Questions tagged [modal-logic]
Questions relating to deductions relating to the expressions "it is necessary that" and "it is possible that"
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show that every S5-satisfiable formula is satisfiable in a universal Kripke frame, i.e. in a frame of the form (X, R) where R = X × X.
Inductive Proof of S5 Model
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ueb3.pdf
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Can you please solve exercise 4 using structural induction?
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Exercise 4 from your document asks to show that every S5-satisfiable formula ...
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Do modal calculi work with possible worlds?
I use a natural deduction calculus for modal propositional logic, but my question eventually is about any (sound) modal calculi with/without axioms.
Just as an example take a proof like $\square$A $\...
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Is the logic $\mathsf{wK4}$ plus Lob's rule the same as $\mathsf{GL}$?
Let $\mathsf{wK4}$ be the logic of weak transitive frames and $\mathsf{GL}$ be the provability logic.
It is well-known that one can define $\mathsf{GL}$ as $\mathsf{K4}$ (the logic of transitive ...
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Axiomatic proof in modal system $⊢_{K5} □(□p → p)$
I’m having a hard time proving $⊢_{K5} □(□p → p)$. Proving validity in Euclidean frames.in this from axiom 5 is valid $◇p → □◇p$
I’m able to derive
$□p → □(□p → p)$
I,m also able to derive from Axiom ...
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Techniques for showing that sole sufficient operators of a given arity do not exist for a given algebra
I came across this question yesterday which is interesting. It asks whether a binary sole sufficient operator for the modal logic K exists. I tried to find an example of a sole sufficient operator for ...
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How to construct filtrations in modal logic that preserve a specific properties to prove finite model properties.
I am following a course on modal logic and I have issues with a specific area, namely filtrations that preserve frame properties. A definition for a filtration is given here: https://en.wikipedia.org/...
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Two questions about the partial order of propositional modal logics
The propositional modal logics $T$, $B$, $S4$ and $S5$ are related as follows: $T$ has the fewest theorems, $S5$ has the most, $B$ and $S4$ are intermediate between $T$ and $S5$, but they are ...
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Can the empty set be defined as a contradiction in logic?
Can the empty set be defined as something like this in logic?
□(∃xEx↔∃x¬x=x)
Also, how should the empty set be defined in logic? Thanks and simultaneously sorry for my amateurish question.
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recommendations for studying modal logic
I'll be reading a textbook on modal logic with my colleagues this winter (for 3-5 months).
I might need to teach them many points from the textbook because I have a basic understanding of ...
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Semantic Deduction Theorem in Modal Logic
The semantic version of the deduction theorem says that $\{A_{1}, …, A_{n}\}\vDash$ B $\iff \vDash A_{1}, …, A_{n} \to$ B. I read that the deduction theorem does not hold in modal logic, but upon ...
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How many modal logics are there? [closed]
How many propositional modal logics exist? Is it a finite number, countably infinite, or continuum-sized? Of course, to answer this question, we would need a definition of a propositional modal logic.
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in modal set theory, why it is issue?
I have been studying the Iterative Set Concept within the context of the paper titled "Modal Set Theory" from Menzel specifically on pages 11-12.
"As we’ve just seen, the iterative ...
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A question about S4 modal logic
I’ve been doing some proofs in S4 recently and noticed that the following holds:
If $\phi$ is a wff and all positive literals $A$ that occur in $\phi$ are prefixed by either $\Diamond$ or $\Box$, ...
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Taking the derived set in a topological space twice
For a topological space $(X, \tau)$ and a set $A \subseteq X$, let $\delta A$ denote the derived set of $A$. That is, $\delta A$ is the set of limit points of $A$. In other words, $\delta A$ contains ...
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A question on identity in quantified modal logic
One can prove that $a=b \to \Box a=b$ by using the substitution axiom for equality. However, since $\neg a=b$ is not of the form $\alpha_1 =\alpha_2$, I don’t see how to prove $\Diamond a=b \to a=b$, ...
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The first source on a modal logic without the replacement property,
The replacement property for a modal logic states that:
$\text{ If } (p \leftrightarrow q) \text{ ,then } (\Box p \leftrightarrow \Box q)$
sometimes modal logics closed under this property are known ...
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What is this sort of signature in model theory?
I was browsing this set of lecture notes for a mathematical introduction to modal logic. I'm not familiar with model theory, and the lecturers (who approach modal logic from a model-theoretic ...
