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Questions tagged [modal-logic]

a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality

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Is the problem of logical omniscience intractable?

Vincent Hendricks and John Symons notes the following about epistemic logic: Epistemic logic gets its start with the recognition that expressions like ‘knows that’ or ‘believes that’ have ...
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Is anything not proven impossible therefore possible?

Is it a truism that, except for that which is proven impossible, everything is or must be considered possible? If so, why? It seems to me to be an argument from ignorance to say that just because we ...
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Does anyone have a proof checker they prefer using for modal logic?

I am looking for a proof checker for modal logic using natural deduction or sequent calculus. I am trying to learn Isabelle, but I think there should be a simpler solution. Although I can rely on ...
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How to prove or disprove validity of ◻◻p → ◻p in the frame (Q,<) of the rational numbers with the usual less-than ordering?

I was wondering if someone can perhaps help with this proving. I am not sure how to handle a temporal frame that is not a set of only natural numbers. Any suggestions? Thank you.
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Prove or disprove ~◇◻p → ◇◇~p in system K

How to start with the following proof? Any help would be appreciated. I have tried by assuming the left side is true, however, I get confused with the negation. ~◇◻p → ◇◇~p
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Is ◻((◻(P → ◻P)) ↔ (◻P v ◻~P)) derivable in S5?

Is ◻((◻(P → ◻P)) ↔ (◻P v ◻~P)) a theorem in S5? If so, is it also a theorem in B or just S5?
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What is reflexivity in temporal logic?

According to Stanford Encyclopedia of Philosophy, reflexivity can be defined by : ∀x(x ≺ x) Where ≺ means "Precedes" : For all x, x precedes x But what does it mean for an instant in time x ...
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Does S4 (and B) hold for the strongest interpretations of ♢ and □?

Suppose these are the interpretations we are working with: ♢Ψ iff no explicit contradiction can be deduced from Ψ in FOL or in other words it's not provable that ~Ψ in FOL. ~♢Ψ iff an explicit ...
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Does ◻(∀x)(Fx ⊃ ◻(E!x & (E!x ⊃ Fx))) entail ◻((∃x)Fx ⊃ ◻(∃x)Fx)?

I was wondering if we can derive ◻((∃x)Fx ⊃ ◻(∃x)Fx) from ◻(∀x)(Fx ⊃ ◻(E!x & (E!x ⊃ Fx)))? (By the way 'E!' is the existence predicate.) I am using the Quantified Free Modal Logic constructed/...
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Prove the rule that proves X(P) from X(a) preserves derivability in modal system K

I'm trying to solve a problem which asks me to show that the meta-rule defined by deriving X(P) from X(a) preserves derivability (i.e. if ⊢X(a) then ⊢X(P) in modal system K, where a is a sentence ...
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S5 proof of ⊢◻(◻P→◻Q)∨◻(◻Q→◻P)

I'm trying to construct an S5 proof of ⊢◻(◻P→◻Q)∨◻(◻Q→◻P). I know that ϕ∨ψ is equivalent to ~ϕ→ψ, and so what I'm really trying to derive is ~◻(◻P→◻Q)→◻(◻Q→◻P) (which is equivalent to ◊~(◻P→◻Q)→◻(◻Q→◻...
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Axiomatic proof of ⊢ □P → □◇□P in S4

As the title explains, I'm trying to give an axiomatic proof of ⊢ □P → □◇□P in S4. This is simple to prove in B, but I'm struggling to see how it's done in S4. I'd really appreciate any help you ...
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Modal validity & vagueness

(2nd version to make explicit my implicit assumptions about A, B and C, and the definitions of the non-logical constants "⊂" and "≡".) Intuitively, the following modal argument seems valid to me and ...
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Equivalence of strings of modal operators in modal logic

I'm trying to solve a question which asks me to show that for any two finite strings O₁ and O₂ of □s and ◊s, (e.g. □□◊□◊□), that i) if O₁≡O₂ then OO₁≡OO₂ and ii) if O₁≡O₂ then O₁O≡O₂O where O is ...
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Logical expression that non-existent things don't do anything?

