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?

[Confused]

Let me interject a fairly well known philospher from Germany: Carl von Weizsacker. He not only worked with philosophy but also with physics with his student being guess who Heisenberg. Weizsacker insisted on the importance of empiricism in science. Matter can be known in two ways. It can be phenomonally given or it can be inferred. His famous example was with an apple. We can see a brown spot in the apple which makes it a phenomonally given (empirical). We can infer that there is a worm in the apple from the brown spot and our general knowledge of apples and worms (metaphysical). Phenomenal empiricism here acts as deductive logic. It is a "proof". Metaphysical inferrences acts a deductive logic, in that we can state with a high probability that a worm is in the apple given our knowledge, but we can't say with "proof" or certainty that a worm indeed is in the apple or made the spot in the apple.

[Furrowed Brow]

Amazing! You're still stuck in your metaphysical hole. You state:

“1 + 1 = 2”: stuff that accumulates includes: area, boundary, the number of exposed frayed denim threads etc

“1 + 1 = 1”: stuff that does not accumulate includes: the word hole.

My response is: try it with the two pieces of silly putty -- as in the example that you seem (conveniently?) to ignore.

Zoro. When are you going to address this point directly. Ok you obviously want to disagree with it, but you keeping deflecting it away with further counter examples. I’m just trying to keep these posts down to a manageable length. Ok I’ll tackle the putty example, in the hope you actually address the point, and take on board what I am saying. Hey you can still disagree with. But frankly at the moment you are just name calling. Stop the evasion now.

You state: "The muddle you [are?] in is that you at heart are an empiricist." Right re. the "empiricist"; wrong re. any "muddle". Look: since "hole" causes you so much difficulty, then forget about it. Never bring it up again. From here on, whenever you think you see the word "hole" read it as "a piece of silly putty" (and similar).

The hole is your example. Now you don’t want to talk about it. Ok then we’ll talk about putty.

For example, read your next sentence "The concept of hole seems to fail a reality check of merging holes" as "The concept of a piece of silly putty seems to fail a reality check of merging pieces of silly putty." Now, to see your error, correct your sentence to read: "The concept of a piece of silly putty fails a reality check of merging pieces of silly putty" -- although it would be more informative if you wrote your sentence something similar to: "I agree with you: when two pieces of silly putty are "smooshed together" [as my grandchildren describe the process!], then using the symbol "1" to mean something exists and is distinct (not to mean mass, volume, number of atoms or any other property characterized by units or dimensions), then 1 + 1 = 1.

Lets play with putty then: you can hold it, play with it into lumps, smoosh the lumps together etc etc.

STEP 1: Take two lumps of putty. They are two distinct lumps. Smoosh them together. Now you have one lump of putty.

So far do good I hope. Agreed?

STEP 2: Now we want to do some philosophy, get a bit more rigorous, and work out how are words have meaning.

So we’ve got the words putty, lump and smoosh. The word smoosh is kinda cute, and indicates the act of smooshing. The word putty can be defined as the stuff kids play with, with a particular kind of malleability, a particular molecular structure and so on. (Ok kids just like the feel of the stuff and don’t worry about the technical gubbinge but its there if you want to dig a bit deeper).

So the act of smooshing ADDS the putty together. You have no more putty after the smooshing than you had prior. Putty is accumulative and follows the general rule “1 + 1 = 2”.

However we have lost two distinct lumps of putty. The distinction however relies on the use of the word lump and not putty. And what is a lump? Well it is a category word. (Sort of word wide open for sorties paradox - but that is another subject.) And how do we come by this category. Well there’s some stuff over there, and some stuff over there…we’ll call those lumps. The word, its meaning only runs as far as the things you can point to that we have decided to call lumps.

Now the next point is where is all smooshes togther - you can’t add lumps together (viz., 1 + 1 = 1) because they have no sense or meaning beyond stuff that is in the world e.g. putty. The category lump has no reality in itself, thus the addition 1 + 1 = 1 has no reality. What this attempt at an addition demonstrates - and only demonstrates - is that categories are not universals. They have no existence/ontology beyond the stuff that exists.

You quote Popper. For brevity I’ll not repeat it in full.

Even where the terms are defined, we never try to derive any information from the definition, or to base any argument upon it.

