Is Logic Empirical?
I tried to read the paper by Hillary Putnam, but there were too many difficult equations, so I'm hoping that someone here can make his case in more ordinary language.
But it seems to me that logic cannot be empirical. If a thought defies a logical rule, then it is meaningless. The sentence "the triangle has four angles" is not a meaningful sentence because the predicate negates the subject. If the triangle has four angles, then it does not have three angles, so it is not a triangle, in which case our proposition does not refer to a triangle, in which case it has no subject and refers to nothing. No empirical observations could confirm a contradiction because contradictions are meaningless. They cannot even be conceptualized, much less observed. How could you observe something of which you cannot form a concept?
Now apparently quantum mechanics gives us reason to alter basic logical concepts. I am happy to admit that we must be careful about how we apply logical inferences to quantum phenomena, but I am doubtful that quantum phenomena defy foundational logical laws such as non-contradiction, identity and excluded middle. And if an interpretation of quantum mechanics does lead to violations of logic e.g. the cat is both alive and not-alive, then I think the problem lies in the interpretation of empirical observation. The conflict is not between observation and logic but between a false or unrefined interpretation and logic.
But it seems to me that logic cannot be empirical. If a thought defies a logical rule, then it is meaningless. The sentence "the triangle has four angles" is not a meaningful sentence because the predicate negates the subject. If the triangle has four angles, then it does not have three angles, so it is not a triangle, in which case our proposition does not refer to a triangle, in which case it has no subject and refers to nothing. No empirical observations could confirm a contradiction because contradictions are meaningless. They cannot even be conceptualized, much less observed. How could you observe something of which you cannot form a concept?
Now apparently quantum mechanics gives us reason to alter basic logical concepts. I am happy to admit that we must be careful about how we apply logical inferences to quantum phenomena, but I am doubtful that quantum phenomena defy foundational logical laws such as non-contradiction, identity and excluded middle. And if an interpretation of quantum mechanics does lead to violations of logic e.g. the cat is both alive and not-alive, then I think the problem lies in the interpretation of empirical observation. The conflict is not between observation and logic but between a false or unrefined interpretation and logic.
Comments (49)
As with most analytical philosophy, it's just a waste of life. Don't be hard on yourself with Putman, analytical philosers have an arrogant style, they just can't help themselves, it makes them feel superior.
I think you have asked an excellent question here.
Here's a link to Putnam's paper, republished as "The logic of quantum mechanics".
Putnam's argument is that the principle of distributivity fails for quantum mechanics. That is, he claims that there are instances where (A and (B or C)) is true, yet ((A and B) or (A and C)) is false.
Putnam gives an example of the double-slit experiment on pp180-181. On his view, the photon goes through (slit A1 or slit A2) and hits region R, yet it is not the case that the photon (goes through slit A1 and hits region R) or (goes through slit A2 and hits region R).
If his view is correct, then that is an example where classical logic fails for empirical reasons.
I don't see how that's possible. "The photon goes through A1 and hits R, or the photon goes through A2 and hits R" is just a less efficient way of saying "the photon goes through A1 or A2 and hits R". Just like "you will eat eggs for breakfast and chicken for lunch, or you will eat pancakes for breakfast and chicken for lunch" is a less efficient way of saying "you will eat eggs or pancakes for breakfast and chicken for lunch". I can't conceive how an observation which violates principle of distributivity could be possible. What would that even look like? Putnam is an accomplished and respected philosopher, so I'm sure there's something I'm missing. I'll try to read the paper again at some point and hope I have better luck with it. But if you think you understand what he's saying, please try to explain.
Addendum: after glancing at the equations on pp180-181, I see that he's using probability. I don't have a good grasp of probabilistic logic, but my understanding is that it adds a number of layers of complexity and uncertainty to classical logic. I may be wrong, but I think that the principles of probabilistic logic are quite contentious among logicians. Perhaps quantum mechanics only violates ordinary principles of probabilistic logic, not classical logic.
That would be an interpretation. As SMBC puts it, 'Sweetie, superposition doesn't mean "and", but it also doesn't mean "or"'.
It's a complex linear combination of going through both slits.
Putnam is saying that the photon going through (slit A1 or slit A2) and hitting region R describes an interference experiment. That is, you don't know which slit the photon went through but, on conventional realist assumptions, it went through one slit or the other. However the photon need not hit region R if you do measure which slit it went through. Now we know this already since this is just what QM predicts. But Putnam's claim is that those two experimental observations are the left-hand-side and right-hand-side of the principle of distributivity, and so violate it.
Quoting Dusty of Sky
Putnam is rejecting classical logic - see statement (10) on p190, where he states the principle of distributivity and says that it fails in quantum logic. Also:
Quoting Quantum Logic and Probability Theory - SEP
Quoting Dusty of Sky
Most quantum physicists accept and use classical logic. But QM can be understood as a generalization of probability theory that, in addition to positive numbers, also allows negative and complex numbers (i.e., probability amplitudes).
Or perhaps in the sense that the photon goes through neither A1 nor A2 because there is never any particle photon at any instant. A wave version follows all paths but can only be realized as a photon particle hit in one of the patterns seen on the detector?
I don't think this necessarily contradicts the principle of distributivity. It seems that measuring which slit the photon goes through affects the conditions of the experiment. So if you don't measure, then both ((A1 or A2) and R) and ((A1 and R) or (A2 and R)) are true. If you do measure, then both are at least potentially false because not-R can be true. Am I still missing something?
Also, here's a derivation of one side of the principle of distributivity. The principle follows from more basic logical principles, so if you reject distributivity, you must also reject at least one of the other principles used in this derivation.
1. Show (A and (B or C)) implies ((A and B) or (A and C))
2. A and (B or C) - assumed conditional derivation
3. Show (A and B) or (A and C)
4. not-((A and B) or (A and C)) - assumed indirect derivation
5. not-(A and B) and not-(A and C) - 4 De Morgan's Law
6. not-A or not-B - 5 simplification and De Morgan's Law
7. not-not-A - 2 simplification and double negation
8. not-B - 6 7 disjunctive syllogism
9. B or C - 2 right side simplification
10. C - 8 9 disjunctive syllogism
11. not-not-C - 10 double negation
12. not-A or not-C - 5 simplification and De Morgan's Law
12. not-A - 11 12 disjunctive syllogism
13. 7 and 12 contradict one another, so the indirect derivation is complete
14. Line 3 is proven, so the conditional derivation is complete
The contention is that the distributive rule is dysfunctional in quantum mechanical descriptions. Putnam rejects the notion that disregarding the distributive rule would amount to an arbitrary change of meaning. I think he is right here; it seems that classical logic cannot be re-interpreted in such a way that it can encompass quantum mechanics.
