Questions relating to deductions relating to the expressions "it is necessary that" and "it is possible that"

learn more… | top users | synonyms

4
votes
0answers
68 views

Is there a logic to formalize the concept of “understanding”

The question may seem little bit weird given that philosophers have been struggling to have a full grasp on the concept of "understanding". But I'm wondering if there are any logics (modal-based or ...
0
votes
0answers
18 views

Deriving $\Box p \rightarrow \Box \Box p$ if we have $\Box(\Box p \rightarrow p) \rightarrow \Box p$ [closed]

Let $F = (W,R)$ - Kripke frame, $AL \rightleftharpoons \Box(\Box p \rightarrow p) \rightarrow \Box p $ Proof if $F \vdash AL$ then $R$ if transitive
2
votes
1answer
38 views

What does not having a first-order frame imply for models?

There are formulas in modal logic which which do not have a first-order frame condition, as stated here (Non-Sahlqvist formulas, Wikipedia). An example is the McKinsey formula for $p$: ...
0
votes
1answer
42 views

problems with validity in type theory

I'm twisting my brains over some simple formulas in intensional type theory. If $\exists x \Box (x=^{\vee}j)$, s.t. $x$ is of type $<e>$ and refers to an entity $e$ and $j$ is of type ...
0
votes
1answer
53 views

Strong Kleene interpretation

Consider: $\\$ $\Box(\phi \wedge \psi) \rightarrow \Box(\phi) \wedge \Box(\psi)$ I guess this yields by the reflexivity axiom for intensional predicate logic? But I was wondering whether it is also ...
0
votes
1answer
28 views

Theorems of GL in modal logic

So I've been reading George Boolos' "The Logic of Provability" and he's explaining different systems of modal logic. He's taken as his basic symbols → (implication), □ (necessity), ⊥ (falsehood), a ...
1
vote
1answer
55 views

Proof of $p\rightarrow (\Box (\Box p \wedge p) \rightarrow (\Box p \wedge p))$

I need to prove: $$p\rightarrow (\Box (\Box p \wedge p) \rightarrow (\Box p \wedge p))$$ The system contains all propostional tautologies and the axiom scheme $\mathbf K$:$ \Box(p \rightarrow q) ...
1
vote
1answer
12 views

Truth Conditions for Modal Logic

I know that $$ 1)\space w ⊩\diamond P\iff there\space is\space some\space worlds\space w' \space such\space that\space wRw': w ⊩ P $$ $$ 2)\space w ⊩\square P\iff for\space all\space worlds\space w' ...
0
votes
1answer
53 views

Why do you only need to show validity in one world when using trees in institutionist/constructivist logic?

Depicted below, my prof used a tree to prove that an argument is valid according to intuitionist logic. However, I can't find a contradiction in world 0. Why is invalidity ascertained when all ...
1
vote
0answers
59 views

A good (and possibly seminal) book on Multimodal Logic?

I'd like to study Multimodal Logics (in the sense of Catach's Normal Multimodal Logics for instance). Some suggestions? Thank you.
1
vote
1answer
41 views

Is there a Fitch style system that works with some of the modal logics?

My prof taught us to use trees to prove modal logic arguments. Trees seem to provide a more efficient way to test arguments than Fitch does. However, I find that trees generally, and alethic (modal) ...
0
votes
1answer
42 views

¬p ⊬ ⎕(p → q): Where's the mistake in my proof?

My professor noted on one of his slides that ¬p ⊬ ⎕(p → q). Intuitively, this seems correct; however, I can only prove that it is false. I suspect I've made a mistake in my proof. Where have I gone ...
2
votes
1answer
54 views

In modal logic, why are models ordered sets?

I just started undergrad math, so I only have a fuzzy idea of what a model is. I'm learning about modal logic in one of my classes. Our text describes modal logic as operating in a model defined as an ...
11
votes
1answer
872 views

Big Bang Theory Reference to Formal Logic

In the second episode "The Junior Professor Solution" of the 8th season of the Big Bang Theory, there exists a brief moment where Sheldon Cooper references one of his boards with what for a brief ...
0
votes
3answers
214 views

Blue Eyes: A Logic Puzzle, has a puzzling solution (a.k.a. What does common knowledge have to do with it?)

In Blue eyes: a logic puzzle (specifically, the follow up questions), the most common answer is that it needs to be common knowledge that someone has blue eyes for all the blue-eyed people to leave. ...
2
votes
1answer
50 views

What does this negation on both sides of K mean: A = ¬ K ¬

What does this negation on both sides of K mean: (A = ¬K¬) ? I'm not sure if it's a typo, as there are some errors in this paper (Hong et al.). Hong, Zhi Ling, and Mei Hong Wu. "Constrained ...
0
votes
2answers
103 views

Help with a modal Hilbert-style proof of (□(a>b)&◊(a&c))>◊(b&c)

Can't grasp how it can be proved. To proof just propositional calculus formula (without modal operators) at first seems rather natural to me. Tried the law of importation scheme but it didn't work ...
2
votes
3answers
100 views

Is there a modal operator which distributes over the implication?

