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

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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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38 views

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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1answer
34 views

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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1answer
39 views

¬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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1answer
51 views

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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1answer
681 views

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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3answers
170 views

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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1answer
48 views

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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2answers
93 views

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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3answers
90 views

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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1answer
28 views

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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2answers
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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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1answer
88 views

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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2answers
63 views

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. ...
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2answers
79 views

Prove that modal logic S4 is properly contained in S5

Is it possible to prove using just the semantics for $S_4/S_5$ that $S_4$ is properly contained in $S_5$? I can see how one could show that there is a theorem of $S_5$ which is not a theorem of $S_4$, ...
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0answers
47 views

Trying to understand Hintikka's logic of Knowledge and belief

I try to understand Hintikka's logic of knowledge and belief but am a bit stumped by it. I study " Knowledge and belief , an introduction to the Logic of the two Notions", (Kings College ...
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1answer
25 views

Can it be the case $p$,$\lnot p$ are true at $\mathbf w$ And $\mathbf w'$ respectively with $\mathbf w$ and $\mathbf w'$ have access to each other?

Given a model $\mathbf{M} = (\mathbf{W}, \mathbf{R}, \mathbf{V})$ for a set of atomic formulae $\Omega$. We have possible worlds $\mathbf{w}, \mathbf{w'} \in \mathbf{W}$, access relation satisfies ...
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2answers
70 views

Universe enlargement and modal logic

In model theory and category theory, we often need to "enlarge" our universe (whatever that means) so that our proper classes become "small" and we can thereby manipulate them in more sophisticated ...
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1answer
36 views

Problems with the usage of Belief and Common Belief operators

I have a problem with the usage of a Belief operator $B_i$ in the derivation of a result on a common belief operator $CB$. First of all, some basic definitions (where $i$ is an individual), that ...
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1answer
47 views

Decidability of normal modal logics

Let's say we have systems of modal logic defined as smallest sets containing propositional tautologies, all instances of schema $\square F \to (\square(F \to G) \to \square G)$ all instances of ...
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1answer
69 views

Book on the first-order modal logic

Is there a book on the metatheory for the first-order modal logic, or do I just need to take FOL as a base and use the standard translation?
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1answer
41 views

Propositional S5: is there a consistent set requiring continuously many worlds?

A recent question asked whether in systems of modal propositional logic having the "finite model property" there are consistent sets of sentences that were not satisfied by a finite model. @Carl ...
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1answer
68 views

Propositional modal logic: infinite models required in systems with finite model property?

A system of propositional modal logic has the "finite model property" if any consistent sentence is satisfiable at a model with finitely many possible worlds. Some systems have this property and ...
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1answer
41 views

Truth Tables for Temporal Operators?

I would like to know whether we can construct truth tables for the following temporal operators in temporal logic as we do in propositional logic . ...
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4answers
166 views

What obstacles prevent three-valued logic from being used as a modal logic?

I am familiar with many of the surveys of many valued logic referenced in the SEP article on many valued logic, such as Ackermann, Rescher, Rosser and Turquette, Bolc and Borowic, and Malinowski. It ...
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2answers
65 views

Proving Gabbay rule for Modal Logic

I'm currently working on exercises of the book "Modal Logic" by A.Chagrov and M.Zakharyaschev (for pleasure, not homework). One exercise asks to prove this version of Gabbay rule (exercise $3.10$): A ...
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Validity of LTL formulas in a given transition system [duplicate]

Say I have the following transition system: I've understood how I can tell if □a and ⟡b are valid (□a is invalid because a is not true is S2 and ⟡b is valid there is a state (i.e. S1) in which b is ...
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2answers
99 views

Validity of LTL formulas in a given transition system

Say I have the following transition system: I've understood how I can tell if □a and ⟡b are valid (□a is invalid because a is not true is S2 and ⟡b is valid there is a state (i.e. S1) in which b is ...
2
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1answer
138 views

Is there a name for this property of a binary relation? $\forall x\forall y(x\mathsf{R}y\to\exists z(x\mathsf{R}z\land y\mathsf{R}z)))$

Consider a binary relation $\mathsf{R}$ such that $x\mathsf{R}y$ is the case only if there is some $z$ such that both $x\mathsf{R}z$ and $y\mathsf{R}z$ are the case. Is there a well-known name for the ...
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0answers
81 views

Prove $\Diamond p \rightarrow \lnot\Diamond\lnot\Diamond p$ in modal logic

I need to prove $\Diamond p \rightarrow \lnot\Diamond\lnot\Diamond p$ in B axiomatic, which contains next conversion rules: 1.$(p\land q)\rightarrow(q\land p)$ 2.$(q\land p)\rightarrow p$ ...
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1answer
69 views

Is it possible to prove $p\rightarrow\diamond (p\land q)$ in modal logic?

