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What is modal logic? Modal logic is an extension of classic propositional and predicate logic that allows the use of modal operators. In others words, modal logic is everything classic logic is + modal operators. Modal operators express modality, such as: Necessity (denoted by □) Possibility (denoted by ◇) The above possibilities are the only operators ...


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The box and the diamond are duals (in the usual systems), so if you have the box, you can define: Definition 1. (Possibility)   ♢φ     =def    ¬▢¬ φ. If you have the diamond as primitive, you can define the box in the same way. Now, suppose we take the box as a primitive. Then the ...


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As you know, the two terms mean something different, as spatio-temporal relations aren't in themselves causal relations. Being spatio-temporally isolated simply means not standing in spatio-temporal relations like “before,” “10 minutes after,” “beside,” “10 meters below.” Being causally-isolated just means not having any causal relations, such that nothing ...


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A very simple explanation: "Possibly P" means "P might be true"; "Not necessarily P" means "P might be false". These are not equivalent. For example, if P is always true, then "Possibly P" is true but "Not necessarily P" is false; if P is always false, then "Possibly P" is false but "Not necessarily P" is true. On the other hand, "Not necessarily not P" ...


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On the prevailing extensional interpretation of modality the difference between possibility and probability is the diffference between quality and quantity, possibility is the quality quantified by probability, see Probability Distributions Over Possible Worlds by Bacchus. This interpretation can be traced back to Leibniz's determinate possible worlds, but ...


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Extending previous answers by ChaosAndOrder and Dennis… You seem to appreciate why "pure" logic (I take it that you mean classical first order logic) is useful in the context of mathematical logic, but you don't see the point in formalizing other modal notions in ordinary language. While presenting you the many applications of modal logic might convince you,...


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Shortest answer: If you have defined a proper model, you know that R(w1, w2) if in your model the pair (w1, w2) is in the extension of R. If you do not have a model you may be able to infer this from the presumed properties of R, for example from R(w0, w1) and R(w0, w2) you can infer R(w1, w2) if R is Euclidean. You define this relation in the way that ...


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It is a central feature of all the main formal systems that when a statement is provable, then it is provably provable. Indeed, this feature is one of the derivability conditions that is commonly used in the proof of the incompleteness theorem, and it is central to Goedel's proof of the second incompleteness theorem. But also, I might add, this principle ...


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So the three valued logic of Łukasiewicz has three truth values {1,i,0}. Łukasiewicz was trying to solve the problem of future contigents with this logic. His view is that statements about the past and present have an unalterable truth value, so if they are true they are necessarily true, if they are false they are necessarily false. Future contingents ...


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Logical omniscience was always only a technical problem related to formalization of epistemic logic in terms of possible worlds. Since classical possible worlds are supposed to be consistent and deductively closed they must include all the consequences along with their premises, and nothing contradicting the premises. So if we are describing acquisition of ...


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So, I will tailor my response to your comment to ChaosAndOrder. The reason we want to utilize modal logic is to precisify ordinary language. Ordinary language is notoriously ambiguous and the analysis of ordinary language modal operators is fraught with difficulty. By regimenting our discourse into formal (quantified) modal logic we can eliminate some of ...


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Preliminaries Priest's presentation of variable domain modal logic in An Introduction to Non-Classical Logic does utilize a free logic base. Check out ch. 15 if you can get your hands on a copy. You ask if there is a flaw in your reasoning: Then I suppose you can use Existential Instantiation at that world such that there is some constant a such that □...


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The first one (often called semantic brackets) is mostly found in formal semantics, and it's the name of the evaluation function, which maps expressions in a formal language to objects in the model of evaluation. Suppose A is the sentence "snow is white." Here's how semantic brackets are used: [["A"]] is true ≡ snow is white The second one (often ...


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Under the usual definitions, there is absolutely no possibility that 7 is a prime number, in this or any other universe. The usual definitions are, say (among many equivalent ways of stating them), that a positive integer n is prime if n ≠ 1 and n has no positive divisors other than 1 and n, and an integer d is a divisor of an integer n if there exists some ...


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I think that you are correct, every mathematical object has a tacit context within which it is being considered. How about geometry, a triangle in euclidean, hyperbolic & elliptic spaces are different. Presumably this can be generalised to manifolds with a metric (patches are locally euclidean, hyperbolic & elliptic). This can't be standard smooth ...


