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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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How to prove this sequent system is sound and complete for S4 Modal Logic?

Take LK sequent calculus without cut, but change the $\to R$ rule to the following: $\cfrac{\Gamma’, A \vdash B}{\Gamma \vdash A \to B, \Delta}$ where $\Gamma’=\{C \in \Gamma|C=(D \to E)\}$ for well-...
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Why Does This Proof Hold?

I'm currently reading "Mathematics Without Numbers" by Hellman, G., and I'm on pages 26-27. It seems like Hellman is discussing opposition to viewing mathematical proofs solely through the ...
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Is $\Diamond \Box p$ "implies" $\Box \Diamond p$ a theorem in any modal logic?

I'm new to modal logic. Is it a known theorem in (one of the many) modal logic(s) that $$\Diamond \Box p \quad \text{"implies"} \quad \Box \Diamond p\qquad?$$ More precisely, is either $$\Diamond \Box ...
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Would this logic be considered constructive?

I have asked about similar logics before, but this one is different. The logics that I’ve asked about in the past take the Gödel-McKinsey-Tarski translation for Intuitionistic Propositional Logic to ...
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What is the relation between formal logic and grammar?

Does anyone know where I might find set-theoretic (or broadly formal) explications of grammatical categories? I’m interested in something like what Brandom describes here One of the reasons I went to ...
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Validity and substitution

In Fitting and Mendelsohn's textbook on First order modal logic, the concept of validity is defined only for sentences (i.e, formulas without free variables). How usual is this? Doesn't it have a ...
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Studying modal logic using category theory

When reading about modal logic, namely general frames and algebras, I've been seeing a lot of potential functors and/or universal properties. Like constructing an algebra with Boolean operators from a ...
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Prove in K modal logic $\square p , \neg \square q \vdash_K \neg \square (p \rightarrow q) $

I need to prove in K modal logic the following $\square p , \neg \square q \vdash_K \neg \square (p \rightarrow q)$ $1. \square p \qquad premise$ $2. \neg \square q \qquad premise$ $3. \square (p \...
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Is there a constructive proof of the 4 axiom in $\sf Grz$?

Willem Blok managed to prove that the 4 axiom of modal logic $\Box \phi \to \Box \Box \phi$ is provable in $K+ \Box (\Box(\phi \to \Box \phi) \to \phi) \to \phi$. Let $\psi=(\phi \land (\Box \phi \to \...
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Motivation behind axiom 4 in Gödel's ontological proof. [closed]

Axiom $4$ states: $$\textbf{Ax. 4.}\quad P(\varphi) \implies \square P(\varphi)$$ I don't understand the logic behind this axiom. In general, I don't understand the distinction between $p$ and $\...
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Prove that if a frame is dense then $\Box \Box A \to \Box A $ is valid [duplicate]

I am having trouble with the following question on modal logic. A modal logic frame $M =\langle W, R\rangle$ is dense whenever $$\forall x\forall y(Rxy\rightarrow\exists z(Rxz \land Rzy))$$ Prove that ...
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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.)...
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Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos?

I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the ...
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Is "If ⊨□φ then ⊨φ" true in modal logic?

(Under propositional modal logic, system K) There's seems to be an obvious counterexample: A model with one world w s.t. $\phi$ is not true in $\phi$ and $\lnot R(w,w)$. But consider this argument: ...
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Prerequisites for Cocchiarella modal logic book?

I’m hoping at some point to begin working through Cocchiarella and Freund’s book Modal Logic: Its Syntax and Semantics. Have any of you used this book before? What would you say are its prerequisites, ...
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Proving Completeness of Modal Logic (K) with Minimal Assumptions

I'm reading van Bentham's Modal Logic for Open Minds (section 5.7), where he proves the (weak) completeness of K. I'm trying to work out each step, but I seem to need to use an intuitionistically ...
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Does Hennessy-Milner Theorem hold when I weaken its condition (image-finite) a little?

