# Questions tagged [modal-logic]

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### Sum and Product game

Two perfect logicians Steve and Pete, who have never met, before are imprisoned by an eccentric villain. "I have two positive integer numbers x and y" he says to them. "I will tell Steve the sum x+y, ...
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### Expressing a model transformation by using monads in the simply-typed lambda calculus

In https://link.springer.com/content/pdf/10.1007/s10670-019-00128-z.pdf , page 16, the following clause is given for a modal operator $\langle R_k \rangle$ (see definition 4.2 for the definition of a ...
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### Connection between Provability Logic (GL) and geometry?

In Provability Logic (aka GL) we have The Beth definability theorem and De Jong-Sambin Fixed Point Theorem The former has a vague similarity to the implicit function theorem in that you can loosely ...
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### Superintuitionistic logics which are not hereditary/monotonic: impossible or possible?

An intuitionistic Kripke model is a triple $\langle W,\leq, \Vdash \rangle$, where $\langle W,\leq \rangle$ is a preordered Kripke frame, and $\Vdash$ satisfies the following condition of ...
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### What is the dual of generating Boolean subalgebra by subexpressions of a modal formula?

I am supposed to be answering this question rather than asking it but I really cannot figure out. There is a variation on Stone duality linking algebraic and (descriptive) Kripke semantics for (...
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### Modal Principles of Field Extensions

In 2007 (with more work done later), J. Hamkins and B. Löwe found that the ZFC provably valid principles of forcing are the assertions of S4.2. In the introduction, they mention field extension as a ...
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### Looking for help in defining a new epistemic logic

I'm looking for some guidance in defining a new epistemic, temporal logic. I am looking to extend a logic called Sequential Epistemic Logic (SPAL): https://pdfs.semanticscholar.org/dae6/...
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### How can you formalize the metamathematics conventionally used to state Godel’s theorem?

Gödel’s incompleteness theorem states that for any sufficiently strong formal system $T$ there exists a statement $G$ such that if $T$ is consistent, then $G$ is true but not provable in $T$. But I’m ...
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### What is the modal logic of outer multiverse?

The mathematical multiverse could be viewed as a gigantic Kripke model with models of $ZFC$ as possible worlds connected to each other via a certain accessibility relation. The modal logic associated ...
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### Axioms for modal logics based upon counterfactuals

Suppose we have a logic for counterfactuals as with David Lewis. I here use $\Rrightarrow$ for the counterfactual conditional. So suppose we have: Rules: (1) If $A$ and $A\rightarrow B$ are ...
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### Limits in subcategories of Powerset-coalgebras

Let $F:Set\to Set$ be a functor. An $F$-coalgebra is a pair $\mathcal{A}=(A,\alpha)$ where $\alpha:A\to F(A)$ is arbitrary map. Given $F$-coalgebras $\mathcal{A}=(A,\alpha)$ and $\mathcal{B}=(B,\beta)$...
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### Modal vs First-Order Logic on finite models

It is known that Modal Logic can be interpreted in First-Order logic via Standard translation. However, this translation needs a unary predicate for every propositional variable. It is also known that ...
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### Question on deriving $\diamond \alpha \rightarrow \diamond \diamond\alpha$ in modal logic S4-system [closed]

I can't seem to achieve this derivation. The other way around, so $\diamond \diamond \alpha \rightarrow \diamond\alpha$, I did. But could someone helpt me with this part?
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### Modal logic in combination with automata theory

I'm planning to write a paper about the possibility of describing modal logic and the multiple world aspect of it with techniques of automata theory. To not duplicate my work does anyone have more ...
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### Literature on Kripke models

Which is the best introduction to Kripke-models for modal logics? I am a M.Sc in mathematics and know predicatlogic.
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### What kind of set theory is obtained from the canonical models of K?

Consider the minimal normal modal logic $K$ (axioms = classical propositional logic + $(\Box(p\land q)\leftrightarrow\Box p\land\Box q)$ + $(\Box\top)$, nothing else). Its canonical model with no ...
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### Does being an Euclidean relation imply being a Shift-Reflexive relation? [closed]

Let me start with a few basic definitions. Let $X$ be any set. A relation $R\subseteq X\times X$ is: Euclidean when $$\forall x,y,z\in X: (x,y)\in R\,\wedge\,(x,z)\in R\, \rightarrow\,(y,z)\in R$$ ...
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### Proving satisfiability in modal logic [closed]

So I've been doing some self study on Modal logic and I would like some external input on how to present my proofs for some of the axioms 1) say for example I am told to prove that □phi implies ♢psi ...
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### How to get $\omega$-regular expression from buchi automaton

Is there an algorithm or a trick on how to get $\omega$-regular expressions from Buchi automatons? If yes, is there also some way to do create minimal such regular expressions? It is extremely ...
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### Minimum regular open set containing a given set in a T0 Alexandrov topological space

What is known about the minimum regular open set containing a given set in a T$_0$ Alexandrov topological space? I'm particularly interested in the condition for the minimum set happening to be first-...
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### A question on two modal formulas

I want to find out the correspondences for the following two formulas or whether they are already derivable in the modal logic $KD4.2$, i.e. whether the formulas are valid in serial, transitive and ...
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### Is there a good list of nomenclature for modal axioms?

I would like to see what names that has been suggested for useful modal axioms. By name here I mean some abbreviation such as $T$, $K$, $4$, $.2$, $E$ and so on. In particular I am interested in ...
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### This modal logic semantics is not S5, but is it something else well-known?

The short form of the question is this: Is there a model of modal propositional calculus that gives the modal operators the meanings ...
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### Kripke frames as classes of partitions

Here's something I've been playing with off and on for a bit; I'm curious if anyone has seen it before. For this question, a Kripke frame $K$ is a finite reflexive directed graph. (Reflexivity isn't ...
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### A specific Model of ZFC

In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a ...
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### A question on Carnap's modal semantics on the basis of Cochiarelli's primary semantics

I believe I learned that Carnap's state description semantics for propositional modal logic suffered from validating $\lozenge p$ for all atomic variables p. Re-reading Nino Cochiarelli's primary ...
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### Interesting fragments of first-order logic induced by sorting?

In first approximation, modal logic (I'm using the term loosely) can be understood as an interesting fragment of first-order logic (for simplicity I ignore e.g. how modal logic relates to ...
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### Is this system incomplete?

Let $\mathbf{SBM}$ be the normal modal logic system defined as $\mathbf{T}$ plus the following two axioms: $$\mathrm{SB}: \Box(\Diamond p \rightarrow p)\rightarrow (p \rightarrow \Box p)$$ \mathrm{...
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### Question on deriving $\alpha \rightarrow \Box \alpha$ in modal logic KTU

Let K and T be the usual modal logical principles $\Box (\alpha \rightarrow \beta) \rightarrow (\Box \alpha \rightarrow \Box \beta)$ and $\Box \alpha \rightarrow \alpha$. Let U be the modal logical ...
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### On a modal correspondence

Is there an intuitive characterization of the correspondence for the modal logical formula $\square (\alpha \rightarrow \square \alpha) \rightarrow (\square \alpha \vee \square \lnot \alpha)$? In ...
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### Is a computer program for correspondence theory available?

In the 1990s I some times used a computer program with the Max Planck Institute which helped with calculating complicated correspondences for modal logical formulas. Is some program like that ...