Questions tagged [kripke-models]

This tag is for questions relating to "Kripke’s models" for modal logic (or variants thereof) are the basis for many modern approaches to reasoning about knowledge and belief. For philosophers, by far the most important examples are ‘Kripke models’, which have been adopted as the standard type of models for modal and related non­classical logics.

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On provability in modal logic

Consider the classical modal logic $\mathsf K$, given by the following axioms and rules over a language containing the standard propositional connectives and $\Box$: A complete set of axiom schemes ...
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characterization of non-reflexive Kripke frames

Can the class of non-reflexive (that is not being reflexive of course) be characterized with a set of modal formulae? What about irreflexive (having only non-reflexive points)?
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Counterexample of Goldblatt-Thomason theorem

The Goldblatt-Thomason theorem states that a class of first-order definable Kripke frames is modally definable if it is closed under disjoint unions, is closed under generated subframes, is closed ...
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283 views

How to prove in K $ \vdash_{p \rightarrow \lozenge \square p} (\square p \rightarrow \lozenge p)$?

Let us denote $C$ the modal system obtained by adding the axiom $ \alpha \rightarrow \lozenge \square \alpha$ to the axiom $K$ and all the propositional tautologies. As said in the title, I'm looking ...
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What does the category of RDF models look like in Institution Theory? [closed]

This question has been represented on cstheory.stackexchange.com where it is getting answers. The Question in short Here is the question in its pure form. Details of my reasoning can be found below. ...
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$\mathcal{M} \models \Box \phi \rightarrow \Box \Box \phi$ for all $\phi$ if and only if $\mathcal{M}$ is transitive

Exercise 1.8.2 in Fitting and Mendelson's "First Order Logic" asks to show that $\mathcal{M} \models \Box \phi \rightarrow \Box \Box \phi$ for all $\phi$ if and only if the accessibility relation of $\...
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Understanding The First Axiom Of Gödel's Ontological Proof

On Wikipedia, the first Axiom of Gödel's ontological proof is $$(P(\phi)\land\square\forall x(\phi(x)\Rightarrow\psi(x)))\Rightarrow P(\psi),$$ I assume there are implicit quantifiers present for $\...
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Proving the Glivenko theorem via Kripke models

We'll prove it in just one direction, since the other one is obvious. So, assume $\psi$ is a theorem of classical propositional logic. Prove that $\lnot \lnot \psi$ is a theorem of intuitionistic ...
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Proving $\lnot \lnot (\psi \lor \lnot \psi)$ is a theorem of intuitionistic propositional logic

Here, $\psi$ is some arbitrary formula. The proof I've come up with is as follows. Assume $\lnot \lnot (\psi \lor \lnot \psi)$ is not a theorem of IPL, which means there exists some Kripke model ...
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What is the Upper-Bound for a Kripke Model in Normal Modal Logics?

I am currently looking for a paper for a proof about the upper-bound on the size of a Kripke model in modal logic, no matter the axiom considered. I am considering only the propositional modal logic ...
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144 views

Determine a modal logic formula which a connective that is not valid but is true

I'm trying to understand how a formula can not be valid but also true in the above question.
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Proove that all finite non-standard Kripke frames are standard

In Harel's book on PDL (Propositional Dynamic Logic) I've learnt that Kripke frames are pairs such as: K = (K; $m_k$) Where $K$ is a set called states and $m_k$ is a meaning function assigning ...
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259 views

Counter examples in modal logic

As I'm new to modal logic, I wanted to check whether my counter examples for the given formula is right. $$ \Box A \rightarrow \Diamond B \Rightarrow \Box(\Box A \rightarrow \Diamond B) $$ First I ...
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How to show that someone beleive someone don't beleive $\phi$ with Kripke models?

I want to model a logic of beliefs with two agents where : $$M,w^*⊨\phi\wedge Ba\phi\wedge Bb\neg\phi\wedge BaBb\neg\phi$$ I think I'm using Kripke semantics. I think that i.e. it describe a world ...
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135 views

Minimal Modal Logic and the Class of all Kripke Models

It's well known that minimal modal logic (i.e. propositional tautologies and axiom $K$ together with modus ponens and necessitation rule) captures the validities of the semantic class of all Kripke ...
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126 views

Church-Rosser property and normal modal logic

I am trying to prove soundness and completeness for S4.2 and I am considering Kripke frames which are reflexive, transitive and have the Church-Rosser property. Now, there is one thing that really ...
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408 views

Modal logic Box and diamond T

I didn't understand what is mean Box T, diamond T in this example: what is mean box True exist in world v?
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Are there any invalid $S5$-formulas $\psi$ such that $\diamond\psi$ is valid?

