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Recently, I am reading course materials on intuitionistic and modal logics. I have two questions about the notion of accessibility relation in Kripke semantics for intuitionistic and modal logics. When we give a Kripke semantics for intuitionistic propositional logic (IPL), the accessibility relation (usually written as $\leq$) is said to be hereditary: If $w\Vdash p$, then for $w\leq u$, $u\Vdash p$ as well. My questions are as follows:

(1) First, isn't the accessibility relation, usually written as $R$, in the Kripke semantics for classical modal logics, say K or S4, also hereditary?

(2) If not, how should I understand $R$ in modal logics and also the property of being hereditary?

Thank you, guys!

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    $\begingroup$ Hereditariness isn't a property of the accessibility relation on its own, but rather the accessibility relation together with the valuation in question. (Also, what text are you reading?) $\endgroup$ Commented Mar 24, 2020 at 22:07
  • $\begingroup$ @NoahSchweber I am reading a paper on constructive S4 (link here: citeseerx.ist.psu.edu/viewdoc/…). I am a little bit confused now. So, what does it mean that a world $u$ is accessible from $w$. Doesn't it mean that we can properly extend $w$ with $u$? $\endgroup$
    – ferdinand
    Commented Mar 24, 2020 at 22:12
  • $\begingroup$ I don't know what "we can properly extend $w$ with $u$" means. $\endgroup$ Commented Mar 24, 2020 at 22:12
  • $\begingroup$ Note that that paper is talking about mixing modal and intuitionistic logic, so before reading it I'd first focus on understanding each part separately. $\endgroup$ Commented Mar 24, 2020 at 22:18

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No, there is no requirement of hereditariness in general modal logic (and per my comment above that wouldn't even make sense - hereditariness is a property of the accessibility relation and the valuation together, not merely one or the other). Specifically, any pair $(W,\leadsto)$ with $W$ some (nonempty) set of "worlds" and $\leadsto\subseteq W^2$ a binary relation on $W$ is a Kripke frame. Moreover, given a frame $(W,\leadsto)$ and a set $P$ of propositional atoms, any map $v:W\rightarrow\mathcal{P}(P)$ (sending $w$ to the set of propositional atoms true at $P$) is an appropriate valuation. The modal logic K is sound and complete with respect to the class of all Kripke frames (and many other classes of Kripke frames besides). Special subclasses of Kripke frames then often correspond to stronger modal logics: e.g. demanding that $\leadsto$ be an equivalence relation corresponds to S5.


Re: your second point, this reflects the much broader scope of modal logic in general, at least from the perspective of frame semantics. When we set up Kripke semantics for intuitionistic logic, the basic idea is that worlds represent states of partial information. Accessibility corresponds to possible extension: $w$ is accessible from $u$ if all the information $u$ possesses is also possessed by $w$ (that is, $u$ and $w$ don't disagree on anything and $w$ hasn't forgotten anything $u$ knew). In light of this it makes sense to only pay attention to valuations with some appropriate consistency property - and this yields exactly the notion of hereditariness.

In general, though, we're much more permissive. For example, we could think of the worlds as moments in time, and the accessibility relation as "is in the future of." Here it wouldn't make sense to demand anything like hereditariness: "Today it is Tuesday" would be true at some world, false at a later world, true again at a yet later world, and so on.

More broadly, the most common interpretation of Kripke frames is as "possible worlds semantics:" the worlds of the frame are just imaginable universes, and the accessibility relation is "is plausible from the perspective of." For example, in the current world $w_0$ I'm not eating frosted flakes right now, but it is totally plausible that I would be - so world $w_1$, which looks exactly like $w_0$ except that I'm eating frosted flakes now, is accessible from $w_0$. However while it is imaginable it's not at all plausible that I would be the pope, so that's a whole bunch of imaginable worlds which aren't accessible from $w_0$. We can even argue that we should allow non-reflexivity: "truth is stranger than fiction!"

But there are tons of other kinds of modal reasoning where Kripke frames are useful - deontic logics, doxastic logics, epistemic logics, ... - and in general we don't make any assumptions at all on the accessibility relations or the truth valuations we'll allow until we specify a further context.

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  • $\begingroup$ Thank you so much, Noah. I have to say that perhaps I was influenced by the constructive notion of possible world, which is given by Per Martin-Lof in his 1988 lecture, and also by Aarne Ranta in his 1991 paper. That's why I took it for granted that $\leq$ is interpreted as some sort of world/contextual extension. But I see the point now, these are not completely equivalent. Thanks again! $\endgroup$
    – ferdinand
    Commented Mar 24, 2020 at 22:51
  • $\begingroup$ @ferdinand it's been a while, but what's the reference for Martin-Löf's 1988 lecture? $\endgroup$ Commented Jul 14, 2022 at 4:16
  • $\begingroup$ @theHigherGeometer Hi, it's called "mathematics of infinity", it's a 1988 lecture and later published in 1990. Here's the link to all the papers of Martin Löf: github.com/michaelt/martin-lof, or you can also take a first look at Ranta 1991 ''constructing possible worlds'', and find the reference there. $\endgroup$
    – ferdinand
    Commented Jul 15, 2022 at 10:13
  • $\begingroup$ @ferdinand OK, thanks! Here's the direct link to the lecture: raw.githubusercontent.com/michaelt/martin-lof/master/pdfs/… $\endgroup$ Commented Jul 15, 2022 at 22:21

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