Classical propositions are simply true or false, although you may not know which. They do not change from false to true or vice versa, and classical logic is grounded in this property. “Propositions” such as “today is Monday” are true at some times and false at other times, and hence are not propositions of classical logic.
If you want a “proposition” that depends on time or location, then what you need is a predicate—essentially, a template that yields different specific propositions depending on what values you substitute into the open slots. “Today is Monday” corresponds to the predicate A(t), where
A(t)≜(dayOfWeek(t)=Monday).
The closest we can come to an actual proposition meaning “today is Monday” would be
∀t.memories(t)=y⇒A(t)
where y is some memory state and memories(t) means your memory state at time t.
In any particular structure, each proposition is simply true or false. But one proposition can be true in some structure and false in another structure. The universe could instantiate many structures, with non-indexical terms being interpreted the same way in each of them, but indexical terms being interpreted differently. Then sentences not containing indexical terms would have the same truth value in each of these structures, and sentences containing indexical terms would not. None of this contradicts using classical logic to reason about each of these structures.
I’m sympathetic to the notion that indexical language might not be meaningful, but it does not conflict with classical logic.
The point is that the meaning of a classical proposition must not change throughout the scope of the problem being considered. When we write A1, …, An |= P, i.e. “A1 through An together logically imply P”, we do not apply different structures to each of A1, …, An, and P.
The trouble with using “today” in the Sleeping Beauty problem is that the situation under consideration is not limited to a single day; it spans, at a minimum, both Monday and Tuesday, and arguably Sunday and/or Wednesday also. Any properly constructed proposition used in discussing this problem should make sense and be unambiguous regardless of whether Beauty or the experimenters are uttering the proposition, and whether they are uttering it on Sunday, Monday, Tuesday, or Wednesday.
That’s not how I understand the term “classical logic”. Can you point to some standard reference that agrees with what you are saying? I skimmed the SEP article I linked to and couldn’t find anything similar.
You run into the same problems with any sort of pronouns or context-dependent reference, and as far as I know most philosophers consider statements like “the thing that I’m pointing at right now is red” to be perfectly valid in classical logic.
The main point of classical logic is that it has a system of deduction based on axioms and inference rules. Are you saying that you think these don’t apply in the case of centered propositions? Does modus ponens or the law of the excluded middle not work for some reason? If not, I’m not sure why it matters whether centered propositions are really a part of “classical logic” or not—you can still use all the same tools on them as you can use for classical logic.
Finally, if you accept the MWI then every statement about the physical world is a centered proposition, because it is a statement about the particular Everett branch or Tegmark universe that you are currently in. So classical logic would be pretty weak if it couldn’t handle centered propositions!
At least according to SEP classical logic includes predicates. But in any case if you want to do things with the propositional calculus, then I see no difference between saying “Let P = ‘Today is Monday’ ” and “Let P = ‘Sleeping Beauty is awake on Monday’ ”. Both of them are expressing a proposition in terms of a natural language statement that includes more expressive resources than the propositional calculus itself contains. But I don’t see why that should be a problem in one case but not in the other.
There is a relevant distinction: the machinery being used (logical assignment) has to be stable for the duration of the proof/computation. Or perhaps, the “consistency” of the outcome of the machinery is defined on such a stability.
For the original example, you’d have to make sure that you finish all relevant proofs within a period in Monday or within a period in NotMonday. If you go across, weird stuff happens when attempting to preserve truth, so banning non-timeless propositions makes things easier.
You can’t always walk around while doing a proof if one of your propositions is “I’m standing on Second Main”. You could, however, be standing still in any one place whether or not it is true. ksvanhorn might call this a space parametrization, if I understand him correctly.
So here’s the problem: I can’t imagine what it would mean to carry out a proof across Everett branches. Each prover would have a different proof, but each one would be valid in its own branch across time (like standing in any one place in the example above).
I think a refutation of that would be at least as bizarre as carrying out a proof across space while keeping time still (note: if you don’t keep time still, you’re probably still playing with temporal inconsistencies), so maybe come up with a counterexample like that? I’m thinking something along the lines of code=data will allow it, but I couldn’t come up with anything.
