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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"

Modal logic is an extension of propositional and predicate logic that expresses modalities, which are qualifications to a statement. The most commonly used modalities in mathematics are "possibly," "necessary," and "impossibly."

For more information, see these links:

566 questions
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Prove if $\Box \Box p \to \Box p$ is valid on Frame $F=\langle W,R \rangle$ then $R$ is dense

Prove if $\Box \Box p \to \Box p$ is valid on Frame $F=\langle W,R \rangle$ then $R$ is dense. Where dense means $\forall w,v \in W: wRv \implies \exists u \in W : wRu\ \&\ uRv$. I know we are supposed to come up with a valuation such that when…
CHTM
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Prove that the following argument is invalid

$□p$, $□q$ therefore, $□(p→q)$ according to the K system of modal logic, the argument is invalid. I tried proving it using a truth tree, but all the branches unfortunately close, I don't know how to proceed.
Shreya Mishra
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The definition of the semantic consequence relation for modal logic

I have been unable to find a general definition of semantic consequence ($\vDash$) for modal logic, so I would appreciate if you commented on this speculation of mine: Definition. Let $M = (W, R, V)$ be a model of modal logic and $X$ a set of…
God bless
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The exponential gap between the first-order quantifier rank and modal quantifier rank

In the proof of van Benthem/Rosen characterisation theorem that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation, we can use Ehrenfeucht-Fraisse games. One of the consequences of that proof is…
user1141222
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First steps in infinitary modal logic

In modal logic, we have $\Box(p\wedge q)\leftrightarrow\Box p\wedge\Box q$ but only $\Box p\vee\Box q\rightarrow\Box(p\vee q)$, where $\Box$ is the operator of necessity. Do the same relations hold in infinitary modal logic, i.e. do we…
Hugh
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Translation valid formulas neighbourhood semantics to kripke semantics

I have been tasked with an exercise which I cannot figure out and thus I require some assistance. The question is as follows: Let $\mathsf{EM}$ be the set of formulas valid on all neighbourhood frames $\mathcal{F} = (W,N)$ in the unimodal language.…
Tungsten
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Proving axiom M in modal logic from K plus A → ◊A

I'm slowly working my way through Garson's Modal Logic for Philosophers. I'm stuck on exercise 2.1c. It asks me to prove axiom M (= ◻A → A) in system K (propositional logic + necessitation and distribution) with the added axiom A → ◊A. I've tried a…
dfi6ju
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Prove of K ⊢ A ↔ B then K ⊢ C [A/q ] ↔ C [B/q ]

I am currently reading "Boxes and Diamonds: An Open Introduction to Modal Logic". Here is the PDF to the book: https://builds.openlogicproject.org/courses/boxes-and-diamonds/bd-screen.pdf One of the exercises reads the following: Prove Proposition…
Kylie
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Prove that if $\Box p \rightarrow \Box\Box p$ is valid on a frame $F=\langle W,R\rangle$ then $R$ is transitive

Prove that if $\Box p \rightarrow \Box\Box p$ is valid on a frame $F=\langle W,R\rangle$, then $R$ is transitive. Suppose $F\vDash\Box p\rightarrow \Box\Box p$, where $F=\langle W,R\rangle$. Let $u,v,w\in W$ be arbitrary worlds such that $Ruv$ and…
beachtrees
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Show $K_\rho$ is a proper extension of $K_\eta$

Show $K_\rho$ is a proper extension of $K_\eta$: $K_\rho$ is an extension of $K$ where $R$ is reflexive. And $K_\eta$ is an extension of $K$ where $R$ is extendable. So, I reasoned as follows: $K_\rho$ is a proper extension of $K$ because it can…
beachtrees
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Frame class characterizing S4W5

Let W5 be the basic modal formula $\diamond \square \phi \rightarrow (\phi \rightarrow \square \phi)$ and let S4W5 be the smallest normal modal logic generated by the S4-axioms and W5. I want to find a frame class characterizing S4W5 but have failed…
sequitur
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What is required to believe one is not a Situationist?

It has been somewhat humorously claimed that No person who calls themselves a Situationist is! If I think about this informally for a second, it would seem to reason that anyone who believes this claim must believe they are not a Situationist. If…
Sriotchilism O'Zaic
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Validating both K and its converse

Consider a system of modal logic that validates both of the axioms below. : ◻(→)→(◻→◻) ′: (◻→◻)→◻(→) Could you please explain in simple words what would this system look like? I guess there would be some kind of trivialization going on.
Mijito
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Modal logic and validity (Hughes and Cresswell)

I'm trying to do question 16.3 on page 310 of Hughes and Cresswell for any of you that have access to the text. (This is not for a class, just for my own personal studies.) 16.3: Using the semantics discussed on p291 in which $V_\mu^*(\forall…
emdash
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Irreflexivity ($\neg Rxx\ \forall x$) is not modally definable. Why are these frames bisimilar?

I was reading a book about modal logic and in the book I read: Irreflexivity ($\neg Rxx\ \forall x$) is not modally definable They prove it with the frames $A$ and $B$, where $A = (\{a\}, \{(a, a)\})$ and $B = (\{b1, b2\}, \{(b1, b2),(b2, b1)\})$,…
jennifer ruurs
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