Why does the world conform to logic?

[TheVat]

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?

[Lozza]

Logic and math are merely tools we humans use to describe that which we perceive. Whether or not maths and logic have any bearing upon objective reality, cannot be known, for we cannot know objective reality...we are not gods. But they are extremely useful tools that enable us to function as we do, explore ideas and develop technologies.

A little spoken aspect of logic, using words as communication, is the emotional content of language, which more often than not, is fraught with the bias of the reader/listener as much as the bias of the writer/speaker. Personally, I find this more prevalent in the written word than in the spoken word due to the lack of visual queues/expressions and vocal modulation that is absent in the written word, but abundant in the spoken word.

I don't believe that the world "conforms to logic" at any stretch of the imagination, but that it's the best tool we have devised so far to communicate an idea, provided that the person(s) proffering the idea keep emotive language to a minimum and the recipients of the idea do the same, attempting to only pay attention to the facts within the logic, ignoring most of the emotive descriptors that are often included.

For example, in a theological discussion, there is NO "logic" between a theist and an atheist, there is merely subjective reasoning, justifying an emotionally based stance....both stances are emotional stances, for neither can evidence their stance, but stand by it based upon emotional terms....many theists will openly admit that there's is an emotional experience, but I haven't encountered an atheist that will admit that. I find that very amusing, since atheists like to proffer their "logical" reasons for their stance.....without an iota of evidence. Personally, I'm ignostic...based upon the evidence I can draw no conclusion outside of "I don't know, and by the way, what are you defining as a 'god?'"

Another example, sometimes when speaking to someone that has had an idea, their emotional connection to the idea is the impasse, and no amount of good logic and reasoning can dissuade them of their idea. The idea itself in being relayed may have had no emotive language, but it's the emotional connection to the idea that becomes the problem. We often call this "ego", which to a certain extent it is, but ego is founded upon the emotional connection.

My final example is this; even seemingly innocuous words hold emotional content. If I'm a capitalist, the word "socialism" has a negative emotional feeling to it, and vica versa, if I'm a socialist, the word "capitalism" has a negative emotional content to it. Yet to a like minded person, each of those words has a positive emotional content to them.

And that's the main problem in communication...we don't know what emotional content any word we use has upon the person(s) we are communicating with...until they respond.

[-1-]

"Why does the world conform to logic?"

Why is this asked as if the cart pulls the horse?

Logic conforms to the world. Our entire system of logic, expressed by language, is based on how we see things happen in nature. Thus, a cart can't pull a horse, despite the cart exerting the same amount of force on the horse equal and opposite to the force that the horse is exerting on the cart.

[A_Seagull]

Why does the world conform to logic?

The world has no other option of making sense. Logic is mathematics in words; it is our innate (a priori) means of “making sense”. Without it, we are all fools.

There is no way we can invalidate logic without invalidating our own invalidation.

The basis of the question is partly noting posters like RJG, who seems to recommend logic...

Yes, I am of the opinion that Logic always trumps Science, and therefore our truths should be dictated by Logic, and not necessarily Science.

Logic can overturn Science, but Science can never overturn Logic. If something is logically impossible, then all the science in the world cannot make the impossible, possible.

- The truths of Science are constantly evolving and changing. The truths of Science are fallible.
- The truths of Logic/Math are constant; never changing. The truths of Logic/Math are non-fallible.

I agree with you that logic is essential for making sense of the world. But which logic? Logic can only operate within a logical system. Something that many people seem to ignore.

A logical system requires elements and rules of inference. Then it can be shown how a theorem can be generated from those elements and rules of inference. That is the only way logic can work.

All to often people conflate logical systems and language, without any formal exposition of how those words are to be handled within the logical system.

The result is chaos and paradoxes and hand waving arguments. It is simply not proper logic. And hinders rather then helps in an understanding of the world.

[edy420]

All logical calculations are an approximation when applied to the real world.

I remember as an electrical apprentice, we could calculate volt drop over distance if we knew the resistance in the conductor. The mathematical calculation was close but never consistent with and actual measurement. The calculation and the measurement were plus or minus in accuracy.

It turns out, this happens in any application of logic when compared to real world measurement. This effect seems universal. I wonder if it's due to unknown variables or if it's just that the world has some kind of filter between logic and application. (Perhaps constants are not really constant or we live in a dream of God etc)

Sure 2+2=4 But 2 (object) + 2 (object) does not equal 4 (objects)
In our minds it works nicely, but 2 objects will differ in atomic structure and mass, making the logical truth an impossibility in application. So much so that the logical truth we cling on to is not absolute.

This is consistent with the logical truth, that there is no absolute truths. Hah!

[A_Seagull]

All logical calculations are an approximation when applied to the real world.

I remember as an electrical apprentice, we could calculate volt drop over distance if we knew the resistance in the conductor. The mathematical calculation was close but never consistent with and actual measurement. The calculation and the measurement were plus or minus in accuracy.

It turns out, this happens in any application of logic when compared to real world measurement. This effect seems universal. I wonder if it's due to unknown variables or if it's just that the world has some kind of filter between logic and application. (Perhaps constants are not really constant or we live in a dream of God etc)

Sure 2+2=4 But 2 (object) + 2 (object) does not equal 4 (objects)
In our minds it works nicely, but 2 objects will differ in atomic structure and mass, making the logical truth an impossibility in application. So much so that the logical truth we cling on to is not absolute.