It's a pretty good argument against conventionalism, understood as the notion that logical rules are a tradition or habit.
However I will maintain that there is a sense in which we choose, amongst possible logical systems, that which best fits what it is we are trying to explain.
SO what might very loosely be described as the empirical component of logic is no more than choosing a logic that fits our purpose, when our purpose is describing how things are.
Although we may have to modify the way we apply logic depending on what purpose we are employing it for, it seems to me that the most basic principles of logic, as well as the principles which can be derived from them, such as distributivity, must always remain the same. Isn't a proposition or inference which violates a basic principle (e.g. identity, non-contradiction, excluded middle) nonsensical and impossible for us to conceptualize. What purpose could a logic which deals in incomprehensible nonsense possibly serve?
If you don't measure, you don't know whether those propositions are true. So it would be an interpretation.
Alternative interpretations are (A1 and A2 and R) and ((neither A1 nor A2) and R) as suggested by @Pfhorrest and @magritte earlier.
Instead of trying different combinations of classical conjunctions and disjunctions, Putnam instead reinterprets them based on a non-distributive lattice. Here's an example of how it works:
The rule for disjunction is that while there is no common node, go up the lattice. The rule for conjunction is that while there is no common node, go down the lattice.
So for the photon going through (slit A1 or slit A2) and hitting region R, we have:
Whereas for the photon (going through slit 1 and hitting region R) or (going through slit 2 and hitting region R), we have:
For a nice explanation of this, see Alex Wilce's talk, A Gentle Introduction to Quantum Logic. At 34:20, Wilce connects this to von Neumann and Birkhoff's quantum logic - if subspaces are ordered by set inclusion, we have a non-distributive lattice.
Quoting Dusty of Sky
See 37:00 where Wilce mentions non-unique complements which affects negation. Also see Disjunction in quantum logic.
I don't think that this proves that the principle of distributivity fails. It may be useful to not apply distributivity when dealing with quantum phenomena, but that doesn't mean that the principle is false. It is inconceivable for the principle to be actually false. If R is true and A1 or A2 is true, then either R and A1 is true or R and A2 is true. That's a simple tautology. Just because we can discover more in quantum mechanics by not applying a principle does not necessarily mean that the principle is false. And if we have reason to believe that the principle is necessarily and universally true, as I think we do in the case of distributivity, then its usefulness in quantum mechanics should make no difference. Even if we treat it as false in quantum mechanics, I don't think we must interpret this as invalidating the principle's universality. Perhaps it is true that either (A1 and R) or (A2 and R), but since we can verify neither disjunct, we treat it as false, not because it is false in reality because our measurements fail to demonstrate it. (Or, if our measurements in fact demonstrate the contrary, that the photon passed through neither, then we would have to interpret the act of measurement as affecting the photon).
You don't have to use quantum logic in quantum mechanics either; classical logic works perfectly well there - it's just one way of thinking about QM that some find useful or entertaining. Which goes to show that "laws of thought" - including the principle of distributivity - don't have to be as rigid and universal as people often assume. We can adopt different logics for different uses.
There might be certain contexts (although I can't think of any) in which it might be useful to assign certain contradictions as true. But I still think there is a law of though which makes contradictions inconceivable. It seems to me that a violation of the principle of distributivity is likewise inconceivable. If I turned on my blinker and turned either left or right, then does it not necessarily follow that I turned on my blinker and turned left or turned on my blinker and turned right? Can you conceive of the former as true and latter as false? Would that not violate the laws of thought?
Yes.
Quoting Dusty of Sky
They can be verified as false. In this case, we observe the photon pass through either A1 or A2 (since we place detectors there), but we never observe it hit R (i.e., we observe it hit some other region). Which is to say, no interference occurs in this case.
Quoting Dusty of Sky
OK, so it's interesting to consider Putnam's argument here. He notes that you could says exactly the same thing about Euclidean geometry. It might be considered necessarily and universally true, but it nonetheless fails to describe the world we live in. Whereas if you drop the parallel postulate you get non-Euclidean geometry which does describe the world we live in. That relegates Euclidean geometry to a special case of non-Euclidean geometry that just so happens to approximate what we observe in everyday experience.
Similarly, if you drop distributivity you get quantum logic. Classical logic is a special case within that more general logic that approximates what we observe in everyday experience. That is, the Boolean lattice that characterizes classical logic emerges as a special case within a more general non-distributive lattice. So, for example, if you fired bullets at a suitably robust double-slit apparatus, they would clump behind each slit whether or not you measured which slit they went through. Interference is rarely observed at a macroscopic level, so classical logic closely approximates what we normally observe there. As Putnam puts it:
Quoting Putnam: The logic of quantum mechanics, p184
In a sense, we can ask what the world would look like if distributivity was generally false, but was approximated in everyday macroscopic experience. Well, it would look just like our world.
Quoting Dusty of Sky
If one of the disjuncts were true in reality, it would be a hidden variable. But that would require a non-local interpretation, per Bell's Theorem.
But otherwise, yes, it comes down to the measurement problem.
I just refreshed my memory on the double slit experiment, and it seems practically certain that setting up a device to determine which slit the particle goes through will affect where it lands. So if you place a detector on A1 or A2, then it is certain that the particle will not hit R. If you turn off the detector, then the particle will hit R, but you won't know whether it went through A1 or A2. The particle behaves like a wave if you don't track its motion and like a particle if you do. So since the detector clearly affects the conditions of the experiment, then I think it makes sense to assert that the principle of distributivity holds when the detector is off. The statement "(A1 and R) or (A2 and R)" is true, we just don't know which disjunct is true and which disjunct is false. When we turn the detector on, both disjuncts are false because tracking the particle changes the way it moves. It seems arbitrary to me that we should make the realist assumption that (A1 or A2) is true, even though this assumptions can't be empirically verified, but not also assume that the principle of distributivity holds just because we can't empirically verify either (A1 and R) or (A2 and R). In fact, I think rejecting the realist assumption would be preferable. I can conceive of a particle teleporting straight onto R without passing through either slit. This would only violates the nomological law of continuous motion. But I can't conceive of the principle of distributivity as being false because it is a logical tautology. Like I said before, distributivity follows necessarily from more primitive logical axioms, so the principle's universality can't be denied without also undermining the universality of some of the most basic laws of thought.
