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jamespotter

Russell K

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Claims:

1. Naive set theory fails. This means you can't group things and then make rules about how these groups function.

2. Any attempt to rule out this paradox relies on making a group (inside and outside) and then making a rule about how these groups function (the inside is allowed, and the outside is not)

3. Given that Russell's Paradox is "only those barbers who shave men who do not shave themselves", one can restate Russell's Paradox as the axiom "only those axioms which rule in axioms which rule themselves out are allowed". To the answer "your axiom is silly" I say "says who? some more ruley rule? Guess what I'm going to say"

4. Rinse and repeat as they make rules about their rules, etc. Say "we said rules are paradoxical, so what did they do? Make a rule, which isn't responsive."

5. ECQ-prove a paradox, and you can prove anything. https://en.wikipedia.org/wiki/Principle_of_explosion

6. That includes time travel. Yet I don't have a time travel machine, which is not predicted by the principle of explosion

7. Something is rotten in the state of Denmark, from a proof-theoretic standpoint, if not according to generally accepted wisdom. See: no time travel machine. (you don't have to make points 6 and 7, the K functions fine without them)

8. "This Logic is false according to this logic" is a Godel statement. Godel statements are real so there is no perfcon. (is "this statement is a lie" true or false?)

Pull it off and it's kryptonite to policy wonks with a judge that votes for the team that won the debate

Questions?

Edited by jamespotter

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"Russell's paradox is based on examples like this: Consider a group of barbers who shave only 
those men who do not shave themselves. Suppose there is a barber in this collection who does not 
shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave 
themselves.)"

 

??? why, "by the definition of the collection" must he shave himself? The collection says he can only shave men who don't shave themselves, not that all men who don't shave themselves must be shaved by him.

Edited by TheSnowball

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6 hours ago, TheSnowball said:

"Russell's paradox is based on examples like this: Consider a group of barbers who shave only 
those men who do not shave themselves. Suppose there is a barber in this collection who does not 
shave himself; then by the definition of the collection, he must shave himself. But no barber in the collection can shave himself. (If so, he would be a man who does shave men who shave 
themselves.)"

 

??? why, "by the definition of the collection" must he shave himself? The collection says he can only shave men who don't shave themselves, not that all men who don't shave themselves must be shaved by him.

i don't see how this pertains to anything in debate though, is it about like the data in the affs evidence?

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2 hours ago, AlistairTheKDebater said:

i don't see how this pertains to anything in debate though, is it about like the data in the affs evidence?

Yeah, one logical paradox =/= logic doesn't exist.

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22 hours ago, TheSnowball said:

Yeah, one logical paradox =/= logic doesn't exist.

In paraconsistent logic it doesn't. 

"In classical logic, the logic developed by Boole, Frege, Russell et al. in the late 1800s, and the logic almost always taught in university courses, has an inference relation according to which

A, ¬A  B

is valid. Here the conclusion, B, could be absolutely anything at all. Thus this inference is called ex contradictione quodlibet (from a contradiction, everything follows) or explosion (emphasis mine). Paraconsistent logicians have urged that this feature of classical inference is incorrect.

...

So if it is invalid to infer everything from a contradiction, then this rule, called disjunctive syllogism,

A  B, ¬A  B,

must be invalid, too."

https://www.iep.utm.edu/para-log/

An example of disjunctive syllogism:

The breach is a safety violation, or it is not subject to fines.
The breach is not a safety violation.
Therefore, it is not subject to fines.

https://en.wikipedia.org/wiki/Disjunctive_syllogism

So you have your response when you get your next speeding ticket:

"You are not speeding, or you are subject to the fine.

Your are speeding.

Therefore you are subject to the fine."

Is wrong because one logical paradox doesn't equal logic doesn't exist! Ironclad. 

The description of Russell's paradox comes from this article: https://www.scientificamerican.com/article/what-is-russells-paradox/

 

 

Edited by jamespotter
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See, thing is, that's a lot of fancy notation you just used the meaning of which I won't investigate.

But using logic to disprove logic seems to already require logic to work.

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1 hour ago, TheSnowball said:

See, thing is, that's a lot of fancy notation you just used the meaning of which I won't investigate.

But using logic to disprove logic seems to already require logic to work.

