The modal-logic tag has no wiki summary.
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A specific Model of ZFC
In his paper "Some Second Order Set Theory", Joel Hamkins asked whether there is a model of set theory $V$ that is elementary equivalent to $V[G]$, Whenever $G$ is $V$-generic for the collapse of a ...
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1answer
89 views
A question on Carnap's modal semantics on the basis of Cochiarelli's primary semantics
I believe I learned that Carnap's state description semantics for propositional modal logic suffered from validating $\lozenge p$ for all atomic variables p. Re-reading Nino Cochiarelli's primary ...
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0answers
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Interesting fragments of first-order logic induced by sorting?
In first approximation, modal logic (I'm using the term loosely)
can be understood as an interesting fragment of first-order logic
(for simplicity I ignore e.g. how modal logic relates to
...
2
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1answer
483 views
Is this system incomplete?
Let $\mathbf{SBM}$ be the normal modal logic system defined as $\mathbf{T}$ plus the following two axioms:
$$\mathrm{SB}: \Box(\Diamond p \rightarrow p)\rightarrow (p \rightarrow \Box p)$$
...
2
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1answer
115 views
Question on deriving $\alpha \rightarrow \Box \alpha$ in modal logic KTU
Let K and T be the usual modal logical principles $\Box (\alpha \rightarrow \beta) \rightarrow (\Box \alpha \rightarrow \Box \beta)$ and $\Box \alpha \rightarrow \alpha$. Let U be the modal logical ...
4
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2answers
312 views
On a modal correspondence
Is there an intuitive characterization of the correspondence for the modal logical formula $\square (\alpha \rightarrow \square \alpha) \rightarrow (\square \alpha \vee \square \lnot \alpha)$?
In ...
4
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1answer
306 views
On directedness, transitivity and ancestral directedness
Let $\textit{C}$ be the modal logical schema $(\square (\square \alpha \rightarrow \alpha) \wedge \square (\square \lnot\alpha \rightarrow \lnot \alpha))\rightarrow (\square \alpha \vee \square \lnot ...
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0answers
60 views
A question on completeness for quantified temporal logics
Quantified temporal logics have the Barcan formulas and its converses for both G (it will always be the case that) and H (it has always been the case that), so that both $\forall x G \alpha ...
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1answer
192 views
Is a computer program for correspondence theory available?
In the 1990s I some times used a computer program with the Max Planck Institute which helped with calculating complicated correspondences for modal logical formulas. Is some program like that ...
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0answers
39 views
Completeness results for quantified tense logics with BF?
Modal tense logics or temporal logics are important in that they correspond with partial orders and their extensions.
Are there completeness results for quantified temporal logics with the Barcan ...
3
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0answers
54 views
A question on the incompleteness of quantified K.2 and S4.2 with the Barcan formula
I have been attempting to come to grips with Max Cresswell's account of this in Journal of Philosophical Logic 24 (4):379 - 403 (1995) where he presents proofs of the incompleteness of QK.2BF as well ...
2
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1answer
157 views
A question on the modal logic S4.2
The modal logic S4.2 with the characteristic axioms
4: $\square \alpha \rightarrow \square \square \alpha$
and
.2: $\lozenge \square \alpha \rightarrow \square \lozenge \alpha$
and
T: ...
5
votes
1answer
381 views
Is this system identical to S4.4?
Consider the normal modal logic system $\mathbf{TAR1}$ given by $\mathbf{T}$ plus the following axiom:
$$\mathrm{AR1}: \lozenge \square p \rightarrow (\square p \lor \square (p \rightarrow \square ...
3
votes
1answer
105 views
Soundness of modal logics which contain the reflection rule
Let $ML$ be a modal logic which contains the Reflection Rule (from $\vdash\Box F$ infer $\vdash F$). For a modal formula $F$, let $H(F)=\{\ \Box G\rightarrow G~|~\Box G$ is a subformula of $F\}$. A ...
6
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1answer
563 views
Non-iterative modal logics
Let S be a propositional modal logic system (extension of K, or even E) with a single unary modal operator and defined by a single non-iterative axiom (i.e. of modal degree 1).
