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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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Modal logic with an omniscient actual world

(This post is similar to but different to another post I have since deleted because it actually digressed on a topic tangential to what I originally wanted to ask and which is what this post is about.)...
noname_lonestar's user avatar
1 vote
1 answer
81 views

False statements in intuitionistic logic

In the explanations of intuitionistic logic I've been reading (1, 2, 3), especially in the explanation of the semantics, I don't understand how a proposition being false influences the situation. ...
bobismijnnaam's user avatar
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125 views

Describe the set of Kripke scales

Describe the set of Kripke scales in which the formula $\square(\square p \to p) \to \square p$. is generally valid. it seems that this is just a set of scales in which a loop necessarily comes out of ...
Quark's user avatar
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1 answer
100 views

How to prove that a formula is intuitionistically valid using Kripke semantics?

I want to know how to use Kripke semantics so that I can prove that a formula is intuitionistically valid. I think that all others cases will clear out if I understand the case of implication. Let's ...
Νικολέτα Σεβαστού's user avatar
2 votes
1 answer
83 views

Intuitionistic proof of $((p\rightarrow q)\rightarrow p)\rightarrow\neg \neg p$

I need to prove that the $\psi=((p\rightarrow q)\rightarrow p)\rightarrow\neg \neg p$ is intuitionistically valid. I tried using the topology of open sets of $\mathbb{R}$ and an arbitrary valuation, ...
Νικολέτα Σεβαστού's user avatar
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24 views

Related work for a 'compatibility' modality in modal logic meaning ◊(𝜙∧𝜙′)

Fix some set of propositional atoms $\mathsf{Atoms}$ and consider a (very simple) propositional logic with a binary compatibility modality $\mathsf{C}$ as follows: $$ \phi,\phi' ::= A,B,C\in\mathsf{...
Jim's user avatar
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1 answer
121 views

Linear temporal logic question

I am curious about what if we have this Kripke model and I am trying determine the set of worlds where $\Box\Box p$ is satisfied This is the part I am confused about when we have $\Box$ I am ...
zellez11's user avatar
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1 answer
105 views

Is there a simpler Kripke counter-model for this formula?

$\forall x \neg \neg \phi(x) \to \neg \neg \forall x \phi(x)$ is not intuititionistically valid. I can come up with a complicated Kripke counter-model as follows: Let there be a countably infinite ...
PW_246's user avatar
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2 votes
1 answer
110 views

Constructing a Kripke model where $p \rightarrow \Box \Diamond q$ is false.

I have constructed the following Kripke model for this problem: My idea is the following: Implication is false iff we have $ \top \implies \bot$. For world $0$, we have that $p$ is true. Now we need ...
l0ner9's user avatar
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Apparent counterexamples to claims of tautologies in modal logic

I'm reading the paper "A fixpoint semantics and an SLD-resolution calculus for modal logic programs" by L.A.Nguyen, and in the paper he asserts that certain tautologies hold, but it seems ...
Apollo's user avatar
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If you have a tableau proof for $\Box A$, show that there is also a tableau proof for A.

If you have a tableau proof for $\Box A$, Show that there is also a tableau proof for A. Here is my attempt but I'm not sure if it's correct: If we have a tableau proof for $\Box A$, it means that all ...
Len's user avatar
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1 answer
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Tableau Calculus for Reflexive Frames

How would you create a tableau calculus for modal logics that have reflexive frames? What rules would need to be added to the existing system K? My attempt is to refer to the axiom scheme for ...
lowlypalace's user avatar
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0 answers
73 views

The semantics of until operator in linear temporal logic

According to the definition of until operator from Wiki: $w \models \varphi~\text{U}~\psi$ if there exists $i \geq 0$ such that $w^i \models \psi$ and for all $0 \leq k < i, w^k \models \varphi$. ...
maplgebra's user avatar
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Requirement of the Second part of PDL Filtration Lemma

I was reading this filtration lemma of PDL in David Harel's book Dynamic Logic. The Filtration Lemma: Let $\kappa = \langle W, \mathcal{R}, V\rangle$ be a Kripke model of PDL and let $u, v\in W$: (i) ...
Avijeet Ghosh's user avatar
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2 answers
255 views

Intuitionistic logic, tree-like Kripke model

There is a tree-like Kripke model in which the set of worlds $\mathfrak{W}$ is ordered as a tree: (a) there is a smallest world $W_0$ (b) for any $W_i \ne W_0$ there is a unique preceding world $W_k: ...
Ray Cox's user avatar
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3 votes
1 answer
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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 ...
Paul's user avatar
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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 ...
Paul's user avatar
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2 votes
2 answers
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About interpretation of accessibility relations in Kripke like structure

A Kripke like structure is something like tuple $(S, R, V)$ in which $S$ is a set of states, $R$ is the relationship between states and $V$ is an evaluation function that maps propositions to some ...
Doralisa's user avatar
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1 answer
319 views

Find and prove a necessary and sufficient condition for a Kripke Frame

I am studying Kripke frames in modal logic and I am trying to understand how to solve the task below (the task comes from a workbook and this particular question lacks a conclusion). I know that ...
kabin's user avatar
  • 391
4 votes
3 answers
280 views

Why is Normality an axiom in Modal Logic?

