added (here) the (co)joins to the table of (co)units of the (co)modalities in quantum modal logic

]]>added (here) quotes from Aristotle matched to this formalization of “potentiality”.

]]>have made more explicit (here) how to read the argument that “potential data is possibility-modal data” namely as: “potential data is data whose possibility entails its actuality, consistently”.

]]>have touched wording, typesetting, and hyperlinking in the section *Via S4 modal logic*, in the hope of streamlining a little more

added pointer to

- Steve Awodey, p. 279 in:
*Category theory*, Oxford University Press (2006, 2010) [doi:10.1093/acprof:oso/9780198568612.001.0001, ISBN:9780199237180, pdf]

for the observation that S5 Kripke semantics is given by base change adjoint triples.

]]>With the expected arrival of directed HoTT next week, what can we say of how the analogue of our possible worlds story should go?

So we have in the ordinary case the adjoint triple generating the necessity and possibility (co)monads:

$(\exists_W \dashv W^\ast \dashv \forall_W) \;\colon\; \mathbf{H}_{/W} \stackrel{\stackrel{\forall_{w \colon W}}{\longrightarrow}}{\stackrel{\stackrel{W^\ast}{\longleftarrow}}{\underset{\exists_{w\colon W}}{\longrightarrow}}} \mathbf{H} \,.$Presumably in the directed case we have similar maps between a slice of some $(\infty, 2)$-topos and itself.

Let’s keep things simple. So a classic example of the undirected case sees a set of worlds, $W$, and then $W$-dependent propositions. We also might consider $W$-dependent sets and look at sections or the total space.

In the directed case, we might take $W$ to be a poset of worlds. A $W$-dependent proposition is presumably an upper set. Then adjoints to base change, $W^{\ast}$, are the limit and colimit over $W$, I take it. And we might consider presheaves of $(\infty, 1)$-categories over $W$.

This set-up then generates 2-(co)monads which resemble necessity and possibility, and so on right up to dependence on an $(\infty, 1)$-category or a morphism between two such. I take it there wouldn’t be anything new here, but considering things modally might be interesting.

]]>He definitely wrote something about that idea but I believe in Croatian only. Maybe with some effort some summary can be found in English somewhere.

]]>If there is any reference, I’d be interested in having a look.

]]>191

Aristotle’s actuality and potentiality

Late Ivan Supek (Croatian theoretical physicist, writer and a bit of philosopher and political activist), who was a student of Heisenberg, often emphasized that because of this aspect of Aristotle’s picture of the world, Aristotle’s point of view would better suit quantum world than Platon’s which is otherwise more dominant in traditional scientific Weltanschauung. I listened to one lecture of him as a student dedicated to this topic, at a Croatian Academy of Sciences workshop related to Ruđer Bošković (whose theory of forces influenced Bohr according to Bohr himself).

]]>added missing cross-link with *epistemic modal logic*

and analogously for the linear/quantum version (here)

]]>I have polished up the diagram (here) showing the four modalities of $B$-measurements induced by $B$-dependent type formers.

The text around it deserves to be streamlined, too, but I leave it as is for the time being.

]]>On the other hand, given that people got used to the word “coinvariant”, maybe “coindefiniteness” is not such a stretch, in particular since both are aspects of left base change.

In any case, I have now typeset the full Eilenberg-Moore-factorization in the dependent linear case: here

]]>I find “$potential \Rightarrow indefinite$” works well in itself, but it clashes badly with “$definite \Rightarrow potential$”.

Maybe we need to abandon “definite” as the linguistic value of the $\star$-comonad. The problem is that “definite” equals “in-in-definite”, but we need a word for “co-in-definite”, instead. (Like the necessary is the co-im-possible :-)

]]>Or should we be returning to Hegel, Transition to Actuality? Very little about potentiality there.

Too much for my poor brain at this moment.

]]>Yes, I was wondering about that. Maybe ’indefinite’ is better. “The potential is indefinite”.

