# 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 this an example of an infinite regress, an infinite loop, or both? [closed]

Consider a logician who thinks the following thought: whatever I assume or expect is false or will never happen. But then the logician thinks this itself is an assumption, so it will not happen either....
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### What are some noteworthy consequences of a deontic logic extended with the axiom “Ob(A) → Ob(◊A)”?

I think one unsuccessful attempt to construct a form of deontic logic in which the “ought” modal operator implies the “can” modal operator was to include the axiom OBA → ◊A, for an “obligation” ...
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### Challenging the distinction between a validity and a tautology

I know that there is a formal difference between a validity and a tautology. On the one hand, a validity is any formula of First-Order Logic that is true in all interpretations. A tautology is a ...
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### what's means scope of further modal operators?

I am reading page 315 of Parsons' Sets, Classes, and Truth. He presents the comprehension principle in the following form, but at the same time, he argues that this does not prevent Russell's paradox....
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### What is a condition on accessibility relation corresponding to ◊□(A→B)→(◊□A→◊□B)?

Recently, I am struggling with finding specific semantic conditions on frames in which the following axiom holds (the analogue of the distribution axiom for ◊□ instead of □): ◊□(A→B)→(◊□A→◊□B) I know ...
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### Suppes–Lemmon-Style ◇-Introduction and -Elimination Rules for Modal Logics?

I'm trying to find natural-deduction introduction and elimination rules for the diamond (◇) in popular modal logics (e.g., K, T, S4, and S5) in the style of Suppes and Lemmon, where on each line of ...
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### Are there any known precendents of philosophers using modal logic (or any other theory of math) to formalize works of other philosophers?

Like maybe philosopher Anna wrote a long and complicated treatise on some topic. Then comes philosopher Bob who interprets Anna's treatise in some way and writes down his interpretation of Anna's ...
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### Why is this argument valid?

I m reading Linnet's paper 'pluralities and set' where his claim said that collapse principle lead contradiction if we didn't assume 'it is possible to quantify over absolutely everything' He uses ...
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### Why is modal logic not focused more on strict implication?

Modal logic is designed to capture various notions, such as necessity, time, provability, etc. These notions are captured via a box operator or sometimes the letter L. For example, in systems capable ...
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### Are there logical facts absolutely necessary in modal logic across possible and impossible worlds?

If we consider weird universes or ones in which impossibilia exist alon others. Are there any logical things or laws of thought which have to exist across all types of wild conceptions? Roujd squares ...
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### what is the definition of a necessary fact in the contingency argument?

how can we define a necessary fact in the contingency argument in a way which does not lead us to the ontological argument? (exists in all possible worlds) the contingency argument is: A contingent ...
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### could the set of all contingent facts be necessary?

I was thinking about the PSR. when it comes to the set of all contingent things, it seems that the set must also be contingent and could fail to exist because each member could fail to exist. but ...
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### What papers or books should I read in order?

I have been reading literature on modal set theory and am currently reading Putnam's "Mathematics Without Foundations," which is known for being one of the earliest presentations of this ...
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### What is Dummett's narrow/wide scope objection to Kripke's modal argument against descriptivism?

I just cannot wrap my head around this concept, if anyone can make it clear for me I'd be greatly appreciate it. I've tried reading the literature, but the papers I read invariably start putting the ...
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### What does it mean to say "possible in reality?" [closed]

Consider the following truth table. I am hungry My eyes are closed 0 0 0 1 1 0 1 1 Each row is possible. But that seems to me like an incomplete statement. Instead, I think it makes more sense ...
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### If a proposition is necessarily true, does it follow that it's a tautology?

If □P, does it follow that P is a tautology? I know in K modal logic, the law of NEC states ⊢ P; therefore □P. The corresponding conditional of the previous argument is If ⊢ P then □P. Now ⊢P iff P is ...
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### How do you prove transitivity of equality?

There are only two axioms of equality in SQML, reflexivity (RX), and substitution (II). Obviously you can prove transitivity of equality, but how does the proof go? Reflexivity (RX) ∀x[x = x] ...
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### Is there a counterpart of "merely possible"/"merely permissible" in temporal logic?

It might not be too bad to say that, in modal and deontic logic, possibility and permissibility are like "ground states," having the lowest "logical energy." At least we think ...
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### Can 𝐅𝐑 be taken for a deontic negation operator (rather than just a specified negation of 𝐎𝐁)?

Presuppositions of the question: beliefs about the ambient structure of negation: I was rethinking the following in light of questions about supervenience, grounding, alterity, and identity: A ...
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### In the usual modal logics, are there tautologies of the form ◊¬X or ¬☐X?

And not when, "Possibly not X," or, "Not necessarily X," are implied by, "Impossibly X," already. But so is it possible to have a tautology be a statement of mere ...
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### Question regarding modal logic version of disjunctive syllogism

Is the following theorem provable in SQML? ⊢ (A ∨ B) ∧ ~◊A → ☐B Here's how far I got. ⊢ (A ∨ B) ∧ ~A → B [Prop.] ⊢ ☐(A ∨ B) ∧ ~A → B [1;NEC] ⊢ ☐[(A ∨ B) ∧ ~A] → ☐B [2;Dist.,MP] ☐(A ∨ B) ∧ ☐~A → ☐B [3;...
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### Was Gödel actually convinced that his ontological proof was correct?

The proof is obviously logically valid, but it is as obvious that it isn't logically sound. For instance, the second axiom states that ¬P(φ) ⟺ P(¬φ), take φ(x) ⟺ x is a male human being. Then either ...
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### Can the modal logic S5 be reduced to Rosser's system for a first order function calculus?

