# 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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### Why is □p→◇p not a theorem of K? [closed]

¬(□p→◇p) □p ¬◇p □¬p □p ∧ □¬p As long as □ ranges over ANYTHING, □p ∧ □¬p is going to result in a contradiction.
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### Could there be a universe where the concept of order and logic and numbers and objects and space and time don't exist?

Could there be a universe where the concept of order and logic and numbers and objects and space and time don't exist ? This would preclude math as we know it. Maybe there is something better than ...
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### What is basically at stake in contemporary discussions regarding essentialism and essential properties?

Modern discussions of " essentialism" ( 17th century) can be associated with the empiricism / rationalsit debate. By denying that we have acces to " real essences" , Locke also denies that we can know ...
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### How does the proof of ◻(φ → ψ) → (◻φ → ◻ψ) not presuppose ◻φ → φ?

To be honest, I don't quite follow what happens after 5., and how they conclude 8. without ◻φ → φ. I'm guessing that because ¬◻ψ, they can do ◇¬ψ, so they pick a world where ¬ψ is true. And because ◻φ,...
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### How does identity work in modal predicate logic?

Namely how can we can have a correspondence between objects in different possible worlds if they are different? If we have two worlds in S5 aw0(P(x)) = 1 and aw1(P(x)) = 0 How can we 'identify' x ...
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### What is the distinction between A-intension and C-intension?

I'm having a really hard time understanding a concept discussed in “Qualia and Analytic Conditionals” by Braddon-Mitchell and "Why We Need A-Intensions" by Jackson. Here's my extraction of these ...
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### Alphabetic Substitution, Barcan, and Strict Implication

Context: I'm stuck on Axiom 8 from the introduction to Barcan 1946, "A Functional Calculus of First Order Based on Strict Implication." My instinct is that I'm missing a basic, perhaps obvious concept-...
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### What does □□p mean?

The premises of the S5 system are: □p → □□ p ◊p → □ ◊ p (Note that □ is an actual square, not the missing-symbol placeholder). What does the first one mean? If □p is what is necessary in all ...
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### What does the "⊢" symbol mean in ⊢ p ⇔ (p & p)?

⊢ p ⇔ (p & p) what means the right part of these Entailments?
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### what is meaning of accessibility of possible worlds?

I have a question about the notion of possibility in modal logic. There are systems and worlds with this notion. They say that a world w1 is accessible to an other world w2 if and only if for any true ...
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### Did Quine change of opinion towards quantified modal logic?

Willard Van Orman Quine was a strong opponent to quantified modal logic calling it unreasonable and useless. But, did he always think like that? Or did he relax his attitude towards it with time? Did ...
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### Having S5 system Why we need to use WPSR in Gale Pruss New cosmological arguement?

As you know Gale_ Pruss improved The cosmological argument using WPSR instead of Strong PSR. They in their 1999 article said if it be possibly that there be a God who freely created the world using ...
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### Proving the Modal Logic S5-Theorem ♦□ϕ ↔ □ϕ

As above (http://gki.informatik.uni-freiburg.de/teaching/ss15/ml/script.pdf page 59), and in many other sources, the formula ♦□ϕ ↔ □ϕ is considered a theorem in modal logic S5. However, the paper ...
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### Modal logic question about possibility (axiom?)

I'm not sure if PN(A) implies N(A) is an axiom or if it follows from the definition of P(possibility).
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### Where is the fallacy in Seth Yalcin's counterexample to the modus tollens?

Where is the fallacy, do you think, in Seth Yalcin’s argument (2012) that the Modus Tollens is not a generally valid form of argument? Seth Yalcin’s counterexample to the Modus Tollens (MT) https://...
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### Does it hold that everything exists necessarily?

In Quantified modal logic, "constancy’s defenders can point to certain powerful arguments in its favor. Here’s a quick sketch of one such argument. First, the following seems to be a logical truth: ...
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### Does Kripke hold a view of free logic?

if Kripke doesn't want to accept Barcan's formula（the changed form in free logic） - given his essentialism - one solution is free logic. So does Kripke say that he accepts free logic or?
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### Inference Rules of Modal Logic

I'm currently reading the book "An Introduction to Non-Classical Logic." Currently, I'm being introduced to modal logic for the first time. This book seems to prefer to present the reader with the ...
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### From Ignorant to Researcher in Modal Logic

I am a student in a French university in pure mathematics. I recently came across modal logic and more particularly doxastic logic. But, the only masters of logic offered are reserved for students of ...
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### In Quine's ontology, why does a 'recognition' of something lead to ontological commitment while a 'feeling' does not?

We are discussing Quine's On What There Is in a metaphysics class I am in. I felt like I understood what he meant, that if something has to be predicated for in a sentence, we are ontologically ...
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### Boethius, Logical Necessity, and Accidental Necessity: A Solution to Free Will and Foreknowledge?

In his classic book, the Consolation of Philosophy (Book V), Boethius attempts to make an argument that libertarian free will and [divine] foreknowledge are not incompatible. His argument goes ...
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### Is God's existence either impossible or necessary?

This is meant as a supplement to a modal ontological argument to show that God's existence is either logically impossible or necessary. Am I committing a fallacy or a logical error of some kind or ...
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### Modal Logic: One non-transitive frame where schema 4 is invalid?

I know that schema 4 defines the class of all transitive frames, meaning that it is not valid in at least one model that is non-transitive. However, I am not sure how we would go about proving that if ...
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### Do possible worlds partition the set of all possible states of affairs?

