That’s fine, I think–state spaces and observables aren’t propositions, so I don’t particularly expect a distributive property to apply to them.
What it means for something to “violate” a logic is a bit up for debate in this thread, so I can’t say for sure what it means. But what I’m claiming (which I’m offering to illustrate and make plausible through the aforementioned “challenge”) is that there is no triad of propositions which fails to satisfy distributivity. A proposition, for our purposes, is basically something that can be asserted–not just “said” but actually “said to be the case.”
I’d have to know more about the correspondence you’re setting up between statements and quantum states before I could say much more about your proposal. (Intuitively, it sounds to me (fwiw) like there might be something to it.)
I can’t argue you out of this way of looking at all assertions, because I don’t think it’s “wrong”. But I don’t think it’s right either.
More seriously, you are free to take whatever perspective you like for how you organize your understanding of assertions, but there are a number of assertions for which I don’t think this is a useful way of looking at things. Here is what I would like to say (this has nothing to do with quantum mechanics specifically; it’s just my view on assertions/propositional logic in general):
A) A proposition (or, if you like, an “assertable”, or whatever word is best suited to this purpose) is something for which there are rules under which one may, in some circumstances, achieve the license to assert it (the rules governing the achievement of this license in some sense comprising the definition of the proposition).
For example, the rule may be that you have license to assert “Ooga-booga!” upon enumerating the digits of π via some fixed algorithm and discovering a hundred 7s in a row. This defines “Ooga-booga!” as a meaningful proposition, by giving its truth-conditions, which is to say, the rules under which one is licensed to assert it.
B) Proposition do not come with complementary propositions by default. Just because there are rules governing the warrant to claim “X” does not mean there are thus created other rules governing the warrant to claim some other statement “NOT X”. “NOT X” is not automatically a meaningful proposition.
Nowhere above have rules been given which tell us circumstances under which we are licensed to claim “NOT ooga-booga!”. One could imagine various propositions which one might, in certain circumstances, consider to fill this role (e.g., you might in some context take “NOT ooga-booga” to be claimable whenever you find a proof in Peano Arithmetic of “There is no sequence of a hundred consecutive 7s in the decimal digits of π”, though in other contexts, this would be markedly less than satisfactory), but there is no general means of taking a proposition, governed by arbitrary rules, and producing from this an “opposite” proposition governed by “opposite” rules.
C) Many, but not all, propositions do happen to be of the form “[These rules determine for this variable one of these two particular values]. I assert that this variable takes on this particular value” (which one is licensed to assert precisely when one has, by following the rules governing the variable, arrived at the variable taking on that particular value). These are Boolean propositions, and do have complementary propositions in the obvious fashion: we can take the negation of such a proposition to be governed by the rules describing when that variable takes on the complementary value. But not all propositions are like this.
Again returning to the Brouwerian example of the hundred consecutive 7s, I would claim that I have defined there a proposition without defining it as a Boolean proposition; I have given no rules for when its negation triggers, and indeed, there is no automatically meaningful notion of its negation. One might metaphysically claim that this proposition does have a negation, just not one whose truth is tied to human-followable* rules in the same way… and you are free to take this perspective, but I think this perspective leads to a mass of needless clutter. I find my perspective more personally useful, and I see no non-question-begging* reason to be swayed away from it.
[*: One might say “NOT ooga-booga!” is true whenever the rules governing “Oooga-booga!” are NOT capable of producing license to assert it, but A) this would be begging the question as to whether there is a general notion of NOT applicable to arbitrary propositions, and B) This would not be a genuine, human-followable rule; I would never be aware of when this “rule” was kicking in and licensing me to claim “NOT ooga-booga!”. And thus, I’d hardly like to call it a rule at all (at least, not in the sense of “rule” which I have in mind in this post)]
Sounds pretty similar to what I said, so far so good…
To me this amounts to saying that “Ooga-booga” means “I have enumerated the digits of pi via the algorithm and have discovered a hundred 7s in a row.” Is that not what you intend, though?
I guess what you’re illustrating is the fact that being unlicensed to claim X is not the same as being licensed to claim not-X, right?