So what is going on with the “1 + 1 = 1” argument? You are taking words like hole or putty and then trying to show how these concepts fail reality. But you are taken these words too seriously by trying to show who they fail to bear a particular kind of information. The point I keep try to make - when we are relying on a theory of reference - is that reality is the arbiter, and lays down the limits, of meaning - not our definitions. If reality adds together “1 + 1 = 2” then so must our words - and any concept that lead us otherwise is nonsense.

This is why our terms make so little trouble. We do not overburden them. We try to attach to them as little weight as possible. We do not take their ‘meaning’ too seriously.

Are you reading him Zoro. Because Wittgenstein attends to words you seem to mistakenly think this is just word games. A semantics which takes words too seriously. The whole point to Wittgenstein's later philosophy was to attend to how we use words and to show why we should not take them too seriously. It was an attempt to break us away from old near unconcious philosophical habits. Habits like……and the next point is key.

There could hardly be a greater contrast than that between this view of the part played by definitions, and Aristotle’s view.

For Aristotle’s essentialist definitions [i.e., in which a word is burdened with “capturing the essence” of some thing or process] are [imagined to be] the principles from which all our knowledge is derived; they thus [are imagined to] contain all our knowledge;

You addition of “1 + 1 =1” shows perfectly why Aristotle’s essentialist definitions are no good. If essentialism were true then when you add the essence of one lump to another lump you should have the answer 2, because the essence of being a lump is being distinct. But the view I’m offering you is the polar opposite of essentialism.

2. You ask for the area of that hole. I left it as "an exercise for the reader" to see that the hole in your semantic argument could be substantially enlarged by showing a Wittgenstein disciple word usage -- but of course, it would still be one hole, as in 1 + 1 = 1.

Arrggh! That addition relies on a twist in logic. It relies on taken an essentialist view of a category, so when you add categories 1 + 1 you would hope to get the answer 2. But you’ve provided real examples where this does not happen. So the right conclusion should be…..Aristotle’s essentialism is false. So what gives our words and categories meaning…..reality. Nothing else. So if we are talking about lumps of putty or holes, the meaning of these words is determined by reality. A reality that makes our words true or false. Thus true/false logic - if it is to be fully consistent -requires the dumping of essentialist thinking. Your “1 + 1 = 1” argument is an argument for true/false logic - not against.

[Furrowed Brow]

Oops! Couple of typos in that last post. Hope it still makes sense.

[zoro]

To Confused: With good feelings, I started to attempt to correct your mistaken ideas about systems and entropy that one of your posts revealed. Such mistakes didn't concern me much (because I found that even in the qualifying exam of some of my Ph.D. candidates in the physical sciences, they still had trouble understanding entropy), but I was somewhat concerned that someone who claims some knowledge in "social evolution and genetics" doesn't seem to appreciate the difference between open and closed systems; after all, two of the most important open systems (from an anthropological perspective) are "open" societies and living systems. After reading your later posts, however, I changed my plans: although your incompetence in dealing with entropy and systems didn't bother me, you're competence as a hypocrite "turned me off". Consequently, I'll respond only to your question: No: not zorro; zoro, which I chose as an abbreviation for A. Zoroaster (the pseudonymous author of the book at http://zenofzero.net/) and which, from among the many possibilities, I choose to mean "seed of the stars" -- since evidence suggests that's what we are.

To Others: Should you be wondering if there's a resolution to the other matter that was addressed, I'd say that it's left for you to decide. I continue to maintain that 1 + 1 = 2 only if the conditions of (simple) logic aren't violated [viz., that the things considered must 1) exist and 2) be distinct], but the Wittgenstein-disciple ("Furrowed Brow") apparently continues to maintain otherwise, apparently deeply committed to finding what he will consider to be acceptable definitions of words such as 'piece' and 'lump' and 'hole'.

For example, I continue to maintain that

1 (piece of silly putty) + 1 (piece of silly putty) = 2 (pieces of silly putty)

provided all pieces remain distinct, while

1 (piece of silly putty) + 1 (piece of silly putty) = 1 (piece of silly putty),

if the pieces (or 'lumps' or 'clumps' or whatever word is desired) are "smooshed" together, i.e., if the condition of 'distinctiveness' is violated. Even in the latter case, however, of course the mass (and similar) of putty is conserved (e.g., 1 gram of putty + 1 gram of putty always = 2 grams of putty -- if none of the mass is turned to energy!), but again, the number of pieces needn't be conserved. Similarly, the number of molecules in most chemical reaction isn't conserved, e.g., each case of 2H2 + O2 = 2H2O reduces the number of molecules from three to two.

