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Questions tagged [modal-logic]

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1answer
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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 ...
3
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1answer
130 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 (...
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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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0answers
64 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/...
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3answers
324 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 ...
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1answer
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 ...
3
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1answer
149 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 ...
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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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1answer
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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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0answers
51 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
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1answer
163 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
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1answer
128 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 ...
7
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0answers
297 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/...
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A bi-modal logic related to determinacy

Short version: there are a couple natural (I hope!) bi-modal logics associated to games on $\omega$. How much does AD determine about these logics? The motivation for this question ultimately comes ...
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0answers
40 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
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1answer
731 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 ...
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1answer
352 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 ...
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0answers
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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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0answers
111 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\...
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1answer
406 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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0answers
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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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1answer
169 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 ...
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2answers
245 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.
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1answer
190 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 ...
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1answer
214 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
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1answer
121 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
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1answer
392 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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0answers
165 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
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1answer
216 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 ...
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2answers
406 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 ...
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0answers
201 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 ...
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0answers
182 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 ...
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2answers
778 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
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1answer
114 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 ...
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0answers
139 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
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1answer
537 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
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1answer
161 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
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2answers
335 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
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1answer
343 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 \...
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0answers
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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 \...
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1answer
276 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 ...
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0answers
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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 ...
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0answers
69 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 ...
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1answer
628 views

A question on the modal logic S4.2

The modal logic S4.2 with the characteristic axioms 4: $\square \alpha \rightarrow \square \square \alpha$ and .2: $\lozenge \square \alpha \rightarrow \square \lozenge \alpha$ and T: $\...
5
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1answer
430 views

Is this system identical to S4.4?

Consider the normal modal logic system $\mathbf{TAR1}$ given by $\mathbf{T}$ plus the following axiom: $$\mathrm{AR1}: \lozenge \square p \rightarrow (\square p \lor \square (p \rightarrow \square p))...
3
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1answer
154 views

Soundness of modal logics which contain the reflection rule

Let $ML$ be a modal logic which contains the Reflection Rule (from $\vdash\Box F$ infer $\vdash F$). For a modal formula $F$, let $H(F)=\{\ \Box G\rightarrow G~|~\Box G$ is a subformula of $F\}$. A ...
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1answer
628 views

Non-iterative modal logics

Let S be a propositional modal logic system (extension of K, or even E) with a single unary modal operator and defined by a single non-iterative axiom (i.e. of modal degree 1). Is it true that for ...
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2answers
240 views

Validity in Kripke frames whose points are finite or infinite sequences

Suppose $D$ is a non-empty set and $\{ R_i : i \in \mathbb{N} \}$ is a family of binary relations on sequences over $D$ so that $R_i \subseteq D^i \times D^i$. Let $R_\omega \subseteq D^\omega \times ...
2
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2answers
213 views

On the Combinatorial Classification of Modal Kripke Frames

We have that S5 modal logic is characterized by the modal axioms $K$, $M$ (reflexive), $4$ (transitive), and $B$ (symmetric). That is, an equivalence relation on a set of possible world (which can be ...
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2answers
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Adjoint of Pushout as Modal Operators in Internal Logic

Regarding the internalization of mathematics to a particular category as in the nLab article: Internal Logic, there is a peculiar table mentioned in the section on Categorical Semantics in which there ...