Of course, this depends on what “not” means in the language. In propositional logic, I think (but confess I’m not sure), the dash’s meaning is such that any time we’re not licensed (by the rules of license relevant to PL) to claim X, we are licensed to claim -X. Doesn’t this follow from the completeness and consistency of PL?
What’s the mass of needless clutter you’re talking about?
And is it possible, in your opinion, to give three propositions expressed in English (i.e. the applicable rules of interpretation and license are at least those that apply to some subset of the English language) for which distributivity fails?
That is basically what I intend, sure. (Though, with this particular example, I would like to note that it is furthermore apersonal in the following sense: any one person who has found license to claim “Ooga-booga!” can share that license with others, by simply walking them through the same computation.)
That’s right.
[I mean, I should acknowledge, you are free to take “NOT X” to be governed by the rules “I am licensed to claim NOT X whenever I am aware that I am unaware of any license to claim X” or such things. Then, (assuming you are such a creature as is always precisely one out of {aware of A, aware that you are unaware of A}), you may recover all the Boolean properties for NOT which you like. (At the cost of being able to destroy license-apersonality through negation: My having license to claim “NOT Ooga-booga” on this scheme wouldn’t come with any means to share that license with others; my lack of finding a hundred 7s can’t induce the same lack in others). You are free to do lots of things, as I say. If you want to always think about propositions within a ubiquitous Boolean framework, you are free to do so.
But very often, there is some other concept of negation around governed by different rules which one would prefer to attach the ordinary language word “NOT” to. Which is also fine. The word “NOT” means many different things, just like “+” or “times” or “and” and “or” or any of the rest of it.]
I don’t see how that would follow from the completeness and consistency of (Boolean) propositional logic (for interpretations valued in the 2 element Boolean algebra). Other, non-Boolean propositional logics have completeness and consistency results as well; what would follow analogously for those?
Suppose I tell you I have a propositional variable p in my propositional language, and say nothing more. Does PL entail that p = True (i.e., are you right now licensed by the rules of PL to claim p, simply on the grounds that it is a proposition)? Does PL entail p = False (i.e., are you therefore licensed by the rules of PL to claim -p)?
In this particular example: the supposition of some external boolean variable off in some Platonic realm whose value is fixed regardless of our ability or lack thereof to access it. (That is, the variable whose value is “true” just in case we may find a hundred 7s in the expansion of π, and in any case, one of “true” or “false”. To suppose such a variable has a fixed value, whether or not this value can be determined through simply following the defining rules we gave, seems to me a bit like supposing there is some external fact of the matter as to the value of Sherlock Holmes’ blood type, regardless of whether it can be found in any of the canonical stories written about him)
Potentially, but only because English, like any human language, is quite malleable. There are certainly contexts in which one uses English assertions and the language of “and” and “or” and so on in a Boolean manner; these are quite common, of course. At the same time, I can picture contexts in which one does not… I don’t have a specifically non-distributive one in mind, though, other than the jargon of quantum mechanists of a certain bent. But, I will happily claim, and join you in forcing anyone else to acknowledge, that there is nothing quantum mechanists speak about which you couldn’t speak about in a perfectly Boolean way (speaking within Boolean logic about non-Boolean lattices).
[Just to make it absolutely clear, I have never disagreed that you could look at everything that happens in QM in a manner in which one takes all statements to live in a Boolean algebra, interpreted exactly in accordance with all the rules of Boolean algebras, including distributivity. I just think it might also be useful to use that same language of quantum mechanical statements about particles and positions and “and” and “or” and so on in a different way, which happens to violate distributivity. On the other hand, it might just be more distracting than useful. This is something I do not have the ability to make an informed judgement on, for lack of experience working with quantum mechanics.
At times, as a mathematical logician trying to understand quantum mechanics, I have felt the perspective of “quantum logic” helpful; at other times, I have felt it misleading. My current naive presumption is that it is ideal for a physicist to be able to work natively within, and fluently translate between, both the “Everything is still ultimately Boolean” and the “Let’s speak about things using a nondistributive logic” perspectives in quantum mechanics, as long as they understand what kind of language/propositional framework they are using at any particular moment.]