In addition, moving on from the condition of 'distinctiveness' (a concept different from the concept of 'existence'), I continue to maintain that if you divide a piece of wood in half, you'll get two pieces of wood (or even divide an earthworm in half to get two earthworms, if what I've read is correct), but if you divide an elephant in half, you won't get two elephants -- because the premiss of existence has been violated. Similarly, if you electrolytically divide a gram of water in half, don't expect to find two half-grams of water. Thus, I continue to maintain that the basic meaning of any (dimensionless) number is 'existence', and if existence is violated, then a fundamental premiss of (simple) logic (and therefore of algebra) is violated, namely, existence (which, again, is different from 'distinctiveness' -- and perhaps a good case could be made for the idea that 'existence' is more fundamental than 'distinctiveness').

But in any event, it's your logic; you decide.

[Furrowed Brow]

To Others: Should you be wondering if there's a resolution to the other matter that was addressed, I'd say that it's left for you to decide. I continue to maintain that 1 + 1 = 2 only if the conditions of (simple) logic aren't violated [viz., that the things considered must 1) exist and 2) be distinct], but the Wittgenstein-disciple ("Furrowed Brow") apparently continues to maintain otherwise, apparently deeply committed to finding what he will consider to be acceptable definitions of words such as 'piece' and 'lump' and 'hole'.

For example, I continue to maintain that

1 (piece of silly putty) + 1 (piece of silly putty) = 2 (pieces of silly putty)

I maintain this.

1 (piece of silly putty) + 1 (piece of silly putty) = 1 (piece of silly putty),

I say there is no addition going on. (Look at this um again. Keep looking at it and tell me when you see addition going on).

if the pieces (or 'lumps' or 'clumps' or whatever word is desired) are "smooshed" together, i.e., if the condition of 'distinctiveness' is violated.

I say Aristotle’s essentialism or any other ontological theory of distinctiveness is violated. So any such theory was never any good in the first place.

Even in the latter case, however, of course the mass (and similar) of putty is conserved (e.g., 1 gram of putty + 1 gram of putty always = 2 grams of putty -- if none of the mass is turned to energy!), but again, the number of pieces needn't be conserved. Similarly, the number of molecules in most chemical reaction isn't conserved, e.g., each case of 2H2 + O2 = 2H2O reduces the number of molecules from three to two.

In addition, moving on from the condition of 'distinctiveness' (a concept different from the concept of 'existence'), I continue to maintain that if you divide a piece of wood in half, you'll get two pieces of wood (or even divide an earthworm in half to get two earthworms, if what I've read is correct), but if you divide an elephant in half, you won't get two elephants -- because the premises of existence has been violated.

No. The concept elephant and wood work in different ways. The elephant is dead, and elephants stop working when in two pieces. That sad result goes with being an elephant, whilst the same criteria (functioning as a living creature) does not apply to wood, and oddly enough earthworms.

Lets chop this up further. The living organism was a tree. Chop a tree in half and you get two halves of the tree. Freeze an elephant solid and saw it up into planks, and now saw one of those planks in half. You had one, but now you’ve got two elephant planks.

Zoro, I think, believes this is just word games. And in a way he is right. But the lesson to take from the games is that language is mercurial, and play tricks on us if we let it. My point, and opposition, is based in Zoro sleepwalking through the language jungle.

Similarly, if you electrolytically divide a gram of water in half, don't expect to find two half-grams of water.

Ok you get H2 and O2 gasses given off. More complicated rules apply. Rules that apply to the concepts of water, H2 and O2, as defined by physics and chemistry. For most of us we don’t need to know or worry about H20 when thirsty and we reach for a glass of water. But the chemical description of H20 and the physical rules that go with it, are more rigorous description of reality. If we said water is an essence, or an element, or was a manifestation of a platonic ideal, or was a pure abstraction, and then we find we can electrolytically divide it into two separate gases we would have disproved……..all of the aforementioned metaphysical/abstract assumptions.

So lets be clear about what’s be going on here over the last few posts. Zoro has been making a point about the failure of true/false logic because of the failure of “distinctiveness”. All the critiquing of his examples I have given work to one end - to show that the notion of “distinctiveness” he is relying on, relies on some kind of metaphysical presumption about how words and concepts and categories find their form. He’ll deny this - but I’m saying that the implication of his position the likes that implication or not.

