The modal-logic tag has no wiki summary.
5
votes
1answer
259 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
95 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
437 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
185 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
118 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
296 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
258 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
441 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
436 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
339 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
128 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
462 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
914 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
509 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?
...