This is consistent with the logical truth, that there is no absolute truths. Hah!

It is the distinction between pure mathematics and applied maths. The interface between pure mathematics and the real world requires a mapping. Simple arithmetic applies to some objects eg children's building blocks, but not to others eg drops of water.

There is no logical process for determining whether a particular mapping is effective or not other than one of pragmatism.

[edy420]

The interface between pure mathematics and the real world requires a mapping. Simple arithmetic applies to some objects eg children's building blocks

I agree, but that fact is not incorporated into the way we teach. At the very core, we use faith to believe in the correlation between 100% mathematical truths and the real world. And that's perfectly fine. The mathematical fact loses a bit of its truth in real world application, to the point of say 99.9% accuracy. Which is probably why we ignore it. I think of it as the faith/fact ratio.

Bit as we get into more complicated logical truths, we need to keep this factor of faith in mind so that we can take not of how much is logical truth, and how much is faith. Some logical truths fall closer to 92% truth and we happily use 8% faith to fill the gap. Again there us nothing wrong with this.

The problem is that alternate logic may correlate with the same ratio of logical truth but be in direct opposition to another parties logical truth.

One example is fossils and the great flood. Dr Hovind predicts that because of the great flood, he may go dig up the remains of animals that were wiped out. Where as Dr Hawkins believes that due to the Theory of Evolution, he will have the exact same findings. Here we have two opposing hypothesis with roughly the same logical accuracy to faith ratio.

In schools, we aren't taught to recognise this logical truth to faith ratio. Instead we are indoctrinated with absolute facts and logical truths. (Not at the self thinking level, but across the board in general education).

The more complex the logical truth, the higher the chance is that our faith/fact ratio is less than the near perfect ratio of mathematical logic.

[-1-]

Sure 2+2=4 But 2 (object) + 2 (object) does not equal 4 (objects)

Yes, it does.

One basic tenet of math says you can add and subtract only like objects. You can add two apples to three apples, but you can't add two apples to three oranges.

When you come up with the example of drops of water, you change the object in the sum. 2 drops of water added to three drops of water, are five drops of water. When you put them together, however, and they morph, they are not drops of water any more of the kind that they used to be. So you change their quality, and only things of equal qualities can be added.

[-1-]

One example is fossils and the great flood. Dr Hovind predicts that because of the great flood, he may go dig up the remains of animals that were wiped out. Where as Dr Hawkins believes that due to the Theory of Evolution, he will have the exact same findings. Here we have two opposing hypothesis with roughly the same logical accuracy to faith ratio.

This is a great way to show that the Diluvian Theory is 0.0000001 ratio of logical accuracy to faith, whereas evolution is 0.99999999 ratio of logical accuracy to faith.

Your theory is correct, except you erred with the example. However, if you are saying that the same ratio applies to five-year-old children and to mentally challenged adults in our culture, then you are right.

The "same logical accuracy to faith" ratio is not the same for science-literates, and to very religious people, in the case of fossils. Only to those normally developed people who are very gullible and have no background to check for accuracy will believe that the two theories carry the same "logical to accuracy to faith" ratio. For instance, they may be equal to a people living like nomadic hunting-gatherers in the Kalahari. But to 1 billion Christians and to 2 billion Muslims there is no contest. In their perception of reality, the Diluvian explanation wins.

[TheVat]

Posts containing personal remarks, off-topic critique of SCF (which goes in Feedback BTW), or extended digression onto religion (goes in religion forum) will be deleted. Further violations of forum rules may result in thread locking and/or banning.

We will be very short-staffed after I stop admin duties Dec. 15, so moderators who lack time to sort things may choose the quick solutions over arbitration and hand holding. Consider this your one warning.

[edy420]

-1-,

Define science literate. I see it as someone who is capable of hypothesising an outcome, based on solid scientific input.

When evolution is in mind, we look at the evidence. A random series of bones. Cell mutations. Species mutation in adaption to environments. And a lot of theory. If you can look at all the evidence and come to the conclusion that evolution is the most likely, then you would be what I define as science-literate.

But if your just agreeing with what teachers and peers tell you, then you are building your conclusion on faith, more so than fact. The general public knows very little about the scientific fact. They generally don't have the time to study the evidence. What they do, is trust in scientists they have never met.

Me personally, I haven't look into the information on how virus evolve in accordance to evolutionary theory. Nor have I seen the series of bones that theory is built on. Nor have I read the findings on how birds on an isolated island start to mutate. But I know many intelligent people who claim they have. (Now I think about it, I doubt most of them have. They most likely parrot from another source).

I don't have time to study the theory of evolution in it's entirety. But I trust Paralith when she shares her expertise. I have faith that she knows what she's talking about.

I don't know that evolution is how we were created. But I have faith that it's most likely accurate. If we could recreate abiogenisis then I would believe in it based more on logical truth. I like to think my fact to faith ratio on evolution is around 75/25 But if I look at all the study I've done on the topic it's probably more like 55/45

This means that with respect to evolution, I am not science literate.

Did you study the theory of evolution? If not, then does your definition of science literate differ from mine?