Quoting Andrew M
I think I see how this could be problematic. Suppose you did a double slit experiment with two entangled particles separated by a significant distance. Then turning on the detector for one of them would communicate an effect to the motion of the other which would travel faster than the speed of light. Is that what you have mind? So is Putnam's argument that we ought to sacrifice the universality of classical logic in order to preserve realism and locality? If so, I don't think that's advisable. I don't think there's anything wrong with using quantum logic as tool, but I think we should still maintain that the principles of classical logic are always true. Maybe in the future, there will be something to explain how localism and realism are compatible with classical logic. Maybe future experiments will deliver definitive evidence against localism and or realism. I know that some very strange paradoxes would seem to follow if localism were false, but this seems like nothing in comparison with the confusion that would follow if we relativize logic, which is the foundation of all rational thought.
Quoting Andrew M
I don't think Euclidian geometry is necessarily universally true in the same way that classical logic is. A principle like "there is exactly one straight line passing through any two points" is always true with regard to our perception of space. We can't imagine a non-Euclidian realm in which the principle does not hold. But just because non-Euclidian space is unimaginable does not mean it's inconceivable. Non-Euclidian space violates the principles of sensory perception but not the principles of rational thought. My claim is that a logic in which the principle of distributivity is false does violate the laws of thought such that any claim made in such a logic, regardless of its usefulness, amounts to nonsense if we actually try to conceive of its meaning. It is impossible to think the proposition "((A1 or A2) and R) and not ((A1 and R) or (A2 and R))". You can write it out in symbols and claim that it is true, but you don't actually have a concept of what you are affirming any more than you have a concept of a married bachelor. The only reason that we are even capable of working out the principles of non-Euclidian geometries and non-classical logics is that we are capable of using classical logic. The principles of classical logic underlie everything rational we think and do in science, philosophy and daily life.
Is there some specific axiom of classical logic that you think we can afford to relativize? Distributivity is a theorem, not an axiom, so to reject the theorem would require rejecting an axiom. I know some people point to the law of excluded middle (which allows for double negation) as possibly dubious, but I don't think so.
Quoting Dusty of Sky
But this is if you look at quantum logic as making an absolute metaphysical statement about quantum mechanics, rather than simply treating the logic instrumentally, or as usefully capturing some aspect of the phenomenon without pretending to the ultimate truth.
Quoting Dusty of Sky
Logic without distributivity is not as problematic as you think. You may find this recent article interesting: Non-distributive logics: from semantics to meaning.
From a creationist perspective:
[quote=Britannica]God on the sixth day [the last day] of Creation created all the living creatures and, “in his own image,” man both “male and female.” [/quote]
Here there's some uncertainty. Did God create us, our brains/minds, to be compatible with the universe or did God create the universe to be compatible with our brains/minds? Did God create the universe for us or were we created for the universe? If the former then logic isn't empirical and if the latter it is.
I read the paper's introduction, and two pieces jumped out at me:
1) "For instance, a natural question is whether relational semantics of (some) non-distributive logics can provide an intuitive explanation of why, or under which circumstances, the failure of distributivity is a reasonable and desirable feature; i.e. whether a given relational semantics supports one or more intuitive interpretations under which the failure of distributivity is an essential part of what ‘correct reasoning patterns’ are in certain specific contexts. Perhaps even more interestingly, whether relational semantics can be used to unambiguously identify those contexts. Such an intuitive explanation also requires a different interpretation of the connectives ? and ? which coherently fits with the interpretation of the other logical connectives, and which coherently extends to the meaning of axioms in various signatures."
If conjunction and disjunction (? and ?) are interpreted differently than in classical logic, then it does not seem so surprising that the principle of distributivity might fail. But this does not entail that the principle does not hold universally. The principle does hold universally (it seems to me) so long as we interpret the conjunction and and disjunction symbols (and whatever other symbols might also be relevant) to mean what they mean in classical logic. If we change their meanings, then it makes (classically) logical sense that we'd get a different set of theorems.
2)
"the graph-based semantics supports a view of LE-logics as hyper-constructivist logics, i.e. logics in which the principle of excluded middle fails at the meta-linguistic level (in the sense that, at states in graph-based models, formulas can be satis?ed, refuted or neither)"
So it seems like the law of excluded middle might also be rejected in these logics. There's nothing strictly illogical about operating within a system where true and false aren't the only possible values a statement can have. It might sometimes be useful make 'unknown' a third option, and maybe other logics can incorporate probability such that a statement can have any value between 0 and 1. But I still think it is true that, in reality, every meaningful proposition must be either true or false. The proposition either corresponds to reality (e.g. the Eiffel Tower is in France) or it does not correspond to reality (e.g. the Eiffel Tower is not in France). There is no third option.
But I admit that much of what I read in the introduction went over my head. So perhaps I'm misinterpreting the quotes I pulled out.
Quoting SophistiCat
I agree that if we look at logic as an instrument, then it does not matter whether its axioms and theorems correspond to objective truths about reality. But I think that we should look at the principles of classical logic as being objectively true.
Quoting TheMadFool
To be clear, I think that, phenomenologically, logic is at least somewhat empirical. We come to understand logical truths, like the principle of non-contradiction, by abstracting from various experiences, and all experience involves sensory phenomena, whether it's directly received from the senses or reproduced in the imagination. And we communicate logic using words and symbols, which we process via our senses. But we should distinguish the cause of our coming to understand logical principles from the cause of the certainty of logical principles. Once we understand logical principles, it becomes apparent that their certainty does not depend on the specific experiences whereby we came to understand them. I may have realized that non-contradiction is true by considering that a triangle cannot be a square, but I could have just as easily realized it by considering that a bachelor cannot be married or by considering any other contradiction. If the principle were empirical, then I'd have to admit the possibility that non-contradiction may only apply to my specific set of experiences. But once I admit this possibility, the entire structure of thought is undermined. If contradictions are possible, then perhaps experiential evidence against contradictions (e.g. I've never observed a square triangle) might really be evidence for contradictions. I don't have empirical evidence to tell me how I ought to interpret empirical evidence, so if contradictions are possible, perhaps I should interpret my experiences to signify the opposite of what they seem to signify. We arrive at our knowledge of logic through a combination of sensory experience and intuition. But once we grasp logic adequately, it becomes a necessary and indispensable foundation for our understanding of the world. So when I say that logic is not empirical, what I mean is that we should not seek to adjust the fundamental principles of logic in order to accommodate empirical evidence. We need logic to make sense of empirical evidence, so, as long as we are committed to the search for truth, we must regard logical axioms and theorems as a priori and self-evident.