How could we express the idea that "something that doesn't exist cannot possibly do anything" using a logical argument? And if so, it is at all possible to prove that kind of proposition?
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Semantic expressiveness of modal logic

I am wondering how much of the semantic of basic philosophical questions can be expressed by formal arguments in modal logic. Here is one argument I formalised myself: P1 ◇ ∀a, ∃x // GNB(x, a) ∧ ...
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Questions from a new introduction to modal logic by Hughes and Cresswell

I just want to see if I got this correct. Show that the following wff is valid in every seating arrangement: Mp -> (Lq->Mq). Here's my argument: Assume player A sees at least one player B that ...
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Going from ◊(∃x)◻Fx and ◻(∀x)(Fx --> E!x) to ◊(∃x)◻(Fx & E!x)

Is it a valid inference (in, say, any quantified modal logic S5 system) to go from ◊(∃x)◻Fx and ◻(∀x)(Fx --> E!x) to ◊(∃x)◻(Fx & E!x)? (E!x is the existence predicate, by the way.)
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Validity of a modal argument?

Would you say that the following argument is valid? And if not, why exactly? For all we know, A may be the state of B; What C does is determined by the state of B; Therefore, for all ...
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A problem with Modal Logic (necessary truths)

I have this problem with Modal Logic that I am not sure I see which propositions in this logic are necessary besides tautological propositions. I mean metaphysically, something is necessary if it's ...
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Negation of a E quantifier in a Modal Logic T System?

In System T its possible to conclude: (1) □Ex->Ex Therefore its possible to conclude: (2) Ex->◊Ex But whats the result, if I negate the existence quantifier? Is it: (3) ~Ex->~◊Ex or is it: (4) ~Ex->◊~...
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On that p does not imply necessarily p

Am I right to assume that in no modal logic, whether in K or in a logic where the accessibility relation is specified as either reflexive, symmetrical or transitive, does ”p implies necessarily p” ...
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How to prove or disprove “Necessarily P entails P” in Kripke Modal Logic?

I'm doing the body of exercises in Priest's Introduction to Non-classical logic, and got weirdly stuck on 2.12 (o). (I'm on Problems 2.12, if anyone has a link to the answers that would also be much ...
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Axioms for modal logics based upon counterfactuals

Suppose we have a logic for counterfactuals as with David Lewis. I here use ⇛ for the counterfactual conditional. So suppose we have: Rules: (1) If A and A→B are theorems, then B is a theorem. (2)...
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Conditional logic - how to apply a conditional with complex antecedent in tableaux?

I'm referring to the conditional logic of C+ as described Graham Priest in "An introduction to non-classical logic" chapter 5, where the strict conditional is enhanced with ceteris paribus, and a ...
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What's the difference between logical modalities and physical modalities?

I am just wondering what's the difference between the two. I would say that there is something different, but honestly I can't define what it is exactly. What do you think?
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How do pragmatists avoid this modal argument against their view of truth?

I am reading Harry Gensler's Introduction to Logic (Routledge, 2002) and doing exercise 1 of 7.3b in the "Basic Modal Logic" chapter. I think I follow the steps. The answer is in the book. My ...
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Propositions that can't be used to distinguish possible worlds

Are there known ways of formalizing the notion of propositions that can't be targeted by counterfactuals in a coherent way? Or of propositions that are outside the scope of the framework in question? ...
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Are all logically necessary statements self identity claims?

When we say that x is necessarily P, are we not asserting that this is the case regardless of all contingent facts, so regardless of what x and P are? And if x was P regardless of what x and P are, ...
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What the main reason for Bertrand Russell's skepticism and rejection of modal logic?

It seems to me that Bertrand Russell was somewhat hostile to the idea that modality plays a role in logic.
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What is necessity in modal logic?

I understand that something that is necessary must exist. However, what is the exact definition of necessity? Thank you.
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Can possible-but-non-actual objects have accidental properties?

The modal logic I am considering is the "Simplest Quantified Modal Logic" which combines first-order predicate logic with identity, with S5 in the most straightforward way, described here and slightly ...
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How to denote physical (not logical) impossibility in modal logic?

