## 28 Mar 2018

### Priest (4.1) An Introduction to Non-Classical Logic, ‘Introduction [to ch.4, “Non-Normal Modal Logics; Strict Conditionals”],’ 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 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 I

Propositional Logic

4.

Non-Normal Modal Logics; Strict Conditionals

4.1

Introduction

Brief summary:

(4.1.1) In the following sections of this chapter, we will examine non-normal modal logics. They involve non-normal worlds, which are ones with different truth conditions for the modal operators. (4.1.2) Following that in the chapter is an examination of the strict conditional.

Contents

4.1.1

[Non-Normal Worlds and Non-Normal Modal Logics]

4.1.2

[The Strict Conditional]

Summary

4.1.1

[Non-Normal Worlds and Non-Normal Modal Logics]

[In the following sections of this chapter, we will examine non-normal modal logics. They involve non-normal worlds, which are ones with different truth conditions for the modal operators.]

[Recall from section 3.1 that we have been working with “normal” modal logics, given the name K. The varieties of K logics are made by applying restrictions to the accessibility relation R. Here were some restrictions:

ρ (rho), reflexivity: for all w, wRw.

σ (sigma), symmetry: for all w1, w2, if w1Rw2, then w2Rw1.

τ (tau), transitivity: for all w1, w2, w3, if w1Rw2 and w2Rw3, then w1Rw3.

η (eta), extendability: for all w1, there is a w2 such that w1Rw2.

(p.36, section 3.2.7)

We will now examine non-normal modal logics. Priest writes here that in non-normal worlds, the truth-conditions for the modal operators are different. We learn the details of that in section 4.2. The other notion we should mention here is that these non-normal modal logics are “weaker” than K. I do not know what the terms “stronger” and “weaker” mean yet. Later at section 4.4.4, he writes:

Note that Kρστ(Kυ) is the strongest of all the logics we have looked at: every normal system that we looked at is contained in Kρστ (3.2.9), and every non-normal system that we looked at is contained in the corresponding normal system (4.4.1, 4.4.2). N is the weakest system we have met. It is contained in every non-normal system, and also in K, and so in every normal system.

(p.68, section 4.4.4)

If we take into consideration what we said in section 3.2.8, and combine it with what is said here, perhaps we can assess the meaning of these terms, but this is all my guesswork (sorry). What is said above suggests that when a certain logic is contained within another, then it is weaker than that other one. But I am not sure what containment is. It could be having a set of interpretations that is a subset of that of another logic, or it could be having a set of valid inferences that is a subset of that other logic (again, see section 3.2.8). But from what is written above, it would seem that Kρσ is contained within Kρστ, and thus Kρσ is weaker. With regard to subsets, I think the set of valid inferences of Kρσ is a subset of that of Kρστ. So my guess is the following. To be a weaker system means that it has fewer valid inferences. Non-normal modal logics are weaker than normal logics, thus they would have fewer valid inferences. Sorry for not knowing for sure; please consult the quotation below.]

In this chapter we look at some systems of modal logic weaker than K (and so non-normal). These involve so-called non-normal worlds. Nonnormal worlds are worlds where the truth conditions of modal operators are different.

(64)

[contents]

4.1.2

[The Strict Conditional]

[Following that in the chapter is an examination of the strict conditional.]

After that, we will examine strict conditionals.

We are then in a position to return to the issue of the conditional, and have a look at an account of a modal conditional called the strict conditional.

(64)

[contents]

From:

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

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