Questions relating to deductions relating to the expressions "it is necessary that" and "it is possible that"

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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 ...
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Model-theoretic characterization of local modal correspondence

I've been reading van Benthem's dissertation (available on ILLC's website) on modal correspondence theory. In Section I.3, he develops a model-theoretic characterization of modal formulas having ...
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Examples of logical possibility

According to Wikipedia, something is logically possible if it doesn't imply a contradiction. In that case, how could a mathematical statement be false but possible? Wouldn't a false statement be false ...
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Is $\Diamond (p \rightarrow q) \rightarrow (\Diamond p \rightarrow \Diamond q)$ valid in K?

The modal logic K is the weakest normal modal system, comprised by classic logic augmented by (K), the necessity distribution axiom schema: $$\Box (\alpha \rightarrow \beta) \rightarrow (\Box \alpha ...
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How can temporal and epistemic logic be combined?

Recently I read all kinds of work from logic scientist in which epistemic logic was the main topic. Where epistmic change refers to change in knowledge of some agent in a multi-agent system (in a ...
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Modal Logic: ◊-Distribution

It's a theorem of K that $\diamond$ distributes to disjuncts and vice versa: $$\diamond(p \lor q) ≡ \diamond p \lor \diamond q$$ Does it distribute to negated disjuncts? Is the following a licit ...
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What is the connection between game theory and (modal) logic?

I'm interested in dynamic epistemic logic lately (reasoning about information and change in multi-agent systems). I also like game theory. I'm looking for some good resources about the connection ...
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What is the difference between intuitionistic, classical, modal and linear logic?

I am currently going through Philip Walder's "Proposition as Types" and a passage of the introduction has struck me: Propositions as Types is a notion with breadth. It applies to a range of ...
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Problems with basic proof in modal logic (event based)

I am having trouble deriving the following basic result: $\ast$) For every $\omega \in \Omega, \omega \in P (\omega),$ from the following axioms: A1) $K (\Omega) = \Omega$, A2) $K (A) \cap K (B) ...
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Is there a more useful formulation of the frame condition for the McKinsey axiom?

I am looking for a Kripke frame condition corresponding to the McKinsey axiom M: $\Box\Diamond p \rightarrow \Diamond\Box p$. I read somewhere the following condition: "For every partitioning of the ...
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Deduction theorem in modal logic

I am looking for a semantic for deduction theorem in modal logic,I wanna find a semantic way to prove this theorem,but I wasn't successful.tnx for your help
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How to prove that a set containing G$\phi$ and G$\neg \phi$is inconsistent without completeness but with soundness.

I'm stuck with this problem... The logic is a adaptation to temporal logic from $K_4$ of modal logic. The interpretation of G$\phi$ is always true in the future (now is not included). The axioms for ...
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Why “Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong” is paradoxical?

In a paper(see here) by Adam Brandenburger and H. Jerome Keisler, they give a game-theoretic impossibility theorem akin to Russell’s Paradox: Ann believes that Bob assumes that Ann believes that ...
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What is this symbol ($\Vdash$) called?

This symbol: $$\Vdash$$ What is it called? It is often used in modal logic, like this: $W\Vdash$. I looked for it in wikipedia/modal logic and wikipedia/logic notation, but could not find it.
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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 ...
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$A\land FB\rightarrow F(PA\land B)$ in temporal logic

Temporal minimal logic $\mathbf{K_T}$ calculus is characterised by the following axioms, where $F=_{\text{def}} \lnot G\lnot$ and $P=_{\text{def}} \lnot H\lnot$: $G(A\rightarrow B)\rightarrow ...
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Equivalence between $\mathbf{KT_4}$ and Lewis' $\mathbf{S_4}$

Let us define modal logic system $\mathbf{KT_4}$ by adding the following axioms to classic propositional logic $\diamond A\leftrightarrow\lnot\square\lnot A$ $\square(A\rightarrow ...
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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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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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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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What is the relation between three-valued Kripke frame semantics and modal systems K, T, S4?

I'm looking for literature that discusses three-valued Kripke frame semantics (strong Kleene or Łukasiewicz) and their relation to the (two-valued) normal modal systems K, T, S4 and extensions of S4, ...
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Hilbert style proof for $\Box A \vee \Box B \rightarrow \Box(A\vee B)$ in K.

I have to find a formal Hilbert style proof for $\Box A \vee \Box B \rightarrow \Box(A\vee B)$ on modal logic, K. I can use all classical propositional tautologies, Modus Ponens and Distribution ...
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Finitary assignment functions and typed modal languages

I'm working through Giovanna Corsi's article on Counterpart Semantics for Modal Logic. She is working with a typed modal language $\mathscr{L}^{t}$. Where this differs from usual presentations I've ...
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Temporal Logic Tautology

I have the following question: if it is necessary that p -> p = tautology? I think it's not, and I am showing my example for the contradiction, below: ...
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How do I notate this statement about a state of affairs (similar to a possible world)?

I'd like to notate this statement formally: If any given agent desires that a certain state of affairs obtains, then there is no state of affairs in which she enjoys greater security than that one. ...
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Equivalent formulations of the axiom K in modal logic

In modal logic, the axiom K can be equivalently formulated as: (1) $\Box(p\rightarrow q)\rightarrow (\Box p\rightarrow\Box q)$ (2) $\Box(p\wedge q)\leftrightarrow (\Box p\wedge\Box q)$ (3) ...
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Question on epistemic logic

Is there any epistemic modal logic in which the knowledge of a conjunction is not implied by the knowledge of its conjuncts, i.e. $\Box A\wedge\Box B\not\Rightarrow\Box(A\wedge B)?$
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Models of H and GL

I've been reading The Logic of Provability by George Boolos, and something he said stumped me for a bit. Let us use H (for Henkin) to refer to the system that results when (YS) is added to K, ...
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Can linearity be expressed by a modal logic formula?

