# Questions tagged [modal-logic]

Questions relating to deductions relating to the expressions "it is necessary that" and "it is possible that"

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### Please can someone explain how to translate intuitionistic logic into S4? [closed]

I understand this is an easy question, but I cannot quite get my head around it.
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### How does the technique of 'existence-relativisation' allow us to embed varying-domain semantics in constant-domain semantics?

How can the technique of 'existence-relativisation' embed varying-domain semantics in constant-domain semantics?
1 vote
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### Proving axiom M in modal logic from K plus A → ◊A

I'm slowly working my way through Garson's Modal Logic for Philosophers. I'm stuck on exercise 2.1c. It asks me to prove axiom M (= ◻A → A) in system K (propositional logic + necessitation and ...
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### The question asks about basic tense logics and tried to proof if a schema is valid or not

Consider the following schema: FHA → A Determine whether or not it is valid. If it is valid, prove this; if it isn’t show this by constructing a countermode
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### Proof that M-Trees Can Be Converted into M - Proofs

I'm studying modal logic through Garson's book, and I got to the part of proving that given a K - tree, there's a corresponding K - proof. I did a proof for the minimal system K case, but I can't ...
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### Is the converse of (◇◇A -> ◇A) true too?

Axiom 4 in System S4 is this: ◻A -> ◻◻A Its corollary is this: ◇◇A -> ◇A As I understood with the corollary, it means that "If the fact that A is true in at least one world, true in at ...
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### Is the distribution axiom in modal logic biconditional and does it hold for diamond as well as box?

I'm a new learner of (normal) modal logic. I'm slowly teaching myself and have some questions related (broadly) to the distribution axiom: $\square (p \to q) \to (\square p \to \square q)$. I have no ...
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### Literature for modal logic and coalgebras (focused on logic and category theory)

I've been looking for literature that focuses on coalgebras and modal logic. But all literature I can find is mostly related to automata theory, or otherwise takes heavy influence from computer ...
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### Is modal logic required to handle assumptions in logic? (Tic Tac Toe example)

Suppose the Tic Tac Toe board is like this: \begin{array}{c|c|c} 1 & 2 & 3 \\ \hline 4 & 5 & 6 \\ \hline 7 & 8 & 9 \end{array} and I define a predicate ...
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### Prove that $\mathbf{L}$ is almost a normal modal logic.

Suppose we have a logic $\mathbf{L}$ containing all the propositional tautologies and the formula $\square(p\wedge q)\leftrightarrow(\square p \wedge\square q)$, and it is closed by Modus Ponens, ...
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### Computation tree logic (CTL): Are AG, AF, EG or EF idempotent?

I'm studying CTL and I got stuck on the following problem: Determine if the following operations are idempotent or not: AG, AF, EG or EF. I have the following definitions: Let M = (S,→,L) be a model ... 1 vote
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### Prove of K ⊢ A ↔ B then K ⊢ C [A/q ] ↔ C [B/q ]

I am currently reading "Boxes and Diamonds: An Open Introduction to Modal Logic". Here is the PDF to the book: https://builds.openlogicproject.org/courses/boxes-and-diamonds/bd-screen.pdf ...
1 vote
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### Hughes and Cresswell's definition of consistency

On page 46 of Hughes and Cresswell's A New Introduction To Modal Logic, we have the following definition of consistency. Consistency : We shall say that an axiomatic system is consistent iff not ...
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### Excercise 1.5.1 from Blackburn, de Rijke and Venema's Modal Logic

I've recently started to learn Modal Logic using this book as a reference. At the end of Section 1.5 "Modal Consequence Relations" there are some excercises about the topic. The first one ...
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### Is there a mathematical system that is: complete, consistent, and decidable?

I know very little about modal logic (only some set theory) in mathematics, but I am aware that there exists a completeness theorem, incompleteness theorem, and the axiom choice, and that maths is not ...
1 vote
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### Every normal modal logic Σ contains ¬♢⊥.

The following proposition comes from this PDF: http://builds.openlogicproject.org/content/normal-modal-logic/normal-modal-logic.pdf (p.31) It says: Every normal modal logic Σ contains ¬♢⊥. The problem ...
1 vote
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### Does consistency of Peano arithmetic follow from arithmetical completeness of modal logic GL?

Solovay's Arithmetical Completeness theorem states that if $A$ is sentence of modal logic, then if every realization $A^*$ of $A$ is proved by $\mathsf{PA}$, then $\mathsf{GL}\vdash A$. By ...
1 vote
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### Formulas of basic modal logic involving only $\top$, $\bot$, propositional connectives and modalities

I am given a Kripke model $\mathcal{M}=(W, R, L)$ where $W=\{w_1, w_2, w_3, w_4\}$ $R=\{(w_1, w_2), (w_2, w_3), (w_4, w_1), (w_4, w_3)\}$, and for all $w \in W$, $L(w)=\varnothing$. For each $w \in W$,...
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### Prove that if $\mathcal{C}\vDash C\rightarrow \Box B$ then $\mathcal{C}\vDash C\rightarrow\Box\forall x B$

$\mathcal{C}$ is the class of all variable domain existence models. Further, we are supposing that $x$ does not occur free in $C$. But now consider this counter-model $M=\langle W,R,D,f,V\rangle$ ...
1 vote
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### How to show that the canonical model for $K4.3$ contains proper clusters?

It is a well-known result that the canonical models for some temporal logics, e.g, $K_t4.3$ and $K_tQ$, contain proper clusters (equivalent class of points seeing each other). As a result, we need to ...
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### Constructing the largest bisimulation from matchings

I'm reading the textbook 'First Steps in Modal Logic' by Sally Popkorn (the pen name of Harold Simmons). I got stuck in a transfinite construction of the largest bisimulation from matchings. Here are ...
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### Is $\forall x \square \varphi \leftrightarrow \square \forall x \varphi$ a valid formula for all formulas $\varphi$?

I haven't yet studied formal semantics of a modal logic with quantifiers, so forgive me if I make any rookie mistakes. My thinking goes like this. The modal quantifier $\square$ ranges over the set of ...
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### Rooted Kripke frames connection to $K$

I was reading the Modal logic book from Chagrov and Zakharyaschev. I read the following theorem (generation theorem 3.11): If $N$ is a generated submodel of $M$, then for every point x and every ...
1 vote
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### For every intuitionistic kripke frame $F = (W, R)$ and every formula $\phi$ and points $x,y \in W$ if $x \models \phi$ and $x R y$ then $y\models\phi$

I need to show that: For every intuitionistic kripke frame $F = (W, R)$ and every formula $\phi$ and every points $x,y \in W$ if $x \models \phi$ and $x R y$ then $y \models \phi$ I know it's done by ...
1 vote
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### Show $K_\rho$ is a proper extension of $K_\eta$

Show $K_\rho$ is a proper extension of $K_\eta$: $K_\rho$ is an extension of $K$ where $R$ is reflexive. And $K_\eta$ is an extension of $K$ where $R$ is extendable. So, I reasoned as follows: $K_\rho$...
72 views

1 vote