Is there any notable modal operator $\Box$, so that if $P,Q$ are proposition $$\left(\Box(P\implies Q)\right)\Leftrightarrow\left(\Box P\implies \Box Q\right)$$
1
vote
1answer
30 views

Definition in satisfiability problem

While I was reading the PhD thesis of Balder ten Cate (2005). Model theory for extended modal languages. I found a theorem that says: 2.6.4Theorem. The frame satisfiability problem for modal ...
3
votes
2answers
126 views

Why is $\square\square=\square$ and $\Diamond\Diamond=\Diamond$ in the S5 modal logic?

I'm reading about modal logic on the Stanford Encyclopedia of Philosophy. They define the modal logic S5 as propositional logic augmented with the modal operators $\square$ and ...
1
vote
1answer
94 views

Modal set-theory

In his “The Potential Hierarchy of Sets”, Review of Symbolic Logic 6:2 (2013), 205-28 Øystein Linnebo has proposed a modal set-theory. I was wondering what kind of utility can such a theory have for a ...
1
vote
2answers
71 views

The Entscheidungsproblem (decision problem) for modal logic

The Entscheidungsproblem is identified with the decision problem for first-order logic that is, the problem of algorithmically determining whether a first-order statement is universally valid. ...
3
votes
2answers
108 views

Prove that modal logic S4 is properly contained in S5

Is it possible to prove using just the semantics for $S_4/S_5$ that $S_4$ is properly contained in $S_5$? I can see how one could show that there is a theorem of $S_5$ which is not a theorem of $S_4$, ...
1
vote
0answers
51 views

Trying to understand Hintikka's logic of Knowledge and belief

I try to understand Hintikka's logic of knowledge and belief but am a bit stumped by it. I study " Knowledge and belief , an introduction to the Logic of the two Notions", (Kings College ...
2
votes
1answer
26 views

Can it be the case $p$,$\lnot p$ are true at $\mathbf w$ And $\mathbf w'$ respectively with $\mathbf w$ and $\mathbf w'$ have access to each other?

Given a model $\mathbf{M} = (\mathbf{W}, \mathbf{R}, \mathbf{V})$ for a set of atomic formulae $\Omega$. We have possible worlds $\mathbf{w}, \mathbf{w'} \in \mathbf{W}$, access relation satisfies ...
3
votes
2answers
70 views

Universe enlargement and modal logic

In model theory and category theory, we often need to "enlarge" our universe (whatever that means) so that our proper classes become "small" and we can thereby manipulate them in more sophisticated ...
1
vote
1answer
36 views

Problems with the usage of Belief and Common Belief operators

I have a problem with the usage of a Belief operator $B_i$ in the derivation of a result on a common belief operator $CB$. First of all, some basic definitions (where $i$ is an individual), that ...
1
vote
1answer
55 views

Decidability of normal modal logics

Let's say we have systems of modal logic defined as smallest sets containing propositional tautologies, all instances of schema $\square F \to (\square(F \to G) \to \square G)$ all instances of ...
1
vote
1answer
73 views

Book on the first-order modal logic

Is there a book on the metatheory for the first-order modal logic, or do I just need to take FOL as a base and use the standard translation?
1
vote
1answer
43 views

Propositional S5: is there a consistent set requiring continuously many worlds?

A recent question asked whether in systems of modal propositional logic having the "finite model property" there are consistent sets of sentences that were not satisfied by a finite model. @Carl ...
0
votes
1answer
87 views

Propositional modal logic: infinite models required in systems with finite model property?

A system of propositional modal logic has the "finite model property" if any consistent sentence is satisfiable at a model with finitely many possible worlds. Some systems have this property and ...
4
votes
1answer
51 views

Truth Tables for Temporal Operators?

I would like to know whether we can construct truth tables for the following temporal operators in temporal logic as we do in propositional logic . ...
5
votes
4answers
180 views

What obstacles prevent three-valued logic from being used as a modal logic?