Is it true that for ...
3
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2answers
200 views
Validity in Kripke frames whose points are finite or infinite sequences
Suppose $D$ is a non-empty set and $\{ R_i : i \in \mathbb{N} \}$ is a family of binary relations on sequences over $D$ so that $R_i \subseteq D^i \times D^i$. Let $R_\omega \subseteq D^\omega \times ...
2
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1answer
125 views
On the Combinatorial Classification of Modal Kripke Frames
We have that S5 modal logic is characterized by the modal axioms $K$, $M$ (reflexive), $4$ (transitive), and $B$ (symmetric). That is, an equivalence relation on a set of possible world (which can be ...
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2answers
147 views
Adjoint of Pushout as Modal Operators in Internal Logic
Regarding the internalization of mathematics to a particular category as in the nLab article: Internal Logic, there is a peculiar table mentioned in the section on Categorical Semantics in which there ...
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4answers
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How are Modal Logic and Graph Theory related?
I am currently taking a graduate logic course on Modal Logic and I can't help notice that there are a certain class of graphs characterized by the modal axioms such as (4) $\Box p \rightarrow \Box ...
1
vote
3answers
322 views
Why the preimage rather than image in Stone-type dualities.
I am seeking a deeper understanding of the representation of set-based objects in terms of Boolean algebras.
Let $\wp(A)$ be the set of subsets of a set $A$. A relation $R \subseteq A \times B$ ...
5
votes
3answers
280 views
Why are possibility and necessity dual?
Hello,
Recently, I'm studying modal logic for my master's thesis, and my research background is category theory.
So, I naturally have a question that why it is said that necessity (box) and ...
4
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1answer
598 views
Non-trivial consequences of Lob's theorem
Informally, Löb's theorem (Wikipedia, PlanetMath) shows that:
a mathematical system cannot assert its own soundness without becoming inconsistent [Yudkowsky]
In symbols:
if $PA\vdash$ ...
6
votes
2answers
450 views
A necessary condition for S4-completeness?
It is well-known that the modal logic S4 is complete with respect to the class of all finite quasi-trees (where we interpret the $\Box$ modality as topological interior, and topologize a quasi-tree ...
6
votes
1answer
367 views
Is it possible to define a closure operator in terms of partial ordering?
For boolean algebra, let's take Roman Sikorski's Boolean Algebras as our reference. After giving a set of axioms, he proves (p.9) that the join of A and B is the least element of the algebra such that ...
0
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1answer
141 views
how to determine the condition on frame if some axiom schema is given together with K axiom?
In semantics for modal logic, if a new axiom schema is given together with K in question then how can one find out that what conditions the frame for the new system need to satisfy i.e reflexive, ...
7
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1answer
331 views
Looking for papers and articles on the Tarskian Möglichkeit
Some background: Łukasiewicz many-valued logics were intended as modal logics, and Łukasiewicz gave an extensional definition of the modal operator: $\Diamond A =_{def} \neg A \to A$ (which he ...
0
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2answers
477 views
How many models are there, for a particular propositional modal logic?
Background/motivation: A model for the classical propositional calculus is a boolean function b(S) which assigns 1 or 0 to each (modal-free) sentence S according to the usual rules. I'm looking at ...
0
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3answers
973 views
Modal logic - box rules
Hi guys,
In modal logic i.e. propositional logic with box and diamond, are then any laws to get a box or a diamond from outside a bracket to inside?
I.e. $\Box (x \rightarrow \Box x)$
I want the ...
3
votes
2answers
543 views
Modal logic - satisfiability
Hi there,
Assuming X and Y are modal formulae and diamond X is satisfiable and diamond Y is satisfiable, how would one show that they X AND Y is satisfiable?
I don't think it requires much effort?
...
3
votes
2answers
464 views
Applications of propositional dynamic logic
Propositional dynamic logic (PDL) is an example of a (multi)modal logic with a structure on the set of modalities. In particular, the set of its modalities is indexed by "programs" and one can use ...
