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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:

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Does the dynamic logic page at wikipedia have some mistakes?

I was reading about dynamic logic over at wikipedia as a possible lead on a previous question. However, its not making a lot of sense to me. In particular, wikipedia says that The constant action $0$ ... does nothing and does not terminate, whereas…
goblin GONE
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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 $\vdash$ A that any calculus based on T or stronger…
God
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Modal logic: deriving K* from K' and RM

I'm reading an article on modal logic which says that K* can be derived from K' and RM. (Note that ⊃ is the material conditional symbol and ≡ stands for material equivalence.) [K'] □(p ⊃ q)⊃(□p ⊃ □q) [RM] ⊢p ⊃ q => ⊢□p ⊃ □q [K*](□p ∧ □q) ≡ □(p ∧…
Mijito
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Why is modal logic's S4 Axiom written as '◻A→◻◻A'?

The Stanford Encyclopedia of Philosophy article on modal logic notes that in S4 ◻◻A is equivalent to ◻A and that "any string of boxes may be replaced by a single box". The article expresses S4: '◻A→◻◻A'?. However, if they're equivalent, wouldn't the…
Hal
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Equivalent of Post-completeness for modal logic S5; any stronger system leads to modal collapse.

I've heard before that classical propositional logic is Post-complete, for example, see this answer. This means that given a set of axioms and inference rules $(A, I)$ for classical propositional logic, for any well-formed formula $\alpha$ such that…
Greg Nisbet
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Prove $M\Vdash\Box(\Box p\rightarrow p)\rightarrow\Box p$ for a model $M=\langle W,R,V\rangle$ in which $R$ is transitive and converse well-founded

Prove $M\Vdash\Box(\Box p\rightarrow p)\rightarrow\Box p$ for a model $M=\langle W,R,V\rangle$ in which $R$ is transitive and converse well-founded. I proceed by reductio. So suppose $M\nVdash\Box(\Box p\rightarrow p)\rightarrow\Box p$. Then,…
beachtrees
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Use induction on the complexity of $C$ to prove that if $K\vdash A\leftrightarrow B$ then $K\vdash C[A/q]\leftrightarrow C[B/q]$

Use induction on the complexity of $C$ to prove that if $K\vdash A\leftrightarrow B$ then $K\vdash C[A/q]\leftrightarrow C[B/q]$ This is question 3.3 from OLP's Boxes and Diamonds. I have found proofs for the base cases, where $C\equiv\top$,…
beachtrees
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Satisfiable or valid formulae

(a) $P\rightarrow \diamondsuit Q \wedge \square P \wedge \neg Q$ (b) $\neg \square \neg Q, \diamondsuit \neg P \wedge \square (\square \neg P \rightarrow \diamondsuit Q)$ I want to state for each of these formulae, a and b, whether it should be an…
kabin
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Expressing the truth sets of $\Box\varphi$ and $\Diamond\varphi$ in terms of the truth set of $\varphi$.

Since that the answer to this question will depend on the particular semantics that I'm using, allow me to first define truth of $\Box\varphi$ and $\Diamond\varphi$ at a world $\alpha$ in a model $\mathfrak{M}=(\mathit{W},\mathit{R},\mathit{P})$,…
Modesto Rosado
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Modal Logic: Proof that a Maximally Consistent Set is Complete

To understand the completeness proof for modal logic, I need to show that Maximally Consistent Sets are complete. A set $\Sigma$ is consistent if $\Sigma \not \vdash \bot$. A consistent set is maximal if any set that has $\Sigma$ as a proper subset…
9sven6
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How to Prove $ \Box (A \wedge B) \rightarrow ( \Box A \wedge \Box B) $ in S5

So far I have: $(A \wedge B)\rightarrow (A \wedge B)$ assumption $A \rightarrow (B \rightarrow (A \wedge B)$ from 1 by propositional logic $\Box A \rightarrow \Box (B \rightarrow (A \wedge B)$ from 2 by RM $\Box(B \rightarrow (A \wedge B)…
beachtrees
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How prove or refute $\diamond \Box A$ → A characterizes symmetry

Can some of you nice people help me and show me how to prove $\diamond \Box A$ → A characterizes symmetry. I really appreciate it Bests
Norman
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How to prove that $\vdash\phi\rightarrow\neg\Box\neg\phi$ is a theorem in S5?

I want to prove that $\vdash\phi\rightarrow\neg\Box\neg\phi$ is a theorem in S5 I have S5 definition : \begin{align*} T&:\Box p\rightarrow p\\ 5&:\Diamond p\rightarrow \Box\Diamond p\\ K&:\Box (p\rightarrow q)\rightarrow(\Box p\rightarrow \Box…
Revolucion for Monica
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Duality between necessity and possibility?

Is there any system of modal logic in which the duality between necessity and possibility does not hold? In other words, is there any system of modal logic in which $\Box p = \neg\Diamond\neg p$ and $\Diamond p = \neg\Box\neg p$ does not hold?
user60264
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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 modal statements only, or concerned with both…
user60264
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