I am familiar with many of the surveys of many valued logic referenced in the SEP article on many valued logic, such as Ackermann, Rescher, Rosser and Turquette, Bolc and Borowic, and Malinowski. It ...
1
vote
2answers
69 views

Proving Gabbay rule for Modal Logic

I'm currently working on exercises of the book "Modal Logic" by A.Chagrov and M.Zakharyaschev (for pleasure, not homework). One exercise asks to prove this version of Gabbay rule (exercise $3.10$): A ...
0
votes
0answers
3 views

Validity of LTL formulas in a given transition system [duplicate]

Say I have the following transition system: I've understood how I can tell if □a and ⟡b are valid (□a is invalid because a is not true is S2 and ⟡b is valid there is a state (i.e. S1) in which b is ...
1
vote
2answers
103 views

Validity of LTL formulas in a given transition system

Say I have the following transition system: I've understood how I can tell if □a and ⟡b are valid (□a is invalid because a is not true is S2 and ⟡b is valid there is a state (i.e. S1) in which b is ...
2
votes
1answer
148 views

Is there a name for this property of a binary relation? $\forall x\forall y(x\mathsf{R}y\to\exists z(x\mathsf{R}z\land y\mathsf{R}z)))$

Consider a binary relation $\mathsf{R}$ such that $x\mathsf{R}y$ is the case only if there is some $z$ such that both $x\mathsf{R}z$ and $y\mathsf{R}z$ are the case. Is there a well-known name for the ...
0
votes
0answers
91 views

Prove $\Diamond p \rightarrow \lnot\Diamond\lnot\Diamond p$ in modal logic

I need to prove $\Diamond p \rightarrow \lnot\Diamond\lnot\Diamond p$ in B axiomatic, which contains next conversion rules: 1.$(p\land q)\rightarrow(q\land p)$ 2.$(q\land p)\rightarrow p$ ...
2
votes
1answer
74 views

Is it possible to prove $p\rightarrow\diamond (p\land q)$ in modal logic?

I need to prove $p\rightarrow\diamond (p\land q)$ in B axiomatic, which contains next conversion rules: 1.$(p\land q)\rightarrow(q\land p)$ 2.$(q\land p)\rightarrow p$ 3.$p\rightarrow(p\land p)$ ...
0
votes
0answers
30 views

Problems with a basic proof in Aumann Structures

I am pretty sure this is more than trivial, but I have a problem with the proof of a basic results in Aumann structures (this is related to a more general problem I have with proofs that involve ...
0
votes
1answer
74 views

Can necessity rule be derived from box introduction rule?

I need to find a proof of $\top \vdash \Box \top$ (where $\top$ is the truth constant and $\Box$ is the necessity modal operator) in the natural deduction system of IS4 modal logic. In the axiomatic ...
1
vote
0answers
34 views

Test axiom in PDL

Wikipedia says the axiom for test in PDL is $$ \langle \psi ? \rangle \phi \leftrightarrow \psi \wedge \phi, $$ but why is this right? (i.e. what does it say?) And what is the corresponding relation ...
2
votes
2answers
139 views

Basic Modal Logic question #1

Its about two weeks I have started Cresswell's "A New Introduction To modal Logic". Now I've got a few questions on the text and I would deeply thank you if you help me clarify on them. ...
1
vote
1answer
328 views

Incomplete normal modal logic systems

Apart from the classical example of KH, given by axiom $\Box(\Box p\leftrightarrow p)\to \Box p$, are there any other examples of incomplete propositional normal modal logic systems defined by axioms ...
0
votes
2answers
88 views

Does the dynamic logic page at wikipedia have some mistakes?

I was reading about dynamic logic over at wikipedia as a possible lead on a previous question. However, its not making a lot of sense to me. In particular, wikipedia says that The constant action ...
4
votes
2answers
86 views

Equivalence between temporal logic and notions of forcing

I have come across literature comparing modal logic to forcing (by Hamkins et al). Has anything similar been done showing equivalences between temporal logic and forcing? This would be interesting to ...
55
votes
25answers
9k views

Blue eyes: a logic puzzle

Today I read the Blue Eyes puzzle here. I also read the solution which I find quite interesting. But there are three follow up questions which I don't know the answer to: What is the quantified ...
4
votes
1answer
188 views

What is the definition of algebraic equivalence of formulas?

This question is for the propositional normal modal logic system K (although it may apply to other logics too). I saw a couple of papers which mentioned algebraic equivalence of modal logic formulas ...
3
votes
1answer
52 views

exchange “globally” and “future” in temporal logic

I cannot prove the following theorem given in Schneider's “On Concurrent Programming” as (3.16e): $$\Diamond\Box P \to \Box\Diamond P$$ I was given $$\begin{align} \Box P & \to P\\ \Box P ...
1
vote
2answers
59 views

What is the modal interpretation of the converse to a statement?

I have just begun learning about modal logic and I am trying to get my head around Kripke Semantics. I then began to wonder if there is any connection between convereses of statements and modal logic. ...