Are the rules of logic immutable?

[OccamsRazor]

In the past my philosophical stance was always based on a single overarching truth. This was that the laws of logic and mathematics were immutable and could not be changed for any description of reality that one may provide.

More recently I have been grappling with the question, what if the rules of logic and mathematics are not immutable but subjective or specific to our incarnation of reality?

In Michael Frayn's book The Human Touch he makes the statement:

Logic is just a system we have made up, not an inherent condition of the natural world.

Is this true? Is logic changeable?

In another thread I saw the following statement:

I don't quite know how knowing something about events inside a system from outside of the system is on the same level of impossibility as a logical impossibility. There cannot be a square circle, a rational root of a prime number or the simultaneous existence of an irresistible force and an immovable object. These are logical impossibilities..

Is this true? Could a being outside our own manisfestation of material reality not create such logical impossibilities?

I can see that here many readers of this post would begin to state that logic and mathematics were immutable. That there indeed could not exist a rational root of a prime number and these are objective truths.
This leads on to the question, how may one prove it? Bearing in mind that any proof of the immutability of logic must have its basis in logic. The question is, how can immutable logic prove istelf to objectively exist?

If we then decide that, possibly, logic is not immutable then where does this leave us? Can we ever make a metaphysical argument without firstly assuming that mathematics and logic are immutable?

[skepticFromTX]

Furrowed Brow: "If the rules of logic were not immutable then maybe the argument...
All bachelors ar men.
Socrates is a man
Therefore Socrates is a woman.
...would become valid. Therefore I suggest something is immutable here."

Maybe we're saying "immutable" when we should be saying "self consistent"

"...the kind of logic we humans use easily and daily cannot be reduced to machine code, and is inconsistent if it is to be complete."

To say "the kind of logic we humans use" is to imply that there are different kinds of logic. If that's so, then our variety evolved over the millenia along with our brains. The reason why we can't reduce it to machine code is because it's messy, and consists of a large number of little ad hoc components that our brains originally used to solve survival-related problems.

But then our brains discovered paradox (like a serpent in our cognitive Eden), and started coming up with stuff like: "This statement is false", and chains of reasoning to demonstrate that physical motion was impossible. We invented the propositional calculus and then later demonstrated that the propositional calculus inconsistent. And don't forget that the assortment of dieties that our brains came up with throughout our history is the product of that same talent for ratiocination.

Bottom line is, I don't think we can consider logic separately from consciousness. Even Roger Penrose said that to understand consciousness it might be necessary to understand the fundamental nature of physical existence itself, and to me his assertion makes a lot of sense.

And maybe it'll turn out that our conceptual framework just isn't up to the job. Maybe in trying to understand what we really are we're like parakeets trying to understand long division.

In any case we must never stop trying because if we did, we surely would dissolve into goo.

[Furrowed Brow]

Hello skepticFromTX

You say

Maybe we're saying "immutable" when we should be saying "self consistent"

I think the question is "what are the immutable rules of self consistency?" If they are immutable then my rather poorly constructed argument can never be valid. If they are not immutable then maybe there will be a time when it could be valid.

You also say,

To say "the kind of logic we humans use" is to imply that there are different kinds of logic.

Well there are different kinds of logics. However there is some dispute to just how logical some of them really are. The True/False choice of the propositional calculus and fuzzy logic certaintly appear incongruent.

I said the propositional and predicate calculus are human ways of thinking. And by this I meant we can readily make reality sense of arguments like

All men are mortal.
Socrates is man
therefore Socrates is mortal.

And we seem to do this without multivalued logics or any fuzziness. But maybe the human brain can also cope with fuzzy logic. Maybe it has found ad hoc ways of using all sorts of logics for different purposes? I don't know. It's a fascinating area.

You say

We invented the propositional calculus and then later demonstrated that the propositional calculus inconsistent.

Sorry. The propositional calculus is consistent and complete. I think you mean the predicate calculus. However, things are a bit more complicated than that. The rules of the predicate calculus are consistent. Alonzo Church showed that some arguments cannot be completed because they fall into infinite regress. Thus their validity/invalidity cannot be decided. It is this result which tells us the predicate calculus is not machine calcuable.

The predicate calculus can be translated into the formalism of number theory and taken to be logically equivalent. And Godel's theorems strictly apply to number theory and formal systems which are its logical equivilent.

