Believing Impossible Things

[Dragar]

John is my hypothetical friend, and a rather misguided individual. We were discussing geometry and logic the other day, and I mentioned that a square circle - a shape with all the properties of a square and a circle, at the same time and in the same respect - could not exist, as it was self contradictory.

Indeed, even the phrase 'a shape with all the properties of a square and a circle', or the shorthand 'square circle' are nonsense statements. They do not refer to anything at all.

John, however, was still adamant he believed that they not only could exist, but did exist.

Later on, I managed to point out the error of his ways, and he now agrees with me that 'square circle' doesn't really convey any information after all.

My question is...did John really believe in square circles? I don't mean to ask if he believed in something that he labelled 'square circle'.

I mean to ask if he really believed in the existence of a shape with all the properties of a circle and a square?

[seebs]

I think he may well have. People often hold contradictory beliefs, meaning they generally believe in a universe with contradictory properties. Isolating the contradiction doesn't make it less plausible.

[livius drusus]

Please excuse the ignorance which is sure to follow, but didn't people once think light could only be a particle or a wave when in fact it has properties of both? Were those properties not entirely contradictory in the first place -- at least not to the point of mutual exclusivity -- or could an object conceivably exist with the properties of a square under certain conditions and the properties of a circle under others?

[Dragar]

I think he may well have. People often hold contradictory beliefs, meaning they generally believe in a universe with contradictory properties. Isolating the contradiction doesn't make it less plausible.

So you disagree self contradictory statements are meaningless?

If I say 'I know a married bachelor' or 'I know a married man who is not married', what am I actually saying?

Please excuse the ignorance which is sure to follow, but didn't people once think light could only be a particle or a wave when in fact it has properties of both?

It has some of the properties of both. Really, it's neither. It's a quantum thingie.

Were those properties not entirely contradictory in the first place -- at least not to the point of mutual exclusivity -- or could an object conceivably exist with the properties of a square under certain conditions and the properties of a circle under others?

Well, this is why I specified 'in the same respect and at the same time'. Is it sensible to talk about a circle with four sides?

I don't really want to get bogged down in the details of the specific example, so if it helps, change my example to unmarried bachelors, or triangles with four vertices (sides).

[seebs]

Well, consider the statement W, which is a meta-statement of my beliefs about the world. W is certainly contradictory in some way. So. Is it meaningless? I don't think so, exactly.

[Dragar]

I'm not sure, seebs. For instance, consider statement S:

S = "I did and did not go for a walk in the park, at the same time and in the same respect."

What does S actually say?

Could W be meaningless, though individual statements that are 'included' in W are not?

[viscousmemories]

If I say 'I know a married bachelor' or 'I know a married man who is not married', what am I actually saying?
I think that's a naughtology. :P

[seebs]

I'm not sure, seebs. For instance, consider statement S:

S = "I did and did not go for a walk in the park, at the same time and in the same respect."

What does S actually say?

Could W be meaningless, though individual statements that are 'included' in W are not?

You raise a serious and interesting theoretical question. I don't have an answer.

[John Carter]

Formally, what you are asking about is called the Law of the Excluded Middle, Dragar. It's one of the axioms of classical logic, though in some classical systems it's a theorem instead (either way, it's considered one of the fundamental principles of classical logic). However, not all systems of logic include it, and, for example, Constructivist (also called Intuitionist) Logic specifically denies the LEM.

[justaman]

Does the acknowledgement of meaninglessness actually make a commentary on the existence of the belief of the subject?

[Gurdur]

John Carter said it best, and obviously knows the field; thus asking him is your best bet.
However, despite that, I'm going to give you an answer from a linguist/neuro angle (i.e., the philosophers do this better, but I want to make a point from a different field anyway).

When you say "A square triangle is meaningless and/or illogical",
you have in fact made a circular argument.

