NB: this question is a reference request.

I have already read countless introductions to modal logic (the latest one being Chapter 10 of @PeterSmith's Beginning Mathematical Logic: A Study Guide1).

In all cases, I have not being able to get past the very beginning, because I just cannot stomach the whole idea of "a set of possible worlds," which is foundational to much of modern work on modal logic.

I wholeheartedly agree with Hans Blumenberg's inclusion of "the world" among what he called mankind's "absolute metaphors." As such, the breezy talk about not just "possible worlds", but even "sets of possible worlds", let alone "the set (or class, if you prefer) of all sets of possible worlds", strikes me as an utterly outrageous example of "fools rush[ing] where angels fear to tread."

From this point on, everything else I read about modal logic theory has for me a ring similar to that of doing elaborate computations based on the number of angels that can dance on the head of a pin: i.e. a highly technical exercise on an entirely fictitious subject. Sure, this can be an amusing parlor game for some, but outside of a handful of enthusiast, why would anyone care?

Granted, I understand that the machinery of modal logic can be applied to areas in which the "possible worlds" are replaced with concepts that are entirely unproblematic, such as "states of a computer program's execution." I have no problem with this.

Similarly, I understand that one can replace "set of possible worlds" with an arbitrary graph, so that each of the graph's nodes is now a "world". In this case, the resulting theory is entirely unproblematic.

Indeed it is, but it is also much less interesting.

After all, one point all treatments I have read on modal logic agree on is that the original interest in modal logic is as a guide for human reasoning. Smith's introductory paragraph (p. 120) is entirely representative of this (my emphasis):

A deduction, Aristotle tells us, requires a conclusion which ‘comes about by necessity’ given some premisses. So it is no surprise that, from the very beginning, logicians have been interested in the modal notions of necessity and possibility. Modern modal logics aim, at least in the first place, to regiment reasoning about such notions.

If this is so, the whole apparatus of modal logic is a mere technical triviality that is entirely insignificant in comparison to the problem of specifying a notion of "possible worlds" that is at once sufficiently precise and coherent to allow for unambiguous computation and also sufficiently representative of human experience to serve as a useful guide to human reasoning in actual practice.

What I find most disturbing, by far, about the idea of "possible worlds" is that, so far, I have not yet found an exposition of modal logic that even hints at the possibility that there could be any philosophical objection to the concept. The idea is always introduced as a simple, straightforward, unproblematic one. Something that everyone understands perfectly.

In light of this, I find myself doubting my sanity. How can anyone discuss in such a banal way something that I find so far beyond the reach of formalization?

Therefore, I am looking for an introduction1 to modal logic that gives "possible worlds" its philosophical due. In particular, this treatment should show how one can make the notion of possible worlds sufficiently sharp and coherent to allow for computation and sufficiently rich to be useful as a real, effective guide to human thought. This treatment should also address the question of consistency: how do we know that something analogous to Russell paradox does not lurk inside our definition of "possible worlds."

1 I hasten to add that, even though BML's treatment of modal logic does not address the issues I have with "possible worlds", the rest of the book has been for me a godsend. This is the only book I know of its kind, a (new?) genre of guides to the advanced literature aimed at those interested in serious independent learning. I wish there were more books like this one. A lot more. In fact, I think there is a widespread and urgent need for such book for statistics, but I digress.

2 Here I mean a book, or one or more chapters in a book, or a review article. I.e., I am looking for something far more extensive and thoroughly fleshed out than what can be expected from even the most expansive Philosophy StackExchange answer. In other words, please do not attempt to provide such an introduction here, as an answer to my question; this is not what my question asks for. This question is, as tagged, a reference-request.