I cannot find an example for an invalid $S5$ formula $\psi$ (i.e. $\nvDash\psi$), such that $\diamond\psi$ is valid (i.e. $\vDash\diamond\psi$). If there is none, then $\vDash\diamond\psi\Rightarrow\,\...
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I am looking for a soundness, completeness and consistency proof for this particular $S5$ calculus.

I know that it suffices to add the axioms $(T)~\square\psi\rightarrow\psi$ $(K)~\square(\psi\rightarrow\varphi)\rightarrow(\square\psi\rightarrow\square\varphi)$ $(5)~\diamond\psi\rightarrow\square\...
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Helping me to prove H axiom in S4.3

We have frame $\mathcal{F} = (W,R)$ that $R$ is reflexive and transitive and $\forall x,y,z (xRy \wedge xRz \wedge y\ne z \rightarrow yRz \vee zRy)$ ,prove $\mathcal{F} \models \square(\square p \...
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194 views

Show that (using Kripke models) formula $\phi:p\vee (p\to q)$ is not tautology in intuitionistic logic

Show that (using Kripke models) formula $\phi:p\vee (p\to q)$ is not tautology in intuitionistic logic. My proof* Above, you can see my trial. Check it please and try to help me defeat my ...
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363 views

Proof of the disjunction property

I am trying to prove the disjunction property "if $\,\vdash\phi\lor\psi$ then $\,\vdash\phi$ or $\,\vdash\psi$" for intuitionistic propositional logic. So far I thought about choosing two non-...
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Different ways to define Kripke structure

The Wikipedia page https://en.wikipedia.org/wiki/Kripke_structure_(model_checking)#Example has an example of a Kripke structure $M = (S,R,L)$; however, others define Kripke structures as $M = (S,R,V)$ ...
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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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321 views

What is the difference between First-Order Structures and Kripke Structures?

In the SEP article on Model Theory by Wilfrid Hodges (here), he writes: Particular kinds of model theory use particular kinds of structure; for example mathematical model theory tends to use so-...
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515 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 and ...
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58 views

Persistency on formulas in Kripke models

Let $ (W,R,f) $ a Kripke model. I have some trouble proving that the persistency property holds for formulas i.e if $ wRw' $ and $ w \Vdash \phi $ then $ w' \Vdash \phi $ , mainly due to the forcing ...
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146 views

what does the “fixed point” in fixpoint algorithms refer to?

I was reading the following powerpoint here to remember something I studied a long time ago. http://www.cs.cmu.edu/~emc/15-820A/reading/lecture_1.pdf the 12th slide is labeled fixpoint algorithms, ...
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81 views

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 instants....
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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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225 views

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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478 views

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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173 views

Proving the non-derivability of formula using the of the kripke model

I try proving the non-derivability of $(p\to q) \to \lnot p \lor q$, using the of the kripke model. I tried using different combinations of $Wi$, but I get fail.
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The definition of interpretation in a Kripke model collides with my intuition of what it should do

In Lindröm and Segerberg (2007) exposition of a Kripke model, with frame $F= \langle W,D,R,E,w_0\rangle$, they define an interpretation $I$ as a family of functions $I_w$, where $w$ ranges over $...
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1answer
107 views

Propositional S5: is there a consistent set requiring continuously many worlds?

A recent question asked whether in systems of modal propositional logic having the "finite model property" there are consistent sets of sentences that were not satisfied by a finite model. @Carl ...
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323 views

Propositional modal logic: infinite models required in systems with finite model property?

A system of propositional modal logic has the "finite model property" if any consistent sentence is satisfiable at a model with finitely many possible worlds. Some systems have this property and ...
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101 views

Proving Gabbay rule for Modal Logic

I'm currently working on exercises of the book "Modal Logic" by A.Chagrov and M.Zakharyaschev (for pleasure, not homework). One exercise asks to prove this version of Gabbay rule (exercise $3.10$): A ...
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51 views

Explanation of states, worlds and models?

Can someone explain to me how the concepts of states, models and worlds work together in Kripke semantics? I've been trying to piece together how the parts work are linked together but cannot figure ...
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115 views

Question concerning satisfiability in a certain Kripke model

My question concerns the exercise on p.77 of Boolos, Logic of Provability: True or false: if $A$ is satisfiable in some finite transitive and irreflexive [FIT] model and contains at most one ...
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175 views

A question on intuitionistc propositional logic

Prove that: Two finite rooted frames are isomorphic iff they validate the same formulas. (This is an exercise in the book "Modal Logic" by A.Chagrov and M.Zakharyaschev)