So for Aristotle, a wooden bowl can be seen as a piece of wood one of whose potentialities (being a bowl) has been realised. As a piece of unformed wood, it has an array of potentialities (to be a bowl, to be a table, etc.). It is as yet indefinite as to form.

]]>Thanks. I’ll be adjusting the labels of diagram (5) now.

I keep feeling undecided whether “$definite \Rightarrow potential \Rightarrow random$” is linguistically as satisfactory as “$necessary \Rightarrow actual \Rightarrow possible$”.

But I grew fond of “random” for the reader modality, when I realized that the quantum reader, coinciding with the quantum coreader, is really close to what is called *QRAM*.

The diagram (5) needs changing too, so that it’s “the definite becomes potential” and “the potential is random”.

]]>Excellent! And what was I thinking? I’m supposed to be the philosopher. I have a colleague who endlessly goes on about Aristotle’s actuality and potentiality. I’ll add a SEP entry on his now.

]]>With that linguistic stumbling block out of the way (?! :-), back to the math:

I realized that not only are the potential types the possibility-modules in actual types (which holds also classically)

but for linear/quantum types we have the converse: Actual types are the randomness/reader modules in potential types!

As a stand-alone statement this is now recorded at *Reader monad – Examples – Quantum reader monad*. Will now draw the respective diagram here.

I claim that the “dynamic lifting” of the Quipper community is taking the preimage of the necessity counit under the functor which regards potential types as the free randomness-modules.

Will be typing this out now…

]]>It occurs to me that this linguistic issue has a classical resolution: The antonym to “actuality” which we need is “potentiality” as in Aristotle, and Heisenberg had already proposed this as the modality of the quantum state, just what we need in labelling the diagram for Quantum Modal Logic.

I have made corresponding edits now: here.

]]>I’m not appealing to the authority of philosophy. It’s about the meaning of the English word ’contingent’. We use it almost synonymously with ’dependent’, either in terms of some known factor (where generally we don’t yet know the value of that factor “I’m planning for all contingencies”) or in terms of some unknown factor (close to ’random’).

If contingency is anywhere it’s on the side of the slice, $\mathbf{H}/W$ (when the source of dependency is known). Of course, there’s the trivial dependency brought about by context extension, $W^{\ast}$. These constant dependent types, $W^{\ast} X$, are the possibility algebras. So the non-dependent types can give rise to a kind of degenerate dependency, but the paradigmatically contingent concerns genuine dependence.

If this is just a matter of English language use, then sample some other people.

]]>In the section of quantum modal logic, I have added the remark (here) that with the evident notational abreviations ($\mathcal{H} \coloneqq p_! \mathcal{H}_\bullet \,\simeq\, p_\ast \mathscr{H}_\bullet$ and leaving an outermost $p^\ast$ notationally implicit) the quantum writer/reader monads appear as:

$\begin{array}{ccc} \star \mathcal{H} &\simeq& \Box \lozenge \mathscr{H}_\bullet \\ \bigcirc \mathcal{H} &\simeq& \lozenge \Box \mathscr{H}_\bullet \end{array}$These identifications are borne out vividly in the analysis of quantum measurement/state preparation here, where they show that the quantum writer comonad records quantum measurements and the quantum reader monad reads out which quantum states to prepare.

]]>From the position of analytic philosophy, a contingent proposition is (…) a nonconstant map

Yes, and I have argued that the demand of non-constancy does not reflect the everyday use of “contingent”. Then I have provided some non-trivial argument that the everyday word usage of “contingent” is better formalized by possibility algebras.

You may agree with this or not, and I understand that I am being contrarian to a decade-old logic folklore that seems to never have been questioned or scrutinized before.

I may adjust the wording in the article later, when I have the energy. Meanwhile, let’s either discuss the actual arguments I provided or else leave it at that, for otherwise we seem to be going in circles.

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