It starts like so:
we need to categorise the world, just so we don't eat the wrong things, or attempt to mate with the wrong things, or ask by accident a collegue to pass us that stick with Acme Dynamite Company printed in big letters on it, with a fizzing fuse attached, when we really wanted the collegue just to pass us a ciggie.

So we divide up the world into linguistic terms, and along the way, we say if something has 3 corners, it's a triangle, and if it has 4, it's a rectangle.
Thus, if we say "a four-cornered triangle", we are not yet in fact making a nonsense statement about the actual world as yet, we are only contradicting our own terminology.

Thus, when you say "A square triangle is meaningless and/or illogical", all you have actually done is say, "We define anything with 3 corners as a triangle", and then, "If it doesn't have only and exactly three corners, it's not a triangle", and then say, "A four-cornerd triangle is a meaningless concept", and there you see how you have in fact only made a circular argument.

And our terminology never corresponds to the world in a direct 1-to-1 way, and is never completely accurate --- for the very simple computational reason, that for our vocabulary to correspond very exactly to the world, we would need to have a computational grammar and vocabulary that is bigger in computational power terms than the world is big, i.e. our grammar and vocabulary would have to be able to encompass every tiny part of the world and then a tic more.

But then it gets worse: on grounds of sheer practicality, because our brains are not terribly large, because our senses are limited and even worse, our senses have built-in error artefacts (think all types of sensory illusions, think the "blind spot" of vision and confabulation, etc.), then our grammar and vocabulary are not only very limited indeed compared to how big the world is, but more, our grammar and vocabulary will get things occasionally and unexpectedly wrong, because our brains already have inbuilt ways of getting things wrong.

And it just goes quickly downhill from there --- as soon as you allow limited free will and subjective experience, then you must admit subjective experience and choice affect and modify perception as well as cognition in very direct first-order ways as well as indirect processing ways.

And if you try not allowing limited free will and subjective experience, you get really into hot water, because too many facts contradict you.

Our experience of reality is difficult, and reality is a terribly complex thing, even before you get into what are matters of subjective opinion, so very easily it can be possible to believe in impossible things.

For the whole philosophy of this, ask John Carter, Clutch or The Heretic.
Me, I'm just a kibbitzer.

[Sauron]

John is my hypothetical friend, and a rather misguided individual. We were discussing geometry and logic the other day, and I mentioned that a square circle - a shape with all the properties of a square and a circle, at the same time and in the same respect - could not exist, as it was self contradictory.

Indeed, even the phrase 'a shape with all the properties of a square and a circle', or the shorthand 'square circle' are nonsense statements. They do not refer to anything at all.

John, however, was still adamant he believed that they not only could exist, but did exist.

Later on, I managed to point out the error of his ways, and he now agrees with me that 'square circle' doesn't really convey any information after all.

My question is...did John really believe in square circles? I don't mean to ask if he believed in something that he labelled 'square circle'.

I mean to ask if he really believed in the existence of a shape with all the properties of a circle and a square?


Sounds like your friend has an inherent fear of boundaries or limits. He likes to believe that all boundaries can be crossed, with enough energy/imagination/faith.

I leave it as an exercise to the reader why someone would fear or distrust boundaries. May have something to do with a fear of death, or a fear of being boxed in to a certain kind of life.

[Dragar]

When you say "A square triangle is meaningless and/or illogical",
you have in fact made a circular argument.

Of course. Definitions are determined by common usage, after all.

Does the acknowledgement of meaninglessness actually make a commentary on the existence of the belief of the subject?

I think it does. Because if two words placed side by side like 'square circle' actually are not a concept at all, what was my friend thinking of when he told me he believed in 'square circles'?

What are we even saying when we say he believed in 'square circles'? We might say he was very confused, or that he was very mistaken. But whatever he was thinking of, it can't have (by definition) been 'an object with all the properties of a square and a circle' - for such an assemblege of words does not carry meaningful information for humans.

[Clutch Munny]

Dang, shouldn't have glanced at this thread! Interesting topic.