The problem is simply that it’s wrong to say the particle will not be found in ∆x[sub]1[/sub], just as it’s wrong to say it will not be found in ∆x[sub]2[/sub] (given, of course, that the particle’s momentum is in ∆p) – it may be found in either, and it will be found in ∆x. There’s simply no fact of the matter, prior to doing the experiment, in which one it will be found. It’s undetermined, in standard quantum mechanics. In something like the many worlds interpretation, both outcomes will obtain, so there’s not even any post-facto justification of assigning a truth value to 3 or 4. The particle may be in ∆x[sub]1[/sub] when you look, or it may be in ∆x[sub]2[/sub]; but it’s not the case that it will or will not be in either. Asserting this is tantamount to asserting that there exists more information about the particle’s location than is allowed by quantum mechanics.
We can attach the following truth values to the following statements:
The particle will be found in ∆p: true
The particle will be found in ∆p, and the particle will be found in ∆x: true
The particle will be found in ∆p, and the particle will be found in ∆x[sub]1[/sub]: false
The particle will be found in ∆p, and the particle will be found in ∆x[sub]2[/sub]: false
The particle will be found in ∆p, and the particle will not be found in ∆x[sub]1[/sub]: false
The particle will be found in ∆p, and the particle will not be found in ∆x[sub]2[/sub]: false
Of the last four, either must be false, since quantum mechanically, the particle may be found in either interval, and it is prior to the experiment not settled which one that will be – so talking about where the particle will be found if one is talking about an interval smaller than the minimum allowed by uncertainty always yields falsehood.
And yes, I do remember arguing that outside of observation, there is not necessarily anything going on behind the scenes – it’s what I’m arguing now: there are no hidden variables, where the particle will be is not sharply determined prior to measurement. I also remember you being the one opposing that view, claiming that there needed to be some ‘continuity’ in order to grant lawfulness, if I’m not mistaken.
That’s just the thing – it’s not correct to say that the particle had that position prior to measurement. There’s a maximum fidelity, if you will, to which the particle’s position is definite prior to measurement, and 3 and 4 exceed it.
“X’s position is Y” does not have to be interpreted in a way such that Y is a sharp point, it may be an interval – retaining the operational meaning that “when you look, you will find X in Y”. 3 and 4 are then not opposite one another: if position is truly just defined up to a certain interval, then claiming that the position is more sharply defined than that is false in either case, and just a vestige of classical thinking.
As a clarification: it’s true that ‘the particle is not within ∆x[sub]1[/sub]’, on account of its opposite – that the particle is within ∆x[sub]1[/sub] – being false; but this does not entail that ‘the particle will not be found within ∆x[sub]1[/sub]’ – clearly, it may be found just there.
Does this square with what I said, that position is where you’ll find the particle? I think so – if I say: ‘the particle is within ∆x’, then I mean that the particle will be found there; but saying ‘the particle is not within ∆x[sub]1[/sub]’ means that the particle’s location is ‘too big’ to fit into this interval; i.e. it’s ‘not just’ within ∆x[sub]1[/sub], speaking sloppily. If we imagine the particle as some extended thing, whose size is simply bigger than ∆x[sub]1[/sub], then it’s perfectly OK to say that ‘the particle is not within ∆x[sub]1[/sub]’, even though, looking there, one will see it – it’s just that, looking elsewhere, one will see it as well.
Geeze, I shouldn’t even attempt to think when it feels like there’s more mucus than brains in my head right now… I think I got most of what I wanted across, but of course this:
Should be regarded as only an analogy, how one might go about such an ascription of truth values to propositions in a classical case. In the quantum case, to muck up the analogy somewhat more, it’s perhaps closer to the truth to imagine the extended particle ‘collapsing’ down to a sharply defined location when it’s looked at, which may be either in ∆x[sub]1[/sub] or not, meaning that in the quoted paragraph, the ‘wills’ should be replace by ‘mays’.
