# Questions tagged [modal-logic]

The modal-logic tag has no usage guidance.

79
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### Proof of the Local Deduction Theorem, for one of many logics

I'm looking for a proof of the Local Deduction Theorem for any one of many logics. That is, a proof of the statement:
$\Sigma \cup \{\phi\} \models \psi$ iff for some positive $n,$ $\Sigma \models \...

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1
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### An exercise in fuzzy logics built from a t-norm

Consider the following t-norm:
$$
a * b = \begin{cases}
2ab, &\quad\text{if }a, b\le1/2\\
\min\{a, b\} &\quad\text{otherwise}
\end{cases}
$$
We build from it the $\...

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### Logics of proper class sized Kripke frames

The following can be stated as a sentence of Morse-Kelley set theory:
If L is the logic of a proper class sized Kripke frame, then L is the logic of a set sized Kripke frame.
It follows from a $\Pi^...

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### Inconsistency in a modal logic

I need a first order modal logic, where inconsistency between formulas in not binary: a pair of formulas may be more or less inconsistent. The modal operators express uncertainty. So the formulas ...

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### Bimodal determinacy logic for Borel games

This question is intended to be a first step towards answering this old question of mine.
Let $K$ be the set of pairs $(\Sigma,\Pi)$ of quasistrategies, in the usual sense of games on $\omega$, for ...

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1
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### Modal logics which have an algebraic semantics but not a Kripke semantics

A colleague told me that there are modal logics which have an algebraic semantics of some kind but which do not have a Kripke semantics and in which both $\Box$ is not monotonic with respect to $\to$, ...

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### Where are good sources for correspondences in relevant logic? [closed]

Relevant logics for entailment have correspondences for a ternary relation $\textit{R}$ such that $M,a\vDash (A\to B)$ just if $\forall b, c(M\vDash Rabc\Rightarrow(M,b\vDash A\Rightarrow M,c\vDash B)...

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102
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### Counterexample equivalent in relevant logic DL

On page 7 in the article referred to below an axiom $D9$ is stated as follows: $$A\to B\to.\lnot(A \& \lnot B)~\\ (\text{equivalently: } (A\to\lnot A)\to\lnot A)$$
How may one prove the alleged ...

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### Deduction theorem for the modal mu-calculus

Does the modal mu-calculus have a deduction theorem?
If yes, how is it stated? Does it have the 'classical' form (i.e. as in classical propositional logic) or is it more involved?

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373
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### Interpretations of modal logic where $\Box$ means "valid"

Consider the propositional modal language in one propositional letter, $p$.
Recall that a pointed Kripke frame is a Kripke frame $(W,R)$ with a designated world $w_0\in W$, and a sentence is valid in ...

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### The provability logic of $I\Delta_0+\Omega_1 $ and complexity theory

Almost 30 years ago, a number of folks in provability logic tried to show that GL (see for instance the excellent survey by Rineke Verbrugge here) is indeed the logic of $I\Delta_0+\Omega_1$ (in the ...

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260
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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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102
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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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### Are buttons really enough to bound validities by S4.2?

Joel Hamkins recently claimed on twitter that buttons suffice to bound the validities of a potentialist system to the modal logic S4.2 (see here), and that switches are not necessary. We have been ...

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132
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### In modal logic, is there a formula that could express the inverse of accessibility relation?

For example, in S4, is there a formula that corresponds to the proposition "p is true in every world from which u is accessible (but is not accessible from u)"?

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### Definable modal logics in first-order structures

The old version didn't ask the right question and was also terribly written; see the edit history if interested. Also: throughout, formulas are allowed parameters, and when I say "definable subset of $...

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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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1
answer

141
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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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1
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175
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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/...

3
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3
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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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answer

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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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199
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### Translations between S4 and S5 modal logics

$\textbf{Question}$: Is there a translation from $\textbf{S5}$ modal logic to $\textbf{S4}$ such that
$$\text{If} \hspace{0.3cm} \textbf{S5} \vdash F \hspace{0.3cm} \text{then } \hspace{0.3cm} \...

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### Substructural shuffling: can we avoid a modal collapse in a certain Intuitionistic modal logic via making the logic linear?

Consider Propositional Lax Logic ($PLL$)
https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf
The Hilbert system of $PLL$ takes as axiom ...

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1
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286
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### Can we avoid the modal collapse in a certain Intuitionistic modal logic by abandoning ¬◯⊥ but retaining the law of the excluded middle?

Consider Propositional Lax Logic ($PLL$)
https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf
The Hilbert system of $PLL$ takes as axiom ...

3
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2
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### Modal collapse upon addition of the law of the excluded middle to an Intuitionistic modal logic

Consider Propositional Lax Logic ($PLL$)
https://www.uni-bamberg.de/fileadmin/uni/fakultaeten/wiai_professuren/grundlagen_informatik/papersMM/pll.pdf
The Hilbert system of $PLL$ takes as axiom ...

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437
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### The Curry Howard Isomorphism and models for an intuitionistic modal logic and its bimodal translation

My question regards the Curry Howard Isomorphism and how it constrains models in the case of a particular logic.
Consider quantified Lax Logic $QLL$.
https://pdfs.semanticscholar.org/468e/...

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### A bi-modal logic related to determinacy

The short version of my question is as follows. There is a natural (I hope!) way to associate a bimodal theory to a game (two-player, perfect-information, length-$\omega$, on $\omega$); are there "...

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46
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### Is there a restriction of Linear Temporal Logic that has a "Markov" property?

I have a problem, that I can formulate as model-finding in Linear Temporal Logic (via Büchi automata). I also have the additional knowledge, that there is always satisfied a Markov-like property, ...

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### Is the axiom $\Diamond\Box\varphi\to\Box\Diamond\varphi$ in c.c.c. forcing potentialism equivalent to the productivity of c.c.c. forcing?

This question arose in connection with a lecture series on
Potentialism
that I have just completed here in Hejnice in the Czech Republic at
the Winter School 2018 (see
Slides). Several of us discussed ...

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468
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### Substitutional modality

An informal definition of a logical truth is a sentence that's true in virtue of its form alone: $\phi$ is logically true iff all substitutions of $\phi$ that leave its logical vocabulary alone are ...

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### Are all normal modal logics isomorphic to this type of algebras?

Consider a Boolean set algebra on a set $\Omega$. Let $\sigma$ be a set function on $\Omega$ such that for all $m\in \Omega$ $\sigma(m)\subset \Omega$. The operator $\square_\sigma$ is defined by $\...

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### How do I express a second order restriction upon a third order comprehension schema?

I want a third order $\Pi^2_1$-comprehension schema so that $\alpha$ in
$$\forall x_1,\ldots,x_k, X_1,\ldots,X_l,\Psi_1,\ldots,\Psi_m\exists \Upsilon\forall Y(\Upsilon Y\Leftrightarrow\forall \Phi\...

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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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301
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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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1
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510
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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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664
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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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563
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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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2
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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 ...