Commentary by Chris Pressey

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Modal Logic

Modal Logic Playground

• rating: 3

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Introductory Modal Logic

• rating: 3

A good introduction to classical modal logic, including extending a natural deduction proof system with rules to handle modalities, and some philosophy.

First Steps in Modal Logic

• rating: 3

Concentrates on semantics (not proof systems) and multi-modal logics (corresponding to state transition systems).

Dynamic Logic

• rating: 1

Dynamic logic is a kind of modal logic. Inside this book, in Section 3.3 to be precise, there is a very good introductory exposition on equational logic as well.

Modal Logic for Open Minds

• rating: 1

I found it a bit tough to get through. It's clearly a collection of lecture notes (having the same somewhat choppy style as Kozen's "Automata and Computability").

Mathematical Modal Logic: A View of its Evolution

• rating: 1

Only skimmed. Historical overview. Not bad but more information than I probably need.

Topological semantics of modal logic

• rating: 3

Like some beaches I've been at, this starts out really fine and then gets deep, QUICK.

But it probably is a good place to start (and to come back to, and to keep coming back to) if you are interested in the relationship between topology and modal logic.

It is very much an overview of the ideas, and it is very much slides for a talk. If you want to delve deeper, Chapter 19 of "Modal Logic for Open Minds" ("Modal patterns in space") gives an overview in more prose, including the spoon example. "A Modal Walk Through Space" goes even deeper (much deeper). Other Modal Logic resources in this resourcebase may also be of interest, see their entries.

The Relationship between the Topological Properties and Common Modal Logics

• rating: 1

A short paper. For all sets X and all interpretations of the modal necessity operator compatible (in some sense) with its meaning in S4, the image of the interpretation is a topology on X. This makes intuitive sense, but it's good to know.

It still seems to say only a limited amount about which topological properties map to what constraints on modal necessity. They identify 4 "axioms of topology", and show which modal logics map to which of those axioms. But those axioms aren't ones I've seen a lot. They're not seperation axioms, I don't think. They don't seem to have names.

A Modal Walk Through Space

• rating: 2

It's very interesting but there is a lot covered here.

It was probably the best exposition I had read about the connection between S4 and topology, at the time, but as an overview, "Topological semantics of modal logic" is probably better.

Here's a quote:

The logic S4 is defined by the KT4 axioms and the rules of Modus Ponens and Necessitation. In the topological setting, the key principles are as follows, with an informal explanation added:

◻T                     (N) the whole space X is open
(◻φ ∧ ◻ψ) ↔ ◻(φ ∧ ψ) (R) open sets are closed under finite intersections
◻φ → ◻◻φ              (4) idempotence of the interior operator
◻φ → φ                 (T) the interior of any set is contained in the set

These can be compared to the Kuratowski closure axioms (and "Topological semantics of modal logic" does, I believe.)

Interesting concept: the topological bisimulation. It is a coarser notion than a homeomorphism, as I understand it -- all homeomorphisms are topo-bimsulations, but not vice versa. (Is this because homeomorphisms are continuous? Not sure. But bisimulations, in general, are quite lax - as long as you get to the same final destination in all cases, so to speak, they don't care how you got there.)

Topology, Connectedness, and Modal Logic

• rating: 1

It's very interesting but there is a lot in here, probably much more than I personally need. It goes deeper than "A Modal Walk Through Space", examining the computational complexity of various topological models of modal logics.

The Situation Calculus: A Case for Modal Logic

• rating: 0

So the idea is that if you use modal logic in the situation calculus, it becomes more useful for stating some things. No doubt, but in the longer view I'm skeptical. One of the strengths of the situation calculus is that it can be stated in a bog-standard logic (FOL) and that it has an efficient "resolution algorithm" for planning. While I can definitely see why one might be tempted to use modal logic here, I'm not convinced that it's an improvement, at least not for the use cases that interest me.

Modal Mu-Calculi

• rating: TODO

mu-Calculus can express CTL and related temporal logics.

This PDF of a book chapter is available on the first author's website. I like how it's described as "a chapter for the (very expensive) Handbook of Modal Logic, published by Elsevier in 2006".

Modal Logic (Stanford Encyclopedia of Philosophy)

• rating: 2
• useful: true

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history of philosophy - Are there any known precendents of philosophers using modal logic (or any other theory of math) to formalize works of other philosophers? - Philosophy Stack Exchange

• rating: 0

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