From the SEP In propositional logic, a valuation of the atomic sentences (or row of a truth table) assigns a truth value ( T or F ) to each propositional variable p . Then the truth values of the ...
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### Question regarding conjunction and necessity in modal logic

I've been learning SQML, and I have a question regarding conjunction and necessity. How do you prove the following theorem, or is it not generally true? ⊢ □(A ∧ B) → □A ∧ □B
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### Question regarding disjunction and necessity in modal logic

I have a question regarding disjunction and necessity. Is the following theorem provable in any system of modal logic, or is it not generally true? ⊢ A ∨ B → □A ∨ □B I was thinking about using the ...
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### Can natural deduction be incorporated into SQML?

EDIT - From the SEP 3.4 Propositional Modal Logic (S5) (1) □(φ → ψ)→ (□φ → □ψ) (2) □φ → φ (3) ◇φ → □◇φ Rule of Necessitation (RN): □φ follows from φ (3) is precisely what I have in my demonstration, ...
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### How can I derive ~a=b→☐~a=b in SQML?

I have been thinking about this question for a long time but didn't seem to make any progress. Here are the axioms of SQML:
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### Can modal logic be used to define the notion of an “arbitrary constant” in FOL?

I was wondering if first-order logic can be reduced to propositional calculus if we eliminate quantification. For example, instead of saying “for all x in a domain D, P(x)”, we could state “P(x)” for ...
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### What is the reason behind the fourth axiom in Gödel's ontological proof?

In Gödel's ontological proof, axiom 4 goes like this: And I'm not sure about what it means. If that P(φ) is true, then isn't it necessarily true as well? There's some basic concept about modal logic ...
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### Do universals exist in all possible worlds?

Exactly what it says on the tin: Do abstract objects, like universals for instance, necessarily exist in all possible worlds? To my knowledge, David Lewis held to the opinion that they did (And that ...
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### Is B(p) V B(~p) an instance of LEM in doxastic logic?

So in classical logic either p is T or p is F. But is it same in doxastic logic, ie, is B(p) V B(~p) an instance of LEM? And the second issue, is it equivalent to B(p) V ~B(p)?
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### Understanding possible world semantics and time

In possible world semantics, statements of the form "It is possible that P" are interpreted as meaning "There is some 'possible world' in which P is true". And if you're a modal ...
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The masked man fallacy (or paradox) is roughly: Premise 1: I don't know who the man wearing the mask is. Premise 2: The man wearing the mask is my father. Premise 3: I know who my father is. ...
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### When is the proposition expressed by "I am here" necessary?

I'm currently reading Kaplan's On The Logic of Demonstratives (1979). He considers the example (1) I am here now. and on page 84 he argues that (b) In almost (if not all) contexts, an utterance of (...
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### Proof that the single variable fragment of first order logic is equivalent to an S5-like modal logic

I think I read that a single sorted logic is a logic in which there is only one unique variable symbol permitted. In other words, there is only one “parameter of variation” amongst all sentences of ...
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### Can there be nested possible worlds semantics?

Fairly straightforward question, I'd think: Usually, when we do Modal Logic, we think of propositions as sort of embedded within a framework of possible worlds. What, then, do we make of propositions ...
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### What is the modality of a statement that follows from a necessary statement?

Let □P. Suppose □P => Q. What can be said about the modality of Q? □P <=> P holds in every possible world. Thus it is available as a premise to derive Q in every possible world. Suppose Q is ...
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### Questioning the category of the “moral”

Briefly: it occurs to me that taking as given the pre-existence of the terms “morality” and “ethics” structures our thinking preemptively and heavily. In the manner of discursive analysts like ...
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### Do the incompleteness theorems need the provability predicate to be expressed, or can they be expressed via just ⊢?

In his "Epistemic Set Theory," William Reinhardt says: It is the purpose of this paper to formulate axioms for Gödel's modal operator B for provability (see [3], [8]) in the context of set ...
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### Modal dualism: is there a combination of counterpart and transworld-identity theory classifying objects based on which relation they enter into?

Theorists love to be competitive, but often enough we find out that they don't have to be like that. The SEP article on infinitesimals, for example, notes at one point: It is of interest to note that ...
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Are '□(p → q)' and 'p → □q' semantically equivalent? Specifically, does eliminating parentheses in the former gives us the latter?
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### Trying to avoid a modal explosion: if anything can be obligated, and ought-implies-can, then would everything be possible?

Where by "anything"/"everything" I mean atomic propositions (and I am quantifying over atomic propositions). The argument would seem to be something like: ◊OBA, ∀anyA OBA → ◊A ◊◊A ...
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### Is Fermat's last theorem a logical necessity or a different kind of necessary truth?

Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than 2. The question was, is this a logically necessary ...
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### Proof verification of god existence theorem

NB: My question was closed on math stack exchange. They advised me to post it here, but due to the lack of LaTeX formatting, I had to upload it as images. Apologies for that. I am a first year student ...
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You have probably seen these types of arguments before on incompatibility of omniscience and free will. The question is are these arguments valid and what can be a good refutation? Let G= x is known ...
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### Does Necessity Mandate Uniqueness?

Suppose there is a nonzero number of contingent entities and a nonzero number of necessary entities. For example, the law of non-contradiction is probably a plausible candidate for something that ...
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### Is necessary existence a property?

If existence is not a property then doesn't it follow that necessary existence is also not a property? If it is then why?
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### Completeness theorem for QML. A doubt about the relation R in the canonical model. (constant domain)

I dont understand this script: wRv iff □−w ⊆ v, where w is a word of W, that is an Lc-saturated set (maximal consistent with the ∀-property, (C is the set of constants that we use to amply the set of ...