I need to prove $p\rightarrow\diamond (p\land q)$ in B axiomatic, which contains next conversion rules: 1.$(p\land q)\rightarrow(q\land p)$ 2.$(q\land p)\rightarrow p$ 3.$p\rightarrow(p\land p)$ ...
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0answers
30 views

Problems with a basic proof in Aumann Structures

I am pretty sure this is more than trivial, but I have a problem with the proof of a basic results in Aumann structures (this is related to a more general problem I have with proofs that involve ...
0
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1answer
69 views

Can necessity rule be derived from box introduction rule?

I need to find a proof of $\top \vdash \Box \top$ (where $\top$ is the truth constant and $\Box$ is the necessity modal operator) in the natural deduction system of IS4 modal logic. In the axiomatic ...
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0answers
34 views

Test axiom in PDL

Wikipedia says the axiom for test in PDL is $$ \langle \psi ? \rangle \phi \leftrightarrow \psi \wedge \phi, $$ but why is this right? (i.e. what does it say?) And what is the corresponding relation ...
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Transitive temporal filtration of a model.

Exercise. Given a transitive model $\mathfrak{M} = (W, R, V)$, show that it has a transitive temporal filtration $\mathfrak{M}^f = (W^f, R^f, V^f)$. Progress. Given a subformula closed set $\Sigma$ ...
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2answers
133 views

Basic Modal Logic question #1

Its about two weeks I have started Cresswell's "A New Introduction To modal Logic". Now I've got a few questions on the text and I would deeply thank you if you help me clarify on them. ...
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1answer
312 views

Incomplete normal modal logic systems

Apart from the classical example of KH, given by axiom $\Box(\Box p\leftrightarrow p)\to \Box p$, are there any other examples of incomplete propositional normal modal logic systems defined by axioms ...
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2answers
85 views

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 ...
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2answers
85 views

Equivalence between temporal logic and notions of forcing

I have come across literature comparing modal logic to forcing (by Hamkins et al). Has anything similar been done showing equivalences between temporal logic and forcing? This would be interesting to ...
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24answers
8k views

Blue eyes: a logic puzzle

Today I read the Blue Eyes puzzle here. I also read the solution which I find quite interesting. But there are three follow up questions which I don't know the answer to: What is the quantified ...
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1answer
162 views

What is the definition of algebraic equivalence of formulas?

This question is for the propositional normal modal logic system K (although it may apply to other logics too). I saw a couple of papers which mentioned algebraic equivalence of modal logic formulas ...
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1answer
52 views

exchange “globally” and “future” in temporal logic

I cannot prove the following theorem given in Schneider's “On Concurrent Programming” as (3.16e): $$\Diamond\Box P \to \Box\Diamond P$$ I was given $$\begin{align} \Box P & \to P\\ \Box P ...
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2answers
58 views

What is the modal interpretation of the converse to a statement?

I have just begun learning about modal logic and I am trying to get my head around Kripke Semantics. I then began to wonder if there is any connection between convereses of statements and modal logic. ...
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1answer
63 views

Weak classical Deontic Logics

I am writing a paper at the moment and an area of Deontic Logic has cropped up in it. I know very little about the area and I was wondering if people could give me opinions on the axiomatic system ...
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2answers
199 views

Books about modal logic?

I've just approached modal logic reading "An Introduction to Non-Classical Logic" of Graham Priest. I am looking for some books that treat this argument in a more extensive way than the book I am ...
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32 views

Explanation of states, worlds and models?

Can someone explain to me how the concepts of states, models and worlds work together in Kripke semantics? I've been trying to piece together how the parts work are linked together but cannot figure ...
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1answer
70 views

Modal logics and logics with integer or even rational numbers - is that possible?

I am trying to find some research trends (publications, keywords for futher search, etc.) on logics and modal logics whose predicates can containt integer or even rational numbers (like x>2, x^2>3, ...
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1answer
62 views

Analog of modus ponens for semantics

To pose my question, I first must first quickly define a language, a model, semantics for such models, and a logical system called S4O. Consider a language $L$ with a set $PV$ of propositional ...
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1answer
110 views

Proof that Scheme T implies reflexivity

In my modal logic book it's written that, for each frame $F(S,R)$ the accessibility relation $R$ is reflexive IF AND ONLY IF the scheme T:$\square A \implies A$ is valid in $F$. Even if I can easily ...
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2answers
404 views

The “set of all possible worlds”, etc.

The following is an excerpt from a highly-respected paper (Angelica Kratzer, Modals and Conditionals: New and Revised Perspectives, chapter 1 "What Must and Can Must and Can Mean") (my emphasis): ...