

...the official position... — Agent Smith

That's not a thing.

As Dana Scott has put it, “It is to be understood from the start that Russell’s paradox is not to be regarded as a disaster. It and the related paradoxes show that the naïve notion of all-inclusive collections is untenable. That Is an interesting result, no doubt about it”. — SEP, Russell’s Paradox

Are you, if I may ask, trying to say that all paradoxes can be reduced to a negative self-referential paradox?

I'd like to see you do that with Zeno's paradoxes if you don't mind that is. Can you? — Agent Smith

I didn't notice this earlier, so I'll try to give an answer to this. I'm no mathematician, so the answer can be quite difficult to understand. Hopefully I make sense to you.

As we know, Zeno's paradoxes are about infinity. Modern math has "no problem" with Zeno's paradoxes of Dichotomy, the Arrow or the Tortoise as it uses limits (or the infinitesimal). And modern math just takes infinity as an axiom, and axioms don't have to be explained. Hence we still have a lot of questions about infinity, because we don't have an understanding about it. (The people who say we do, then should answer the Continuum Hypothesis for us)

Now when you think about infinity, the self reference should be obvious. It's pretty hard going from the finite to the infinite: you cannot just add finite numbers to other finite numbers and get infinite. Or, you add them an infinite times (hence the self reference). Cantor himself did understand the paradoxical nature of infinity, but could make something about with the proof of there being more reals as natural numbers. Although you can show in another way, Cantor's diagonal argument uses negative self-reference, proves by reductio ad absurdum that not all reals simply can be put into 1-to-1 correspondence with the natural numbers (hence it would have the same aleph).

Why isn't it a paradox? Well, if we would assume that all numbers can be well-ordered/put into 1-to-1 correspondence with the natural numbers, then it would be a paradox for us! Yet as we don't take as an axiom that all numbers can be well-ordered, we don't have a problem with this, just like we don't have any problems with irrational numbers.

Yet there is the link to Zeno's paradoxes as they are about infinity.

I'm not sure if I've been able to show the connection to you, but well, all I'll say in the end that paradoxes for me aren't questions to be answered or solved, but more like answers that should be understood.

Gracias for letting me know about the position that Russell's paradox isn't as damaging to math as iniitially believed. Didn't know that. It's a moving target I suppose. What's Dana Scott's argument? Do you know?



I like what you said there.

I can't add more black to black and hope to get white!

As for your attempt to show Zeno's paradoxes are self-referential negations, I'm sorry I don't follow. Let's keep it simple, use one of Zeno's many paradoxes (your choice) and demonstratehow it is self-referential negations. Danke!

Curry's paradox does not directly involve negation.
Yablo has a paradox that explicitly does not involve self-reference.

Thanks.

Perhaps it would be proper to say that the set of paradoxes has more paradoxes than just one's with negative self-reference.

As we know, Zeno's paradoxes are about infinity. Modern math has "no problem" with Zeno's paradoxes of Dichotomy, the Arrow or the Tortoise as it uses limits (or the infinitesimal). — ssu

The paradox lies in the infinitesimals.

Then set it out.

The paradox is that infinitesimals have zero length.

infinitesimals have zero length. — Hillary

Then why does Wolfram say

...they are some quantity that is explicitly nonzero and yet smaller in absolute value than any real quantity. — Infinitesimal