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

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Exact scope of modal logic?

Is modal logic the logic of necessity, possibility, and impossibility alone, or the logic of truth, falsity, necessity, possibility, and impossibility? In other words, is modal logic concerned with ...
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Show that, at each possible world $\Gamma$ of a modal model, $\Gamma \Vdash \square X \equiv \sim \diamondsuit \sim X$

Show that, at each possible world $\Gamma$ of a modal model, $\Gamma \Vdash \square X \equiv \sim \diamondsuit \sim X$. I'm not exactly sure how to proceed here. I know that a modal model ...
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49 views

Gödel's Incompleteness theorem in modal logic

I think I once taught that the statement of the Gödel's Incompleteness theorem is equivalent to : $ \diamond \neg \square \square p $, if p is a complex enough system. Is this right?
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How to deal with propositional First-Order-Logic Dynamic Logic formula

Let p be an atomic proposition, let a be an atomic program, and let $π = (K, M)$ be a Kripke frame with $K = \{u, v, w\}$ $Mπ(p) = \{u, v\}$ $Mπ(a) = \{(u, v), (u, w), (v, w), (w, v)\}.$ The ...
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Proving $\Box A \rightarrow \Box\Box A$ from $KG_r$

I'm trying to prove $\Box A\rightarrow \Box\Box A$ from $KG_r$ where $G_r$ is the axiom $$\Box[ \Box A \rightarrow A ] \rightarrow \Box A$$ I'm given the hint that $$ A \rightarrow ((\Box\Box ...
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dynamic logic: how to understand the truth value?

how to understand the truth value of a dynamic logic formula? for example: $S = \{u_1,u_2,u_3,u_4\}$ is the set of four states $R(a) = \{(u_1,u_1), (u_2,u_1), (u_4,u_1)\}$ is after action a, like ...
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Show that the canonical modal for the modal logic s4.3 has no branching to the right

$$ S4.3 = S4 + \Box(\Box p \to q) \lor \Box(\Box q \to p) $$ We may use that the canonical modal of S4.3 is reflexive and transitive. A reflexive frame has no branching to the right if $$ \forall x ...
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Proving completeness of modal systems

In reading through Chellas' Modal Logic I'm trying to understand his proof of completeness. Here is my best outline of his proof: If $\Sigma$ is any normal system of modal logic then it has a proper ...
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Canonical models for systems

I've been struggling with understanding canonical models for systems. I'll state my question here, but at the end I define some terms in case they're not uniformly defined in Logic. My text claims ...
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Validity of $ F \supset \Box F$

I started to study propositional modal logic and Kripke semantics. I learned that for any Kripke interpration $\mathcal{M}$, we have that, if $\mathcal{M} \models A$ then $\mathcal{M} \models \Box A$. ...
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Nested Sets of Points

This question is inspired by modal logic, but reduces to a basic set theory problem. David Lewis in Counterfactuals claims that the answer to my question is "easily verified" but I can't figure it ...
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Finding a condition on models that validate theorems

I want to find a condition on standard models for which $\Diamond A \rightarrow \Box A$ is a theorem. I can see that the condition $\forall w_1,w_2,w_3\in W$ if $w_1Rw_2$ and $w_1Rw_3$ then ...
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1answer
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Replacing equivalent formulas and Rule N in epistemic logic

I am studying Epistemic Logic on my own from different books and articles. I have some knowledge of Modal Logic. In one of the basic steps in epistemic logic, I am having some difficulties. Please ...
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Validity at state w

Book called 'Modal Logic' has an definition for validity in page 125. It says in the first part: "A formula $\phi$ is valid at a state w in a frame F if $\phi$ is true at w in every model (F,V) ...
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Modal logic of contingency and necessity operators?

Well let me say that this is a challenging question, I am stuck in it myself :( I am fully aware of modal systems for necessity, i.e. being true in every accessible possible world, and possibility; ...
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Proof set of a sentence (error in textbook?)

I'm reading the Chellas book on Modal Logic and in one place he defines the proof set of a sentence relative to a system $\Sigma$, denoted $|A|_\Sigma$ to be the set of a $\Sigma$-maximal sets of ...
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Is intuitionistic logic translatable into modal logic S3?

