# Questions tagged [modal-logic]

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### Normal modal Logic with finite proposition letters

Assume our modal language $L$ has only diamonds, and the set of proposition letters $Prop$ is finite. The deduction rules are the same as normal modal logic. Now consider $M$ is a finite model of this ...

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### An extension of the disjunction property in modal logic

A normal modal propositional logic $\Delta$ has the disjunction property if and only if
For any formulas $A_1,\dotsc,A_n$, if $\Box A_1 \vee \dotsb\vee \Box A_n \in \Delta$ then $A_k\in \Delta$ for ...

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### Can this order relation, defined in terms of all topological spaces, be defined in terms of the reals alone?

Let $K$ be the operator monoid under composition of Kuratowski's $14$ set operators generated by topological closure $k$ and complement $c.$ Kuratowski's 1922 paper gives the poset diagram of the ...

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### A formula which is true in all possibilities for variables in IPL

Let $\mathcal{F}(n, 2^m)$ be an intuitionistic Kripke in Fig. 1, which is formed by the set
$$
\left\{(i, j)\in \omega \times \omega \mid (0 \leq i \leq n-3, 0 \leq j \leq 1) \vee (i= n-2, 0 \leq j \...

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### A class of Kripke frames which preserves validity

The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame.
For $1\leq s\leq n-2$, the frame $\mathcal{C}_n(s)$ denotes the frame which is ...

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189
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### Preserve validity between the two Kripke frames

The background of our discussion is intuitionistic logic, i.e. the following definitions are intuitionistic Kripke frame.
For $n \geq 1$, let $\mathcal{C}_n$ denote the frame which is shown in Fig.1. ...

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### Descriptive general frames without differentiation?

Descriptive general frames are usually defined as general frames that are tight, compact, and differentiated. On p.91 of this paper by Skvortsov and Shehtman 1993, the authors omit the third condition,...

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### Do the transitive models of ZFC form a canonical Kripke model for the Gödel-Löb axioms?

Let $\mathcal{C}$ be the class of all transitive models of ZFC, i.e., sets $S$ such that $S$ is downward closed ($x \in S \to x \subseteq S$) and $(S, \in)$ is a model of ZFC (where $\in$ is set ...

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### Existence of certain formulas in modal logic K

Does there exist a modal formula $φ$ with the following properties?
for every finite Kripke frame $F$ there is some ground substitution $\sigma$ such that for every point $w \in F$ we have $F, w \...

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### How complicated are 3-player clopen determinacy facts?

Say that a clopen 3-player game is a well-founded tree $T\subseteq\omega^{<\omega}$; intuitively, starting with player $1$ and continuing cyclically, the players $1,2,3$ alternately play natural ...

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### Finite axiomatisation of an extension of $T$ complete w.r.t. Neighbourhood Semantics and incomplete w.r.t. Kripke

A modal logic is normal if and only if $\Box (p \to q) \to (\Box p \to \Box q)$ is provable and the rules of modus ponens and necessitation holds.
Let $T$ be a smallest normal modal logic in which $\...

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### Modal logic of "mostly-satisfiability"

For $n\in\omega+1$ let $\mathsf{ZFC}_n$ be $\mathsf{ZC}$ + $\{\Sigma_k$-$\mathsf{Rep}: k<n\}$. Let $\widehat{\mathsf{ZFC}}$ be the strongest consistent theory $\mathsf{ZFC}_n$ (so if $\mathsf{ZFC}$ ...

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### Algebraic logical structure

Let $M=(W,R)$ be a Kripke frame, $A=(f_1,...,f_m)$ is a tuple of operations $f_i:W^{n_i}\to W$, and $\Phi=(\varphi_1,...,\varphi_m )$ is a tuple of first-order logic formulas in vocabulary $\sigma=\{=...

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### Self-referential Quinean proof of Löb's Theorem

Given its relevance for Open-source game theory, Dr. Andrew Critch asks the following about provability logic:
We conjecture that Löb’s Theorem can be proven without the use of the
modal fixed point $...

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### A modal logic with two diamonds, one is interpreted as the complement of the relation corresponding to the other one

Suppose our language has two diamond operators $\Diamond$ and $\overline{\Diamond}$ and, over a Kripke model whose relation is $R$, we have the following semantics:
$w\models\Diamond\varphi$ if there ...

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### Doing reverse mathematics by regarding modal logic as weak first-order logic

Reverse mathematics seeks to find subsystems of second-order logic that are equivalent to certain mathematics theorems, say over $\mathsf{RCA}_0$.
Modal logic can be regarded as a weak version of ...

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### How strong is this "modal definability principle"?

Throughout, we work in the class theory $\mathsf{MK}$ (although I'm open to tweaking this), "logic" means "set-sized logic whose semantics is definable over $V$," and "$\...

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801
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### Difference between provability and the existence of a proof?

In provability logic, $\square X \rightarrow X$ is not a theorem.
In my head[1] this reads as "if X is provable you don't necessarily have a proof of X".
This has lead to the question, what ...

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### Question regarding ultrafilter extension of $\tau$-model

Since I'm not native speaker, my writing is probably difficult to read. Hence please point out any mistakes.
I'm reading page 96 and 97 of Modal Logic written by Patrick Blackburn.
$\textbf{...

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166
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### Predicativity and axiom $\mathbb{R}\flat$ in real cohesive homotopy type theory

In Mike Shulman's article Brouwer’s fixed-point theorem in real-cohesive homotopy type theory, the fundamental axiom adopted for his real-cohesive homotopy type theory (axiom $\mathbb{R}\flat$), which ...

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### Initial reference on Gödel-Löb axiom in Kripke semantic of $GL$

It seems well known in modal logic society that $\Box(\Box p\to p) \to \Box p$ in Kripke semantics of $GL$ implies well-foundedness of the relation i.e. no infinite ascending chains are allowed.
And ...

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### Does the following variant of common belief exist?

Let $A$ be a finite set of agents and $\mathtt{B}_a$ a modal operator where $\mathtt{B}_ap$ means agent $a$ believes proposition $p$. For now I don't assume any properties of $\mathtt{B}_a$, though ...

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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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### An exercise in fuzzy logics built from a t-norm [closed]

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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219
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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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### 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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### 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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### 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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### 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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### 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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### 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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157
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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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### 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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315
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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 ...

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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 ...