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this is a bare list of references, to be !include
-ed into the References-section of relevant entries (notably at possible-worlds semantics and at many-worlds interpretation of quantum mechanics)
I don’t recommend actually reading these references, this is just to record that and where the thought has occurred to people of relating the “possible worlds” of modal logic to the “many worlds” of quantum philosophy.
What do you make of the connection? Is it just a formal affair, one formalism is the linear version of the other?
Just to note the the “many-worlds interpretation” has remained far from being a formalism. The body of literature on the topic is of vague verbosity (“many words”).
But I think that linear epistemic logic clearly is the missing formalism to make sense of the “Everettian many-worlds”.
It’s curious how in each case what is just some variation over a type gets blown up into ’worlds’ talk.
That’s the power of mathematical formalization: To map grandiose-sounding ordinary language to a small repository of fundamental mathematical concepts.
(For instance, much of standard-model cosmology is essentially a game of mapping creation stories of cosmogeny to a small class of ordinary differential equations.)
I used to be unhappy about the “worlds”-terminology, but maybe more for the ill effect it has on many authors than for the term in itself.
Now, I also used to be unhappy about the terminology “universe” in set/type theory — but in combination these make some sense (maybe more due to luck than brains): With dependent type theory viewed as epistemic modal type theory, its universes are indeed “universes of worlds”. This is maybe not too bad.
Sadly, the metaphysicians using modal logic (not sure why you add ’epistemic’) take ’world’ to mean something like self-contained universe. According to David Lewis (a hugely influential philosopher), a world is made up, and unified, by all the things which are spatiotemporally connected, and each possible world is spatiotemporally isolated from every other world. So all possible worlds make up a pluriverse.
not sure why you add ’epistemic’
Seems to be just the right word.
Starting all the way back with Bohr 1949, the issues of measurement in (quantum) physics which S5-type modal logic applies to are referred to as “epistemological”.
each possible world is spatiotemporally isolated from every other world. So all possible worlds make up a pluriverse.
Not according to quantum physics with state collaps and a classical observer. Here in a single universe (“ours”) we still have “many worlds” of measurement outcomes, all spatiotemporally isolated from each other.
I know this is confusing people, but to a logical observer internal to a tangent $\infty$-topos it is completely clear. :-)
The first point, I’m just observing that many philosophers would take S5 to apply more unproblematically to an ’alethic’ modal logic, how things are in the realm of metaphysical possibility and necessity. It’s seen as a substantial thesis that we should only understand possibility and necessity epistemically. That the linear version ends up talking about measurement in quantum mechanics which can be understood epistemically (again seen as a substantial thesis) is unlikely to persuade.
But perhaps we should say they have things the wrong way around. We live in a quantum world – this must come first.
Then even adopting this thesis that modal logic is to be understood epistemologically, there’s then the objection to taking epistemic logic as S5. One standard objection is to the KK-principle that knowing implies knowing that one knows. But maybe again the order should start with the quantum world.
Hmm, so how does that work in the linear case? If the necessity counit is quantum measurement, what corresponds to the comultiplication? Back in the nonlinear case I treat in my book it was about sections of sections of a bundle reducing to sections of the bundle along the diagonal.
Again, with the second point you prioritise the quantum. Something again arises for me about the inclusion of the classical observer. Is it that this is inevitable? Physics could only hope to represent what is measured by a classical observer?
I gather this is literally an argument about words?
For what it’s worth, the Wikipedia entry on Alethic modality consists mostly of a paragraph saying that it’s hardly distinguishable from “Epistemic modality”:
Alethic modality is often associated with epistemic modality in research, and it has been questioned whether this modality should be considered distinct from epistemic modality which denotes the speaker’s evaluation or judgment of the truth. The criticism states that there is no real difference between “the truth in the world” (alethic) and “the truth in an individual’s mind” (epistemic).[3] An investigation has not found a single language in which alethic and epistemic modalities would be formally distinguished
But I don’t want to argue this either by authority, nor by majority consensus, nor by Wikipedia quotes.
I want to argue by inspection and here I observe that if $W$ is the parameter space to be reasoned about then the operator $\Box P \,\colon\, \underset{w \in W}{\forall} P(w)$ is well-described by expressing necessity in the epistemic sense.
