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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Axiom 4 in epistemic logic
In epistemic logic, axiom 4 says that if I know p, then I know that I know p. What is the philosophical value of such an axiom?
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Does ◇◇A mean ~◻~~◻~A? If so is it by definition or it requires a proof?
In system K, ◇A is defined to mean ~◻~A. Therefore, it is very tempting to conclude ◇◇A means ~◻~~◻~A. But I am not certain whether this is valid conclusion to make, because in ◇◇A, the main operator ...
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Proving ~◻p → ~p in System K [closed]
I am working on a proof of ~◻p → ~p in System K. It says "If it is not the case that p is necessarily true, then p is not true". I have turned all the abbreviated symbols into their ...
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Proving validity/invalidity of a modal argument
□(A v B) → (□A v □B) ...(1)
This symbolic argument is intuitively invalid. In (1), if we replace B with ~A, then we see that though the antecedent is necessary, the consequent is a contradiction since ...
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What is the best place for a layman to learn about Modal Logic?
As stated in the title, I am interested in learning about Modal logic as a layman in the subject.
I would appreciate any books, videos, articles, ect.
Thanks!
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can S5 be the weakest logic?
If we were to prove that an argument is a logical truth only in S5 logic out of (K, T, S4, and S5). does that make S5 the weakest of these four logics in which the argument is a logical truth?
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How to define ‘impossible’ using propositional modal logic?
I am trying to define impossibility using the symbols we have in propositional modal logic. I got ‘negation diamond alpha’ in mind as equivalent to ‘it is impossible that alpha’. It that correct and ...
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Can Fregean sense of proper names be described in terms of intension?
If I am not wrong, the old Fregean distinction between sense and reference can be read in terms of a distinction between intension and extension:
The star of the night and the star of the morning are ...
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Lewis argument to defend modal realism
In the fourth chapter of "Counterfactuals", David Lewis tries to justify his positions about modal realism. He claims that:
"We might take them [modalities] as metalinguistic predicates ...
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Stances on possible worlds
Modal realism is the belief that all possible worlds actually exist. Actualism is the belief that possible worlds don’t exist at all. What are some examples of modal-metaphysical views which try to ...
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Natural Deduction in S5 Modal Logic - Introduction and Elimination Rules
Are there natural deduction rules for the S5 modal operators that mirror the introduction and elimination rules for quantifiers in predicate logic? I recall seeing somewhere rules like the following:
...
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Can you give me some concrete example, so that I could understand these modal logic sentences
So there is these simple modal logic sentences:
□(a → b) and a → □b
Can anyone help me with some real-life examples, because I have troubles grasping the difference?
edit
The simpler question is this: ...
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Is it possible to find a counter model for epistemic closure in Nozick's system?
The epistemic closure is that:
If S knows (if p then q) then (If S knows p then S knows q).
In Nozick's Truth-Tracking Analysis
S knows p if and only if
p is true
S believes that p
If p were false, ...
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Necessary possibility
One subsequence in the argument I'm working on goes something like this:
♢A → □♢A.
¬□♢A.
∴ ¬♢A.
This seems valid (it's modus tollens, no?) but it seems to make the actual argument too "easy&...
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"Where" does the counterpart relation subsist?
AFAIK, according to the counterpart theory, it is true of me that I could have lived a different life, if my counterpart in another possible world did live a different life. But where is it true that ...
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Does the forcing phenomenon prove some sort of set-theoretic multiverse?
It seems that, "A can be forced to equal B," allows, "A is possibly equal to B." In possible-worlds lingo, this gets us, "There is a possible world where A = B." Since ...
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Can truths about the natural numbers vary across possible worlds?
The truths of logic are the same in all possible worlds. However, what about truths about natural numbers? Like, for instance, is there a world where there are only finitely many primes, or a world ...
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How do you reduce repeated modality in S5?
Kenneth Konyndyk's Introductory Modal Logic claims (p. 55) that all formulas in S5 with a modal degree greater than 1 can be reduced to degree 1. By degree, he means the maximum number of modal ...
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Is modal logic too coarse-grained? [closed]
Modal logic has "necessary" operators (true in all worlds) and "possible" operators (true in some world). Compare this to probability, where only probabilities of 1 would be "...
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Possible worlds semantics for quantifiers
In possible worlds semantics, sentences are associated with propositions, i.e. a set of possible worlds in which the sentence is true.
For sentences like, "There is an x such that x is F", &...
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What is 'expendable' in logic and how to explain 'tautology' given this image?
This image is from http://www.nfillion.com/index.php/teaching/9-logic-112. According to this, a proposition can have 4 basic properties: (1) necessarily, (2) not possibly, (3) missing, and (4) ...
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Is it consistent to say "X is possible but false"?
Is it consistent to say something like "Possibly there is a cat in my room, but in fact there is not"? Basically, is it consistent to assert that something is possible but in fact not the ...
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How to make sense of " I know that p but I could be wrong as to p"? ( Faillibilism)
There is a well known modal fallacy regarding knowledge which says that if some subject s knows that p, then p cannot be false, and therefore , p is a necessarily true proposition.
Source : [ by ...
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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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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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