In reply to your first point, the only notation I'm using is the disjunctive syllogism, which I gave an example of. The principle of explosion is based solely on disjunctive syllogism. The solution is the law of the excluded middle: "Either Socrates is mortal, or it is not the case that Socrates is mortal." has a third answer. What is it? 

https://en.wikipedia.org/wiki/Law_of_excluded_middle

The reason you have to affirm the excluded middle is:

"a or not a

not not a 

a"

doesn't make sense in any paraconsistent framework!

In reply to your second point, that is what I used to think. Why isn't "logic is false according to logic" just a Godel statement? Surely you don't rule out Godel statements (this statement is a lie)?

Finally, you still have to rule out disjunctive syllogism. My argument, in a nutshell, is that you don't get rules. 

To answer Alistair, it pertains to debate because of the links in the files. You can also link the short story to IR epistemologies (I attached some more Bleiker cards) and the Lacan seminar to just about anything. 

bleiker_cards.doc

Edited by jamespotter

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https://www.quora.com/How-was-Russells-paradox-resolved

ZFC deals with this problem.

The Library of Babel doesn't seem to have any relevance here, unless it's just as an illustration of what the principle of explosion looks like.

Quote

But using logic to disprove logic seems to already require logic to work.

Not true. You're thinking as if "logic" is one big thing, when really it's a bunch of little axioms and intuitions all grouped under the same label. If we accept some axioms, we can use them as a reason for rejecting others. Using the principle of explosion as a reason that certain systems of logic are invalid is fine, and it is correct to say that if Russell's paradox can be stated in some system of logic then that system of logic is inconsistent.

Different argument, if you don't like that one. Using logic to disprove logic already requires logic to work, fine. But using logic to disprove logic also already requires logic to not work. That's a deadlock. You don't resolve a deadlock by choosing one side of the deadlock to emphasize and then walking off satisfied. You resolve it by using some different tool.

When OP talked about paraconsistent logics they were deferring to your assertion by pointing out one way in which the principle of explosion might not hold, and so a logical paradox wouldn't imply logic doesn't exist. Without that, though, the argument that a logical paradox implies the unsoundness of all of logic is sound.

Edited by CanIGetAFavor

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Thank you. I appreciate your comment. 

ZFC is post-hoc, and the axiom "the axiom that rules in only those axioms that rule themselves out" requires an axiom about axioms, which results in an infinite regress. The axiom of comprehension doesn't answer "axioms fail".

The Library of Babel is just a description of explosive logic, with a history of how people might react to it. Also it synergizes with the Bleiker evidence.

Girard's Paradox seems to answer ZFC, but I'll be damned if I understand how. My understanding is "what type is 'type'", which if ZFC is a simplification of type/category theory is responsive. Is "type" type "true"? Then does type "true" contain itself? 

BTW, if they read Lacan defense, ask whether the set of all sets contains itself (there is no universal prediction engine if it doesn't, because that engine would have to account for itself. https://philosophy.stackexchange.com/questions/30326/impossibility-of-a-laplace-kind-demon )

http://liamoc.net/posts/2015-09-10-girards-paradox.html (def cut this article. I can't believe that I haven't)

"Musings and Speculation

I find it very curious that two very different approaches to formalising mathematics end up with much the same stratified character, and for different reasons. Perhaps this Russell-style heirarchy is, kind of like the Church-Turing thesis, a fundamental characteristic of any sufficiently expressive foundation. Something discovered rather than invented. In the words of Scott [1974]:

The truth is that there is only one satisfactory way of avoiding the paradoxes: namely, the use of some form of the theory of types. That was at the basis of both Russell’s and Zermelo’s intuitions. Indeed the best way to regard Zermelo’s theory is as a simplification and extension of Russell’s. (We mean Russell’s simple theory of types, of course.) The simplification was to make the types cumulative. Thus mixing of types is easier and annoying repetitions are avoided. Once the later types are allowed to accumulate the earlier ones, we can then easily imagine extending the types into the transfinite — just how far we want to go must necessarily be left open. Now Russell made his types explicit in his notation and Zermelo left them implicit. [emphasis in original]"

 

Edited by jamespotter

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I haven't taken a class on any of this, but my understanding is that Russell's paradox happens because of the idea that we can define sets in terms of properties that objects have. ZFC gets rid of the idea that objects having some property is sufficient to make it coherent to talk about a set with that property. That idea is the axiom of unrestricted comprehension, and ZFC gets rid of it. In ZFC, you can't start with properties and then make sets of all objects with such properties.