Let S be the set of all ( logically) possible states of affairs ( I could have said " events" or " propositions" maybe). Let R be the relation : state of affairs x is compossible/ compatible with ...
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### How to prove rigorously that " necessarily ( A OR ~A) " does not imply " necessarily A OR necessarily ~A"?

In De Interpretatione , Aristotle criticizes logical fatalism ( a metaphysical doctrine professed in the Megarian School, in particular by Diodorus Cronus). Aristotle reconstructs the reasoning of ...
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### Modal Logic: Proving Schema Validity

When faced with a question of the sort, "Is schema X valid in class of frames C?", we usually go about proving or disproving this by assuming the antecedent of X and showing that the consequent ...
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### Trying to formalize a certain sentence in predicate modal logic

I have the following sentence which I want to formalize in predicate modal logic. The sentence is: What is good for you is not necessarily good for others" My attempt at formalizing this is as ...
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### Modal logic: Why is K5 determined by the class of all Euclidean frames?

I am interested in proving that K, the smallest normal modal logic, in combination with the formula 5 (Possibly A then Necessarily Possibly A) is determined by the class of all Euclidean frames. For ...
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### What kinds of proofs can be given for axioms, e.g. the modal axiom S5?

From John Bigelow and Robert Pargetter's book, titled, 'Science and Necessity', they assert the following: . . . . The resulting system, S5, contains all the theorems of S4 and all the theorems of ...
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### Modal Logic: Why are Universal frames a subset of Equivalence frames?

I'm looking through the lecture notes for my course on modal logic and am having a hard time understanding why it is that U, the class of all Universal frames, is a subset of E, the class of all ...
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### Modal Logic: Point-wise equivalence vs. (simple) equivalence

I am reading Brain Chellas' Modal Logic and have a question regarding point-wise and (simple) equivalence: what is a scenario in which two models are equivalent but not point-wise equivalent? Say, ...
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### Is this valid in first-order modal logic?

(∀x)(∀y){(xEy) --> (∀z)[Tz --> (C(x,z)Ey)]} (Premise) {(∀y)[Ty --> (∃x)(Tx & xEy)]} --> {(∀y)[Ty --> (∃x)(Tx & xEy & ◻(Tx --> Ty))]} (From (1)) I'm trying to work this out in a quantified ...
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### Did Carnap propose some kind of Modal Realism?

I have read in several pages that Rudolf Carnap, just as David Lewis (The creator of Modal Realism philosophical hypothesis) proposed that every logical proposition exists as a universe But is this ...
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### David Lewis and naturalism

I am studying Lewis and I am wondering what are the cornerstones of his philosophical perspective. In particular, I am wondering if Lewis is a naturalist like his supervisor Quine.
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### What are the various interpretations of modal logic?

Wikipedia lists the following interpretations of modal logic: Alethic: fundamental conditions of possible worlds, causality, time-space parameters, and the action capacity of persons. Indicates the ...
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### What are good sources on vivid designators?

Wikipedia lists a vivid designator as the following: Vivid designator: In modal logic and the philosophy of language, a vivid designator is a term which is believed to designate the same ...
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If A, then B ~ A So, possible that B Valid or not? My take: Not valid. Reason: Valid means if all the premises are true, the conclusion must be true That means adding new information should not ...
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### What is the Barcan Formula?

can someone please help explain this Barcan formula to me? (In English translation and maybe with an example?) (◊ ∃x Fx) ↔ (∃x ◊ Fx) And if there is only one possible state of the world, would it ...
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### Can a Rigid Designator still exist if there is only one possible world?

According to Kripke, a rigid designator is a pronoun (but not all pronouns are rigid designators) and they pick out the same unique individual in each possible world. I understand this, however, if ...
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### How and why Aristotelian essentialism is a problem for Quine?

I cannot understand how and why the Aristotelian essentialism is problematic for Quine. I have tried to read articles on the theme but probably I am not smart enough to understand them. Could you ...
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### Modal logics - philosophical paradoxes using modeling by possible worlds

I'm searching for "paradoxes" in classical modal logics, meaning lines of reasoning which give a counterintuitive conclusion if performed in classical logic, which can be modelled by the (semantical) ...
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### De Re, Counterfactuals, and rigidity

This is going to come off as vague or obscure; but, I hope the idea is performatively expressed: Two questions: Do you think that Kripke would argue that the impossibility of de re counterfactuals ...
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### Does ◻(∀x)[Px ≡ ◻(E!x → Qx)] and ◻(∀x)(Qx → ◻E!x) entail ◻(∀x){Px ≡ ◻[E!x & (E!x → Qx)]}?

Say we are working in a free quantified modal logic system S5. Would the following argument be valid: ◻(∀x)[Px ≡ ◻(E!x → Qx)] (Premise) ◻(∀x)(Qx → ◻E!x) (Premise) ◻(∀x){Px ≡ ◻[E!x & (E!x → Qx)]}...
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### How to prove □Q from P→Q and □P?

Given P→Q, and □P, in what systems can we conclude □Q? I want to know in which system the consequent is derivable from the premises and in which it is axiomatic (or perhaps we can include it as an ...
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### Should truth entail possible truth?

It is a well-accepted axiom of modal logic that truth implies possible truth. Is there any philosophical argument against this conclusion? In other words, should truth entail possible truth?
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### Is □(□A→B) v □(□B→A) provable in S5? If it's not, I am having trouble coming up with a countermodel

Modal Logic question about provability/ countermodels.
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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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### 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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### 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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### 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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