Number theory can be consistent as long as some of its arguments are not provable. If all arguments in such a formal system are provable then the system is inconsistent.




You also say

Bottom line is, I don't think we can consider logic separately from consciousness.

I'd put it the other way around. I think logic is a necessary condition of some large part of human thought. I am not so sure, though could be persuaded, that consciousness is a necessary condition for the rules of logic.
You could be right. But this is cloudy sky. I' don't know which way to fly.

Pretty Polly. Chirp. Squawk!

[skepticFromTX]

Hello skepticFromTX

You say

Maybe we're saying "immutable" when we should be saying "self consistent"

I think the question is "what are the immutable rules of self consistency?" If they are immutable then my rather poorly constructed argument can never be valid. If they are not immutable then maybe there will be a time when it could be valid.

You also say,

To say "the kind of logic we humans use" is to imply that there are different kinds of logic.

Well there are different kinds of logics. However there is some dispute to just how logical some of them really are. The True/False choice of the propositional calculus and fuzzy logic certaintly appear incongruent.

I said the propositional and predicate calculus are human ways of thinking. And by this I meant we can readily make reality sense of arguments like

All men are mortal.
Socrates is man
therefore Socrates is mortal.

And we seem to do this without multivalued logics or any fuzziness. But maybe the human brain can also cope with fuzzy logic. Maybe it has found ad hoc ways of using all sorts of logics for different purposes? I don't know. It's a fascinating area.

You say

We invented the propositional calculus and then later demonstrated that the propositional calculus inconsistent.

Sorry. The propositional calculus is consistent and complete. I think you mean the predicate calculus. However, things are a bit more complicated than that. The rules of the predicate calculus are consistent. Alonzo Church showed that some arguments cannot be completed because they fall into infinite regress. Thus their validity/invalidity cannot be decided. It is this result which tells us the predicate calculus is not machine calcuable.

The predicate calculus can be translated into the formalism of number theory and taken to be logically equivalent. And Godel's theorems strictly apply to number theory and formal systems which are its logical equivilent.

Number theory can be consistent as long as some of its arguments are not provable. If all arguments in such a formal system are provable then the system is inconsistent.

But doesn't "proving" some proposition mean deriving it from other propositions within the same system?




You also say

Bottom line is, I don't think we can consider logic separately from consciousness.

I'd put it the other way around. I think logic is a necessary condition of some large part of human thought. I am not so sure, though could be persuaded, that consciousness is a necessary condition for the rules of logic.
You could be right. But this is cloudy sky. I' don't know which way to fly.

Pretty Polly. Chirp. Squawk!

[QED]

I would like to know how we reconcile the differences in logic as they occur between the classical and quantum worlds. Reading a mention of Penrose reminded me of the Elitzer-Vaidman bomb testing problem. In the classical world it is defined that the only way to test the bomb to see if it's a dud or not is to trip the trigger mechanism. This obviously can't leave us with a "known to be good" bomb. But if the trigger mechanism is operating at a quantum level then it is possible "to test something that might have happened but did not happen". This is a property of the world no matter how unfamiliar and if the formal logic being discussed here operates under a different set of rules then I think we can say that they clearly are not immutable.

[Enrique]

This leads on to the question, how may one prove it? Bearing in mind that any proof of the immutability of logic must have its basis in logic. The question is, how can immutable logic prove istelf to objectively exist?

The problem you are stating is, how do you prove logic without being circular? Because if you use logic to prove logic, the vicious circle is before us. But that can't be used as a disproof either, because you would also need to use logic to disprove logic. So, we see that both the affirmation and the denial of logic reguires the affirmation of logic to make either proposition. So the proposition that affirms logic is at least coherent and cohesive with the principles that all the statement to need in order to be made. The negation is self stultifying, in other words the negation, if true, negates the principles that the proposition is based on.

The problem could be futher inflated by pointing out that you can not form a thought without the use of logic. So, for any disproof to be of value, it must be presented without any form of thinking being involved.