The only reason for you to think that our capacity for logic is a product of evolution or intelligent design is that these explanations are (or seem) reasonable. But if the implication of these explanations is that logical principles are mere tools for interpreting the world or contingent structures of thought that might have been created otherwise, then the foundation of reasonable thought is undermined. In that case, you would have no rational justification for believing in either evolution or intelligent design. I think the best explanation for why we use logic is that logical axioms correspond to basic features of all existence: things are what they are, they are not what they are not, and for any given property, a thing either has or does not have that property.
If you identify logic with classical logic, or something with a close family resemblance, then yes. But formal logic in general is less specific than classical logic, even though it still has to do with reasoning, with inference. Which is to say that the patterns of reasoning that are available to us go beyond those that are covered by classical logic. With this general sense of logic, it is indeed possible to have a logic in which conjunction and disjunction mean something different than what they mean in classical logic, but play broadly similar roles. Perhaps the example on pp. 12-13 will help to illustrate the point, though admittedly, taken in isolation it may not look very convincing.
Quoting Dusty of Sky
Yeah, I am out of my depth here as well. Perhaps one of our resident mathematicians will come along and enlighten us :)
:ok:
Quoting Dusty of Sky
First thing to notice here is we're worried about logic and reality not in agreement. The one thing that gives us sleepless nights is the law of non-contradiction, the possibility of it being violated, and for good reason I suppose. If contradictions are ever allowed, then the specter of ex falso quodlibet threatens to cause havoc:
[quote=Google]Ex falso quodlibet is Latin for “from falsehood, anything”. It is also called the principle of explosion. In logic it refers to the principle that when a contradiction can be derived in a system, then any proposition follows. In type theory it is the elimination rule of the empty type.[/quote]
However, there are systems of logic that allow contradictions, and such systems are designed in such a way as to forestall ex falso quodlibet. Honestly, sometimes I feel all this is just a game. Anyway, the point is there are logics that can handle contradictions pretty well or so some tell me.
However, what I can't wrap my head around is how contradictions can be thought of as possible. Suppose there's this blank space: (.......). I now write in this blank space a proposition, say about god: (God exists). Then I contradict myself and say god doesn't exist: ([s]God exists[/s]). In effect, I'm back to square one, the blank space: (.......). After all, ([s]God exists[/s]) = (.......). There's no net effect on the blank space i.e. a contradiction is not a proposition at all ( :chin: ). How then can it be true? Unless of course, in the case of my example, God doesn't exist is not [s]God exists[/s]. As you can see, we need to look at "not" or negation differently.
Yes, although the problem can be demonstrated with just the standard double-slit setup. Just place a single detector on one of the slits, say A1. That is sufficient to make the interference pattern disappear, even for just those photons that go through A2 that one might think should not be physically disturbed from their path.
Quoting Dusty of Sky
We can observe the surface of a sphere which is a non-Euclidean surface. Consider a geodesic such as the Earth's equator. A geodesic in non-Euclidean geometry generalizes the notion of a straight line from Euclidean geometry to instead be the shortest path between two points on a surface. In the special case where a surface is flat, the geodesic is a straight line.
Similarly, consider a flipped coin where the result is heads or tails but we don't know which. The coin is definitely either heads or tails, which is a classical disjunction. Now consider a quantum coin that is in a linear superposition of heads and tails. Quantum logic generalizes classical logic to include superpositions. In the special case where the state of the coin has collapsed to a definite state, quantum logic just reduces to classical logic.
Quoting Dusty of Sky
I'm not sure if Putnam was aware of Bell's theorem at that time. But his position can be seen as rejecting realism in Bell's sense, i.e., as rejecting counterfactual-definiteness. To give a tangible/visualizable sense of what quantum logic is, I'll outline a geometric proof-of-concept in a separate post.
The quantum superposition state (psi) in the double-slit experiment (where A1 and A2 represent the photon going through each respective slit) is
where the probability of the photon being measured at either slit is the square of 1/sqrt(2), i.e. 1/2.
This can be geometrically represented by the following state space where the potentially definite states A1 and A2 are orthogonal axes of unit length, and psi is a 45° diagonal line of unit length (imagine the plus signs form a solid line to the origin).[*] Psi is essentially a North-East arrow which has a North component (A2) and an East component (A1).
We can see here that the superposition state psi is not the same as either of the axis states A1 or A2. So, absent a measurement at the slits, the question of which slit the photon goes through has no definite answer (i.e., it's counterfactually indefinite). On a measurement at the slit, psi is projected (collapses) onto one of those axes, at which point the question of which slit the photon goes through becomes definite.
Now we can ask what is the smallest subspace that contains both axis lines. It is the 2D plane. The 2D plane is the span of the union of the two axis lines. However the plane contains any line that lies on it, and we can see that psi is another line on the plane.
So, in quantum logic, that span of the union is what is meant by disjunction. Intersection is what is meant by conjunction (in this case, the lines intersect only at the origin). The complement of a line is what is meant by negation (in this case, each line has multiple complements - every other line on the plane - which makes the logic non-distributive.)
Now, absent a measurement at the slits, if we ask whether the photon goes through slit A1 or A2, we can see that psi is contained in the span of the union of A1 and A2, so the answer is yes. That is, the disjunction itself is definite even though the question of which disjunct is true is indefinite.
There is a special case where psi may already be on one of the axes when it is measured. This is the case if a second detector is placed at the slits. If it is measured by the first detector as having gone through slit A1, then it will be measured by the second detector as having gone through slit A1 as well. The fact that no interference pattern is observed on the back screen just is a consequence of that. It's a consistent history, so to speak.