Why are we having $\square(A → B) → (\square A → \square B)$ as an axiom when we can prove* that if we have $\square(A → B)$ then we will have $(\square A → \square B)$? *$B$ is true in every world by ...
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1 answer
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Exact definition of SAT problem in S4 [closed]

I kind of know what SAT problem is and I need the exact formal description of this problem for S4 logic. I am trying to prove this problem is PSPACE-Complete! Any help would be very much appreciated!
Senmorta's user avatar
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2 answers
357 views

Why S4 logic should have kripke models with reflexive and transitive relations?

From here and here you can see that in S4 modal logic we should have Kripke models (S,R,V) which have a relationship R on their set of states S so that R is reflexive and transitive on S! I know that ...
Senmorta's user avatar
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0 answers
152 views

Necessity and sufficiency on dead end worlds in Kripke's model

Suppose we are evaluating $\square p$, $\diamondsuit$p, $\square\square$p, $\diamondsuit\diamondsuit$p, $\square$$\diamondsuit$p, $\diamondsuit$$\square$p on a dead end world (the world that can't ...
Glavatar's user avatar
2 votes
1 answer
108 views

Semantic explanation for converting intuitionistic logic into classical logic by adding LEM as an axiom

I have a question about converting intuitionistic logic (IL) into classical logic (CL) by adding LEM as an axiom. IL is usually understood as a logic without LEM. $$\textrm{LEM}:=A\vee\neg A.$$ In ...
ferdinand's user avatar
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1 vote
1 answer
67 views

Accessibility relation in non-classical logics: hereditary or not?

Recently, I am reading course materials on intuitionistic and modal logics. I have two questions about the notion of accessibility relation in Kripke semantics for intuitionistic and modal logics. ...
ferdinand's user avatar
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0 answers
93 views

Proofing transitivity of simulation relation in Kripke structures

I would like to proof transitivity of simulation in Kripke Structures. Proof Sketch: Let A, B, C be Kripke Structures, with A=(Sa,S0a,Ra,AP,La) B=(Sb,S0b,Rb,AP,Lb) C=(Sc,S0c,Rc,AP,Lc) H ⊆ S × S′ ...
Pawn Powl's user avatar
1 vote
2 answers
178 views

Why is $\lozenge (p\to p)$ not valid in system $K$ of modal logic?

Why is $\lozenge (p\to p)$ not valid in system $K$ of modal logic? How could this formula be false in any accessibility relation or setting of values in worlds? Even if it was a dead end world, ...
whatsgoingon's user avatar
1 vote
1 answer
65 views

Is the statement $(\exists x\psi(x)\rightarrow\forall x\theta(x))\vdash \forall x(\psi(x)\rightarrow\theta(x))$ true in intuitionistic/minimal logic?

I know that this sequent is true for classical logic. But is it true in minimal or intuitionistic logic? What would be a Kripke model that doesn't satisfy the property? I would appreciate some (as ...
H. Walter's user avatar
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2 votes
0 answers
86 views

How does the problem of quantified modal logic relate to Kripke's model

Throughout multiple papers and reviews, Kripke's models are viewed as a countertheory to the work of Quine on the three grades of modal involvement. I understand how the modal is set up: I consider a ...
Maurice's user avatar
  • 594
2 votes
1 answer
151 views

Is Kripke/relational/possible world semantics the special case of the the set semantics or first order logic?

I am reading https://www.springer.com/gp/book/9783319225562 about expression of modal operators as the predicates in the first order (and possibly higher order) logic. My question is - is the modal ...
TomR's user avatar
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3 votes
0 answers
93 views

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 ...
blub's user avatar
  • 4,813
2 votes
0 answers
67 views

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)?
AnikiAliz's user avatar
2 votes
1 answer
206 views

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 ...
Vsotvep's user avatar
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2 votes
1 answer
377 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 ...
Bijco's user avatar
  • 188
4 votes
1 answer
166 views

$\mathcal{M} \models \Box \phi \rightarrow \Box \Box \phi$ 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 $\...
DSM's user avatar
  • 375
2 votes
0 answers
137 views

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 $\...
fweth's user avatar
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1 vote
0 answers
280 views

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 ...
0xd34df00d's user avatar
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3 votes
1 answer
149 views

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 ...
0xd34df00d's user avatar
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0 votes
0 answers
81 views

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 ...
Valentin Montmirail's user avatar
0 votes
1 answer
415 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.
Daryn Wilkinson's user avatar
0 votes
1 answer
739 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 ...
user3285532's user avatar
1 vote
0 answers
40 views

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 ...
Revolucion for Monica's user avatar
0 votes
1 answer
204 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 ...
LoMaPh's user avatar
  • 1,371
1 vote
1 answer
337 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 ...
D.Q.'s user avatar
  • 358
0 votes
1 answer
821 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?
Amit's user avatar
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6 votes
1 answer
224 views

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\,\...
xamid's user avatar
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1 vote
0 answers
271 views

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\...
xamid's user avatar
  • 258
0 votes
1 answer
134 views

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 ...
theSongbird's user avatar
1 vote
1 answer
122 views

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 \...
haleh's user avatar
  • 133
1 vote
1 answer
477 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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