ZFC is post-hoc, that's true, but it maintains consistency, which is enough to motivate it as an improvement over naive set theory even if it's a little weird and unnatural to approach sets through it.

Girard's paradox applies to type theory, but ZFC is not a simplification of type theory as you claim. Type theory is higher order logic, and ZFC is explicitly first order.

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From your Quora link: "How was “solved” the paradox? Dropping one of the previous items/assumptions. 1. (ii) “every multiplicity can be captured by a set” — In the standard ZFC ( Zermelo Fraenklen + Choice) axiomatic system."

"In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that the Cartesian product of a collection of non-empty sets is non-empty"

https://en.wikipedia.org/wiki/Axiom_of_choice

Translation: We banned Russell's Paradox, which is what my argument "the axiom that rules in only those axioms that rule themselves out" answers. 

“The joke is really about the Banach-Tarski theorem, which says that you can cut up a sphere into a finite number of pieces which when reassembled give you two spheres of the same size as the original sphere. This theorem is extremely counterintuitive since we seem to be doubling volume without adding any material or stretching the material that we have.

The theorem makes use of the Axiom of Choice (AC), which says that if you have a collection of sets then there is a way to select one element from each set. It has been proved that AC cannot be derived from the rest of set theory but must be introduced as an additional axiom. Since AC can be used to derive counterintuitive results such as the Banach-Tarski theorem, some mathematicians are very careful to specify when their arguments depend on AC.”

https://math.stackexchange.com/questions/6489/can-you-explain-the-axiom-of-choice-in-simple-terms

“It was questionable, therefore, whether this axiom is even consistent with the rest of the axioms of set theory. Gödel proved this consistency in the late 1940's while Cohen proved the consistency of its negation in the 1960's (it is important to remark that if we allow non-set elements to exist then Fraenkel already proved these things in the 1930's).”

https://math.stackexchange.com/questions/132007/why-is-the-axiom-of-choice-separated-from-the-other-axioms

MBG differentiate between "classes" and "sets", so one speaks of the class of all sets. Begs the question of what type class is, class or set? The former just defers the contradiction, the latter doesn't solve it. Also, has axioms.

Edited by jamespotter

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Arguments about the AoC have nothing to do with arguments about the meat of ZFC. ZF without choice is a thing. Trust me, you're overconfident here. You're disagreeing with well-established math. It's widely agreed that ZFC has not been proven inconsistent. Someone other than you would have noticed this first.

ZFC doesn't ban Russell's paradox. It removes the axiom that gives rise to Russell's paradox. That's a valid choice, even if it's hard to motivate "in advance" of Russell's paradox. Clearly Russell's paradox and the problem it poses serves as sufficient motivation in itself for using a different system of axioms.

Edited by CanIGetAFavor

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A. Disagreeing with well-established x is the engine of knowledge. Otherwise we would know everything! My arg is that axioms (in logic, rules) per se=naive set theory, and no axioms=quodlibet. BTW, I'm disagreeing with well-established most things. Like Aristotle, who is kind of important to the history of Western philosophy, with his radical idea that we can explain things. 

B. I hate arguing with people about this. In response to "ZF without choice" or "other mathematics", I refer you to A. I'm not going to answer every derivation, in round you would have to commit to something, at which point I would apply the top-level. You may not win the round (because ZFC is in point of fact well-established), but in my opinion you would be rightish. Worth something.

C. You can't formulate axioms without recourse to the axiom of unrestricted comprehension because axioms are naive sets (see A)

D. Group any and all objections. See A. Read the file or don't.

E. I respect you. 

Edited by jamespotter

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AT-Paraconsistency:

"either paraconsistency or not paraconsistency

not paraconsistency

Ergo, neither paraconsistency nor not paraconsistency"

recurse that formula and repeat for each individual element of paraconsistent logic.

I fail to see how explosion doesn't == paraconsistency. 

Edit: Added a link to "Affirm the Other" to Lacan.

 

 

spivak.doc

Edited by jamespotter

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On 12/13/2018 at 9:36 PM, jamespotter said:

Disagreeing with well-established x is the engine of knowledge. Otherwise we would know everything! 

"[Thought] is having the courage to not know what others do."

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