The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
63 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
vote
0answers
68 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 ...
1
vote
0answers
236 views

Is this system incomplete?

Let $\mathbf{SBM}$ be the normal modal logic system defined as $\mathbf{T}$ plus the following two axioms: $$\mathrm{SB}: \Box(\Diamond p \rightarrow p)\rightarrow (p \rightarrow \Box p)$$ ...
1
vote
1answer
94 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 ...
5
votes
1answer
289 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
97 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
473 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
votes
2answers
190 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
123 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 ...
10
votes
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
305 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
270 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
476 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
438 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
347 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
133 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
326 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
466 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
933 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
519 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? ...