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
16.
Necessary Identity in Modal Logic
16.1
Introduction
Brief summary:
(16.1.1) We turn now to identity in modal and tense logic. There are two kinds of modal logic semantics for identity: {1} necessary (or “world-invariant”) and {2} contingent (or “world-variant”). (16.1.2) “If S is any system of logic without identity, S(NI) will denote the system augmented by necessary identity, and S(CI) will denote the system of logic augmented by contingent identity” (349). (16.1.3) “We will assume, first, that the Negativity Constraint is not in operation. We will then see how its addition affects matters” (349). (16.1.4) “Next, we will look at the distinction between rigid and non-rigid designators, and see how non-rigid designators can be added to the logic” (349). (16.1.5) At the end of the chapter “there is a short philosophical discussion of how this distinction applies to names and descriptions in a natural language such as English” (350).
[Identity in Modal/Tense Logic. Necessary (World-Invariant) and Contingent (World-Variant) Identity]
[S(NI) and S(CI)]
[The Issue of the Negativity Constraint]
[Rigid and Non-Rigid Designators]
[Names and Descriptions]
Summary
[Identity in Modal/Tense Logic. Necessary (World-Invariant) and Contingent (World-Variant) Identity]
[We turn now to identity in modal and tense logic. There are two kinds of modal logic semantics for identity: {1} necessary (or “world-invariant”) and {2} contingent (or “world-variant”).]
[(ditto)]
In this chapter we will start to look at the behaviour of identity in modal logic. (Henceforth, I use ‘modal logic’ to include tense logic.) There are, in fact, two kinds of semantics for identity in modal logic: necessary and contingent.1
(349)
1. The terminology is not entirely happy. The distinction turns on whether identity statements can have different truth values at different worlds. For this reason, it might be more appropriate to call the identities world-invariant and world-variant. Later in the book, we will be concerned not only with possible worlds, but with impossible worlds of various kinds. It is therefore entirely possible for identity statements to change their truth values, but only at impossible worlds. If this is the case, then true identity statements can still be necessarily true (i.e., true at all possible worlds) even though identity is world-variant. However, since the terminology is standard, I employ it.
(349)
[S(NI) and S(CI)]
[“If S is any system of logic without identity, S(NI) will denote the system augmented by necessary identity, and S(CI) will denote the system of logic augmented by contingent identity” (349).]
[(ditto)]
Where it is necessary to distinguish between the two notions of identity, I will use the following notation. If S is any system of logic without identity, S(NI) will denote the system augmented by necessary identity, and S(CI) will denote the system of logic augmented by contingent identity. In this chapter we will deal with necessary identity, which is simpler; in the next chapter, we will turn to contingent identity.
(349)
[The Issue of the Negativity Constraint]
[“We will assume, first, that the Negativity Constraint is not in operation. We will then see how its addition affects matters” (349).]
[(ditto)]
We will assume, first, that the Negativity Constraint is not in operation. We will then see how its addition affects matters.
(349)
[Rigid and Non-Rigid Designators]
[“Next, we will look at the distinction between rigid and non-rigid designators, and see how non-rigid designators can be added to the logic” (349).]
[(ditto)]
Next, we will look at the distinction between rigid and non-rigid designators, and see how non-rigid designators can be added to the logic.
(349)
[Names and Descriptions]
[At the end of the chapter “there is a short philosophical discussion of how this distinction applies to names and descriptions in a natural language such as English” (350).]
[(ditto)]
Finally, there is a short philosophical discussion of how this distinction applies to names and descriptions in a natural language such as English.
(350)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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