# Tagged Questions

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

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### Formalizing a self referential sentence

In The logic of provability, by G. Boolos, we are asked to ponder about this statement: If this statement is consistent, then you will have an exam tomorrow, but you cannot deduce from this ...
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### Is the 1-consistency of $PA$ necessary to prove that $\diamond^{m} \top\implies \diamond^{n} \top$ is false if $m<n$?

In The logic of provability, by G. Boolos, there is a remark in chapter 7 saying that $\diamond^{m} \top\implies \diamond^{n} \top$ is false if $m<n$ (unless $PA$ is 1-inconsistent). Now, it seems ...
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### Valuation in the proof decidability of normal systems of modal logic

In The logic of provability, by G. Boolos, it is stated in the chapter 5 that if L is a formal system between $GL$, $T$, $B$, $K$, $K4$, $S4$, $S5$; then $L\vdash D$ iff $D$ is valid in all models of ...
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### On the truth of $GLS$ and Löb's theorem

Consider the formal system $GLS$, whose axioms are the theorems of $GL$ plus all sentences of the form $\square A\rightarrow A$. A translation maps a sentence of modal logic to a sentence in the ...
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### Categorical semantics for dynamic epistemic logic

Dynamic epistemic logic tries to reason about knowledge that certain actors (people, machines, etc.) have and how it can change in response to outside events. It is usually possible to discuss such a ...
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### The positive introspection axiom

I am studying modal logic with the textbook 'Reasoning about Knowledge' Fagin et al. 1995 The positive introspection axiom is taken as something that can be proved with the possible worlds model of ...
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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 ?...
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### Find an equivalent standard translation for FO formula

Let $\psi(x):=\forall y\exists z\forall u(\neg R(x,y)\vee (R(y,z)\wedge \neg R(z,u)))$. We must find a equivalent form of $\psi(x)$ such that we can apply standard translation on it and find modal ...
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### Does the fact that a modal operator distributive over disjunction imply that a modal operator is distributive over conjunction?

If L is an arbitrary operator on two propositions p and q: Does L(p $\vee$ q) $\Rightarrow$ Lp $\vee$ Lq imply L(p $\land$ q) $\rightarrow$ Lp $\land$ Lq?
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### Why is “necessary p” true in a world when there is no world accessible from it?

So my question is situated in modal logic and everything is defined as usual. I'm reading volume 2 of logic, language, and meaning and on page 24 it says: $V_{M,w_3}(\square p)= 1$ So the valuation ...
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### Use of □ in Modal Logic, Translated into English

For □, do you read it as “for all states. . .”, that which one, maybe, cannot withdraw from.? Does this, □, not request proving of itself? □P↔¬◇¬P How do you use it in English? If, for all ...
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### (Green/blue)-eye logic puzzle. Statement validation

There is a logic puzzle aiming on freeing same-color-eyed people from an island. The thing is that they must be certain of their own eye color so that they can leave. For that reason an external party ...
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### Solovay's Arithmetical Completeness Theorem

Solovay's Arithmetical Completeness Theorem affirms that: Let $A$ be a sentence of modal logic. If every realization $A^\phi$ of $A$ is proved by $PA$, then $GL\vdash A$. I am troubled by this ...
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### Gödel's Incompleteness theorem in modal logic

I think I once taught that the statement of the Gödel's Incompleteness theorem is equivalent to : $\diamond \neg \square \square p$, if p is a complex enough system. Is this right?
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### How to deal with propositional First-Order-Logic Dynamic Logic formula

Let p be an atomic proposition, let a be an atomic program, and let $π = (K, M)$ be a Kripke frame with $K = \{u, v, w\}$ $Mπ(p) = \{u, v\}$ $Mπ(a) = \{(u, v), (u, w), (v, w), (w, v)\}.$ The ...
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### Proving completeness of modal systems

In reading through Chellas' Modal Logic I'm trying to understand his proof of completeness. Here is my best outline of his proof: If $\Sigma$ is any normal system of modal logic then it has a proper ...
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### Canonical models for systems

I've been struggling with understanding canonical models for systems. I'll state my question here, but at the end I define some terms in case they're not uniformly defined in Logic. My text claims ...
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### Validity of $F \supset \Box F$

I started to study propositional modal logic and Kripke semantics. I learned that for any Kripke interpration $\mathcal{M}$, we have that, if $\mathcal{M} \models A$ then $\mathcal{M} \models \Box A$. ...
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### Nested Sets of Points

This question is inspired by modal logic, but reduces to a basic set theory problem. David Lewis in Counterfactuals claims that the answer to my question is "easily verified" but I can't figure it out....