Thus, I continue to maintain that the basic meaning of any (dimensionless) number is 'existence',

This is why I am saying you are sleep walking. The notion of existence as you are deploying it, relies on a notion of distinctiveness that does not belong to reality, but to metaphysics.

But as I have said before - true/false logic relies on a theory of reference. That is - how do our propositions refer to objects in the world. The first part of that answer is to accept that the objects give our propositions their meaning. Think like an empiricist, and stay an empiricist whilst following through on that point. Our best physics says the world is made of atoms. Ok run with that thought. Putty, elephants, wood, H20 are all made from atoms. They all behave as atoms en mass behave in their various forms. The distinctiveness of our propositions about the world, separate and divide and merge with the same distinctiveness as atoms; because our words can mean no more than how those atom come together.

Here we specially talking about how our propositions refer to things - of course language can be use in other ways, but words put into propositions that refer can only follow and be limited by reality. If you have followed that train of thought, and stay onside with empiricism there is no problem with true/false logic in the way Zoro is trying to depict. Our propositions either do or do not model reality. If I say there are two lumps of putty, whilst those lumps have been smooshed togther, my proposition is false. Reality being the final arbiter of the truth or falsity of what I say.

But in any event, it's your logic; you decide.

Grrrrr. Zoro, you stopped addressing me directly and seem to be making an appeal to other readers of this thread. To characterise the form of the debate we have had. You have made a point, I have critiqued it, so you put up another alternative and I critiqued that and so on. It seems to me you’re hoping that you’ll throw me a bone I’ll choke on. This games coming to a close now I guess, as I don’t seem to look at things the way you are looking at them, so you’re kind of giving up on me. But hang on a second….what has not been going here. I’ve offered criticisms you have hardly responded to - if at all. You actually have not addressed anything I’ve said by analysing what I’ve said. Where you have asked me to address a point I have done that. We’ve done holes, putty, elephants and now water. I have explicitly asked you twice to address my point about meaning, but you have systematically avoided doing that. You seem to think that simply by labelling my position Wittgensteinian that is sufficient to dismiss it.

Your appeal reveals the conceit that the debate we have had has amply demonstrated the strength of your position and the weakness of mine. I think you need to read again.I’ve said before - you have a blind spot. This is evidenced by your systematic evasion, and resistance to engage with the points I have made by way of analysis. So once again. I’ve played with holes, elephants, putty and water. Why will you not try my list exercise and tell me what is wrong with it.

[Confused]

zoro:

To Confused: With good feelings, I started to attempt to correct your mistaken ideas about systems and entropy that one of your posts revealed. Such mistakes didn't concern me much (because I found that even in the qualifying exam of some of my Ph.D. candidates in the physical sciences, they still had trouble understanding entropy), but I was somewhat concerned that someone who claims some knowledge in "social evolution and genetics" doesn't seem to appreciate the difference between open and closed systems; after all, two of the most important open systems (from an anthropological perspective) are "open" societies and living systems. After reading your later posts, however, I changed my plans: although your incompetence in dealing with entropy and systems didn't bother me, you're competence as a hypocrite "turned me off". Consequently, I'll respond only to your question: No: not zorro; zoro, which I chose as an abbreviation for A. Zoroaster (the pseudonymous author of the book at http://zenofzero.net/) and which, from among the many possibilities, I choose to mean "seed of the stars" -- since evidence suggests that's what we are.

If indeed I am incorrect about the previous zorro then I humbly apologize. Your arguments tend to follow the same pattern of his, in terms of the comments made to furrowed brow about his inability to "dig out of his hole". However, it is a battle best left to him as he is more than competent to do so.

Now, in regards to me being a hypocrite, I am unsure where in any of my posts I alluded to that. Perhaps you might point it out.

In regards to entropy, once again, I state physics isn't my speciality and I don't pretend for it to be so, however, what I do understand of it has never been quite regarded as incompetent. Perhaps, you being of such a higher level of education could in fact point my errors out. The point of this site is to learn IMHO. If you care not to correct me, then I am destined to continue in the same erroneous pattern, something I prefer not to do. I welcome your criticism, but ask that with it, you do the courtesy of pointing out where I am wrong. As you yourself have admitted, entropy in closed vs open systems is a very difficult concept that changes within the parameters. As stated before, entropy has a natural inclination to increase, but with the addition of energy in can decrease within a closed system such as the universe or even a bedroom, it can also decrease with the addition of gravity. Exceptions always exist. Nothing represents entropy better than the black hole-the most extreme form of gravitational clumping and squeezing of the universe. Nothing has more disorder than such. Now in the universe, uniformly distributed gas has high entropy but when gravity is added with its universally attractive force, it will cause fragments of gas to form droplets, ie less entropy. Here, with gravity, clumpiness, not uniformity, is the norm- the state in which gas tends to evolve which allows structures such as stars and planets to form.