Which brings us to classical observations. Why, when an experiment is performed firing bullets instead of photons, and we don't measure which slits the bullets go through, is no interference pattern observed?
The reason, in effect, is that the measurement has already been done by the environment (termed decoherence). And the rest of what is observed follows consistently from the initial measurement. Thus what we are doing when we ordinarily observe something is to measure a value where psi has already been projected onto one of the axes. The worst case is that we don't know which axis it was initially projected onto, which is just a case of ordinary probability (e.g., whether a hidden coin is heads-up or tails-up). So, even absent our measurement, the disjuncts of which slit the bullet went through are definite. And that special case, which is always and everywhere observed except in quantum interference experiments, is the empirical and intuitive basis for classical disjunction.
So based on this state space geometry, quantum logic is the general case and classical logic is the special case (where states are definite and have unique complements).
See here for a helpful definition of quantum logic and the axioms.
--
[*] As a fun exercise for those who don't have degrees in this sort of thing, see if you can figure out where the 1/sqrt(2) comes from in the geometry.
It seems that disjunction in quantum logic has a different meaning than in classical logic. In classical logic, A or B means either A is true or B is true. In quantum logic, A or B means either A is true or B is true or it is indefinite whether A or B is true. You pointed out that this indefiniteness is not merely epistemic (at least according to the Copenhagen interpretation). It might be epistemically indefinite i.e. uncertain, whether a coin landed on heads or tails, but we know that it actually did land on one or the other side. But in the case of the photon, it is metaphysically indeterminate whether it went through slit A1 or A2. Disjunction in quantum logic can express this state of metaphysical indeterminacy.
But I don't think that metaphysical indeterminacy proves that there are exceptions to the laws of classical logic. We can discuss indeterminacy using a classical disjunction. For instance, suppose that process P has two possible outcomes, A or B. P is either indeterminate or determinate. This is a classical disjunction. If P is determinate, then either A obtains or B obtains. If P is indeterminate, then it is undetermined whether A obtains or B obtains.
If P is indeterminate, then the proposition "A or not A" does not make sense, for the same reason that the proposition "the present king of France is bald or not bald" does not make sense. There is no present king of France, so it's neither quite correct to say he is bald nor that he is not bald. Likewise, there is no determinate outcome of P, so it is neither quite correct to say A obtains nor not A obtains. Neither example proves that the law of excluded middle has exceptions. All existing subjects either have or lack a given predicate, but if the subject does not exist, then it does not make sense to assert that the subject lacks the predicate. It does not make sense to assert that the present king of France lacks baldness because this implies that he has hair, which he does not because he doesn't exist. Likewise, it does not make sense to assert that the outcome of P is not A, because this implies that P has a determinate outcome. For a more concrete example of indeterminacy, take the statement "Bob will leave his house tomorrow". Assuming that Bob has free will and the future does not yet exist, it is undetermined whether he will leave his house tomorrow. So it is neither quite true to say that he will leave his house nor that he won't leave his house because both statements falsely imply that his future is already determined.
Just because classical disjunctions don't express indeterminacy doesn't mean that indeterminacy defies the laws of classical logic. We can still reason about indeterminate states of affairs using classical logic. For instance, we can conclude that, if A obtains, then P is not indeterminate.
So I don't think that we should think of quantum logic as a deeper form of logic and classical logic as merely a special case. It may be true that the physical world is fundamentally indeterminate, meaning that determinate processes such as coin flips are a special case in relation to the indeterminate subatomic processes which underlie them. And it does seem to be true that quantum logic is often more useful than classical logic when it comes to describing quantum phenomena. But this is only because quantum logic is specifically designed to express indeterminacy, not because classical logic is violated by indeterminacy.
Quoting TheMadFool
It seems that we are free to invent logics of many different kinds. Some may be useful and others may be invented as games. But it seems to me that principles such as identity, non-contradiction and excluded middle must hold within any logic whose theorems are conceivable to us.
It appears that what we have to keep a close watch on the law of noncontradiction. It's the cornerstone of [first order] logic. If contradictions are allowed then identity and excluded middle are no longer safe in a manner of speaking:
1. X = X and ~(X = X)
2. p v ~p and ~(p v ~p)
Yes, that's right. But note Putnam's analogy with non-Euclidean geometry. Geodesic has a different meaning to straight line. However a geodesic on a flat surface is a straight line. Similarly, quantum disjunction has a different meaning to classical disjunction. However a quantum disjunction of measured quantities is a classical disjunction.
Quoting Dusty of Sky
Agreed. Along the same lines, consider Aristotle's future sea battle scenario:
Quoting Aristotle, On Interpretation, §9
Aristotle's proposal was that the sea battle propositions had the potential to be true or false, but weren't actually true or false until the event occurred. Similarly, "the present King of France is bald" has the potential to be true or false, but isn't actually true or false without a present King of France. (In Peter Strawson's terminology, "the present King of France is wise" represents a presuppositional failure and therefore isn't truth-apt.)
So the same idea can be applied to QM. There is a potential answer to the question of which slit the photon goes through, but no actual (or definite) answer in the absence of a measurement. As physicist Asher Peres put it, "unperformed experiments have no results".
Quoting Dusty of Sky
Yes, one approach here is to say that classical logic applies when things are definite, e.g., when a measurement has been performed, or the subject being predicated exists, or the contingent event has occurred. But it does not apply outside that context. So it's not that classical logic is violated by indeterminacy, it's that the preconditions for its use have not been met. Garbage in, garbage out.
In a similar way, Euclidean geometry is applicable when a surface is flat. We know that the angles of triangles will sum to 180°, the Pythagorean Theorem will hold, and so on. But it is not applicable (or needs to be applied in a different way) to curved surfaces.
If classical logic doesn't apply to indefinite scenarios, then this would seem to be due to a limitation of the applicability of the law of excluded middle. A statement is indefinite if neither it nor its negation is true. For instance, it is neither true nor false that the photon passed through A1.
