# Questions tagged [modal-logic]

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69
questions

**9**

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195 views

### 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, ...

**0**

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79 views

### 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 ...

**0**

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11 views

### Are all simple models from modal logic ($M_1 p \to M_2 p$) canonical?

I'm reading an interesting on modal logic by Timothy Surendonk [1], where he defines the concept of canonical logic as the following.
A logic $L$ is defined to be canonical if there's a frame $\...

**15**

votes

**1**answer

749 views

### 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 ...

**4**

votes

**1**answer

104 views

### 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)"?

**5**

votes

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159 views

### 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 $...

**10**

votes

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292 views

### 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 ...

**0**

votes

**1**answer

104 views

### 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 ...

**2**

votes

**1**answer

163 views

### 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 (...

**3**

votes

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117 views

### 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 ...

**1**

vote

**0**answers

70 views

### 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**

votes

**3**answers

457 views

### 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 ...

**16**

votes

**1**answer

1k views

### 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 ...

**4**

votes

**1**answer

163 views

### 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 ...

**1**

vote

**0**answers

82 views

### 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 $\mathcal{B}=(B,\beta)$...

**1**

vote

**1**answer

139 views

### 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} \...

**2**

votes

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64 views

### 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 ...

**2**

votes

**1**answer

211 views

### 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 ...

**2**

votes

**1**answer

152 views

### 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 ...

**9**

votes

**0**answers

366 views

### 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/...

**8**

votes

**0**answers

194 views

### 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 "...

**1**

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**0**answers

41 views

### 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, ...

**22**

votes

**1**answer

792 views

### 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 ...

**8**

votes

**1**answer

369 views

### 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 ...

**1**

vote

**0**answers

88 views

### 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 $\...

**1**

vote

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118 views

### 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\...

**6**

votes

**1**answer

587 views

### 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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79 views

### 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?

**1**

vote

**1**answer

193 views

### 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 ...

**5**

votes

**2**answers

269 views

### 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.

**3**

votes

**1**answer

194 views

### 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 ...

**1**

vote

**1**answer

310 views

### 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 $$\forall x,y,z\in X: (x,y)\in R\,\wedge\,(x,z)\in R\, \rightarrow\,(y,z)\in R$$
...

**0**

votes

**1**answer

151 views

### 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 ...

**3**

votes

**1**answer

480 views

### 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 ...

**0**

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**0**answers

169 views

### 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-...

**0**

votes

**1**answer

244 views

### 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 ...

**3**

votes

**2**answers

491 views

### 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 ...

**1**

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**0**answers

208 views

### 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
...

**1**

vote

**0**answers

189 views

### 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 ...

**6**

votes

**2**answers

796 views

### 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 ...

**2**

votes

**1**answer

119 views

### 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 ...

**1**

vote

**0**answers

149 views

### 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
...

**2**

votes

**1**answer

547 views

### 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{...

**2**

votes

**1**answer

170 views

### 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 ...

**4**

votes

**2**answers

338 views

### 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 ...

**4**

votes

**1**answer

352 views

### On directedness, transitivity and ancestral directedness

Let $\textit{C}$ be the modal logical schema $(\square (\square \alpha \rightarrow \alpha) \wedge \square (\square \lnot\alpha \rightarrow \lnot \alpha))\rightarrow (\square \alpha \vee \square \lnot \...

**2**

votes

**0**answers

78 views

### A question on completeness for quantified temporal logics

Quantified temporal logics have the Barcan formulas and its converses for both G (it will always be the case that) and H (it has always been the case that), so that both $\forall x G \alpha \...

**7**

votes

**1**answer

282 views

### 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 ...

**2**

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**0**answers

46 views

### Completeness results for quantified tense logics with BF?

Modal tense logics or temporal logics are important in that they correspond with partial orders and their extensions.
Are there completeness results for quantified temporal logics with the Barcan ...

**3**

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76 views

### A question on the incompleteness of quantified K.2 and S4.2 with the Barcan formula

I have been attempting to come to grips with Max Cresswell's account of this in Journal of Philosophical Logic 24 (4):379 - 403 (1995) where he presents proofs of the incompleteness of QK.2BF as well ...