Formally, what you are asking about is called the Law of the Excluded Middle, Dragar. It's one of the axioms of classical logic, though in some classical systems it's a theorem instead (either way, it's considered one of the fundamental principles of classical logic). However, not all systems of logic include it, and, for example, Constructivist (also called Intuitionist) Logic specifically denies the LEM.

No, he's asking about the Law of Non-Contradiction.

LEM (in quantified form) says that for any proposition, either it or its negation must be true. The doxastic issue there, then, is that rationally you have at least to believe one of them. If LEM is a law, that is. (My view: it's actually more of a municipal by-law, holding in some regions but not all.)

LNC says that any proposition and its negation cannot jointly be true; you cannot rationally believe both. Those are the cases that Dragar is exploring.

Out of interest: the sense in which an intuitionist must deny the EM schema does not extend to accepting any instance of ~(Pv~P). It's just that Pv~P isn't a constructive law. But that's consistent with demurring from denying any instance of it -- a distinction of particular importance to constructivists.

I wish I had time to discuss the main thread issue!

[Dragar]

I wish I had time to discuss the main thread issue!

Me too. You sound like you'd be a great help! :)

If you do get a chance, chime in. :)

[Dragar]

You raise a serious and interesting theoretical question. I don't have an answer.

Bother. I was rather hoping you would.

[Gurdur]

Of course. Definitions are determined by common usage, after all.
Are they ? Not always. Sometimes it's individual definition, a very important point.
Moreover, you've passed over the chance to discuss the theory of inherent properties and abilities.
Are our brains born with set definitions ? Set ways of looking at the world ?

.... it can't have (by definition) been 'an object with all the properties of a square and a circle' - for such an assemblege of words does not carry meaningful information for humans.
This falls down as soon as you get to complex subjects; it only works for the nicely simple and very limited subjects.

[Dragar]

Are they ? Not always. Sometimes it's individual definition, a very important point.

If that's the case, then they were not telling me what I thought they were telling me. Perhaps they really did believe a coherent concept, but I misundertood them.

Are our brains born with set definitions ? Set ways of looking at the world ?

I really have no idea. It's quite possible. I'll see how my reading on the evolution and function of human cognition progresses.

This falls down as soon as you get to complex subjects; it only works for the nicely simple and very limited subjects.

I agree - to an extent.

Words are tools for communicating concepts and likely carry more than their set defintions. This is why poetry works so well; words carry meaning all over the place and in different ways, giving rise to various different connotations and emotional reactions.

I can use the words 'square circle' because they likely do convey some sort of meaning - albeit not the meaning of 'an object with all the properties of a circle and a square'.

This is what you pointed out in your post with regards to a lack of '1-to-1' mapping between words and concepts.

And yet, when we specify set definitions of words to communicate the concept, and those set definitions are self contradictory, what is the set concept I am communicating?

[John Carter]

No, he's asking about the Law of Non-Contradiction.

Thanks for the clarification. It's been a while.


Out of interest: the sense in which an intuitionist must deny the EM schema does not extend to accepting any instance of ~(Pv~P). It's just that Pv~P isn't a constructive law. But that's consistent with demurring from denying any instance of it -- a distinction of particular importance to constructivists.


Yes. As I understand it, the objection here is that constructivists insist you have to know the truth value of P before you can make assertions about Pv~P.

[Clutch Munny]

No, he's asking about the Law of Non-Contradiction.

Thanks for the clarification. It's been a while.

Welcome! I'm no logician myself, but I rub elbows with the stuff a fair bit. (It sticks to elbows, turns out.)


Out of interest: the sense in which an intuitionist must deny the EM schema does not extend to accepting any instance of ~(Pv~P). It's just that Pv~P isn't a constructive law. But that's consistent with demurring from denying any instance of it -- a distinction of particular importance to constructivists.


Yes. As I understand it, the objection here is that constructivists insist you have to know the truth value of P before you can make assertions about Pv~P.