Hennessy-Milner Theorem says that For two image-finite models M,N, we have that the pointed models M,w and N,v are equivalent in semantics (all holds on M,w holds on N,v and vice versa) iff M,w and N,...
Cleanlee's user avatar
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why this procedure of proof valid?

I READ A new introduction to modal logic at page 176-177 it said in modal logic, irreflexive frame preserves the one rule, i.e. Gabb. and proof's content is as follow : I am having trouble ...
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Admissibility of Löb's rule in basic modal logic K

While I was preparing a talk on the admissible rules of modal logic, I found the following fact in Wikipedia (see https://en.wikipedia.org/wiki/Admissible_rule#Examples). It says that Löb's rule $(\...
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Which logic is most fundamental? [duplicate]

A couple of my introductory logic books appeal to modal and set-theoretic notions in building up first-order logic. (They explicitly acknowledge these connections and say, for example, that validity ...
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Modal logic validity of axiom K - proof

I'm studying modal logic and in Chapter 3 of this book, one of the proposed exercises (3.4) is to show that distribution of necessity operator over implication is valid, that is, \begin{equation} \...
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Doing an modal logic exercise. The tableau for both the argument and its negation are open. Am I doing something wrong?

I am doing an exercise from Graham Priest's textbook "An Introduction to Nonclassical Logic", and I think I'm doing something wrong. The exercise is to determine the validity of α=β, ◇Pα, ⊢ ◇...
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S5 as the adjunction of possibility and necessity

I found this statement in the wikipedia article for the modal system S5: I looked at the reference but it just states the claim without any further commentary, just like the wikipedia article. Can ...
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Is there a way to define classical implication in this logic?

I’m asking this question so that I may provide my own answer to it and share what I’ve discovered. I’ve already posted about a logic that results from modifying the Gödel-McKinsey-Tarski translation ...
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Simple modal logic with one possible world and one impossible world. Which logic is it?

I've heard of impossible worlds being used in modal logic before. (This question is one such question with an impossible world in it.) I don't understand exactly how they're intended to be used in the ...
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Proofs of Theorems with naive Modal Logic to motivate their incorporation into formal Modal Logic

I want to show why Modal Logic tries to incorporate the two following implications $\square A \to \square \square A$ $\diamond A \to \square \diamond A$ as valid sentences by showing that a natural ...
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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 ...
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adding axioms to K logic [closed]

Let $K$ be the modal logic extending classical propositional logic by adding the necessitation rule N: if $\vdash A$, then $\vdash \square A$ and the distribution axiom K: $\square(A \rightarrow B) \...
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Is this restriction on Modus Ponens sound for Visser’s Basic Propositional Logic?

I’ve been looking into some different logics recently, and I think I’ve figured out a way to do Modus Ponens in systems like Visser’s Basic Propositional Logic (BPL) and Formal Propositional Logic (...
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Do modal calculi work with possible worlds?

I use a natural deduction calculus for modal propositional logic, but my question eventually is about any (sound) modal calculi with/without axioms. Just as an example take a proof like $\square$A $\...
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Is the logic $\mathsf{wK4}$ plus Lob's rule the same as $\mathsf{GL}$?

Let $\mathsf{wK4}$ be the logic of weak transitive frames and $\mathsf{GL}$ be the provability logic. It is well-known that one can define $\mathsf{GL}$ as $\mathsf{K4}$ (the logic of transitive ...
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Axiomatic proof in modal system $⊢_{K5} □(□p → p)$

I’m having a hard time proving $⊢_{K5} □(□p → p)$. Proving validity in Euclidean frames.in this from axiom 5 is valid $◇p → □◇p$ I’m able to derive $□p → □(□p → p)$ I,m also able to derive from Axiom ...
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Techniques for showing that sole sufficient operators of a given arity do not exist for a given algebra

I came across this question yesterday which is interesting. It asks whether a binary sole sufficient operator for the modal logic K exists. I tried to find an example of a sole sufficient operator for ...
Greg Nisbet's user avatar
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How to construct filtrations in modal logic that preserve a specific properties to prove finite model properties.

