Why does the world conform to logic?

[Braininvat]

Agree. I was wondering if there might be a parallel relation to what this thread is working on, so I tossed this in. Admittedly, a speculative and not well informed toss.

[Asparagus]

Agree. I was wondering if there might be a parallel relation to what this thread is working on, so I tossed this in. Admittedly, a speculative and not well informed toss.

I agree with you that paradoxes aren't informative and so are in the category of 'language on holiday.' I'm getting ready to go to Florida, so I'll see if I can throw some paradoxes into mix.

RCA dog!

[hyksos]

The issue is that set theory leads to a contradiction. R is one example of it. I think Russell was saying that the contradiction is the result of the mistake of treating a collection of classes as one class.
But isn't that pretty close to what set theory is doing with natural numbers? I think Russell was questioning the basic assumptions of set theory, not saying that set theory is fine except R can't exist. But I'm totally open to being corrected about that.

These issues have all been completely resolved in a contemporary context. Any system of first-order logic cannot contain statements that refer to other statements in the system from a "meta-lingual" perspective. (Talking about the fact that you are talking.) So you cannot have elements of a set which are little pointers to the "set itself". You cannot have barber's who shave themselves and constructions which go,

Sentence s = "Sentence s is false."

Once a paradox is permitted in a FO logic system, the paradox begins to corrode the entire system. You can utilize a paradox to prove any statement, and then disprove it.

But consider the attitude that R can't exist because it's contradictory. Would it follow from that that logic reliably guides us in understanding the world?

I would not say this is a matter of existence. The modern answer would contend that R is not a set, because the defining characteristic of a set is that "it can be determined if an element, e, is either in the set S, or not in S." Let S be the set of all sets. Let e be S. Is e contained in S? If you can show that e is not in S, and on another chalkboard show that e is in S, then the criterion for containment in S is not "well formed". Ergo S is not a set. More explicit ,

S = {all e such that P(e) ; }.

In the case of Russel's R, the P() given is not sufficient to allow R to be a set.

[Asparagus]

@hyksos
Hi! As it relates to the OP, the point is that neither the Liar nor RP are examples of nonsense.

RP is excluded artificially, not because it conflicts in any way whatsoever with the concepts of set theory.

[hyksos]

Are you contending that R is definitely a set?

Are you claiming R is forbidden from mathematics due solely to convention and convenience?

[Asparagus]

Are you contending that R is definitely a set?

In naive set theory it is. It's not allowed in ZFC.

Are you claiming R is forbidden from mathematics due solely to convention and convenience?

It's excluded axiomatically. Again, the point is that the pardox associated with R is not nonsensical. It's just paradoxical. That remains true regardless of your stance on the ontological status of set theory.

[Lomax]

Hyksos, putting aside that sets are usually undefined in set theory (that is to say, they are "primitives"), I have a problem with your "defining" characteristic, "it can be determined if an element, e, is either in the set S, or not in S". Actually the set of {all sets which do not contain themselves} does meet this condition, by virtue of being both. Do you rather mean to say that "an element, e, cannot both be in the S and not in the set S"? Because if so, that's what's being asked. Set theory doesn't "solve" this problem by just insisting on an answer. It just avoids it by ad hoc means. Which is the most common criticism of Russell's own proposed "solutions".

[RJG]

Actually the set of {all sets which do not contain themselves} does meet this condition, by virtue of being both.

This is as nonsensical as positing the set of {all sets}.

Since it is logically impossible for sets to contain themselves, then {all sets which do not contain themselves} = {all sets}.

[hyksos]

Hyksos, putting aside that sets are usually undefined in set theory (that is to say, they are "primitives"), I have a problem with your "defining" characteristic, "it can be determined if an element, e, is either in the set S, or not in S". Actually the set of {all sets which do not contain themselves} does meet this condition, by virtue of being both. Do you rather mean to say that "an element, e, cannot both be in the S and not in the set S"? Because if so, that's what's being asked. Set theory doesn't "solve" this problem by just insisting on an answer. It just avoids it by ad hoc means. Which is the most common criticism of Russell's own proposed "solutions".

We have machinery today that Russell did not have during his lifetime. During his lifetime, his solutions may have been completely ad-hoc, sure.