I'm familiar with the translation of intuitionistic propositional logic into modal logic S4, summarized here and I've looked at one of the original sources for this: J. C. C. McKinsey and Alfred ...
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Fixed Point equivalence proof

Given two finite-state LTSs $L$ and $M$ (where $M$ is a "specific" subgraph of $L$) and the Hennessy-Milner-Logic monotone interpretation function $\Theta_F(S)$, where $F$ is the HML formula and $S$ ...
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$S5$ proof of $\diamond \square p\to \square p$

I am trying to solve exercises from the book Modal Logic by Patrick Blackburn, Maarten de Rijke and Yde Venema. I am having a problem to solve one of the exercises in the section 1.6, it is the last ...
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Modal Logic backward looking modality

For an exercise in Modal Logic I have to solve the next problem, can someone please help? Use generated submodels to show that the backward looking modality (that is, P of the basic temporal ...
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Frames characterised by the formula $ p \to \square_2 \diamond_1 p$

I am trying to solve exercises from the book Modal Logic by Patrick Blackburn, Maarten de Rijke and Yde Venema. I am having a problem to solve one of the exercises in the section 3.1, the exercise is ...
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What kind of proof system have Zalta used in “Basic Concepts in Modal Logic”?

I have read that text and I'm so interested in the proof theoretic style (as also claimed by Zalta that it is used in modern approaches to modal logic) in it: That is both more mathematically rigorous ...
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How - in a Kripke model - to define a world by modal formulas true only at them?

I'm currently using van Benthem's "Modal logic for open minds", ed. 2010. In page 16 (and later in exercises), he considers a model whose relations are shown by directed graphs (the so called process ...
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What the definition of validity of a formule in a possible Kripke-world in Modal Logic?

Basic question here but I cannot find the definition: Given a modal logic and a set of propositions $P$, a model $M=(W,R,V)$ where $W$ are possible worlds, $R$ an accesibility relation and $V$ a ...
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Express $V(\diamond \alpha)$ set theoretically in terms of $V(\alpha)$

I am reading Modal Logic. While going through the basics of the subject I am having problem in a place. Please help me. Say we are dealing with a frame $(W,R)$ and defined a model $M$ using a ...
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45 views

Properties of transitive modal frames

I am working through Fitting and Mendelsohn's First Order Modal Logic and have come across the following exercise: Prove that a frame $\langle \mathcal{G}, \mathcal{R} \rangle$ is transitive if ...
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Model-theoretic characterization of local modal correspondence

I've been reading van Benthem's dissertation (available on ILLC's website) on modal correspondence theory. In Section I.3, he develops a model-theoretic characterization of modal formulas having ...
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Examples of logical possibility

According to Wikipedia, something is logically possible if it doesn't imply a contradiction. In that case, how could a mathematical statement be false but possible? Wouldn't a false statement be false ...
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Is $\Diamond (p \rightarrow q) \rightarrow (\Diamond p \rightarrow \Diamond q)$ valid in K?

The modal logic K is the weakest normal modal system, comprised by classic logic augmented by (K), the necessity distribution axiom schema: $$\Box (\alpha \rightarrow \beta) \rightarrow (\Box \alpha ...
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How can temporal and epistemic logic be combined?

Recently I read all kinds of work from logic scientist in which epistemic logic was the main topic. Where epistmic change refers to change in knowledge of some agent in a multi-agent system (in a ...
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Modal Logic: ◊-Distribution

It's a theorem of K that $\diamond$ distributes to disjuncts and vice versa: $$\diamond(p \lor q) ≡ \diamond p \lor \diamond q$$ Does it distribute to negated disjuncts? Is the following a licit ...
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What is the connection between game theory and (modal) logic?

I'm interested in dynamic epistemic logic lately (reasoning about information and change in multi-agent systems). I also like game theory. I'm looking for some good resources about the connection ...
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What is the difference between intuitionistic, classical, modal and linear logic?

I am currently going through Philip Walder's "Proposition as Types" and a passage of the introduction has struck me: Propositions as Types is a notion with breadth. It applies to a range of ...
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Problems with basic proof in modal logic (event based)

I am having trouble deriving the following basic result: $\ast$) For every $\omega \in \Omega, \omega \in P (\omega),$ from the following axioms: A1) $K (\Omega) = \Omega$, A2) $K (A) \cap K (B) ...
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Is there a more useful formulation of the frame condition for the McKinsey axiom?