Application of entropy in genetics as to trace back their origin: In a closed system such as the universe, the addition of energy allows for entropy to decrease so the increase in order could allow for the first random assembly of micromolecules to form without violating the 2nd law of thermodynamics. This is only theoretical, as we cannot discern the exact origin of life, it does show how life could have began without the insertion of a creator while still adhering to the rules of physics. Collins uses ARE's as primary examples of entropy at work at the DNA level. ARE's are essentially junk genetic codes that if they have a purpose, we don't know what it is. However, we see that as evolution occurred, we can see the amount of ARE's increasing and we can predict with the evolution of each species exactly where these ARE's will occur and when seen in that species, how it evolved into another species. Mammals have the highest rate of ARE's (entropy). Tracing back the ancestral tree we can see where these ARE's changed to point to the common ancestry of species such as the wolf evolving into the tame form of dog we see today. This is all I meant to point out to here.
Application to social evolution: In a closed, isolated physical system, such as a tribe that is isolated from the general population both physically and reproductively, we see rapid genetic changes (higher entropy) even though there are less variables available to cause disorder when compared with the larger population. We see this because of Punctuated Equilibrium. This is all I meant to point out here.

Once again, it is not my contention to be an expert in physics. My applications are actually quite limited and I in no way intended to offend you. As stated earlier, the previous zorro (with a second r) made many fallacies in his arguments and was extremely offensive to most. He would simply say we were to ignorant to comprehend him and not enlighten us on where we were wrong. If you would care to enlighten me, I would most welcome it.

[zoro]

Furrowed Brow:

In response to your "Grrrrr", which I'm sorry to see, I'll it put it this way. Normally I agree with (and, of course, admire) Robbie Burns, but for me, he went too far with: “Ah, would some gift the Giftie gie us to see ourselves as ithers see us." That could be hell! Wiser, I think, is what seems to be a Native American expression advising one to "walk a mile in another's moccasins." If you would consider and act on that advice, then I hope (and maybe even expect) that your anger would vanish.

In particular, the mile that I suggest you consider walking would start with your looking at the Preface and the first chapter of my (once again!) free book at (once again) http://zenofzero.net. Then maybe at least glance at some additional chapters. From such glances, surely you'll see that my goal (on which I've been working so hard for so long -- 12 hours per day for 12 months per year for 12 years!) is to try to help youngsters break free from their indoctrination in the god idea. Surely you'll see, also, that I consider it to be an extremely important goal -- for the future of each youngster (especially Muslims) and, thereby, for humanity.

In addition, from what you see is still to be posted of the book, try to gauge the scope of the work still in front of me. Next, consider what you would do to try to "promote" the book, with no publisher and with no money to mount an advertising campaign -- and not wanting to engage either, because you want youngsters to be able to access the book on the internet at no cost (beyond time invested). Then, what would you think of the idea (as I mentioned in my first post) of "wandering" around the internet to try to find where you might find some people, especially youngsters, who might be wondering about the reasonableness of the god idea? Factor in, too, the likelihood that I have little time left to live.

Now, suppose someone wanted to engage you in arguing about the width of a cobweb attached to a twig on a tree in some huge forest. In addition and as an example, suppose you thought that, far more important to finding a path through the forest (to help the intellectual development of youngsters) is to determine whether the entropy of the universe is exactly zero (just as is the total electrical charge, total linear and angular momentum, and total energy, as is outlined in the first chapter of my book). You would know, of course, that such an idea about entropy "flies in the face of established physics". But suppose you recalled the "hand-shake" idea (to explain otherwise inexplicable results in quantum mechanics) and contemplated the possible resolution (to non-zero entropy) that, while time goes in the opposite direction in the chalk-full negative energy portion of the universe that we call "space" or "the vacuum", space's negative entropy (an otherwise foreign concept) might become increasingly negative, thereby holding the total entropy at the same value it had before the Big Bang, i.e., zero. Just think how such a demonstration would strengthen the otherwise almost solid idea that this universe was created by some symmetry-breaking fluctuation in a total void, separating "zero" into positive and negative "components" -- thereby strengthening the ability of youngsters to say: "How silly! The probability that any god ever existed is so small that it's laughingly ridiculous!"

