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 there a non-transitive frame in which schema 4 is true? Or an irreflexive frame in which schema T is true?

So, I know that I can construct a frame {W, R, I} which is not transitive and in which schema 4 is not true (more specifically, Axiom Schema K and Axiom Schema 4 are not both true). I also know that I ...
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Yablo's condition on “Truth about a subject matter”

In section 2.4 of "Aboutness" Yablo offers the following analysis of what does it mean that a statament is true about a certain subject matter/topic: So, what is the proposition we are ...
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Are there any philosophers who are experts on conceivability and have written texts about it?

I recently asked about a definition of conceivable. Now I am asking a slightly different question. I want to know if there are philosophers who have written texts clarifying (and perhaps even defining)...
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Prove that p → □p is not derivable in system K?

The only way I could think of doing this is to show that p → □p with the K axioms would imply a contradiction, but I don't think that's true. Not sure how to get started on this. Do I just have to ...
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Possibly necessarily P → Necessarily P?

So I saw this proof for ◊□p → □p but I don't know if it's true. ◊~p → □◊~p (5 axiom) ◊~p → ~◊~◊~p (Definition of □) ~~◊~◊~p → ~◊~p (Contraposition) ◊~◊~p → ~◊~p (Double negation) ◊□p → □p (Definition ...
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Modal Logic Proof in System T

I need to provide an axiomatic proof of the following formula in System T of modal logic: ◇(A→□B)→(□A→◇B). Any advice on how to start would be great!
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In the ontological argument, can the existence of an MGB be rejected as provably false?

There are a lot of slightly different formulations of the ontological argument for God, but I'm going to use William Lane Craig's phrasing of Plantinga's, because that's the version I first heard. His ...
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Is every world accessible to itself?

I just realized that for the proposition "If p is necessarily true then p is true", i.e. "box p implies p", to be a tautology, we need the condition that every world is accessible ...
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How to do indirect proof (reductio ad absurdum) using natural deduction for modal logic?

I have been using Garson's Modal Logic for Philosophers, 2nd edition, to learn how to use natural deduction with modal logic. (BTW, does anyone know where there's an answer key for chapters 1 and 2 of ...
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Is there a moral to the deontic-logic story?

In alethic modal logic, it is arguably conventional (or metaphysically thematic) to start with a possibility or a necessity operator. For example, you can start with possibility and get not-possibly (...
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How to prove that the axiom H is valid in the following frame

To prove incompleteness of KH, I have to prove that the axiom H is valid on the following frame: Axiom H goes as follows: □(□p↔p)→□p I don't know how to prove this, but here's one idea that's half ...
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Proof that uniform substitution is validity preserving in modal system K?

I'm reading "A new Introduction to Modal Logic" by Hughes and Cresswell. I've encountered this proof and I can't make sense of it: I'll try to break down what I don't understand about it. ...
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What is the literal meaning of “The only thing that I know is that I know nothing”? (Is not knowing anything a knowledge?)

If a person says, "The only thing that I know is that I know nothing." What exactly does that mean (not metaphorically), literally? If the only thing they know is that they know nothing, ...
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Proving □(□A→□□B) in K5

Question 1: Like the title says, i want to prove □(□A→□□A) in K5 which is just a euclidean frame but I don't think the argument is valid in K5 since we need transitivity for the argument to be valid. ...
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How to read modal logic's countermodels?

I'm new to Modal Logic and currently playing a tree proof generator just to see how some stuff work, but I can't read the countermodels that the algorithm gives me when my proposition is invalid. I (...
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Question about fitch 6.19 proving A or C from premises A or B and -B or C

How to prove A or C from premises A or B and -B or C. Am using fitch and have been stuck on this for an hour
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Are KB5 and S5 identical or sublogic (i don't think so) but what about the reflexive relation in this case?

Let W={w,u,v} and let the relation R on W be euclidean and symmetric; Suppose R(w,u) and R(u,v). -by symmetry we get R(u,v) -by euclidean we get R(v,w) and R(w,v). Similarly, by euclidean we get R(u,u)...
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Requesting help with designing curriculum (Logic)

I am currently a math major in university who wants to design my own area of study/major, specifically in Logic. I was wondering if I could get some help on how Logic sets itself apart from philosophy?...
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A possible formal failure made by Kripke?

In Kripke 's Naming and Necessity, there is a footnote says that "Lewis's elegant paper also suffers from a purely formal difficulty: on his interpretation of quantified modality, the familiar ...
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Garson 2.7 (Tense Logic)

Exercise 2.7 wants me to prove PGA -> A in Kt. Summarily speaking, G/H (in/out) is exactly like [ ]in and [ ]out. [pp.50-51 lays out the rules in detail] Kt=PL + G/H (in/out) + GP +HF GP= A -> ...
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Are there famous unsolved problems in logic akin to the Millenium Prize problems?

Are there major theorems that logicians have yet to tackle? And I don't mean any problems that pertain to the philosophy of logic (i.e. logical pluralism, the nature of logical consequence, etc), but ...
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Show S5 contains S4; Garson 2.4

<> = possible [ ]=necessary Hey all, I am trying to show the following axiom is provable in S5: [ ]A -> [ ] [ ]A = (4) The hint says to prove: [ ]A -> [ ] <> [ ]A first, which ...
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Provided that we add an axiom □□(p=>p) to S1, how can we prove the rule of necessitation?

Since we don't have N, we can't use DR1, DR2, DR3 because they were all derived from N. In system K, K was an axiom, so we can't use K either without proving it first. Here are the axioms and ...
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Does the modal system S1 include the rule of necessiation?

I shall first post a couple of screenshots to make it clear what I'm talking about. I am reading A New Introduction to Modal Logic by Hughes and Cresswell. They answer the question in title very ...
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Deductive reasoning & conditionals

What would be a good example of explicit deductive reasoning that doesn't seem to be possibly interpreted correctly as a conditional (If A, then B)?
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On which frames is the modal system KW valid?

KW is defined as K + the axiom W: □(□p→p)→□p. It is said to be valid on all finite transitive and irreflexive frames. Another way to interpret my question is, what exactly does finite mean here? Let's ...
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How to prove the completeness of S5?

I am reading New Introduction to Modal Logic by Hughes and Cresswell, and I don't quite understand the proof described on pages 105-108. I follow up to the point where they prove that for every of WFF ...
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Hey all can you help prove the following from Garson's ML for Philosophers: [closed]

Exercise 1.7 (e) Modal Logic For Philosophers 2nd edition: []p v []q/[](p v q) {hint: set-up vout first} I would appreciate it if you can solve it using the methods laid out by Garson (PL+[]in+[]out)...
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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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modal quasi-conventionalism (Ted Sider)

So, I've been reading a bit on Ted Sider's modal quasi-conventionalism (elsewhere he calls his view modal Humeanism), and I was wondering if anyone had commented directly on it in the literature. I ...
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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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Does S5 modal logic system Entails contradiction?

Almeida in his article A new cosmological argument undone writes: Assume for reductio ad absurdum that q is a contingently necessary proposition. (5∗) MLq & M∼Lq. Assumption But it follows ...
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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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Is contingency necessary or possibly necessary have defined in S5?And is S5 entails contradiction?

Almeida in his article A new cosmological argument undone writes: Assume for reductio ad absurdum that q is a contingently necessary proposition. (5∗) MLq & M∼Lq. Assumption But it follows ...
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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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