Z

Logic is some kind of fundamental rock. Like someone stated, is what rules thought. Humans are very fond of thought, is what makes them who they are over other species. Thought generates knowledge that makes human survive and gain happiness. Humans like and need to think. (I'd like to state that world comes first before mind, that order doesn't need to be proven. World exists without us or our mind. The world doesn't need our mind or thought in order to exist. Thinking is just an activity, something that takes place in the universe like any other physical event). Here the thought has a problem on his own. A problem of identity. He doesn't know where he comes from. Very like to us. He does exist indeed as we had accepted initially, but he doesn't know what he is. He is the cause of our knowledge, something very important to us, but the identity problem that thought has is transferred to our so appreciated knowledge. Thought is the cause of knowledge but he doesn't know who gave him birth. He looks everywhere and finds the universe and himself. He only see this two possible causes. Then there are two alternatives. The universe creates him or he has the property of self existence.
1)If he's self existence then his consecuences can't cause him. We can't pretend cause the thought with his consequences, which is what we're tying to do here. "how can immutable logic prove istelf to objectively exist?" doesn't need to be proven. The action of "proof" was born from thought and not backwards. BUT THIS LEADS TO THE CONSEQUENCE OF DUALISM. The UNIVERSE and THOUGHT interact in us.
2)If universe cause thought or logic then it's just another physical process. We see thought is in humans then we can be more specific and say it's a feeling.

3)This third possibility is what we discard initially, that only thought is self existence and universe a consequence of it.

Note also that what i and you have been doing here is existing then thought comes from existence, much specific our feelings.

I wish someone would answer me what 'IS' is? PLEASE!!!

My first approach is that 'IS' (or existence) is the first cause. The cause of universe. And because Logic messed around innocently with 'IS' is because is having all this problems. Scientists took shelter under Logic, as a way to escape from 'IS'. They thought that Logic was stronger than IS but it isn't. I think that get on IS side could be a better.

[Furrowed Brow]

Zorro1 said,

The problem could be futher inflated by pointing out that you can not form a thought without the use of logic. So, for any disproof to be of value, it must be presented without any form of thinking being involved.

I'd say that is about right.The early Wittgenstein traveled this way in his Tractatus Logic Philosophicus, a book never to be picked up and read by the unwary.

We use logic, can prove arguments are valid invalid etc but cannot prove the axioms of logic. That's why they are axioms.

It gets worse. If you start talking about the axioms of logic and try to explain them you end up talking nonsesne.

This is Wittgenstein's second to last point in the book.

6.54 My propostions serve as eluucidations in the follwing way: anyone who understands me eventually recognises them as nonsensical...

However that does not mean logic is illogical; like seeing the back of your eyeball there are somethings you just can't do.

If this is the right view of logic it cannot be a human construction. If it was then its axioms could be proved because they would have some real and tangible origination; rather than having that puzzling quality of self evidence they seem to have.

I think the more scientific question is whether the quantum world follows a different kind of logic to that deployed in human thinking?

[Enrique]

However that does not mean logic is illogical; like seeing the back of your eyeball there are somethings you just can't do.

If this is the right view of logic it cannot be a human construction. If it was then its axioms could be proved because they would have some real and tangible origination; rather than having that puzzling quality of self evidence they seem to have.

I think the more scientific question is whether the quantum world follows a different kind of logic to that deployed in human thinking?

If it's a human construction, I think the real and tangible origination it's just there, what you feel when you think or the electro-chemistry of your brain. There's no need to go further. What it's hard to accept for us is the transition between both natures. The further we track the causes of an effect, the larger the change in the nature. Sooner or later the thought should find its roots in the phisical world (supposing there's no dualism).
Maybe finding a direct univoque relationship or pattern between chemistry in our brain and a logical thought could be a proof?
If there exists such another quantum logic, could our brain process it? It could be like imagine a 4 dimention phisical world. And if we could proccess this quantum logic, it could be using our logic, couldn't it?

[OccamsRazor]

After reading the flow of this I am presented with a problem in the reasoning that there are multiple approaches to logic. My problem is that all of the approaches presented here including fuzzy logic, predicate calculus and quantum mechanics are mathematically complete.
Therefore all of our approaches to logic are based on a predicate axiom that mathematics and mathematical proofs are immutable.

However that does not mean logic is illogical; like seeing the back of your eyeball there are somethings you just can't do.

If this is the right view of logic it cannot be a human construction. If it was then its axioms could be proved because they would have some real and tangible origination; rather than having that puzzling quality of self evidence they seem to have.

I disagree with this conclusion. I don't completely accept that because we cannot wrap our thought around the idea of describing our own approach to logic then such logic must simply be.

