## 2 Sep 2017

### Priest (8.1) An Introduction to Non-Classical Logic, ‘Introduction [to first degree entailment]’, summary

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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 distracting mistakes, because I have not finished proofreading.]

Summary of

Graham Priest

An Introduction to Non-Classical Logic: From If to Is

8. First Degree Entailment

8.1. Introduction [to first degree entailment]

Brief summary:

In first degree entailment (FDE), interpretations are not formulated as functions that assign truth values, standard or not, to propositional parameters. Rather, in FDE, interpretations are formulated as relations  between formulas and standard truth values. In this chapter we examine FDE, along with an alternate possible world semantics for it, and we discuss the issues of explosion and disjunctive syllogism.

Summary

8.1

[In first degree entailment (FDE), interpretations are not formulated as functions (assigning truth values, standard or not, to propositional parameters) but rather as as relations  between formulas and standard truth values.]

[First recall from section 1.3.1 the notion of interpretation in classical logic:

An interpretation of the language is a function, ν, which assigns to each propositional parameter either 1 (true), or 0 (false). Thus, we write things such as (p) = 1 and ν(q) = 0.

(5)

We also saw in section 7.3 examples of three-valued logics whose interpretation function assigns in addition to 1 and 0.] We will examine first degree entailment (FDE), which “is formulated, first, as a logic where interpretations are relations between formulas and standard truth values, rather than as the more usual functions” (142).  We well also see connections between FDE and many-valued logics (see section 7).

8.2

[We also examine alternate possible-world semantics for FDE. This will introduce a new kind of semantics for negation.]

Another thing we do in this chapter is examine “an alternate possible-world semantics for FDE, which will introduce us to a new kind of semantics for negation” (142).

8.3

[We also examine the issue of explosion and disjunctive syllogism.]

“Finally, we look at the relation of all this to the explosion of contradictions, and to the disjunctive syllogism” (142).

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

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