But the purpose of the present king of France analogy was to expose what I think may be a confusion on the part of some (perhaps not Putnam) who assert that quantum mechanics proves that the laws of classical logic are limited in their application and open to empirically informed revision. The reason the law of excluded middle doesn't apply to the photon's position is the same as the reason it doesn't apply to the present king of France. Neither the photon's (definite) position nor the present king of France exist, and the absence or presence of a property can only be meaningfully predicated of something which exists. No empirically motivated insight is required to grasp this principle. As you brought up, it was grasped at least to some extent by Aristotle. The empirically motivated insights of quantum theories have certainly led us to revise our understanding of physics. Whereas it was previously believed that all spatial entities had definite positions, it is now widely held that subatomic particles exist in superpositions. This notion, although well supported, is highly counterintuitive. I think the confusion occurs when people make something along the lines of this argument:
(1) QM has led us to revise our intuitive understanding of space.
(2) QM has led us to develop new logical languages (with different connectives).
(3) Therefore, QM has led us to revise our intuitive understanding of logic.
This syllogism is obviously unsophisticated and invalid, so it might be fair to call it a straw man, but I think it illustrates the way people often think about this topic. "If quantum mechanics can defy our intuitions about space, then why not our intuitions about logic," the reasoning goes. I think the conclusion is false because the idea that excluded middle doesn't always apply didn't originate with quantum logic. The fact that quantum logic is non-distributive, as I far as I can tell, follows directly from the fact that excluded middle doesn't apply to quantum superpositions i.e. we can neither say that the photon passes through A1 nor that it does not pass through A1 because it doesn't have a definite position.
Although the law of excluded middle may not be universal in as obvious a sense as the law of non-contradiction, I still think we can truthfully call it universal. Reality consists of things which exist, and as long as excluded middle applies to all things which exist, it applies to all things in reality. Therefore, it is universal. Since no definite position of the photon exists, the law of excluded middle does not apply to it. But the photon itself exists, so the law applies to the photon. For instance, it is either true or false that the photon has a definite position.
I'm not sure if we have a substantial disagreement at this point. I agree that the law of excluded middle cannot be carelessly applied to all propositions, and you've shown that quantum disjunctions can describe superposition in a way which classical disjunctions can't. But I don't see this as evidence that the law of excluded middle is contingent or that logic is open to empirical revision (I'm not sure if you ever intended to make that argument, but my original purpose in this post was to argue against it). We develop new logical languages to more effectively describe new facets of reality which we discover. But the fundamental logical laws of reality, or at leasts the laws of our capacity to think about reality, remain the same regardless of what we observe. One point I will concede is that quantum disjunctions are no less valid than classical disjunctions. They are equally applicable to reality, or at least to the parts of reality for which they are intended.
To understand this, we must distinguish what Henry Veatch calls "intentional logic" from modern logic, which is quite different. The kind of logic I am discussing is the art and science of correct thinking, not a set of rules for symbolic manipulation.
The rules of modern logic cannot be applied without thinking an Aristotelian syllogism:
Every case with these characteristics is a case in which this rule applies.
This case (the one I am thinking about) has these characteristics.
Therefore, this is a case in which this rule applies.
This is simply Aristotle syllogism in Barbara, and is what we must think to apply any scientific knowledge we have. Hence, despite the vigorous protestations of modern logicians, they have not done away with Aristotle's logic, but rely on it whenever they apply the rules of manipulation they have developed.
So, we need to consider how it is that we think when we think correctly. Robert Boole, the founder of Boolean logic, entitled his masterwork The Laws of Thought, but it you reflect, there is no law preventing us from thinking "square circle," or "triangles have four sides." It is only if we want our thought to apply to reality, to what is, that we should not think these kinds of thoughts.
So, let me suggest that we abstract from our experience an understanding of what it means to be -- an understanding of the nature of existence. And, implicit in this a posteriori understanding are laws of being that must be reflected in our thought, if our thought is to apply to what is. For example, we come to understand that nothing can both be and not be at one and the same time in one and the same way, and, from this we derive the logical rule that we cannot both assert and deny the same thing at the same time. So we come to grasp the logical principles of Identity, Contradiction and Excluded Middle from the corresponding ontological principles -- and we abstract those from our experience of being.
In counting different kinds of things (pennies, apples, Legos), abstraction allows us to see that counting, and so the relationship between numbers, does not depend on what we count. In the same way, we can see from our experience with different kinds of being, that some relations (the ontological principles above) do not depend on what kind of being we are dealing with, but are true of being per se.
Agreed.
Quoting Dfpolis
There's no law preventing us from thinking the words square circle, but we can't form a concept corresponding to these words. We can form some coherent concepts which don't seem to apply to reality e.g. wizards and unicorns. So I think it's correct to call the laws of logic laws of thought, since they constrain not only what is possible in reality but what is possible for us to conceive in our own minds.
Quoting Dfpolis
I think that you are correct if we understand a posteriori to mean learned through experience and a priori to mean known without having been learned. A child is not born with a developed understanding of the laws of logic. We must hone our intellects in order to comprehend the truth of a proposition like "a proposition cannot be both true and false". However, if a posteriori means contingent upon experience and a priori means knowable as true or false regardless of particular experiences, then I think you are incorrect. For as long as we possess an intellect capable of abstracting and experiences capable of being abstracted from, we should be able to deduce the same laws of logic. Even if I had no experiences to abstract from but the consciousness of my own existence, I should be able to deduce that I exist, therefore I don't not exist, and since not not existing is the same as existing, my only options are to exist or not exist. Moreover, I should see that these principles must be the same for all existing things due to the nature of existence. Given the weakness of the human mind and its dependence on sensation, a human solely conscious of his own existence would probably not being to perform this sort of deduction. But in principle, I think it is possible for a being capable of ideation and understanding to perform this deduction regardless of his particular experiences. As you say, the laws of logic are implicit in the nature of being, so even if the human intellect, due to its weakness, depends on many particular experiences to become conscious of the laws, the laws are contained implicitly within every possible experience. Perhaps you have reasons for objecting to the word a priori, but I think that if anything is a priori, it is logic.
Oddly, one can say the same about i - the root of negative one. Despite this, we make use of them.
We can't form an image of a square circle, but we can add the modifying concept
Quoting Dusty of Sky
I mean a posteriori with respect to the experiences required to learn, and a priori in is subsequent application, So, I think we agree.
Quoting Dusty of Sky
I think we only learn about ourselves as knowing subjects from reflecting on our acts of knowing. So, I don't think that what you envision is possible.