Can I write a modal logic formula that describes linearity? by linearity I mean the following properties: reflexive transitive $\forall{x,y} \;\; (xRy \lor yRx)$ I'm thinking on it for over a day ...
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Is there a logic to formalize the concept of “understanding”

The question may seem little bit weird given that philosophers have been struggling to have a full grasp on the concept of "understanding". But I'm wondering if there are any logics (modal-based or ...
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What does not having a first-order frame imply for models?

There are formulas in modal logic which which do not have a first-order frame condition, as stated here (Non-Sahlqvist formulas, Wikipedia). An example is the McKinsey formula for $p$: ...
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Problems with validity in type theory

I'm twisting my brains over some simple formulas in intensional type theory. First: If $\exists x \Box (x=^{\vee}j)$, s.t. $x$ is of type $<e>$ and refers to an entity $e$ and $j$ is of ...
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Strong Kleene interpretation

Consider: $\\$ $\Box(\phi \wedge \psi) \rightarrow \Box(\phi) \wedge \Box(\psi)$ I guess this yields by the reflexivity axiom for intensional predicate logic? But I was wondering whether it is also ...
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Theorems of GL in modal logic

So I've been reading George Boolos' "The Logic of Provability" and he's explaining different systems of modal logic. He's taken as his basic symbols → (implication), □ (necessity), ⊥ (falsehood), a ...
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Proof of $p\rightarrow (\Box (\Box p \wedge p) \rightarrow (\Box p \wedge p))$

I need to prove: $$p\rightarrow (\Box (\Box p \wedge p) \rightarrow (\Box p \wedge p))$$ The system contains all propostional tautologies and the axiom scheme $\mathbf K$:$ \Box(p \rightarrow q) ...
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Truth Conditions for Modal Logic

I know that $$ 1)\space w ⊩\diamond P\iff there\space is\space some\space worlds\space w' \space such\space that\space wRw': w ⊩ P $$ $$ 2)\space w ⊩\square P\iff for\space all\space worlds\space w' ...
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Why do you only need to show validity in one world when using trees in institutionist/constructivist logic?

Depicted below, my prof used a tree to prove that an argument is valid according to intuitionist logic. However, I can't find a contradiction in world 0. Why is invalidity ascertained when all ...
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A good (and possibly seminal) book on Multimodal Logic?

I'd like to study Multimodal Logics (in the sense of Catach's Normal Multimodal Logics for instance). Some suggestions? Thank you.
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Is there a Fitch style system that works with some of the modal logics?

My prof taught us to use trees to prove modal logic arguments. Trees seem to provide a more efficient way to test arguments than Fitch does. However, I find that trees generally, and alethic (modal) ...
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¬p ⊬ ⎕(p → q): Where's the mistake in my proof?

My professor noted on one of his slides that ¬p ⊬ ⎕(p → q). Intuitively, this seems correct; however, I can only prove that it is false. I suspect I've made a mistake in my proof. Where have I gone ...
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In modal logic, why are models ordered sets?

I just started undergrad math, so I only have a fuzzy idea of what a model is. I'm learning about modal logic in one of my classes. Our text describes modal logic as operating in a model defined as an ...
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Big Bang Theory Reference to Formal Logic

In the second episode "The Junior Professor Solution" of the 8th season of the Big Bang Theory, there exists a brief moment where Sheldon Cooper references one of his boards with what for a brief ...
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Blue Eyes: A Logic Puzzle, has a puzzling solution (a.k.a. What does common knowledge have to do with it?)

In Blue eyes: a logic puzzle (specifically, the follow up questions), the most common answer is that it needs to be common knowledge that someone has blue eyes for all the blue-eyed people to leave. ...
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What does this negation on both sides of K mean: A = ¬ K ¬

What does this negation on both sides of K mean: (A = ¬K¬) ? I'm not sure if it's a typo, as there are some errors in this paper (Hong et al.). Hong, Zhi Ling, and Mei Hong Wu. "Constrained ...
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Help with a modal Hilbert-style proof of (□(a>b)&◊(a&c))>◊(b&c)

Can't grasp how it can be proved. To proof just propositional calculus formula (without modal operators) at first seems rather natural to me. Tried the law of importation scheme but it didn't work ...
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Is there a modal operator which distributes over the implication?

Is there any notable modal operator $\Box$, so that if $P,Q$ are proposition $$\left(\Box(P\implies Q)\right)\Leftrightarrow\left(\Box P\implies \Box Q\right)$$
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Definition in satisfiability problem

While I was reading the PhD thesis of Balder ten Cate (2005). Model theory for extended modal languages. I found a theorem that says: 2.6.4Theorem. The frame satisfiability problem for modal ...
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Why is $\square\square=\square$ and $\Diamond\Diamond=\Diamond$ in the S5 modal logic?

I'm reading about modal logic on the Stanford Encyclopedia of Philosophy. They define the modal logic S5 as propositional logic augmented with the modal operators $\square$ and ...
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Modal set-theory

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a ...
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The Entscheidungsproblem (decision problem) for modal logic

The Entscheidungsproblem is identified with the decision problem for first-order logic that is, the problem of algorithmically determining whether a first-order statement is universally valid. ...