Today we can say things like : there must exist, at least in principle, an "effective procedure" which can determine if an element e is or is not in S. The effective procedure would be going about checking something by rote to determine if P(e) is true. If true, e is in S. If not true, e is not in S. This procedure could have an infinite number of steps. Furthermore, our fancy modern proofs can even pretend as if the procedure finished (literally got done with an infinite number of steps) . Say it "never" found that P(e) is true --- in principle we would know that e is not in S.

All those involved in this kind of reasoning understand such a procedure could of course never be embodied in real space and time. Nevertheless, this kind of Proof-by-effective-procedure is considered rigorous.

Given the ultramodern context described above, if we can show that no such effective procedure could possibly exist to flesh out P(e) , we can absolutely declare that S is not a set. In many textbooks, the author will say "S is too large to be a set". In other words the proposition P throws such a wide net, that it cannot effectively capture whether an element is inside S or not.

One way of going about this would be showing that a procedure of checking a specific e against P(e) would necessarily contain an uncountable number of steps.

A second way is reductio. Assume R is a set as a premise. Then prove a contradiction on e. Where e is R. (We are allowed to set e equal to R since we assumed it was a set in the premise) We soon reach a contradiction of P(e) and ~P(e).

This is what I was employing in showing that R is not a set.

[Lomax]

Actually the set of {all sets which do not contain themselves} does meet this condition, by virtue of being both.

This is as nonsensical as positing the set of {all sets}.

Since it is logically impossible for sets to contain themselves, then {all sets which do not contain themselves} = {all sets}.

RJG, it is not enough to just say it. We are talking about methods of proof here. The point is that saying "it must satisfy the condition of being one or the other" does nothing to advance the proof, particularly bearing in mind that disjunctions can be inclusive.

[RJG]

RJG, it is not enough to just say it. We are talking about methods of proof here. The point is that saying "it must satisfy the condition of being one or the other" does nothing to advance the proof, particularly bearing in mind that disjunctions can be inclusive.

Okay, ...understood.

[Asparagus]

We have machinery today that Russell did not have during his lifetime. During his lifetime, his solutions may have been completely ad-hoc, sure.

If you're wanting to say that R is organically and intuitively at odds with set theory, the machinery you've mentioned doesn't show that. It simply allows the rejection of R to be formalized.

It was the ambition of some logicians to have set theory accepted as a sort of naturalistic foundation. It's really only in regard to that agenda that it's even meaningful to note that R can only be rejected artificially. To a disciple of Wittgenstein, there would be nothing controversial about saying that R exists as part of a language game, but a certain kind of interest leads to the generation of a second game which excludes R.

IOW, if you want to make a strong claim that there is no such thing as R, you're going to have to bind yourself to some interesting ontological commitments about math that will lead us straight back to the OP.

[neuro]

[quote="[url=http://www.sciencechatforum.com/viewtopic.php?p=333667#p333667]
As for whether it can be rendered by logical notation, I think that depends on the logic. In Fregean/Aristotlean propositional logic we can simply render it as
(i) S
as RJG did earlier.
[/quote]
Lomax, can you axpand on this?
In RIG's post, statement S seems to be referring to the truth/falseness of statement T, not of itself...

We might also render it as
(i) ~i

again, can you expand? what does the notation "i" indicate in this line?

As to whether it can be rendered in predicate logic, I think that would be tougher. We might say we can render it simply as

(i) ExFx

Where the predicate F means "asserts ~ExFx", but perhaps some would argue we've split the statement into two, and changed the nature of the game.

Again: can you expand this notation in words?

Sorry about that! may fault. But other members as well may possibly be interested in understanding this.

[Lomax]

Hey Neuro,

Well propositional logic (a la Aristotle & co) only allows the non-logical notation (the variables, if you will) to stand in for entire propositions; so we could only really represent the statement (assuming it is a proposition, which is debated) as "S" or something simple like that. I think our other option would be something like

Premise1: ~Premise1

Where "~" is just the notation for "not".