Quoting Dusty of Sky
I agree if you mean any specific kind of experience, but not if you mean with absolutely no experience of knowing, as then we would have no means of grasping that we can know, This is because we can only understand what is actual/operational, not what is merely potential.
My objection to a priori, is that see no reason to believe that we know anything prior to all experience.
True, but there's reason we say i is imaginary. It's not real in the same way as other numbers are. that being said, I'd love to hear a mathematician's insight into why i is so useful and why it can contribute to the generation of fractal patterns like the Mandelbrot set.
I'm not sure that can be filled out...
How are numbers real? Apart, of course, from real numbers...
Indefiniteness can apply to existence as well. An electron could be in a superposition of an excited state and ground state (having emitted a photon). In this case the number of photons (0 or 1) is in superposition. That is, what exists or not can be in superposition.
So the issue seems to boil down to definiteness and indefiniteness. In everyday experience, an unobserved flipped coin is definitely heads or tails and we don't have any trouble visualizing that scenario. We can describe the situation formally using the law of excluded middle (with observed anomalies, such as the coin landing on its edge, having straightforward explanations).
QM upends that intuitive picture. You can have a quantum coin which, when flipped twice and then observed, will always be found in the same state it started in (e.g., always heads). There is no straight-forward way to visualize that process without giving up counterfactual-definiteness. And so the LEM is no longer applicable in the obvious way.
Further, since a classical coin flip scenario depends on quantum decoherence, that suggests the LEM is contingent. That is, there is no logical requirement for quantum systems to decohere - a quantum system could conceivably remain coherent indefinitely (i.e., isolated and not interacting with other systems). The universe, taken as a whole, may be an example of such a system.
So there are at least two broad options available. One option is that there is a more fundamental logic (say, quantum logic) that applies universally with classical logic emerging as a special (or approximate) case that applies in decohered environments. A second option is that classical logic is universal, with indefiniteness being just a placeholder for what has not yet been satisfactorily explained in definite terms.
It doesn't seem that either option can be definitively ruled out at present. Michael Dickson discusses the first option in his paper:
Quoting Quantum logic is alive ? (it is true ? it is false) - Michael Dickson
i can be conceptualized as a quarter rotation on the complex plane. So, 1 * i * i = -1. As Gauss said:
Couldn't we say that the electron exists but no definite state of the electron exists and no definite number of photons exists? Saying no definite state exists just means that the state is indefinite. I see why it seems problematic that we can neither affirm nor deny that a photon exists. Could we perhaps resolve this problem by thinking of the photon's indefinite existence as a property of the electron. Existence is only existence as such when it is definite. If something exists indefinitely, it only exists as a property of something which exists definitely. For instance, the position of a particle in superposition exists indefinitely as a property of the particle, which exists definitely.
Quoting Andrew M
I'm not sure I understand this argument. Are you saying that giving up counterfactual definiteness also forces us to give up LEM? I've argued that this isn't the case because you can't meaningfully predicate something of a non-existent subject. LEM even applies to indefinite states of affairs because all states of affairs are either definite or indefinite. It just doesn't apply to particular determinations of indefinite states of affairs because no such determinations exist.
If the unobserved universe exists in an indefinite state at both the micro and the macro level, and the only parts of the universe that are definite are the parts we observe, I see how this could be problematic. If indefinite existence is a property of definite existence, and the unobserved universe exists indefinitely, then the unobserved universe is a property of the observed universe. This would imply that the existence of the universe is dependent upon our observation of it. I think it would be better to avoid this conclusion if possible, but it's still preferable to allowing things which don't definitely exist to have properties. You said, "what exists or not can be in a superposition". This strikes me as not only counterintuitive but inconceivable. If something does not exist, then it is nothing, and it can have no properties. Therefore, a thing can only have properties insofar as it exists. If it is indefinite whether a thing exists or not, then it is indefinite whether it has properties.
Quoting Andrew M
I won't speak to the second option, but if the first option entails that things are fundamentally indefinite, then I think we should reject it. It seems to me that, although we can allow some things to be indefinite, our description of the world must bottom out in something definite. For instance, the particle can only have an indefinite position if it definitely exists. If the particle exists indefinitely, then I argue that we should think of its indefinite existence not as an independent entity but as a property of something definite. For only things which exist can have properties, so if a thing doesn't definitely exist, it doesn't definitely have any properties, so nothing we can say about it is definitely true.
Quoting Quantum logic is alive ? (it is true ? it is false) - Michael Dickson
If the disjunction symbol means what it means in classical logic, the distributive principle is correct. If it means what it means in quantum logic, it is incorrect. The disjunction symbol can mean whatever we want it to mean, so I don't think either application is fundamentally right or wrong.
The superposition can extend to the electron's existence as well. Consider Schrodinger's Cat where there can conceivably be lengthy alternative histories in superposition (and exhibiting interference). This also plays out in the Wigner's Friend thought experiment which describes a scenario where a system's state is definite for one observer (the friend) but indefinite for another observer (Wigner, for whom the friend is in superposition).
Quoting Dusty of Sky
I just meant that we can't assume that if the (unobserved) state of the quantum coin is not heads then it must be tails. A superposition of heads and tails is also a possible state. (And, in this single quantum coin flip example, is the state.)
Quoting Dusty of Sky
OK. But be careful not to read "what exists or not can be in a superposition" as more than a mathematical description (e.g., a line somewhere on a plane that represents the probability of measuring the definite alternatives). It's just that quantum mechanics allows such a state, however that be conceived (which is supplied by the various quantum interpretations).
Quoting Dusty of Sky
Dickson would agree that you can define logical connectives either way. But he argues that only (non-distributive) quantum logic has empirical significance, and thus relates to correct reasoning, since it derives from quantum theory. He discusses this further under "The Motivation" on p3.
This is the first I've heard of Wigner's friend, but I just read that the purpose of the thought experiment is to support the theory that consciousness causes collapse i.e. that everything is in a superposition until a conscious being observes it. But maybe I misunderstood what I read. Theoretically, it is possible that nothing exists definitely until it is observed. In that case, consciousness is necessary for existence. I'll call this theory quantum idealism. If something could be definite for one observer but indefinite for another, then perhaps a many worlds interpretation of quantum idealism would follow. Each consciousness exists in its own world. Where two conscious beings observe the same thing, their worlds converge and where they observe contrary things (e.g. I observe the cat is alive and you observe it's dead) their worlds diverge. Since Wigner's friend observes something definitely which remains indefinite for Wigner, the consciousness of Wigner's friend has diverged into two separate worlds, each with its own version of Wigner's friend. Until Wigner contacts his friend, it is undetermined which of the two divergent worlds Wigner's world will converge with.