Yep. It's also that intuitionistic truth is understood as provability, so negation has a very strong interpretation: ~P constructively means that the assumption P provably entails an absurdity. So to actually deny Pv~P requires a proof that Pv~P leads to an absurdity; and this is a stronger claim than simply that Pv~P isn't a logical law.

[John Carter]

I think one reason I made the mistake of conflating the law of the excluded middle and the law of non-contradiction is that they seem to be equivalent; after all, DeMorgan's Law states: ~(P & ~P) = P v ~P. What am I missing here, Clutch?

[Clutch Munny]

I think one reason I made the mistake of conflating the law of the excluded middle and the law of non-contradiction is that they seem to be equivalent; after all, DeMorgan's Law states: ~(P & ~P) = P v ~P. What am I missing here, Clutch?

The relevant DeMorgan law is:

not-(P and Q) <--> not-P or not-Q

If for Q we take not-P, we get

not(P and not-P) <--> not-P or not-not-P.

And this is indeed classically equivalent to LEM. But Double Negation Elimination is not constructively valid; you can't infer p from not-not-P. In fact, DeMorgan is not constructively valid. I hope later today to say a bit more about this -- at the risk of derailing the thread.

[John Carter]

Yes, I meant equivalent in classical logic.

In the meantime, I look forward to your post. I haven't really looked too closely at constructivism. My knowledge of it is mostly limited to its broad implications for mathematics. And as I'm sure you realize, the vast majority of mathematicians reject it. Much of modern mathematics would have to be thrown out, and what's left becomes so complicated that it loses much of its elegance.

[Clutch Munny]

The clarification I'd wanted to add was this: I over-simplified when I said that DeMorgan is constructively invalid because DNE is constructively invalid. More accurately the step

(*) ~P v ~~P

as it stands is already not a constructively logical truth. This is actually the weakest schema that intuitionism declines. It follows from, but does not imply, Pv~P; so in rejecting (*) one must also reject LEM. And an intermediate schema, sometimes called Dummett's schema, is

(P-->Q) v (Q-->P).

It is implied by, but does not imply, LEM, and it implies, but is not implied by, (*). So the relevant structure of entailments and commitments looks like this:

LEM --> Dummett's schema --> (*)

Intuitionistic logic, by def'n: ~(*)
Hence ~Dummett's schema
Hence ~LEM

[justaman]

And for those of us who speak English? :?

[Clutch Munny]

And for those of us who speak English? :?


Sorry -- I did say it would be a derailment.

Think of it this way. The logic that people learn either when they study intro logic at university or when they read up about it (say, on the internet) is usually Classical Logic. That can mean a few related things, but most often it means logic that includes three basic axioms:

Law of Identity: P if and only if P
Law of Excluded Middle: Either P or not P
Law of Non-Contradiction: Not both P and not P

These, in fact, were once upon a time just called the Laws of Thought.

But wait. You can doubt one or more of these so-called laws -- in least in some contexts -- without being crazy.

Consider the second one, LEM. Picture a colour spectrum, and imagine it blown up so large that just the red-yellow segment takes up one whole wall of a large gymnasium. (It doesn't have to be a gymnasium. It could be an aircraft hanger. I just want to have flexible women stretching on mats in the background while we look at this.) Now, the transition from red to orange to yellow is more or less continuous -- that is, seamless; it doesn't seem that there is any particular point x on the wall at which we can say "Everything to the left of x is red, and everything to the right of x is not red, and x itself is one or the other."

In short, it looks like one could reasonably hold that it's a fact about colour and colour discourse that there are no such sharp boundaries between, e.g., red and non-red. At the very least, it looks like an open question about the metaphysics of colour and the truth-conditions of colour-statements. But now here's LEM -- a principle of logic -- telling us that there must, after all, be some such point x. Because for any arbitrarily chosen point x on the wall, LEM tells us that exactly one of "x is red" and "it is not the case that x is red" is true. Since some of the points are red, and some are non-red, there has to be a transition; but the transition can't be mediated by a section of neither-red-nor-not-red colour; LEM requires that it go directly from red to non-red.

