## 24 May 2018

### Priest (4.9) An Introduction to Non-Classical Logic, ‘Lewis’ Argument for Explosion,’ 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

4.9

Lewis’ Argument for Explosion

Brief summary:

(4.9.1) Strict conditionals do not require relevance, as we see for example with: ⊨ (A ∧ ¬A) ⥽ B. So we might object to them on this basis. (4.9.2) C.I. Lewis argues that (A ∧ ¬A) ⥽ B is intuitively valid, because from A ∧ ¬A it is intuitively valid to infer A and ¬A; from ¬A it is intuitively valid to infer ¬A B, and from A and ¬A B it is intuitively valid, by disjunctive syllogism, to derive B. [Now, if each step has a connection on the basis of its intuitive validity, that means the final conclusion B should have a connection, by extension, to A ∧ ¬A on the basis of the intuitively valid steps leading from the premise to the final conclusion. So despite objections to the contrary, there is a connection between the antecedent and consequent in (A ∧ ¬A) ⥽ B, according to Lewis. (4.9.3) C.I. Lewis also formulates an argument for the connection between antecedent and conclusion for A ⥽ (B ∨ ¬B), but this argument is a bit less convincing than the one for (A ∧ ¬A) ⥽ B.

Contents

4.9.1

[Strict Conditionals as Lacking Relevance]

4.9.2

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in (A ∧ ¬A) ⥽ B by Means of Disjunctive Syllogism]

4.9.3

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in A ⥽ (B ∨ ¬B)]

Summary

4.9.1

[Strict Conditionals as Lacking Relevance]

[Strict conditionals do not require relevance, as we see for example with: ⊨ (A ∧ ¬A) ⥽ B. So we might object to them on this basis.]

[Recall from section 4.5.2 and section 4.5.3 that the strict conditional AB is defined as □(AB).  In previous sections – see for example section 4.6 and section 4.8 – Priest has considered objections for the strict conditional ⥽ as providing a correct account of the conditional. Priest will now consider a final objection to the this claim about the correctness of the strict conditional. He notes that we have the intuition that this definition is inadequate, because we expect in a conditional that there is some kind of connection between the antecedent and the consequent (for otherwise, what is the sense of the conditionality of their relation?). But strict conditionals do not require any such connection. For example, there is no connection between A ∧ ¬A and B, (even though, as we saw in section 4.6.3: ⊨ (A ∧ ¬A) ⥽ B.)]

Let us end by considering a final objection to ⥽ as providing a correct account of the conditional. It is natural to object that this account cannot be correct, since a conditional requires some kind of connection between antecedent and consequent; yet a strict conditional requires no such connection. There is no connection in general, for example, between A ∧ ¬A and B.

(76)

[contents]

4.9.2

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in (A ∧ ¬A) ⥽ B by Means of Disjunctive Syllogism]

[C.I. Lewis argues that (A ∧ ¬A) ⥽ B is intuitively valid, because from A ∧ ¬A it is intuitively valid to infer A and ¬A; from ¬A it is intuitively valid to infer ¬A B, and from A and ¬A B it is intuitively valid, by disjunctive syllogism, to derive B. (Now, if each step has a connection on the basis of its intuitive validity, that means the final conclusion B should have a connection, by extension, to A ∧ ¬A on the basis of the intuitively valid steps leading from the premise to the final conclusion. So despite objections to the contrary, there is a connection between the antecedent and consequent in (A ∧ ¬A) ⥽ B, according to Lewis.)]

[Despite what we said about relevance above in section 4.9.1, C.I. Lewis does see a connection in the strict conditional even in explosive formulas like ⊨ (A ∧ ¬A) ⥽ B. (On explosion and the strict conditional, see section 4.8). Only, the connection here is one obtained by a series of inferences, each of which is presumably intuitively valid. (So if each inference is intuitively valid, then they have a logical connection. And so ultimately the explosive inference is intuitively valid). We begin with a premise that is a contradiction: A ∧ ¬A. We then infer the conjects from this conjunction,  ¬A and A. From ¬A we infer the disjunction ¬A B, which with A and by disjunctive syllogism, we infer B. (The idea might be the following, but I am just guessing here. By modus ponens, from A, A B we can infer B. And, AB is equivalent ¬A B. And as we see, by disjunctive syllogism from A, ¬A B we can infer B. Furthermore, maybe another idea here is that when there are premises validly making some other formula true, then you can make the premises be the antecedents and the conclusion the consequent in another formula that will be valid, but I am guessing. So because A B is equivalent to ¬A B, and because the inference from A ∧ ¬A to B is shown to be valid using disjunctive syllogism on premises validly derived from A ∧ ¬A, that means (A ∧ ¬A) ⥽ B should be intuitively valid. Again, these are guesses. See the quotation below.]

C.I. Lewis, who did accept as an adequate account of the conditional, thought that there was a connection, at least in this case. The connection is shown in the following argument:

xxxxxxxxxxxxxA∧¬A

xxxxxxxxxxxx______

xxxxxA∧¬Axxxxx¬A

xxxxx____xxxxx___

xxxxxxxAxxxxx¬A∨B

xxxxx____________

xxxxxxxxxxxB

Premises are above lines; conclusions are below. The only ultimate premise is A∧¬A; the only ultimate conclusion is B. The inferences that the argument uses are: inferring a conjunct from a conjunction; inferring a disjunction from a disjunct; and the disjunctive syllogism: A, ¬A B B. Of course, all these are valid in the modal logics we have looked at. If contradictions do not entail everything, then one of these must be wrong. We will return to this point in a later chapter.

(76)

[contents]

4.9.3

[C.I. Lewis’ Argument for the Connection between Antecedent and Consequent in A ⥽ (B ∨ ¬B)]

[C.I. Lewis also formulates an argument for the connection between antecedent and conclusion for A ⥽ (B ∨ ¬B), but this argument is a bit less convincing than the one for (A ∧ ¬A) ⥽ B.]

[Priest then notes that “Lewis also argued that there is a connection in the case of the conditional A ⥽ (B ∨ ¬B) as well,” using the following argument. We begin with A. From this we infer (AB) ∨ (A ∧ ¬B) (I am not exactly sure how, but maybe the reasoning is something like the following. Either B or ¬B holds, on account of excluded middle. Since we have affirmed A, then either (AB) or (A ∧ ¬B) holds.) From this we infer A ∧ (B ∨ ¬B) (I am not sure how again, but it seems like we extract the A as being the common affirmed formula in both, leaving (B ∨ ¬B).) And from this we infer (B ∨ ¬B) by pulling it out as one of the conjuncts. So by beginning with A, we can validly infer (B ∨ ¬B), and thus A ⥽ (B ∨ ¬B).) Priest says this argument is less convincing than the prior one, because “the first step seems evidently to smuggle in the conclusion” (77). (But I am not sure how that works other than the fact that the (B ∨ ¬B) that we want to derive is built into (AB) ∨ (A ∧ ¬B) by a sort of distribution.) Please see the quotation below, as I do not know the precise reasoning for each step.]

Lewis also argued that there is a connection in the case of the conditional A ⥽ (B ∨ ¬B) as well. The connection is provided by the | following argument:

xxxxxxxxxxxxxA

xxxxx_________________

xxxxx(A ∧ B) ∨ (A ∧ ¬B)

xxxxx_________________

xxxxxxxxA ∧ (B ∨ ¬B)

xxxxxxx______________

xxxxxxxxx(B ∨ ¬B)

This argument is less convincing than that of 4.9.2, however, since the first step seems evidently to smuggle in the conclusion.

(76-77)

[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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