Sure, but I don’t think anyone was talking about problems arising from Sleeping Beauty needing to do a computation taking multiple days. The computations are all simple enough that they can be done in one day.
I’d say your reply is at least a little bit of logical rudeness, but I’ll take the “Sure, …”.
I was pointing specifically at the flaw* in bringing up Everett branches into the discussion at all, not about whether the context happened to be changing here.
I wouldn’t really mind the logical rudeness (if it is so), except for the missed opportunity of engaging more fully with your fascinating comment! (see also *)
It’s also nice to see that the followup to OP starts with a discussion of why it’s a good/easy first rule to, like I said, just ban non-timeless propositions, even if we can eventually come with a workable system that deals with it well.
(*) As noted in GP, it’s still not clear to me that this is a flaw, only that I couldn’t come up with anything in five minutes! Part of the reason I replied was in the hopes that you’d have a strong defense of “everettian-indexicals”, because I’d never thought of it that way before!
Hmm. I don’t think I see the logical rudeness, I interpreted TAG’s comment as “the problem with non-timeless propositions is that they don’t evaluate to the same thing in all possible contexts” and I brought up Everett branches in response to that, I interpreted your comment as saying “actually the problem with non-timeless propositions is that they aren’t necessarily constant over the course of a computation” and so I replied to that, not bringing up Everett branches because they aren’t relevant to your comment. Anyway I’m not sure exactly what kind of explanation you are looking for, it feels like I have explained my position already but I realize there can be inferential distances.
“the problem with non-timeless propositions is that they don’t evaluate to the same thing in all possible context
It’s more
“the problem with non-timeless propositions is that they don’t evaluate to the same thing in all possible context AND a change of context can occur in the relevant situation”.
No one knows whether Everett branches are, or what they are. If they are macroscopic things that remain constant over the course of the SB story, they are not a problem....but time still is, because it doesn’t. If branching occurs on coin flips, or smaller scales, then they present the same problem as time indexicals.
Right, so it seems like our disagreement is about whether it is relevant whether the value of a proposition is constant throughout the entire problem setup, or only throughout a single instance of someone reasoning about that setup.
Classical propositions are simply true or false, although you may not know which. They do not change from false to true or vice versa, and classical logic is grounded in this property. “Propositions” such as “today is Monday” are true at some times and false at other times, and hence are not propositions of classical logic.
If you want a “proposition” that depends on time or location, then what you need is a predicate—essentially, a template that yields different specific propositions depending on what values you substitute into the open slots. “Today is Monday” corresponds to the predicate A(t), where
The closest we can come to an actual proposition meaning “today is Monday” would be
where y is some memory state and memories(t) means your memory state at time t.
In any particular structure, each proposition is simply true or false. But one proposition can be true in some structure and false in another structure. The universe could instantiate many structures, with non-indexical terms being interpreted the same way in each of them, but indexical terms being interpreted differently. Then sentences not containing indexical terms would have the same truth value in each of these structures, and sentences containing indexical terms would not. None of this contradicts using classical logic to reason about each of these structures.
I’m sympathetic to the notion that indexical language might not be meaningful, but it does not conflict with classical logic.
The point is that the meaning of a classical proposition must not change throughout the scope of the problem being considered. When we write A1, …, An |= P, i.e. “A1 through An together logically imply P”, we do not apply different structures to each of A1, …, An, and P.
The trouble with using “today” in the Sleeping Beauty problem is that the situation under consideration is not limited to a single day; it spans, at a minimum, both Monday and Tuesday, and arguably Sunday and/or Wednesday also. Any properly constructed proposition used in discussing this problem should make sense and be unambiguous regardless of whether Beauty or the experimenters are uttering the proposition, and whether they are uttering it on Sunday, Monday, Tuesday, or Wednesday.
That’s not how I understand the term “classical logic”. Can you point to some standard reference that agrees with what you are saying? I skimmed the SEP article I linked to and couldn’t find anything similar.
You run into the same problems with any sort of pronouns or context-dependent reference, and as far as I know most philosophers consider statements like “the thing that I’m pointing at right now is red” to be perfectly valid in classical logic.