I am looking for a Kripke frame condition corresponding to the McKinsey axiom M: $\Box\Diamond p \rightarrow \Diamond\Box p$. I read somewhere the following condition: "For every partitioning of the ...
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121 views

Deduction theorem in modal logic

I am looking for a semantic for deduction theorem in modal logic,I wanna find a semantic way to prove this theorem,but I wasn't successful.tnx for your help
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How to prove that a set containing G$\phi$ and G$\neg \phi$is inconsistent without completeness but with soundness.

I'm stuck with this problem... The logic is a adaptation to temporal logic from $K_4$ of modal logic. The interpretation of G$\phi$ is always true in the future (now is not included). The axioms for ...
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Why “Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong” is paradoxical?

In a paper(see here) by Adam Brandenburger and H. Jerome Keisler, they give a game-theoretic impossibility theorem akin to Russell’s Paradox: Ann believes that Bob assumes that Ann believes that ...
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108 views

What is this symbol ($\Vdash$) called?

This symbol: $$\Vdash$$ What is it called? It is often used in modal logic, like this: $W\Vdash$. I looked for it in wikipedia/modal logic and wikipedia/logic notation, but could not find it.
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Conditions for total orders in temporal logic

Let $(T,>)$ be a frame of minimal temporal logic, i.e. a frame as defined in Kripke semantics where the relation is a partial order relation $>$ defined on the set $T$ of worlds, called ...
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$A\land FB\rightarrow F(PA\land B)$ in temporal logic

Temporal minimal logic $\mathbf{K_T}$ calculus is characterised by the following axioms, where $F=_{\text{def}} \lnot G\lnot$ and $P=_{\text{def}} \lnot H\lnot$: $G(A\rightarrow B)\rightarrow ...
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Equivalence between $\mathbf{KT_4}$ and Lewis' $\mathbf{S_4}$

Let us define modal logic system $\mathbf{KT_4}$ by adding the following axioms to classic propositional logic $\diamond A\leftrightarrow\lnot\square\lnot A$ $\square(A\rightarrow ...
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Symmetric relations and $\varphi\rightarrow\square\diamond\varphi$

I read that the schema $$\varphi\rightarrow\square\diamond\varphi$$ corresponds to the symmetric property (D. Palladino, C. Palladino, Logiche non classiche, 'non-classical logics', 2007) of the ...
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Euclidean relations and $\diamond P\rightarrow\square\diamond P$

I read* that the formula $$\diamond \varphi\rightarrow\square\diamond\varphi$$is valid in a structure $(W,R)$, intended as in Kripke semantics, -i.e. that it is true for any interpretation $I$ and in ...
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Logical consequence in all structures in Kripke semantics

I read* the following definition of logical consequence in all structures within Kripke semantics:$$X\models A\iff\text{ for every } (W,R),\text{ if }(W,R)\models X,\text{ then }(W,R)\models A$$ ...
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What is the relation between three-valued Kripke frame semantics and modal systems K, T, S4?

I'm looking for literature that discusses three-valued Kripke frame semantics (strong Kleene or Łukasiewicz) and their relation to the (two-valued) normal modal systems K, T, S4 and extensions of S4, ...
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Hilbert style proof for $\Box A \vee \Box B \rightarrow \Box(A\vee B)$ in K.

I have to find a formal Hilbert style proof for $\Box A \vee \Box B \rightarrow \Box(A\vee B)$ on modal logic, K. I can use all classical propositional tautologies, Modus Ponens and Distribution ...
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Finitary assignment functions and typed modal languages

I'm working through Giovanna Corsi's article on Counterpart Semantics for Modal Logic. She is working with a typed modal language $\mathscr{L}^{t}$. Where this differs from usual presentations I've ...
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Temporal Logic Tautology

I have the following question: if it is necessary that p -> p = tautology? I think it's not, and I am showing my example for the contradiction, below: ...
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How do I notate this statement about a state of affairs (similar to a possible world)?

I'd like to notate this statement formally: If any given agent desires that a certain state of affairs obtains, then there is no state of affairs in which she enjoys greater security than that one. ...