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Related work for a 'compatibility' modality in modal logic meaning ◊(𝜙∧𝜙′)
Fix some set of propositional atoms $\mathsf{Atoms}$ and consider a (very simple) propositional logic with a binary compatibility modality $\mathsf{C}$ as follows:
$$
\phi,\phi' ::= A,B,C\in\mathsf{...
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What do algebraic semantics look like for intuitionistic modal logic?
I know that a topological pseudo-Boolean algebra is an algebra ⟨L, I, ¬, ∧, ∨, →⟩ such that ⟨L, ¬, ∧, ∨, →⟩ is an algebra and I an interior unary monotone operator on L, where the operator is defined ...
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Distribution law in infinitary modal logic
This post is related to a question I originally asked on Philosophy Stack Exchange
https://philosophy.stackexchange.com/questions/100055/infinitary-modal-logic
In the modal logic (say, ${\bf K}$) the ...
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Constructing a Kripke model where $p \rightarrow \Box \Diamond q$ is false.
I have constructed the following Kripke model for this problem:
My idea is the following: Implication is false iff we have $ \top \implies \bot$.
For world $0$, we have that $p$ is true. Now we need ...
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Proving $\Box(p \land q) \rightarrow (\Box p \land \Box q)$ in modal system $K$
I need to prove $\Box(p \land q) \rightarrow (\Box p \land \Box q)$.
Currently, I know of a proof that utilizes the tautology $(p \land q) \rightarrow p$ as a first premise, from which we use the ...
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Introductory text on logic for those interested in the intersection of logic, algebra, and topology?
I'm a current MA student doing research in formal semantics, which is an application of, among other things, logic and model theory to the study of the semantics of natural languages. I'd like to ...
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Epistemic puzzles and the general necessitation-iteration rule RK
So I came across epistemic puzzles, or specifically the "Surprise Exam"/"Prediction Paradox" and found some explanation in Wesley H. Holliday's paper "Simplifying the Surprise ...
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Bisimulation games to compare equivalent but not bisimilar BML models
Two BML models M, w and N, w' are given: in M, w has with infinitely many R-transitions of finite length, and in N, w' has infinitely many R-transitions and also includes an infinite-length R-...
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What is the difference between operators and connectives?
In the logic context, both the terms operator and connective is used. I wish to know what their differences are. Do we use an operator when we have something that effect only one formula and ...
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Question on Intuitionistic Modal Logic
In the intuitionistic version of the modal logic K (IK), the following is a theorem:
$(\Diamond p \to \Box q) \to \Box( p \to q)$. However, given the definitions for $\Box$ and $\Diamond$, I think I ...
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Uniform Substitution preserves validity on basic system $\mathbf{K}$
I have a question on the proof for uniform substitution preserving validity, in the basic modal system $\mathbf{K}$.
We are assuming that for an arbitrary Kripke model $\mathcal{M}$ and world $s$ we ...
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How to prove $(\lozenge \square p \land \lozenge \square q) \to \square \lozenge (p \land q)$ in $B$?
I am trying to prove in a Hilbert system $(\lozenge \square p \land \lozenge \square q) \to \square \lozenge (p \land q)$ in $B$. I am stuck with $(\square p \land \square q) \to \square \lozenge (p \...
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How to prove Hilbert-style $(\Box(p\rightarrow q)\wedge\Diamond(p\wedge r))\rightarrow\Diamond(q\wedge r)$ in **K**?
Edit: I figured out how to solve it using the theorem K9: $\Box(p\vee q)\rightarrow\Box p\vee\Diamond q$ on p. 36 and then manipulating it using the rules of propositional logic. If you have any ...
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modal logic, derive diamond from box
I should derive $\lozenge$ from the semantics for $\square, \neg$ with classical reasoning at the meta-level:
Using the definitions:
$v_{\mathcal{M}}(\square F,w) = 1 \text{ if } \forall u (wRu \...
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How do you define Robinson diagrams for modal logic? Can you use them to define homomorphisms &c?
In first-order logic, Robinson diagrams are useful for defining different kinds of maps: homomorphisms, embeddings, elementary embeddings, and isomorphisms:
For example, consider two structures $A$ ...
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Apparent counterexamples to claims of tautologies in modal logic
I'm reading the paper "A fixpoint semantics and an SLD-resolution calculus for modal logic programs" by L.A.Nguyen, and in the paper he asserts that certain tautologies hold, but it seems ...
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Reflexive Frame in Temporal Logic
I have a question as follows:
A temporal frame $(T,\prec)$ is reflexive if $\prec$ is reflexive.
Write down a temporal formula that is valid on all reflexive frames, but invalid on any other frames.
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Does the modal logic S5 have a binary sole sufficient operator?