The main point of classical logic is that it has a system of deduction based on axioms and inference rules. Are you saying that you think these don’t apply in the case of centered propositions? Does modus ponens or the law of the excluded middle not work for some reason? If not, I’m not sure why it matters whether centered propositions are really a part of “classical logic” or not—you can still use all the same tools on them as you can use for classical logic.
Finally, if you accept the MWI then every statement about the physical world is a centered proposition, because it is a statement about the particular Everett branch or Tegmark universe that you are currently in. So classical logic would be pretty weak if it couldn’t handle centered propositions!
If classical logic means propositional calculus, then there are no predicates, and no ability to express time-indexed truths.
At least according to SEP classical logic includes predicates. But in any case if you want to do things with the propositional calculus, then I see no difference between saying “Let P = ‘Today is Monday’ ” and “Let P = ‘Sleeping Beauty is awake on Monday’ ”. Both of them are expressing a proposition in terms of a natural language statement that includes more expressive resources than the propositional calculus itself contains. But I don’t see why that should be a problem in one case but not in the other.
The first case has a truth value that varies with time.
And the second case has a truth value that varies depending on what Everett branch you are in. Does it matter?
There is a relevant distinction: the machinery being used (logical assignment) has to be stable for the duration of the proof/computation. Or perhaps, the “consistency” of the outcome of the machinery is defined on such a stability.
For the original example, you’d have to make sure that you finish all relevant proofs within a period in Monday or within a period in NotMonday. If you go across, weird stuff happens when attempting to preserve truth, so banning non-timeless propositions makes things easier.
You can’t always walk around while doing a proof if one of your propositions is “I’m standing on Second Main”. You could, however, be standing still in any one place whether or not it is true. ksvanhorn might call this a space parametrization, if I understand him correctly.
So here’s the problem: I can’t imagine what it would mean to carry out a proof across Everett branches. Each prover would have a different proof, but each one would be valid in its own branch across time (like standing in any one place in the example above).
I think a refutation of that would be at least as bizarre as carrying out a proof across space while keeping time still (note: if you don’t keep time still, you’re probably still playing with temporal inconsistencies), so maybe come up with a counterexample like that? I’m thinking something along the lines of code=data will allow it, but I couldn’t come up with anything.
Sure, but I don’t think anyone was talking about problems arising from Sleeping Beauty needing to do a computation taking multiple days. The computations are all simple enough that they can be done in one day.
I’d say your reply is at least a little bit of logical rudeness, but I’ll take the “Sure, …”.
I was pointing specifically at the flaw* in bringing up Everett branches into the discussion at all, not about whether the context happened to be changing here.
I wouldn’t really mind the logical rudeness (if it is so), except for the missed opportunity of engaging more fully with your fascinating comment! (see also *)
It’s also nice to see that the followup to OP starts with a discussion of why it’s a good/easy first rule to, like I said, just ban non-timeless propositions, even if we can eventually come with a workable system that deals with it well.
(*) As noted in GP, it’s still not clear to me that this is a flaw, only that I couldn’t come up with anything in five minutes! Part of the reason I replied was in the hopes that you’d have a strong defense of “everettian-indexicals”, because I’d never thought of it that way before!
Hmm. I don’t think I see the logical rudeness, I interpreted TAG’s comment as “the problem with non-timeless propositions is that they don’t evaluate to the same thing in all possible contexts” and I brought up Everett branches in response to that, I interpreted your comment as saying “actually the problem with non-timeless propositions is that they aren’t necessarily constant over the course of a computation” and so I replied to that, not bringing up Everett branches because they aren’t relevant to your comment. Anyway I’m not sure exactly what kind of explanation you are looking for, it feels like I have explained my position already but I realize there can be inferential distances.
It’s more “the problem with non-timeless propositions is that they don’t evaluate to the same thing in all possible context AND a change of context can occur in the relevant situation”.
No one knows whether Everett branches are, or what they are. If they are macroscopic things that remain constant over the course of the SB story, they are not a problem....but time still is, because it doesn’t. If branching occurs on coin flips, or smaller scales, then they present the same problem as time indexicals.
Right, so it seems like our disagreement is about whether it is relevant whether the value of a proposition is constant throughout the entire problem setup, or only throughout a single instance of someone reasoning about that setup.