I’m just remarking that many philosophers think that there is a very important difference as shown for instance by their admission of $\Box p \vdash \Box \Box p$, but non-acceptance of $K p \vdash K K p$ generally, for $K$ some knowledge operator.
So I got that the wrong way in #8 about sections of sections. It’s about a dummy variable.
Last night I didn’t get around to replying here, after I had started to compile references (here) that highlight the use of S5 in practical epistemology (notably distributed agent systems).
I think that the problem that (some) philosophers keep running into is that the try to mix a human factor into mathematical formalism. This does not do justice either to the nature of mathematical formalization nor to the problem of human psychology. (Which connects to our other discussion: Human psychology is a highly emergent phenomenon far remote from what can be grasped with a reductionist system like a formal logic.)
So it is misguided to worry about whether an agent always “knows what they know”. The “knowledge” that we have any hope of formalizing with a logical system (as opposed to an emergence structure such as a neural network) is by necessity an idealized one.
And it is that idealized knowledge which we are really interested in when applying modal logic to nature (while for applications to actual humans no modal logic is an appropriate tool anyway):
For instance when one says that with Newton-Langrange mechanics we know the orbits of the planets, we are certainly not interested in taking account of whether a kid in the street actually knows how to compute orbits, much less whether it knows that it knows them or not. What we mean is that in principle these orbits are known.
Same applies to the epistemological problems of quantum measurement that Bohr et al are concerned with. To get to the bottom of this phenomenon we don’t want to get distracted by imperfections of experiments or experimentors, all that is abstracted away in order to get to the bottom of what is in principle knowledgeable.
Thanks for the reply. I certainly agree that any attempt to capture human knowledge in a modal logical formalism is misguided. I do think there are things to be said about human knowing that have a bearing on our flourishing, and as such must form a part of philosophical discussion, or else philosophy is not what it was thought to be by its whole tradition from Socrates onward. For instance, there are things to say about the responsibilities of those making claims to know something. Then there are things to be said about non-propositional forms of knowing, such as the ability to judge timeliness and appropriateness of actions. But these won’t be representable in any modal logic or other formalism.
By contrast, there’s then this idealised sense of knowing that you’re indicating, and we see just how idealised it is when we consider what the ’Necessitation rule’ means here. It’s saying that for every theorem, $P$, of the logic, then $K P$ is also a theorem. This is ’logical omniscience’. So if it has anything to do with knowledge, it’s much closer to your ’in principle knowledgeable’ and far away from anything about human capacities.
At this point, many philosophers will wonder whether the terms derived from ’know’ are appropriate. What’s the grammar of the idealised ’know’? What kind of things can be said to know? What kind of things can be said to be known? Does a measuring device ’know’ the outcome of a measurement? Why not use some other term with less baggage?
What’s the grammar of the idealised ’know’?
S5 :-)
Why not use some other term with less baggage?
In mathematics (and that should include mathematical logic) it is tacitly understood that the English terms used for mathematical concepts are meant to be defined by the idealized mathematical structure that they are naming and are not to be analyzed socio-linguistically.
When for instance we speak of, say, “perverse Schobers” in mathematics, it is understood that we must disregard all socio-agicultural baggage commonly associated with these words.
This is both the shortcoming and the power of mathematics as an intellectual discipline: That it is detached from all non-Platonic noise.
That said (and given that myself I don’t speak of “knowledge” at all but just of “epistemic logic”, which is what started this discussion), I find the examples, in the references that I listed, of “knowledge” of software agents participating in a distributed network protocol, make a convincing case for this terminology.
Generally, I expect that the logically inclined philosopher will benefit from envisioning a software agent instead of a human being when looking for intuition about modal logic.
added pointer also to
(which does not explicitly mention the term “many-worlds” but is clearly concerned with what it signifies)
For the record, I had also added the recent
Curious to contrast the two styles of writing/content.
I see that the point in my comment #11 has also been articulated by
who notes on pp. 121:
What standard logics of knowledge capture is not actual knowledge, but potential knowledge — what one is entitled to know. The switch to potential knowledge means we drop all considerations of complexity
and then concludes:
It is easy to see that, under such an assumption, a knowledge modality should be a normal modal operator. But, what else should be required? $[...]$ All these together make a knowledge operator obey the S5 conditions.
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