It is well-known that nand and nor are sole sufficient operators for classical logic.
I found a ternary operator earlier today that can express classical modal logic, given below:
$$ J(a,b,c) \;\;\...
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Prove $(p \Box\kern-1.5pt\raise1pt\hbox{$\mathord{\rightarrow}$} q) \to (p \to q)$ is valid on an ordering frame $F=<W,\leq>$ iff $\leq$ weakly center
Ordering model is as defined above.
Prove that $(p \Box\kern-1.5pt\raise1pt\hbox{$\mathord{\rightarrow}$} q) \to (p \to q)$ is valid on an ordering frame $F=<W,\leq>$ iff $\leq$ is weakly ...
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Intuitionistic Logic vs Constant Domains
Quantified modal logic is a controversial field, specifically since it forces one to consider what is meant by “world” in Kripke Semantics. For example, the formula $\Box \forall x \varphi \implies \...
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Ockhamist temporal language - confusion on definitions
What does $T,t',b \models\phi$ mean? Is it the same as $T,t' \models \phi$ after you have fixed the branch $b$ and $t'$ such that $t' \in b.$ If so wouldn't "for all branches $b'$ through $t$: $T,...
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Priorean temporal language - Show that U (the until operator) is not definable
The definitions i am working with are:
U is the until operator:
To show U is not definable i'm trying to come up with two bisimilar models that disagree over some formula involving U.
The textbook ...
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Prove if $\Box \Box p \to \Box p$ is valid on Frame $F=\langle W,R \rangle$ then $R$ is dense
Prove if $\Box \Box p \to \Box p$ is valid on Frame $F=\langle W,R \rangle$ then $R$ is dense.
Where dense means $\forall w,v \in W: wRv \implies \exists u \in W : wRu\ \&\ uRv$.
I know we are ...
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If you have a tableau proof for $\Box A$, show that there is also a tableau proof for A.
If you have a tableau proof for $\Box A$, Show that there is also a tableau proof for A.
Here is my attempt but I'm not sure if it's correct:
If we have a tableau proof for $\Box A$, it means that all ...
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Tableau Calculus for Reflexive Frames
How would you create a tableau calculus for modal logics that have reflexive frames? What rules would need to be added to the existing system K?
My attempt is to refer to the axiom scheme for ...
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Does it follow that □ ~ ( A → B)?
Is this argument valid: 1. □ (B ↔ □ B) 2. ◊ ~ (A → B) 3. Therefore, □ ~ ( A → B).
I would appreciate any explanation as to whether or not it is valid. I think it is valid because if B is possibly ...
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Prove $(\Diamond p \to \Box q) \to (\Diamond p \to \Diamond q)$ in K
Modal logic, axiomatic proofs in K
I started with the tautology $(p \to q) \to (p \to q)$ but i'm a bit stuck.
$1. (p \to q) \to (p \to q)$
$2.\Box(p \to q) \to \Box(p \to q)$
$3. (\Box p \to \Box q) \...
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Modal Logic: Maximally consistent sets of formulas closed under necessitation rule?
If $\Gamma$ is a maximal K-consistent set of formulas, where K is the minimal modal logic, is $\Gamma$ closed under the necessitation rule? That is, if $\phi\in\Gamma$, do I necessarily have $\Box\phi\...
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Do all modal formulae classify digraphs?
I know that some modal formulae do classify digraphs. For example, $\Box \phi \rightarrow \Diamond \phi$ classifies all serial digraphs, i.e. digraphs such that for all vertices $v_i,$ there exists ...
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How do I solve exercise 2.2.4 Blackburn/de Rijke/Venema’s “Modal Logic”
Here’s a transcript of the original exercise. (There‘s even a hint given by the authors in the textbook as you can see. But precisely this hint confuses me).
2.2.4 Consider the binary until operator $...
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Proof in KTB (modal logic) using KTB and propositional logic [closed]
I am trying to prove that $\neg \lozenge p$ from $\square (p \rightarrow q)$ and $\lozenge \square \neg q$ using only the axioms in KTB and propositional logic. I can prove using tableux, but not ...
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In Modal Logic, if something is true, is it necessarily true? $P\implies\square P$ [duplicate]
I'm new to modal logic and I am trying to understand it more intuitively.
If something is true, is it necessarilly true? I.e.
$$P\implies\square P$$
This seems intuitive but it is not an axiom. This ...
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Prove that the following argument is invalid
$□p$, $□q$
therefore, $□(p→q)$
according to the K system of modal logic, the argument is invalid. I tried proving it using a truth tree, but all the branches unfortunately close, I don't know how to ...