Where? How could we determine this? What could it even mean to point to some spot in this seamless transition (on God's say-so, suppose) and say, "There it is -- the red/non-red boundary!"?

Here lots of people will say that a logical law surely shouldn't be foreclosing on otherwise apparently live explanatory options. In general, LEM is hard to square with any domain in which we are inclined to recognize indeterminate cases. In those cases for which it seems unwarranted to assert either P or not-P, even though all the relevant information seems to be in, LEM dictates that one must nevertheless be true. But this looks like more than a topic-neutral law of logic ought to tell us.

Sometimes, of course, LEM looks like a good bet. But if it isn't universally a logically certain rule of inference, then it can't be a Law of Thought; and it's at least worth considering what logic would look like if you tried to do without it. The logic that does this is called "intuitionistic logic", and John Carter and I have been wonking (or wanking) about some of its properties.

The answer to the question of what logic without LEM would look like, it turns out, is "Surprisingly similar in many respects". Some classical proofs are not (yet) intuitionistically recoverable. Lots of key proofs in classical logic and math, though, are also provable intuitionistically. They're just more work. Sometimes way more work. And since logicians and mathematicians love elegance (ie, brevity) above almost all things, any approach that tells them to ditch elegant proofs in favour of either clunky ones or losing the results altogether has got a hard row to hoe. In fact intuitionistic approaches have been much more influential in computer science; truth-as-provability is analogous to computability in key respects.

[justaman]

Thanks Clutch that was great :thumbup:

I suppose what Dragar is ultimately talking about (and what I'm interested in also, since this is an offshoot of a subject we are discussing) is whether we are able to believe in things which violate the laws of logic.

What I mean is, do we use logic to describe the methodology of the universe, or the methodology of our brains?

For instance there are any number of examples in QM which violate the laws of logic as you've described them. Superposition is my favourite. So while the universe seems clearly able to proceed without paying any attention to our laws of logic, our comprehension cannot.

This probably seems a no-brainer of a distinction, since logic by definition is a description of human thought, but what does it say about our ability to be confident in the objective 'truth' of our beliefs?

If the universe is capable of doing things that we are not able to absolutely assimilate because of the limitation of our laws of logic, then we must be able to believe in objectively 'impossible' things. If we believe certain elements of the universe are 'true' - such as superposition - then we are believing in something which is not in agreement with logic.

[John Carter]

The answer to the question of what logic without LEM would look like, it turns out, is "Surprisingly similar in many respects". Some classical proofs are not (yet) intuitionistically recoverable. Lots of key proofs in classical logic and math, though, are also provable intuitionistically. They're just more work. Sometimes way more work. And since logicians and mathematicians love elegance (ie, brevity) above almost all things, any approach that tells them to ditch elegant proofs in favour of either clunky ones or losing the results altogether has got a hard row to hoe. In fact intuitionistic approaches have been much more influential in computer science; truth-as-provability is analogous to computability in key respects.

Historically, Intuitionism in mathematics was the brain child of a German mathematician named Kronecker. He strenuously objected to Cantor' work on Set Theory and Dedekind's work on Irrational Numbers.

The major difference between Intuitionists and other schools of thought in mathematics is the nature of proof. This is especially evident when it comes to existence theorems. To the intuitionist, a proof of existence must include an explicit contruction of the object we are proving exists. This is why Intuitionism is also commonly called Constructivism. One common method of proving existence is to assume that the object in question does not exist and derive a contradiction, then invoke the Law of Non-Contradiction and declare that existence is proven. Intuitionists reject all such proofs.

This leads me to the other, and probably most important grounds upon which most mathematicians reject Intuitionism. One of the axioms upon which modern mathematics is based is called the Axiom of Choice. This axiom is completely non-Constructive; it asserts existence without giving a method of contructing it. The problem is that fundamental results in a large number of branches of mathematics are directly dependent on the Axiom of Choice. Linear Algebra and Ring Theory, to name just two examples, would be practically emasculated if we reject the AC.