  • 1
    The "philosophical" idea is quite old (Aristotle, Leibniz,..). At the same time, Modal Logic has some relevant "opponents": Russell, Quine. Modern Modal Logic semantics can be seen as a mathematical tool only; there are others: algebraic, topological, all validating the same set of logical formulas. Thus, you can use it only as a mathematical model. Apr 22, 2022 at 14:43
  • 2
    Maybe useful Rod Girle, Possible worlds (Acumen, 2003) Apr 22, 2022 at 15:32
  • 2
    What is your view of the idea that there can be truths about mathematical statements independent of whether human beings prove or disprove them? If one adopts a view like truth-value realism or mathematical platonism, then combined with the idea that our world precisely follows some set of mathematical laws, this can be provide a natural way of thinking about other possible worlds (either with the same laws but different initial conditions, or different laws).
    – Hypnosifl
    Apr 22, 2022 at 17:56
  • I'm curious as to your motivation to research something that keeps throwing you off so completely? Perhaps this is simply an area that will not appeal to you even when you do understand it. I've never been interested in sports, for example.
    – Scott Rowe
    Apr 25, 2022 at 0:12
  • @ScottRowe: My problem with modal logic is not simply that I don't like it. My problem with it is that its popularity is a threat to my sanity, because it means that I cannot even begin to see what otherwise seemingly intelligent people take as a matter of course. I want to get over the feeling that I must be going insane. Or that I am a subhuman cretin devoid of the basic perceptual faculties of a normally functioning human.
    – kjo
    Apr 25, 2022 at 9:56

2 Answers 2


Similarly, I understand that one can replace "set of possible worlds" with an arbitrary graph, so that each of the graph's nodes is now a "world". In this case, the resulting theory is entirely unproblematic.

You find this description unproblematic, because that's what possible worlds are, at least in Kripke's treatment of them. The term "world" is evocative, not literal; it does not literally mean e.g. "a set containing every particle in the entire universe" or some such concrete object like that. Worlds are simply opaque atoms which the Kripke frame maps to particular valuations. They have no independent structure, do not even consist of or contain the valuations themselves, and are literally described as "just a graph" over the accessibility relation.

If you want to read more about this version of possible world semantics, the book that you are reading should already do a fine job of explaining the rest. It most likely will not tell you anything about the internal structure of worlds, because in its conception, the worlds have no internal structure to describe.

On the other hand, if you want possible worlds that are more interesting than those described by Kripke, then you might start with the SEP article on possible worlds, and maybe also spend some time reading up on transworld identity and David Lewis's counterpart theory (which is wildly incompatible with Kripke semantics).

  • In general it doesn’t even have to be a discrete graph—it can be continuous/dense (C4). Worlds can be seen as labels for points in a phase space whose topology is determined by the reachability relation. The spooky-sounding notion of a “set of possible worlds” is typically just a subset of this space, where e.g. every point has a valid path from some initial condition. How we know if our definition of validity is reasonable in some model (provability/knowledge/belief/&c.) is exactly the kind of doubt that modal logic is meant for investigating.
    – Jon Purdy
    Jun 4, 2022 at 19:47
  • @JonPurdy: Personally, I have no problem with calling such a space an "uncountably infinite graph," but I recognize that some people might be uncomfortable with such a notion.
    – Kevin
    Jun 4, 2022 at 22:22

Based on science, we divided our surroundings into a system (where the experiment occurs) and an environment (that interacts with the system during the experiment). Then we say okay let's find the biggest system possible so that there is no other things left to interact with. This biggest isolated system is called the world. But this definition is not compatible with our spatial observations in particular the world expansion. Because if the world is expanding right now it was smaller before, up to a point that big bang theory models it. Then we have to spilt the concept of world into the world with the same order and the observing world because we can't measure interactions before 14 billion years ago. So, the observable world started from big bang but the world that has the same law with ours today may be greater or equal to the observing world. This division lead scientists to generalize the concept of world into worlds that may have different laws or the same law on some different space and time. The important thing here is that right now we don't have any idea if there is a way to interact between these hypothetical worlds. That's why some people study them.

  • Thank you, but my question is a reference request. It was already tagged as such, but I have edited my post to make this absolutely clear.
    – kjo
    Apr 24, 2022 at 12:41

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