Here is the root of the problem. Now referring back to a previous statement:

Without the rules of logic we cannot think

I would argue this statement on the basis that the world we can directly observe seems to accurately obey our mathematics. We can therefore say that our principles of logic apply to our local view of reality. We therefore have begun to make a statement concerning the nature of reality without the requirement of our immutable rules.

We may then move to argue that if we can accurately describe the history of our existence with the set of mathematical principles we adhere to then we may say that these rules have remained unchanged during this period.

We have therefore stated that in our local view of reality and within a given time period, our principles of logic and mathematics have remained constant. This reasoning is obviously based on a large weight of assumptions and has some holes. My point here is that I do not agree that if we suggest that logic and mathematics are not set in stone then we are simply struck dumb.

[Furrowed Brow]

Hi OccamsRazor

My point here is that I do not agree that if we suggest that logic and mathematics are not set in stone then we are simply struck dumb.

Let me let on to my philosophical prejudices. Wittgenstein wrote about this. Here is the oft quoted but misunderstood last point of the Tractatus

7 What we cannot speak about we must pass over in silence

When contemplating the nature of logic and thought, and when trying to access its foundations we will be, if we see things right according to Wittgenstein at least, struck dumb.

However that is the case if logic is immutable. If when we look at an axiom like something is P or Not-P which in notation looks like P v ~P, i.e. "It is the case unicorns are real" can be true or false, and there is no middle way. Unicorns don't a little bit exist.

In formal logic it is possible to prove |- P v ~P is an axiom, i.e it is true without any further premises other than itself. It is self evidently true, and it can be proved that it is not contingently true.

This however does not prove the laws of standard logic are immutable. Maybe we could all wake up one day and think differently, and wonder why the law of the excluded middle was ever thought to be self evidently true, because it is self evidently false. What were we thinking?!

I don't think that will ever happen, but I can't prove that because I can't prove an axiom. If I could it would not be an axiom.

Def Axiom: proposition regarded as self evidently true.

Some philosophers like John Stuart Mills who were out and out empiricits would argue that the rules of logic are derived empirically. I wouldn't myself go that way, but you might want to think about the problems on those terms. In which case the laws of logic might not be immutable.

But if logic is immutable then Wittgentein's point is I think (and no one can claim to really understand Wittgenstein), that if you start scrabbling around for a language, or a metaphysics, or whatever to give a further foundation to an axiom like P or not P then you start talking nonsense. The right thing to do is absorb the lesson, and pass over the axiom in silence.

However, not all logic is like that. Take a sentence like - it is raining or it is not raining. Well does one drop of raindrop count as "it is raining". This kind of question is rooted in the kind of question labelled a sorites paradox. Usually it goes like this: take one heap of sand. Remove one grain of sand. Is it still a heap? answer yes. Ok keep removing one grain of sand at a time until to get to the last grain. Is the last grain of sand a heap. Answer: no. Ok when did the bunch of sand stop being a heap. Hmm!

Standard either/or logic just does not cope well with this kind of problem. Hence some guy came up with fuzzy logic.

I don't know, I have not thought deeply enough about non standard logics, but they are not axiomatic in the same way standard logic is. So maybe Wittgenstein's point does not extend out to this kind of logic.

However, when doing standard either/or logic, and I shall also include mathematics I think Wittgenstein makes a sound point. In broad terms I think JS Mills got it wrong. (Though I have to admit I am not a scholar On JS Mills.) Silence is the preffered option.

[Furrowed Brow]

Hi Enrique

You say

Maybe finding a direct univoque relationship or pattern between chemistry in our brain and a logical thought could be a proof?

The person who finds that relationship is gonna get a Noble prize!

However if someone does became a noble laureate, have we only pushed the question back a notch and found a new place for it. Say we work out the physics of the brain and how logical thought maps on to physical interactions we still face a few problems.

Take the law of the excluded middle. P v ~ P, or just plain old P or not P.

Say we find that in Group A the brain has a particular kind of interaction that this piece of logic maps on to.And every brain we check out for in this group of brains seems to be working out the law of excluded middle with the same basic interaction. Then we find another group of thinkers Group B who also find the law self evidently true, but their brains seems to have worked out a different interaction. Their ability to reason logically is the same as the first group, but the physical interaction is different.

Does mapping the logical axioms and connectives (and, or, if then etc ) on to physical interactions really prove the logical axioms and connectives. We've proved how we can do it. But have we really proved them per se, in themselves. I don't think we would have done?