[Clutch Munny]

I suppose what Dragar is ultimately talking about (and what I'm interested in also, since this is an offshoot of a subject we are discussing) is whether we are able to believe in things which violate the laws of logic.

Right. What I'm saying, so far, is that "the laws of logic" is an unhappy way of thinking of the problem, since there's more than one set of laws, more than one logic. In fact there's quite a few, and I mean pretty seriously developed and applicable ones, here.

But suppose we set this aside. We could do so by thinking of a situation in which we generally agree that (say) Non-Contradiction ought to apply, and then asking whether it's possible to hold contradictory beliefs in that context.

Okay then.

First let's consider a familiar distinction, between occurrent and dispositional beliefs. The former are beliefs I'm entertaining at some moment, while the latter are beliefs in a conditional sort of sense. If I'm walking to the fridge expecting to get a beer, it's because I occurrently believe that there's a beer in the fridge; at the same time, there's an intuitively quite different sense in which I believe that no left-handed monkey has ever composed a sonata. The latter sense has to do, not with what I'm thinking right now, but with the way I would think if some situation called for it, given my current dispositions.

It shouldn't be surprising that we have contradictory dispositional beliefs. That is, it might be true of me now that in situation 1 I would say and think P, and that in situation 2 I would say and think not-P; hence we can say that I have a dispositional belief that P and a dispositional belief that not-P. But this does not entail that I have a dispositional belief that P and not-P! For it does not follow that there is any situation in which I would say and think both P and not-P.

So... the upshot is that the interesting cases will be occurrent beliefs. Is it possible to occurrently hold contradictory beliefs?

Yes. But it can be more or less a problem.

First suppose that to have a belief is to have some structured neurological representation, one playing the right functional role to count as a belief. For instance, suppose that there is some neurologically realized "Language Of Thought", and to have a belief is to have a sentence of the LOT written in your neurology. On the face of it, having contradictory beliefs would then be no more puzzling than having a sheet of paper with contradictory sentences written on it. It might make you count as irrational, of course, but there'd be no empirical or metaphysical puzzle in it.

On the other hand, we might ask just how some neurological structures could be interpretable in this way. A line of thought going back to philosophers like Quine and Wittgenstein, but expressed most aptly by Dan Dennett, runs like this: In order to have a belief of the form, "a is F", you have at least to grasp the concepts. That is, somebody might utter the sentence "Elms regenerate", but if we knew that they didn't really understand what those words meant, we would not credit them with the corresponding belief. Nor, obviously, if we thought they were just bullshitting us or being poetic or playful or trying to make some obscure metaphorical point. It's not enough just to say it.

But, look, what better evidence could we have that someone doesn't understand the key concepts of "a is F", or is lying, or fooling around, or doesn't speak English, or is equivocating, than if they immediate go on to say "a is not F"? In other words, when you look at the conditions required for us to ascribe the second belief, they turn out to be the conditions sufficient to retract our ascription of the first belief. Bottom line, for the Quine-Davidson-Dennett view: Ascribing beliefs is a process that requires us to assume at least some minimal idealized rationality towards the agent we're considering. We would, they say, give up the whole idea that the person is A Real Psychological System (at least relative to judgements of the F-ness of a) before we'd say that they hold the contradictory beliefs. (Because, if not, then when the heck should we conclude that someone is confused or insincere?)