In summary and stated differently, it seems clear that your goal is to engage in a lively and productive debate. Great! Go for it. Good luck to you: I hope you find willing and able debaters. In contrast, "Confused" stated that her goal was to learn -- a worthy and admirable goal. Meanwhile, my prime goal is to help kids exterminate the god meme that society has infected them with and is devouring the minds of so many our children.

But, "to each his own". And now, if you please, I need my moccasins back.

[Bugmaster]

All you've proven is that it's impossible to use logic to prove that logic is false. This does not mean that logic is necessarily true; nor does it prove that logic is necessarily universal.

What I've shown is that Logic is literally undeniable regardless of the linguistic group, geographic location, or anything else involved. What you do when you deny it is effectively saw off the very branch you're sitting on...

I think we agree on this...

Using Reason to prove there is no Reason is simply a nonsensical and meaningless endeavor and it doesn't cast any doubt whatsoever on the truth - more than that the necessary truth - of the first principles of reason.

...but not on this. Again, just as you can't use logic to prove that logic is false, you can't use logic to prove that logic is a universal necessity. The best you can do is prove that Reason is required in order to make Reasonable conclusions, but that's a tautology that doesn't buy you anything.

Note, again, that I'm not arguing that logic is false. I'm merely arguing that it is not necessarily a universal constant (or, in fact, an independent property of the universe at all). If you want to prove that logic is necessarily true, you need to use something else other than logic (whatever that may be), otherwise you have a circular argument.

That's a false analogy, Logic is the study of proper reasoning itself, nothing is more foundational or self-evident. Thus, the very truth value of the thesis of this thread - that Logic is merely a projection of human minds - is necessarily determined by the very rules of it purports to deny.

I'm not sure what you mean by "projection", but is it not true that these threads are being interpreted by human minds ?

[Bugmaster]

The whole argument about adding holes or silly putty chunks, etc., is starting to resemble (at least, to me), ye olde argument about angels dancing on the head of a pin. I think it illustrates quite handily what happens when you assume that numbers in particular, and math in general, are some sort of an independently existing entity. You run into problems like these.

[Bugmaster]

The 'unable to comprehend' not being contingent on finding out some new knowledge that will improves our comprehension.

I don't know if this is true or not; it all depends on what you mean by "axiom". To use an extreme example, some people take "god exists" to be an axiom, while many others do not. People used to take it as an axiom that the natural state of all objects is rest, but Newton disproved that. I think our comprehension can, and does, change -- albeit slowly.

At the extremes something can be 100% true, or 100% false. Thus true/false are the limiting conditions of fuzziness. Rather than being contradicted, or found to be irrelevant, the rules of logic that apply to true/false propositions are subsumed into a larger set of rules as the limiting conditions. They just can’t be shaken off - which is what we would expect if they are universal.

I'm not sure I understand what this means. Fuzzy logic doesn't deal with binary truths or falsehoods; it deals with values between 0 and 1, AFAIK.

In fact, our own brains do not think logically at the lowest level...

You’ll need to expand on that point. I’m not sure what you have in mind.

Well, if I were to look at your brain, I wouldn't see logic there. I wouldn't see any propositions, conclusions, modus ponens, or binary true/false values. Instead, I'd see a bunch of neurons connected in parallel, affecting each other's firing rate. That's not logic at all.

However, if they are using the rules of the predicate calculus or fuzzy rules, we could quickly learn how they think - because these rules have a generality/universality.

Well, either that, or because these rules are something we have managed to come up with, as well. What if we meet some culture that uses rules that we have never come up with at all ? Can you demonstrate why such a culture cannot possibly exist ?

To really surprise us they’d have to be using a form of logic we have never thought of before, and could never understand. If that were the case then we would probably not recognise that they were thinking beings; though their existence would prove that logic was not universal.

Yeah, as I said in one of my earlier posts, this is a problem for us. We can only recognize beings as "thinking" if they think in ways similar to ours. This makes sense to me, but I think that it would be rash to conclude that our way of thought is the only universally possible way of thought.