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You can do the semantics for QML in one of two ways, either variable domain or constant domain. To keep worlds isolated (i.e., to prohibit overlap) as Lewis does in his modal realism requires a variable domain approach (since the domain of each world's quantifier is disjoint from any other world). You can read about these two approaches in the second half of ...


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Is it an acceptable idea that each individual carries their own model of reality in their mind? Certainly! To some extent, everyone does. How many people have you come across who believe that their spouse or parent is the best in the world? Or even something as concrete as "I believe I sent you the email last week". Is there a name for the model that ...


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The question: is there a significant geometric form of these logics? Here, "these logics" refers to Boolean algebras, Heyting algebras and modal algebra. The various representation theorems for these algebras as set-algebras related to certain topological spaces seem to provide a positive answer to this question. These representation theorems are non-...


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An understanding of classical propositional logic and first-order logic should suffice. Some notes: Modal languages look very much like non-modal ones. For example, if you have a non-modal propositional language generated by the following grammar: (L1) φ := p | ¬φ | (φ ∧ φ), you can obtain a modal propositional language by adding ...


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Peter Smith—who is or has been a user of this forum—has a discussion article posted called Teach Yourself Logic 2015: A Study Guide (PDF, iv + 94 pp. Last updated 1 Jan 2015). It's on his website. It lays out his informed opinions of the relative merits of the various books and resources for self-study, including the good books mentioned by Mauro Allegranza ...


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The specific quotation you gave about Łukasiewicz refers to the fact there was an attempt to understand intuitionistic logic as a many-valued logic, but this failed because Gödel proved in the early 1930s that intuitionistic logic is not n-valent for any n. To address your last paragraph, truth and certainty are quite different things. To ask, to what ...


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This looks like a "lame terms" redux of Plantinga's "victorious" ontological argument from The Nature of Necessity. Here is Plantinga's explanation of why if a maximally great being exists in some possible world it exists in every possible world. He attributes the idea to Findlay (p. 214): "Those who worship God do not think of him as a being that happens ...


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Plantinga uses the concept of non-trivial properties in his transworld depravity defense of God's benevolence, see How does free will defense of God's benevolence work? Ciprotti in Theological Compatibilism and Essential Properties discusses Plantinga's trivial and non-trivial properties with PDO (power to do otherwise, a.k.a. free will) as central ...


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If we're talking about metaphysical possibility, then normally yes. If you reject the claim that "if P then possibly P", you must also reject the claim that "if necessarily P then P". Proof: suppose we reject truth implies possibility (that is, we reject that for every formula P, if P then possibly P). Then for some formula A, we have A and not-possibly A. ...


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You have to be clear on what Ockham's Razor is and is not. How to understand the principle is still a hotly debated subject. The core idea is that, all other things being equal, the simplest theory is best. That begs for an explanation of the relevant sort of simplicity. Generally it is qualitative and NOT quantitative parsimony that people are interested ...


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The answer to your question is "No", nobody has proposed such a number. Because an infinite number of them exist. 0.1 is an example, so is 0.01, 0.2, etc etc etc. As soon as you limit your "i" number to being a positive integer (and hence being relevant to the definition of prime), it becomes impossible, as there are a finite number of integers between 1 ...


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Accessibility is determined by the axioms of the modal logic you're working within. The axioms of S5, for instance, require the relation to be an equivalence relation and so have the result that every world is accessible from every other world. In a system like K, however, there are no constraints placed on the relation. The only time you can tell if a ...


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While I think Hunan's answer is roughly exhaustive, I'm going to give an answer that proceeds more simply. Consider ◻P. From necessarily P, it follows that ◇P [P is possible]. Or at least it does in every system of modal logic I'm aware of. Given this, it would be strange for ◇P → ~◻P. Since that would mean ◻P ↔ ~◻~P. The issue and the solution is to ...


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We begin by recalling the basic definitions needed to settle the questions: Definitions. R is reflexive       =def   ∀w : wRw; R is symmetric    =def   ∀w, v : (wRv → vRw); R is transitive      =def   ∀w, v, u : (wRv ∧ ...


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Some formal notes to complement Mauro's excellent answer. As one would expect in a discussion of modality, we're going to talk about modal models when defining things. Most will be familiar with logics K, S4, and so on. K and its superlogics are too sophisticated for a discussion of metaphysical modalities, so we'll begin with pre-Kripke modal models, going ...


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