This point is most compelling when we think of someone uttering direct and explicit contradictions with all the hallmarks of sincerity. But even on this second view of belief, of course everyone recognizes that the assumption of idealized rationality becomes less tenable the more we consider occurrent beliefs the inconsistent consequences of which are inferentially quite remote -- perhaps even in cases of a single self-contradictory belief, for that matter! Roy Sorenson offers this example: 'The atheism of my mother’s nieceless brother’s only nephew angers God'. This, or some similarly tortured construction, might well attain the assent of someone whose belief of it is self-contradictory. But working out the contradiction is just hard enough work that we could ascribe the belief to someone, on the grounds that he is merely careless rather than a rational singularity (so to speak). The point is clearer still when we consider people in history who, say, believed the Peano-Dedekind axioms while believing that Fermats' Theorem was false. They believed a contradiction, in some now-obvious sense, but there is no hesitation in attributing the two beliefs to them, because inferentially chasing down the contradiction is such unbelievably hard work that there's no practical irrationality whatever -- not even carelessness, really -- in holding them beliefs.

I'd say, then, that any account of belief must respect the differences of degree, not kind, between contradictory beliefs that do and those that do not suggest some failure of rationality. At one extreme, nothing seems more natural and less puzzling than to ascribe beliefs that are inconsistent. At the other extreme we are sometimes forced to (at least momentarily) regard a speaker as less than a functioning agent, meriting no clear belief ascription at all, rather than clearly ascribing two contradictory beliefs.

[Dragar]

For instance there are any number of examples in QM which violate the laws of logic as you've described them. Superposition is my favourite.

Can you provide an example, Justaman? I'm not aware of any contradictions or even violations of LEM in quantum mechanics.

If you're thinking of the Copenhagen interpretation, a superposition of (say) dead cat and live cat states does not mean the cat is both (or neither) dead or alive - it means that there is a chance of observing the cat alive, or observing the cat dead. Copenhagen interpretation deals with observables, not (as John Bell would put it) with beables.



Thanks for the detailed response, Clutch! That was very informative.

[Clutch Munny]

For instance there are any number of examples in QM which violate the laws of logic as you've described them. Superposition is my favourite.

Can you provide an example, Justaman? I'm not aware of any contradictions or even violations of LEM in quantum mechanics.


I don't think justaman meant LEM; he probably meant LNC. As you observe, it's unclear that the idea of superpositions violates LNC, though I suppose it depends on how strongly one interprets the idea.

Lots of people have thought that QM requires modifying classical logic, but not at the level of the three "laws" quoted above. It's a bit more subtle modification that produces Quantum Logic: unlike classical logic, QL does not include the inference rule of the distributivity of conjunction over disjunction. That is, QL does not accept the inference from

P and (Q or R)

to

(P and Q) or (P and R).

Think of P as a position specification and Q and R as momentum specifications, f'r instance.

For what it's worth, LEM holds in QL; for every p, p or not-p. But the apparent indeterminacy of quantum states can be represented by the failure of LEM in the metatheory in which QL is couched. (The jury is out, I should say, on the extent to which QL illuminates QM.)

[Dragar]

I've looked at QL, and have been exploring the issue somewhat. It's one of the more interesting, if a little vague solutions. One of the problems with QL is that it rests on Copenhagen, which has a whole slew of problems which have been illustrated so well by John Bell (check out 'Speakable and Unspeakable in Quantum Mechanics'), not to mention countless others (Schrödinger, Wigner, and so on).

I rather think Everett, Cramer or Bohmian interpretations are better bets.

[John Carter]

For instance there are any number of examples in QM which violate the laws of logic as you've described them. Superposition is my favourite.

Can you provide an example, Justaman? I'm not aware of any contradictions or even violations of LEM in quantum mechanics.


I'd also like to request an example. Is it possible that you are mistaking counter-intuitive results with proposals that violate some principle of logic? There are also many such examples from mathematics. My favorite is the Banach-Tarski Paradox. Try wrapping your head around that! It's more counter-intuitive than anything I've ever heard of about QM, yet in spite of its name it violates no principles of logic (unless you subscribe to intuitionism) and is a perfectly valid theorem.

[Clutch Munny]

For instance there are any number of examples in QM which violate the laws of logic as you've described them. Superposition is my favourite.