I am following a course on modal logic and I have issues with a specific area, namely filtrations that preserve frame properties. A definition for a filtration is given here: https://en.wikipedia.org/...
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Two questions about the partial order of propositional modal logics

The propositional modal logics $T$, $B$, $S4$ and $S5$ are related as follows: $T$ has the fewest theorems, $S5$ has the most, $B$ and $S4$ are intermediate between $T$ and $S5$, but they are ...
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Can the empty set be defined as a contradiction in logic? [closed]

Can the empty set be defined as something like this in logic? $$ \square\left(\exists x\mid E(x)\iff \exists x: \neg x=x\right) $$ Also, how should the empty set be defined in logic? Thanks and ...
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1 answer
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recommendations for studying modal logic

I'll be reading a textbook on modal logic with my colleagues this winter (for 3-5 months). I might need to teach them many points from the textbook because I have a basic understanding of ...
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Semantic Deduction Theorem in Modal Logic

The semantic version of the deduction theorem says that $\{A_{1}, …, A_{n}\}\vDash$ B $\iff \vDash A_{1}, …, A_{n} \to$ B. I read that the deduction theorem does not hold in modal logic, but upon ...
God's user avatar
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How many modal logics are there? [closed]

How many propositional modal logics exist? Is it a finite number, countably infinite, or continuum-sized? Of course, to answer this question, we would need a definition of a propositional modal logic.
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in modal set theory, why it is issue?

I have been studying the Iterative Set Concept within the context of the paper titled "Modal Set Theory" from Menzel specifically on pages 11-12. "As we’ve just seen, the iterative ...
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A question about S4 modal logic

I’ve been doing some proofs in S4 recently and noticed that the following holds: If $\phi$ is a wff and all positive literals $A$ that occur in $\phi$ are prefixed by either $\Diamond$ or $\Box$, ...
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Taking the derived set in a topological space twice

For a topological space $(X, \tau)$ and a set $A \subseteq X$, let $\delta A$ denote the derived set of $A$. That is, $\delta A$ is the set of limit points of $A$. In other words, $\delta A$ contains ...
Jim's user avatar
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A question on identity in quantified modal logic

One can prove that $a=b \to \Box a=b$ by using the substitution axiom for equality. However, since $\neg a=b$ is not of the form $\alpha_1 =\alpha_2$, I don’t see how to prove $\Diamond a=b \to a=b$, ...
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The first source on a modal logic without the replacement property,

The replacement property for a modal logic states that: $\text{ If } (p \leftrightarrow q) \text{ ,then } (\Box p \leftrightarrow \Box q)$ sometimes modal logics closed under this property are known ...
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What is this sort of signature in model theory?

I was browsing this set of lecture notes for a mathematical introduction to modal logic. I'm not familiar with model theory, and the lecturers (who approach modal logic from a model-theoretic ...
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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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What do algebraic semantics look like for intuitionistic modal logic?

I know that a topological pseudo-Boolean algebra is an algebra ⟨L, I, ¬, ∧, ∨, →⟩ such that ⟨L, ¬, ∧, ∨, →⟩ is an algebra and I an interior unary monotone operator on L, where the operator is defined ...
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Distribution law in infinitary modal logic

This post is related to a question I originally asked on Philosophy Stack Exchange https://philosophy.stackexchange.com/questions/100055/infinitary-modal-logic In the modal logic (say, ${\bf K}$) the ...
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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 ...
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Proving $\Box(p \land q) \rightarrow (\Box p \land \Box q)$ in modal system $K$

I need to prove $\Box(p \land q) \rightarrow (\Box p \land \Box q)$. Currently, I know of a proof that utilizes the tautology $(p \land q) \rightarrow p$ as a first premise, from which we use the ...
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