# 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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### What is modal logic for?

I understand "pure" logic as a structural description of what a valid proof is but I have never understood the reasons for using modal logic. What's an example typical of how modal logic is used?
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### What are the practical applications of modal logic?

I'm a computer science and philosophy double major. I know logic is paramount in computer science, but what about modal logic? Are there any practical applications in computer science and perhaps even ...
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### How do I go from ◊∃x□[∃y(y=x) ∧ Mx] to ∃x□[∃y(y=x) ∧ Mx]?

I've been thinking about the ontological argument recently. I'm trying to go from ◊∃x□[∃y(y=x) ∧ Mx] to ∃x□[∃y(y=x) ∧ Mx] I choose that formulation because that seems to express x having the ...
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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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### How is Kripke-style modal logic distinct from classical propositional logic with additional axioms?

I've been considering the possible-worlds semantics for simple forms of modal logic, such as Kripke modal logic. This reading of modal logic seems to be a reduction to restricted truth-tables, where ...
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### Is there a system of logic which denies DNI?

From what I know, the law of double negation is often simplified as p <=> ~~p. Intuitionist logic splits the biconditional into DNI and DNE. DNI: p -> ~~p DNE: ~~p -> p and denies DNE ...
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### Is there a name for each individual's perceived sphere of reality?

Is it an acceptable idea that each individual carries their own model of reality in their mind? Is there a name for the model that each individual uses to perceive reality? Is there a name for the ...
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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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### What's a good source for refreshing my formal logic skills prior to graduate school?

I need to flex my formal logic muscles prior to graduate school--I've had a dry spell in my logic practices while finishing my mathematics degree, particularly since the logic used in analysis, ...
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### Modal Logic: a question concerning accessibility

I’m reading a lot about modal logic lately, right now Lewis “On the Plurality of Worlds” and Priests “Introduction to Non-classical Logics”. It is postulated that the different worlds have nothing to ...
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### How to Prove "Possibly P if Necessarily P" in Kripke Modal Logic?

I wish to prove the following within Kripke modal logic: □P → ◇P This is not a homework problem, but simply the first thing I'd like to prove. I've been able to prove more complex theorems such as □...
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### Is it possible to not know that one knows p?

An axiom that is often included in standard modal logic is Kp => KKp If we use epistemic modal logic, so that K translates as 'he knows', then recalling p stands for a proposition, we have that Kp ...
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### Is there any relation beetwen justification logic and type theory?

Justification logics, was introduced by Sergei Artemov, are epistemic logics which allow knowledge and belief modalities to be ‘unfolded’ into justification terms: instead of □X one writes t:X, and ...
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### Are there any established logical symbols for merely possible and contingently true?

In modal logic we have: P → ◇P - If something is true, then it is true at some possible world. ◻P → P - If something is necessarily true, then it is true. However, the reversed conditionals don't hold ...
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### Are there mathematical properties a mathematical object might have only contingently?

It is generally assumed that mathematics is necessary, such that any mathematical theorem is necessarily true. This can be read as a de dicto necessity such that for any mathematical proposition p, []...
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### Implicit Models and Probability - are degrees of belief/truth/existence a complete free-for-all?

Or, to put it another way, as long as you model your statements using the grammatical framework of our modern logical idioms, is it appropriate practice to assign a probability to any utterance at all,...
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### Obligation and material implication

Deontic logic often contains the axiom □(p → q) → (□p → □q) where □ is being used for "it is obligatory that". This axiom strikes me as odd. It reads "If it is obligatory that p ...
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### What is the explanation for inferring existence in every possible world from existence in some in Plantinga's Ontological Argument?

I found the following version of the Ontological Argument online (I'm pretty certain its Plantinga's but I couldn't find an exact source). Here is how it goes: It is possible that a maximally great ...
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### Is there modal logic without possible worlds?

Would it be desirable to carry out a deflationary research programme in modal logic? In other words, would it be desirable to re-think modal logic without the possible worlds semantics? The original ...
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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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### 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 is the difference between a probability and a possibility?

I ask this in a fairly naive way. I understand that "probabilities" can be quantified in frequencies, degrees of belief, etc. with some defined "space" of probability.But I know little about modal ...
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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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### 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 ...
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### General sentence operators

There are lots of operators that act on sentences. Here are a few examples: P and Q not P forall x.P necessarily P eventually P x believes that that P it is obligatory that P etc. The first two ...
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### How does Plantinga's free will defense of God's benevolence work?

