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

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Formalizing a self referential sentence

In The logic of provability, by G. Boolos, we are asked to ponder about this statement: If this statement is consistent, then you will have an exam tomorrow, but you cannot deduce from this ...
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Is the 1-consistency of $PA$ necessary to prove that $\diamond^{m} \top\implies \diamond^{n} \top$ is false if $m<n$?

In The logic of provability, by G. Boolos, there is a remark in chapter 7 saying that $\diamond^{m} \top\implies \diamond^{n} \top$ is false if $m<n$ (unless $PA$ is 1-inconsistent). Now, it seems ...
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Valuation in the proof decidability of normal systems of modal logic

In The logic of provability, by G. Boolos, it is stated in the chapter 5 that if L is a formal system between $GL$, $T$, $B$, $K$, $K4$, $S4$, $S5$; then $L\vdash D$ iff $D$ is valid in all models of ...
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On the truth of $GLS$ and Löb's theorem

Consider the formal system $GLS$, whose axioms are the theorems of $GL$ plus all sentences of the form $\square A\rightarrow A$. A translation maps a sentence of modal logic to a sentence in the ...
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Categorical semantics for dynamic epistemic logic

Dynamic epistemic logic tries to reason about knowledge that certain actors (people, machines, etc.) have and how it can change in response to outside events. It is usually possible to discuss such a ...
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The positive introspection axiom

I am studying modal logic with the textbook 'Reasoning about Knowledge' Fagin et al. 1995 The positive introspection axiom is taken as something that can be proved with the possible worlds model of ...
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modal logic - examples for it may be supposed that/it is compulsory that

Could someone give me example with modal logic ? $\diamond X$ it may be supposed that $\Box$ it is neccessary that. I mean some example with worlds and arrows between them. Why am I asking about it ?...
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Consistency Lemma in Lindenbaum's Theorem

Let $\Lambda$ be a modal logic, we say that a formula $\varphi$ is $\Lambda$-inconsistent if $\vdash_\Lambda (\neg \varphi)$ and is consistent otherwise. Similarly we say that a set of modal formulas $...
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Modal extensions (operators) for monoidal (categorical) logics

There is nice generalization of first order logic to monoidal (categorical) logics http://www.springer.com/us/book/9783642128202 which has recently been applied extensively as replacement for deontic ...
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Gödel's ontological proof and “modal collapses”

Recent findings on Gödel's ontological argument allowed to ultimately establish a couple of things: Gödel's original axiomata are inconsistent Scott's variation instead is consistent Scott's axioms ...
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Generated subframe of a frame

Let's define that $F=(W,R) $ is a Kripke frame where $W=\mathbb{Z}$ and $R=\{(u,v)\mid v=u+1 \}$. Then we can have generated subframe $F_{0}=(W_{0},R_{0})$, where $W_{0}=\mathbb{N}$ and $R_{0}=\{ (u,...
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Find an equivalent standard translation for FO formula

Let $\psi(x):=\forall y\exists z\forall u(\neg R(x,y)\vee (R(y,z)\wedge \neg R(z,u)))$. We must find a equivalent form of $\psi(x)$ such that we can apply standard translation on it and find modal ...
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Proof that standard translation of modal formula is equivalent to the FO formula

For example $\varphi:=\lozenge\lozenge p\rightarrow\lozenge p$ defines transitivity and it has a standard translation $St_{x}(\varphi):=\forall P\forall x(\exists y(R(x,y)\wedge \exists x(R(y,x)\wedge ...
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$IC(U\cap V)$ VS $IC(U)\cap IC(V)$

Let $(X, \tau)$ be a topological space, $U,V\subseteq X$ any subsets of $X$ . Let $I(A)$ denotes a interior of subset $A$ and $C(A)$ denotes closure of subset $A$. It is clear that $IC(U\cap V)\...
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Does the fact that a modal operator distributive over disjunction imply that a modal operator is distributive over conjunction?

If L is an arbitrary operator on two propositions p and q: Does L(p $\vee$ q) $\Rightarrow$ Lp $\vee$ Lq imply L(p $\land$ q) $\rightarrow$ Lp $\land$ Lq?
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Why is “necessary p” true in a world when there is no world accessible from it?

So my question is situated in modal logic and everything is defined as usual. I'm reading volume 2 of logic, language, and meaning and on page 24 it says: $V_{M,w_3}(\square p)= 1$ So the valuation ...
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Use of □ in Modal Logic, Translated into English

For □, do you read it as “for all states. . .”, that which one, maybe, cannot withdraw from.? Does this, □, not request proving of itself? □P↔¬◇¬P How do you use it in English? If, for all ...
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(Green/blue)-eye logic puzzle. Statement validation

There is a logic puzzle aiming on freeing same-color-eyed people from an island. The thing is that they must be certain of their own eye color so that they can leave. For that reason an external party ...
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Solovay's Arithmetical Completeness Theorem

Solovay's Arithmetical Completeness Theorem affirms that: Let $A$ be a sentence of modal logic. If every realization $A^\phi$ of $A$ is proved by $PA$, then $GL\vdash A$. I am troubled by this ...
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About the disprovability of the Gödel Sentence

In the last exercise of the last chapter Computability and Logic of Boolos et al we are asked to use $GL$'s arithmetical soundness theorem to find a weaker assumption than the $\omega$-consistency of $...
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Is there a kind of modal logic that models boolean satisfiability?

For example, if $p$ is a boolean proposition depending on truth values $q_1, q_2 \dots ,q_n$, can we add a modal operator such that $\diamond p$ holds if and only if there exists some boolean ...
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How to handle degrees (numerical attributes) in logic? How to model “quantitative changes lead to qualitative changes”?

I am using logics (propositional, predicate, modal) to model one domain, but there are variables that have non-boolean domains, these variables are degrees (it is sufficient that they are degrees, ...
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Interpretation of relations in varying-domain models of F.O. modal logic

I am studying the book "First Order Modal Logic" By Fitting and Mendelsohn. In their definition of interpretation for varying domain models (def 4.7.3 pg 103), the interpretation of a relation in a ...
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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 $\mathcal{...
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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 A\...
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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 $u_2$...
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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 out....
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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 $w_2=...
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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; i....
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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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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 ...