Say I want to note that in this world, it's physically impossible to jump over the Eiffel Tower. I can just write ¬◇x, but this seems to say that in all possible worlds x is impossible, and I want to ...
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introduction to modal logic

I am not entirely sure if we are allowed to ask such questions here but I was wondering if anyone could suggest me a real introductory level textbook on modal logic
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How many equivalence classes does the accessiblity relation have in S5?

I'm a math student taking philosophy classes. So I have some math background but am absolutely unfamiliar with metaphysics&c. Recently I've come around the "Nothing Is Impossible" paper as a part ...
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Logic and resolving a modus tollens

Wasserman presents a common argument against time travel: (P1) If backward time travel were possible, it would be possible to perform a self-defeating act. (P2) It is impossible to perform a self-...
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In modal logic, why not 'possibly p' → 'not necessarily p'?

I'm told that if ◇ means 'possible' and ◻ means 'necessary' and ~ means 'not' and ↔ means 'if and only if', then ◇P ↔ ~◻~P I get that if it is not necessarily not going to be sunny ...
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Establishing validity of ◊(∃x)◻[(E!x ⊃ Ox) & E!x]/ ∴ (∃x)◻[(E!x ⊃ Ox) & E!x]

I was wondering under which sort of quantified modal logic S5 system would the inference from ◊(∃x)◻[(E!x ⊃ Ox) & E!x] to (∃x)◻[(E!x ⊃ Ox) & E!x] be valid. Would this require a constant domain ...
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Justification for axiom OB-RE in deontic logic

Let OB p denote "p is obligatory". Axiom OB-RE is (p ↔ q) → (OB p ↔ OB q). This axiom seems false to me (under the interpretation of obligation). For example, let p denote "don't lie to me" and q ...
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The converse to the C modal logic axiom - has it been studied?

There is a C axiom as mentioned at https://plato.stanford.edu/entries/logic-modal/#MapRelBetModLog in Section 8. My question is: what can be said about the formula which is the converse of the C axiom?...
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How to construct a counter-model of □P --> □◊P in T and K?

I am new to modal logics and would really appreciate it if someone would be able to help me out with this practice question. I’ve established the validity theorem in T with an axiomatic proof, but I ...
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If I cannot draw a world in a truth tree in modal logic does that mean that no other worlds exist for that statement?

Suppose I have a truth tree for a modal logic statement that is closed or is open but for which I cannot continue drawing worlds, does that mean that there are no other accessible worlds in that modal ...
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What is the counterexample in modal system K for “⬜A ➡A”?

As I understand it from Modal Logic 2.1 - the systems M, B, S4 & S5, I should not be able to show "⬜A ➡A" in modal logic K. The following truth tree seems to confirm this, if I did it correctly: ...
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Why can one not draw a world in modal logic given necessity?

In Modal Logic 1.2 -- truth trees for K about 1:30 into the video, the presenter said: If you have possibly A then you need to draw an arrow to a new world and derive A in that world because w0 ...
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What is the justification for closing a whole branch in a modal truth tree diagram if there is a contradiction in only one world?

I am watching "Modal logic 1.2 - truth trees for system K" presented through Kane B's channel. The standard propositional truth tree diagrams are not a problem. Where I am having trouble is opening ...
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Why might truth imply necessity?

It seems well established (or, well-accepted) that necessity implies truth. For instance, in a context of epistemic logic, if one knows A, then A is true. On the other hand, I found on the Internet ...
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Natural deduction introduction and elimination rules for modal quantifiers

I've read several papers on modal natural deduction but I've only been able to find one clear explanation of []-Introduction ([]A if A can be proved from no assumptions/premises). But there were no ...
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Nonexistence and invalid formulas in modal logic

In first-order logic, I can essentially just ignore issues related to nonexistence and invalid formulas, without losing much. There is also free logic, in case I'm not happy with simply ignoring these ...
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How to translate classical logic to basic Kripkean logic?

I'm trying to semantically prove the following argument (sorry about the formatting - I'm new to stackexchange): if P ⊨c Q then ⊨k 򪪪(P → Q) However, I don't know how to translate or relate the ...
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What determines accessibility of possible worlds?

Recently, I have begun studying modal logic, using Brian Chellas's Modal Logic: An Introduction. Something keeping me from fully understanding the material is the idea of a possible world. They seem ...