by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part II:
Quantification and Identity
15.
Variable Domain Modal Logics
15.2
Prolegomenon
Brief summary:
(15.2.1) We might object to constant domain semantics by saying that “Just as the properties of objects may vary from world to world, what exists at a world, it is natural to suppose, may vary from world to world. Thus, I exist at this world, but in a world where my parents never met, I do not exist. Or, at this world, Sherlock Holmes does not exist, but in a world that realises the stories of Arthur Conan Doyle, he does” (329). (15.2.2) Another way to make the point that existence should be able to vary between worlds involves the Barcan Formula and the Converse Barcan Formula. (15.2.3) One solution is to say that the domain of quantification varies from world to world. But this presents a problem for universal instantiation when we universally quantify for a predicate in a world while certain objects are not in that world but are in other worlds. (15.2.4) The best solution for this issue of existence varying from world to world is using free logic and thus to include the existence predicate, ℭ.
[The Motivation to Reject Constant Domain Semantics]
[Showing This Problem with the Barcan and Converse Barcan Formulas]
[One Problematic Solution: Simply Making the Domain of Quantification Vary from World to World]
[Using Free Logic]
Summary
[The Motivation to Reject Constant Domain Semantics]
[We might object to constant domain semantics by saying that “Just as the properties of objects may vary from world to world, what exists at a world, it is natural to suppose, may vary from world to world. Thus, I exist at this world, but in a world where my parents never met, I do not exist. Or, at this world, Sherlock Holmes does not exist, but in a world that realises the stories of Arthur Conan Doyle, he does” (329).]
[(ditto)]
Perhaps the most obvious objection to constant domain semantics is as follows. Just as the properties of objects may vary from world to world, what exists at a world, it is natural to suppose, may vary from world to world. Thus, I exist at this world, but in a world where my parents never met, I do not exist. Or, at this world, Sherlock Holmes does not exist, but in a world that realises the stories of Arthur Conan Doyle, he does.
(329)
[Showing This Problem with the Barcan and Converse Barcan Formulas]
[Another way to make the point that existence should be able to vary between worlds involves the Barcan Formula and the Converse Barcan Formula. ]
[(ditto) (See details below.)]
Another way of making the point is as follows. Consider the following formulas:
BF: ∀x□A ⊃ □∀xA
CBF: □∀xA ⊃ ∀x□A
| These are usually called the Barcan Formula and the Converse Barcan Formula, respectively. Both of these are valid in CK (and a fortiori stronger constant domain logics), as may be checked. But intuitively they are invalid. For the Barcan Formula: Suppose that ∀x□Px holds (at this world). Then every object that exists satisfies P at every (accessible) world. It does not follow that □∀xPx is true. For other worlds may contain objects that do not exist at this world, and they may not satisfy P. Conversely, suppose that □∀xPx is true. Then at every (accessible) world, every object that exists (there) satisfies P. It does not follow that ∀x□Px. For this world might contain objects that do not exist at another world, and there is no reason to suppose that they satisfy P at such worlds.
(330)
[One Problematic Solution: Simply Making the Domain of Quantification Vary from World to World]
[One solution is to say that the domain of quantification varies from world to world. But this presents a problem for universal instantiation when we universally quantify for a predicate in a world while certain objects are not in that world but are in other worlds.]
[(ditto) (See discussion below.)]
The natural response to this sort of criticism is to construct a semantics in which the domain of quantification varies from world to world. This presents a problem, however. Suppose that ∀xPx is true at a world. Then for every object, a, at that world, Pa is true. But suppose that b does not exist at the world. There is no reason to suppose that Pb is true. Universal instantiation will therefore fail.1
(330)
1. It is worth noting that if an axiomatic version of quantified modal logic is based on a classical logic using free variables, the CBF is provable in quantified K, and the BF is provable in quantified Kσ. See Hughes and Cresswell (1996), ch. 13.
(330)
Hughes, G. and Cresswell, M.(1996), A New Introduction to Modal Logic (London: Routledge).
(592)
[Using Free Logic]
[The best solution for this issue of existence varying from world to world is using free logic and thus to include the existence predicate, ℭ.]
[(ditto)]
The simplest and most robust solution to this problem is to base the modal logic, not on classical logic, but on free logic. Thus, as in chapter 13, we will take one of the monadic predicates in the language to be a distinguished existence predicate, ℭ.2
(330)
2. It is more usual, perhaps, to formulate variable domain modal logics without an existence predicate in the language. However, its presence is distinctly useful, and has no effect on the validity of inferences employing only sentences which do not contain it (by the Locality Lemma (15.9.3)).
(330)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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