Can you provide an example, Justaman? I'm not aware of any contradictions or even violations of LEM in quantum mechanics.


I'd also like to request an example. Is it possible that you are mistaking counter-intuitive results with proposals that violate some principle of logic? There are also many such examples from mathematics. My favorite is the Banach-Tarski Paradox. Try wrapping your head around that! It's more counter-intuitive than anything I've ever heard of about QM, yet in spite of its name it violates no principles of logic (unless you subscribe to intuitionism) and is a perfectly valid theorem.


John, could you say some more about this? I confess to much ignorance about the B-T theorem, but always assumed that it was fundamentally just another manifestation of the general weirdness of an infinite collection -- ie, that it can be (within the same sized infinity, at least) equinumerous with a proper subset of itself. It would not surprise me if this was too simplistic or outright wrong though. (If you don't have time to explain it, of course I understand.)

[John Carter]

First, let me once again emphasize that the Banach-Tarski Paradox is not a paradox in the logical sense; it gets its name from the incredibly counter-intuitive result. The proof relies heavily on the Axiom of Choice, and was originally intended by Banach and Tarski as an argument against accepting the AC.

Informally, the theorem states that given a ball* in 3-d Euclidean Space, it is possible to partition the ball into a finite number of pieces and, using only rotation and translation, reassemble the ball into two balls each of which has the same radius as the original. It has since been proven that this can be done with as few as 5 "pieces" (it's also been proven that it can't be done with fewer than 5 pieces).

Formally, we say two subsets A, B of Euclidean Space are equi-decomposable if we can represent each set as finite unions of disjoint subsets A = Ui=1nAi, B= Ui=1nBi, such that for any i, Ai is congruent to Bi.

The paradox can then be stated as: Any ball is equi-decomposable with two copies of itself.

It gets worse; there is a stronger version, which states that any two bounded subsets of 3-d Euclidean Space with non-empty interiors are equi-decomposable.

What this means is you can take any shape, and decompose it into any other shape, regardless of relative size, in a finite number of steps.

Because the decomposition involves a finite number of pieces and isometric mappings only, the weirdness is not related to infinite collections, but rather points out that measure, and hence volume, is much more complex than one would think. Indeed, the partitions involved here are not measurable, and thus do not have clear boundaries or even a volume in the usual sense.


*A ball is a sphere together with its interior.

[John Carter]

What the BTP illustrates is that if we accept the Axiom of Choice, then it becomes impossible to provide a consistent definition of volume so that it applies to all subsets of Euclidean 3-Space. So we have a few alternatives. The most acceptable of these are to either resign ourselves to the idea that there are sets that are non-measurable, and thus we must prove that a particular set is measurable before we can speak of its volume, or we must reject the AC. There are a couple of others, but no mathematician would ever consider them seriously. Anyway, the AC is, in the end, just so damned useful that most mathematicians choose the first path.

[justaman]

I'll stick with one point for now, but this has all been very informative.

My instinct was that superposition would violate both LEM and LNC, however I'm far from married to this idea if you more learned gents disagree. However my problem with the Copenhagen interpretation - in a philosophoical sense - is precisely because it makes no attempt to comment on a variable set outside of observation. The jump into conceivability comes with the strict proviso that no investigation into the previous objective condition of the wavefunction may be conducted or understood.

[Dragar]

The jump into conceivability comes with the strict proviso that no investigation into the previous objective condition of the wavefunction may be conducted or understood.

The idea behind this is that particles really don't have a well defined position/momentum before a measurement. The position or momentum attained is a function of the experiment.

Oh, and logical positivism. But shhh!

The real problem is that nobody knows what a measurement is.

For instance, for a long time the Earth was devoid of anything resembling life. We would have to imagine it in a superposition of all sorts of quantum states. But when did it collapse into one? When the first life evolved? The first conscious animal? Or did it only collapse when someone with a PhD turned up?

It becomes even more embarrassing when you apply this to the entire universe.

[Clutch Munny]

John, thanks.