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2
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2answers
132 views

Is there a good list of nomenclature for modal axioms?

I would like to see what names that has been suggested for useful modal axioms. By name here I mean some abbreviation such as $T$, $K$, $4$, $.2$, $E$ and so on. In particular I am interested in ...
1
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0answers
126 views

This modal logic semantics is not S5, but is it something else well-known?

The short form of the question is this: Is there a model of modal propositional calculus that gives the modal operators the meanings ...
1
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0answers
130 views

Kripke frames as classes of partitions

Here's something I've been playing with off and on for a bit; I'm curious if anyone has seen it before. For this question, a Kripke frame $K$ is a finite reflexive directed graph. (Reflexivity isn't ...
5
votes
2answers
532 views

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 ...
2
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1answer
93 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 ...
1
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0answers
85 views

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
votes
1answer
493 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
votes
1answer
121 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
votes
2answers
313 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
votes
1answer
312 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 ...
1
vote
0answers
63 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 ...
6
votes
1answer
194 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 ...
1
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0answers
40 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
58 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
votes
1answer
185 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
389 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
107 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
votes
1answer
573 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
203 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
votes
1answer
126 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 ...
2
votes
2answers
149 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 ...
10
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4answers
1k views

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
332 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
286 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
votes
1answer
660 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
452 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
373 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
votes
1answer
142 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
votes
1answer
333 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
votes
2answers
486 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
votes
3answers
1k 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
553 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
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2answers
480 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 ...