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Questions tagged [modal-logic]

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

Modal logic is an extension of propositional and predicate logic that expresses modalities, which are qualifications to a statement. The most commonly used modalities in mathematics are "possibly," "necessary," and "impossibly."

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566 questions
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Euclidean frame validity

I have trouble understanding why $$\Diamond P \to \square\Diamond P$$ is valid in Euclidean frames. I found a proof online which is detailed as follows: Proof. Suppose $F$ is a Euclidean frame, and $M$ a model based on $F.$ Suppose $\models_w…
idkla
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Modal Logic Questions Validity

These are some examples in modal logic, I am not too sure why the first two are valid and why the last one specifically is not valid. Hope someone can explain! $\square(P \rightarrow P)$ (VALID) $\square P \rightarrow \square P$ (VALID) $\square P…
Masterminder
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Is the $D$ modal logic necessarily serial?

I'm wanting to show in $D$ that $\sim \square (\square p \wedge \square \sim p)$. Here's my attempt: (1) Show $\sim \square (\square p \wedge \square \sim p)$ (2) $ \square (\square p \wedge \square \sim p) \quad $assumption for indirect derivation…
Squirtle
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writing proofs in modal logic

I'm confused about the expectation of what a proof in modal logic is supposed to look like, because the texts I've seen so far have proofs that look more like a proof out of a math text and not something, say, out of a book on introductory…
Squirtle
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Converse of Axiom K

Just for clarity, this is the specific formulation I am referring to. $K$: $\Box(p\to q)\to(\Box p\to\Box q)$ Naturally, we might consider the converse of this statement: $K'$: $(\Box p\to\Box q)\to\Box(p\to q)$ The question is simple. Given…
Conner N. Howell
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Disjunction and conjunction in modal logic

Is there a system of modal logic in which $\Box (A\vee B)\leftrightarrow\Box A\vee\Box B$, $\Box (A\wedge B)\leftrightarrow\Box A\wedge\Box B$, $\Diamond (A\vee B)\leftrightarrow\Diamond A\vee\Diamond B$, $\Diamond (A\wedge B)\leftrightarrow\Diamond…
user60264
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How do I read double connectives in modal logic?

My book says: $\neg \Box p \rightarrow \Box \neg \Box p$ is not a valid formula and proves it by saying that in Figure 5.3, if we change $L(x_4)$ to $\{p, q\}$, then $\neg \Box p$ is true but $\Box \neg \Box p$ is not. I get that $\neg \Box p$ is…
mdemont
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Is $p \rightarrow \lozenge (q \rightarrow q)$ a tautology in K?

Intuitively, it is a tautology. Imagine two possible worlds $m0$ and $m1$, such that $m1$ is accessible from $m0$, i.e., we have the folowing scheme of possible worlds: $m0 \rightarrow m1$. Whatever is the truth value of $q$ in $m1$, $\lozenge(q…
Walter r
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Do sets have all their members necessarily? [Set theory and modal logic]

Can there be two possible worlds $w_1$ and $w_2$, an object $o$ (set or not) and a set $s$, such that $o\in s$ is true in $w_1$, $o\notin s$ is true in $w_2$ and $s$ is the same set in both worlds? Another way to ask the question is whether the…
Seb
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modal logic diamond T meaning

I have been learning about the modal operators of modal logic and it's truth evaluations in different contexts. For instance: $$\Box A \to A$$ Would be false in Deontic logic, but true in Alethic logic. However, I am struggling to understand the…
SilverTear
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modal logic - examples for it may be supposed that/it is compulsory that

Could someone give me example with modal logic ? $\diamond X$ it may be supposed that $\Box$ it is neccessary that. I mean some example with worlds and arrows between them. Why am I asking about it ? Simply, I can't understand it.
user343207
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Finding a condition on models that validate theorems

I want to find a condition on standard models for which $\Diamond A \rightarrow \Box A$ is a theorem. I can see that the condition $\forall w_1,w_2,w_3\in W$ if $w_1Rw_2$ and $w_1Rw_3$ then $w_2=w_3$ would be a sufficient condition but it seems…
Addem
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Truth Conditions for Modal Logic

I know that $$ 1)\space w ⊩\diamond P\iff there\space is\space some\space worlds\space w' \space such\space that\space wRw': w ⊩ P $$ $$ 2)\space w ⊩\square P\iff for\space all\space worlds\space w' \space such\space that\space wRw': w ⊩ P $$ Now…
tom tronbone
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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 true. Please correct me if I'm wrong). But how do…
conectionist
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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.$(q\land p)\rightarrow p$ 3.$p\rightarrow(p\land p)$ 4.$p\land(q\land…
romtsn
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