Predicate logic (a la Russell & co) allows us to break things down a bit more and talk about the objects within the propositions, but since the object in this case is the proposition we have a problem. Maybe a logician will have a better idea than mine but the closest thing I can think of is to say

Premise one: ExFx

"E" in predicate logic just means "there exists" (it should be a back-to-front E but I'm being lazy). "x" is just "something". F isn't part of the logical language; we use it to stand in for a predicate. So I'm suggesting maybe we let "F" mean "asserts than ExFx is false". So ExFx would mean "there exists some x such that x has the property of asserting 'ExFx is false'". But I feel like this is cheating.

Either way, my point is that logical symbolism is something we've made up and we can reinvent it for our purposes; one way or another we can express anything in a formal language. These different languages are useful for different things, and they differ in their scope, complexity, completeness, consistency, power, and so on. I don't see any need to confine ourselves to one formal language. This isn't to say that logic cannot express truth or falsehood; but rather that those truths belong, if you like, to the "universal grammar" rather than to the logic itself. Something we haven't tapped into, and perhaps can't* - Mossling seems to make a similar point, in a more poetic way, on page 4. And regarding the OP, I see no reason to believe that the world "conforms" to those truths, any more than it "conforms" to the fact that Jupiter is bigger than Earth. If that makes sense. I have problems in general with the necessity/contingency distinction, because I don't think it can possibly - to take it on its own terms - be supported with evidence.

__________

* With a nod to Godel: any axiomatic system capable of generating Peano arithmetic cannot be both complete and consistent. A consistent system will always entail a statement which it cannot prove - structurally similar to the liar paradox. Our access to truth through reasoning seems, in this way, to be inexorably limited. But in other ways, we already knew that.

With a nod to Tarski and Dennett: to understand the deep, foundational structure of our thought, we should have to be smarter than what that very structure allows. Or as Wittgenstein put it: "‘in order to draw a limit to thinking, we should have to think both sides of this limit."

[hyksos]

We have machinery today that Russell did not have during his lifetime. During his lifetime, his solutions may have been completely ad-hoc, sure.

IOW, if you want to make a strong claim that there is no such thing as R, you're going to have to bind yourself to some interesting ontological commitments about math that will lead us straight back to the OP.

Ah.. but there is your error.

I did not claim "R does not exist." I only demonstrated it was not a set. It could still exist, as some other entity, such as a class.

Consider :

The paradoxes of naive set theory can be explained in terms of the inconsistent assumption that "all classes are sets". With a rigorous foundation, these paradoxes instead suggest proofs that certain classes are proper (i.e., that they are not sets). For example, Russell's paradox suggests a proof that the class of all sets which do not contain themselves is proper, and the Burali-Forti paradox suggests that the class of all ordinal numbers is proper. The paradoxes do not arise with classes because there is no notion of classes containing classes. Otherwise, one could, for example, define a class of all classes that do not contain themselves, which would lead to a Russell paradox for classes. A conglomerate, on the other hand, can have proper classes as members.

more : https://en.wikipedia.org/wiki/Class_(set_theory)


!Edit : My apologies up front. On a 2nd reading, it seems like I have not entirely fleshed out my position on this. According to the article linked, the Russell Paradox is strong enough to chase its way up the ladder to classes too. SO that there could be a "Russell Paradox of classes". Hmm...

[henriette]

Dear,

As a phenomenon, logic is a social fact, according to Durkheim. Because communication benefits from abstract, impersonal and stable concepts, what a society considers causal is communicated using implication.
The implication is indeed communicating a causal fact, in a way that is both impersonal (no space) and stable (no time). Maybe this contributes to why logic conforms to the society and hence to the known world.

[ontological_realist]

A is not -A

I can not see how can this be untrue.

[someguy1]

I did not claim "R does not exist." I only demonstrated it was not a set. It could still exist, as some other entity, such as a class.

Perfectly correct.

!Edit : My apologies up front. On a 2nd reading, it seems like I have not entirely fleshed out my position on this. According to the article linked, the Russell Paradox is strong enough to chase its way up the ladder to classes too. SO that there could be a "Russell Paradox of classes". Hmm...

Not sure whether that's true. I didn't see it expressed that way in the Wiki article. I'm not sure there's a Russell paradox for classes. Can you explain?