The purpose of the defense is to show that omniscient, omnipotent and benevolent God is consistent with the existence of evil in creation. The most popular version of the defense is due to Alvin ...
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### Assuming P means the same as Q and Bob believes P and is aware that P means the same as Q, can we conclude he believes Q?

Assuming Bob is a fairly rational person. If this is not the case, then is there a way to modify it? Also, is this the argument that Frege is making in "On Sense and Reference" that "the morning star" ...
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### Can logic be significantly geometrised?

Descarte has been lauded for putting together geometry and algebra, and his achievement allowed the invention of calculus by Leibniz & Newton and allowed its efficacious and explosive development ...
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### Why substitutivity doesn't work in an intensional context?

I was trying to grasp some more insights on the difference between intensional and extensional. I started reading this article by Melvin Fitting on intensional logic. It seems interesting but I ...
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### Why is Hesperus necessarily Phosphorus?

"Hesperus (the evening star) is Phosphorus (the morning star)" is one of Kripke's examples of necessary aposteriori, statements that are true necessarily if true at all, even if their truth can only ...
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### Does modal realism dispense with Occams razor or embodies it?

Modal realism posits the reality of all logically possible worlds. This seems to radically dispense with Occams razor by allowing the reality of all logically feasible explanations. But on reflection ...
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### Has any philosopher ever claimed that possibility can't really be tensed?

Has any philosopher ever claimed that possibility can't really be tensed? So that whatever will be possible is possible now. I really have no idea, and would love an answer. I'll add my motive, so ...
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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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### Indiscernibility of possible worlds

According to Lewis, possible words are indiscernible from each other, that is, an individual in a possible world cannot say in which world it resides. It's somewhat similar to the indiscernibility of ...
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### Is Modal Logic Logic?

What makes "Modal Logic" Logic? Why are symbols that stand for "necessary", for example, taken as symbols of Logic (of the same level of symbols that stand for "exists")? What are the limits that ...
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### What's the difference between being spatiotemporally isolated and causally isolated?

In this Wikipedia article on modal realism, section "Main tenets of modal realism", there's a list of six tenets. Here are the fifth and the sixth of them: 5.Possible worlds are unified by ...
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### Is there such a thing as provability of provability?

Gödel says that there are true statements that can't be proved, given a sound axiomatic system. Does anyone say anything about the provability of the provability of statements? Is it still an open ...
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### Does the finitary proof of the consistency of relevant PA shows that first order PA is irrelevant?

Relevance logic takes a closer look at the implication operation in first-order logic. It suggests that implications such as: p and not p -> q cannot hold; in ordinary English, an example of this ...
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### If everything is possible, is it possible for something to be impossible?

If everything is possible, is it possible for something to be impossible? Possibility and impossibility are modal notions; and are dual in the usual formulation; the SEP remarks: It would seem to ...
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### Classical possible worlds semantics

It looks like to me that possible worlds semantics are closely associated with propositional modal logic (or interior/closure algebras). Is there any literature where possible worlds semantics is ...
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### What are the prerequisites for studying modal logic?

A book I'm currently reading briefly mentioned epistemic logic, but didn't say what it was. Since I've never heard of this logic I decided to look it up and saw that it was a type of modal logic. I ...
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### Identity in Quantified Modal Logic

Why is ¬(◇(a=b)∧◇¬(a=b)) a validity in Quantified Modal Logic (QML)? For example, let a:=“the present King of France” and b:=“the richest bald person alive”. Then, it seems ◇(a=b)∧◇¬(a=b) is not a ...
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### What are the objections to the axioms of modal logic?

I am looking for good/classical references on objections/criticism of modal logic. I am a bit familiar with the work of Quine but find his objections around the paradoxes of material implication or ...
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### Many-worlds interpretation of QM and modal realism

Does the difficulty of making sense of quantum mechanical phenomena, (i.e. arriving at a nice philosophical interpretation of QM) taken together with Everett's many-worlds interpretation, constitute ...
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### Why is it wrong to deduce determinism from a logical law?

In Aristotle's famous sea battle argument, he argues from bivalence (or something like it) to determinism. Stalnacker has an argument to determinism using standard logical laws, as well. My question ...
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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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### Does semantic externalism imply modal realism?

Semantic externalism, or more broadly externalism about mental content, is the idea that propositions, intentions, attitudes, etc. derive their meaning from the external world as opposed to internally ...
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### According to modal realism, how many possible worlds are there?

Modal realism tells us there is an infinity of possible worlds, but how many are there exactly? Is it countable infinity ℵ₀, uncountable infinity 𝖈, or some other, bigger uncountable infinity?
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