I don't necessarily endorse either a many worlds or a single world version of quantum idealism, but I think this is one possible way to account for lengthy alternative histories existing in superposition without contradicting LEM. As long as reality bottoms out in definite facts, such as being x observes y, LEM remains in tact. Whatever is indefinite exists only relation to what is definite. The cat is only indefinitely alive or dead in relation to the definite fact that it is in the box.
Quoting Andrew M
All the math in this stuff goes over my head, especially given the limited amount of time I have to peruse it. It seems to me like the main difference between quantum and classical logic is that "a or b" in quantum logic means "a or b or it is indefinite whether a or b". So is the reason he sees quantum logic as more empirically significant that physical states of affairs can be indefinite? I see how this would make quantum disjunctions more useful to apply in certain contexts, but I don't think it makes the classical disjunction false. And if the classical disjunction is not inherently false, then neither is the principle of distributivity. And if I am right that reality bottoms out in something definite, then the classical disjunction applies to what is most fundamental.
While Wigner held that "consciousness causes collapse", the thought experiment is independent of that interpretation. The key issue is that there can be two apparently incompatible reports - one by the friend who reports having measured a definite value (i.e., collapse has occurred) and another by Wigner who reports an indefinite value (i.e., collapse has not occurred). Those reports could be generated by unattended measuring devices, perhaps time stamped and read by someone at a later time.
That aside, your interpretation is a possible one, something like a many-minds interpretation.
Quoting Dusty of Sky
OK. But note that per the Wigner's Friend thought experiment above, that box can itself be in an indefinite state with respect to some further observer or measurement device. In this sense, such definite facts assume a measurement or decohered context.
Quoting Dusty of Sky
Yes. More precisely: per quantum theory, the world can be represented as a non-distributive lattice. So a logic that seamlessly captures that structure will also be non-distributive.
There are two separate claims there. The first is about the geometry of the world (analogous to whether spacetime is fundamentally curved or flat). The second is about the propositional logic that naturally applies to that geometry.
Quoting Dusty of Sky
That is true for quantum logic as well. Any claim about the universe as a whole is definite since it's a claim about the whole geometrical space. Any claim in a measurement or decohered context is also definite (at least relative to that context) since, on measurement, an indefinite state collapses to a definite state.
The difference between a classical disjunction and a quantum disjunction only emerges in those cases where two orthogonal subspaces in a higher-level space (such as two orthogonal lines on a plane) don't exhaust all the possibilities. Which is to say, those situations characterized by indefinite states such as the dual-slit experiment or the quantum coin.
If it does work, then the physical world can be divided into observed (consciously or mechanically) parts, which are definite, and unobserved parts, which are indefinite. If it doesn't work, then perhaps we must adopt a many-worlds or many-minds interpretation. Either way, definite parts of the universe (whether it's only one observer's universe or the universe for all observers) exist definitely and indefinite parts exist indefinitely. I think we must posit that the indefinite parts are grounded in the definite parts. And I think we must do so regardless of what portion of the physical universe is indefinite. Here's an argument:
Firstly, it seems impossible for the definite parts to be grounded in the indefinite parts. For if what exists most fundamentally exists indefinitely, then it does not definitely exist. But we exist definitely, and our existence can't be grounded in things which don't definitely exist. Our existence might be grounded in things with some indefinite properties, but these grounding things must at least definitely exist and possess some properties definitely.
Secondly, it seems necessary that the indefinite is grounded in the definite. For if a thing exists indefinitely, it does not definitely exist, hence it does not definitely have any properties, hence nothing definitely true can be said about it. Perhaps we can only make probabilistic claims about the indefinite, but even these claims are definite insofar as the probabilities we predicate of something indefinite are definitely its probabilities. So unless we want to treat indefinite things like Kantian noumena about which nothing can be meaningfully asserted, we must posit that what exists indefinitely is grounded in what exists definitely, even if what exists definitely has no other definite properties than set of definite probabilities.
Certainly that was Wigner's conclusion:
Quoting Wigner's friend - Wikipedia
Quoting Dusty of Sky
As Wigner implies above, it doesn't work if you want to retain standard physics. Changing the physics leads to theories such as objective collapse and de Broglie-Bohm (which each introduce non-locality to make the standard predictions).
Note that Wigner still observes interference after the friend's measurement. So if the friend's result "becomes definite for all potential observers", then it is a hidden variable. That amounts to a rejection of locality, per Bell's Theorem.
Quoting Dusty of Sky
The way I would put it is that what is indefinite has the potential to be definite (for an observer).
Quoting Dusty of Sky
That seems right to me. Or, in alternative language, that the potential is grounded in the actual.
Note that I haven't argued that what exists depends on what does not exist (or exists indefinitely). I don't think that is true. I've only argued that part of the universe can be indefinite, or lack definite form, for a particular observer. It takes an interaction or measurement to actualize that potential, so to speak.
I like that. It sounds very Aristotelian.
Quoting Andrew M
Then I think we may have reached a satisfactory point of agreement. I still think that certain logical principles must be regarded as necessarily and universally valid, but I've gained a greater appreciation for how QM forces us to modify how we apply and qualify these principles. A universe which contains indefinite portions is significantly different from the universe Aristotle conceived of, so it makes sense that we've had to develop new methods of logic to describe it. I just don't think this amounts to a revolution in logic and an overthrow of the old paradigm (and it doesn't sound like you do either). Methodologically, logic has had to adapt and innovate to keep pace with the natural sciences, but I think the deepest truths of logic have remained the same. The natural sciences have inspired us to develop a more nuanced understanding of the principles of logic and reject certain false conclusions which follow from a lack of nuance (e.g. that superpositions are impossible). But the underlying, conceptual structure of logic does not depend on how we conceive the natural world.
:up:
And thanks for your comments throughout - it's been a good discussion.
Agreed. And thank you as well. This has been educational for me, and I think my point of view on these topics has become more sophisticated as a result.