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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 I:
Propositional Logic
1.
Classical Logic and the Material Conditional
1.10
Arguments for ⊃
Brief summary:
(1.10.1) Even though the material conditional, ⊃, is not properly suited to describe the functioning of the English conditional, it had come to be regarded as such on account of there only being standard truth-table semantics until the 1960s, and the only plausible candidate in that semantics for “if” formations would be the material conditional. (1.10.2) However, there are notable arguments that the material conditional can be used to understand the English conditional, and they construe that relation in the following way: “‘If A then B’ is true iff ‘A ⊃ B’ is true.” (1.10.3) ‘If A then B’ is true then ¬A ∨ B is true. (1.10.4) Suppose A and ¬A ∨ B are true. By disjunctive syllogism: A, ¬A ∨ B ⊢ B. This fulfills (*), when we take ¬A ∨ B as the C term. [Now, since A, ¬A ∨ B ⊢ B fulfills the definition of the English conditional, and since A, ¬A ∨ B ⊢ B also gives us the (modus ponens) logic of the conditional (given the equivalence of A ⊃ B and ¬A ∨ B), that means the logic of the English conditional is adequately expressed by A ⊃ B.] (1.10.5) Suppose A and ¬A ∨ B are true. By disjunctive syllogism: A, ¬A ∨ B ⊢ B. This fulfills (*), when we take ¬A ∨ B as the C term. (Now, since A, ¬A ∨ B ⊢ B fulfills the definition of the English conditional, and since A, ¬A ∨ B ⊢ B also gives us the (modus ponens) logic of the conditional (given the equivalence of A ⊃ B and ¬A ∨ B), that means the logic of the English conditional is adequately expressed by A ⊃ B.) (1.10.6) What later proves important in the above argumentation is the use of disjunctive syllogism.
[The Prevalence of the Mistaken Identification of the English Conditional with the Material Conditional as Resulting Historically from Limitations in Semantics]
[Defining the English Conditional Using the Material Condition as “‘If A then B’ is true iff ‘A ⊃ B’ is true”]
[¬A ∨ B from ‘If A then B’]
[Disjunctive Syllogism and the English Conditional]
[Disjunctive Syllogism and the English Conditional]
[Noting Disjunctive Syllogism in the Argumentation]
Summary
[The Prevalence of the Mistaken Identification of the English Conditional with the Material Conditional as Resulting Historically from Limitations in Semantics]
[Even though the material conditional, ⊃, is not properly suited to describe the functioning of the English conditional, it had come to be regarded as such on account of there only being standard truth-table semantics until the 1960s, and the only plausible candidate in that semantics for “if” formations would be the material conditional.]
[Recall from section 1.8 and section 1.9 that there are a number of ways that the material conditional does not function exactly like the English conditional ‘if’. Priest now explains why many thought it could be adequate despite such problems. He says that this resulted from a historical factor, namely, that standard truth-table semantics were the only semantics we had until the 1960s, and, of the options they provided, “⊃ is the only truth function that looks an even remotely plausible candidate for ‘if’” (15).]
The claim that the English conditional (or even the indicative conditional) is material is therefore hard to sustain. In the light of this it is worth asking why anyone ever thought this. At least in the modern period, a large part of the answer is that, until the 1960s, standard truth-table semantics were the only ones that there were, and ⊃ is the only truth function that looks an even remotely plausible candidate for ‘if’.
(15)
[Defining the English Conditional Using the Material Condition as “‘If A then B’ is true iff ‘A ⊃ B’ is true”]
[However, there are notable arguments that the material conditional can be used to understand the English conditional, and they construe that relation in the following way: “‘If A then B’ is true iff ‘A ⊃ B’ is true.”]
[I might be mistaken about the following, so please consult the quotation below. The idea might be now that we will examine an argument that in fact the material conditional can be used to understand the English conditional. Here it is something like saying: “‘If A then B’ is true iff ‘A ⊃ B’ is true.”]
Some arguments have been offered, however. Here is one, to the effect that ‘If A then B’ is true iff ‘A ⊃ B’ is true.
(15)
[¬A ∨ B from ‘If A then B’]
[‘If A then B’ is true then ¬A ∨ B is true.]
[Priest next will show that if ‘‘If A then B’ is true’ then ¬A ∨ B is true. He does this with the following reasoning. First we suppose ‘‘If A then B’ is true, and then we consider two further possibilities. He notes that either ¬A or A is true. Take the first possibility, that ¬A is true. That means ¬A ∨ B is true. (I am not entirely sure why, but it might be something like disjunction introduction. See Agler’s Symbolic Logic section 5.3.8. But I am guessing.) Or take the second possibility, that A is true. That means, by modus ponens, B is true. For, we are affirming the antecedent and thus thereby the consequent. So again, we have that ¬A ∨ B is true. (I am again guessing it is something like disjunction introduction.) Thus in either case, ¬A ∨ B is true.]
First, suppose that ‘If A then B’ is true. Either ¬A is true or A is. In this first case, ¬A ∨ B is true. In the second case, B is true by modus ponens. Hence, again, ¬A ∨ B is true. Thus, in either case, ¬A ∨ B is true.
(15)
[C and A Entailing B for “If A then B”]
[“(*) ‘If A then B’ is true if there is some true statement, C, such that from C and A together we can deduce B” (15).]
[I do not follow the next part, so please see the quotation below. First Priest speaks of the “converse argument”. I do not know what that is. Is it that from ¬A ∨ B we can derive ‘If A then B’? I do not know. And even if it were, I do not understand Priest’s point that the converse argument appeals to the claim that “(*) ‘If A then B’ is true if there is some true statement, C, such that from C and A together we can deduce B.” In fact, I do not know what it means to “appeal to a claim.” But the example makes sense: “Thus, we agree that the conditional ‘If Oswald didn’t kill Kennedy, someone else did’ is true because we can deduce that someone other than Oswald killed Kennedy from the fact that Kennedy was murdered and Oswald did not do it.” Yet I am not sure if and how to relate this to ¬A ∨ B. Perhaps I could follow this if I had access to the Faris text that Priest later cites for this issue (‘Interderivability of “⊃” and “If” ’), but currently I do not have it. I will continue this poor explanation in the next section.)]
The converse argument appeals to the following plausible claim:
(*) ‘If A then B’ is true if there is some true statement, C, such that from C and A together we can deduce B.
Thus, we agree that the conditional ‘If Oswald didn’t kill Kennedy, someone else did’ is true because we can deduce that someone other than Oswald killed Kennedy from the fact that Kennedy was murdered and Oswald did not do it.
(15)
[Disjunctive Syllogism and the English Conditional]
[Suppose A and ¬A ∨ B are true. By disjunctive syllogism: A, ¬A ∨ B ⊢ B. This fulfills (*), when we take ¬A ∨ B as the C term. (Now, since A, ¬A ∨ B ⊢ B fulfills the definition of the English conditional, and since A, ¬A ∨ B ⊢ B also gives us the (modus ponens) logic of the conditional (given the equivalence of A ⊃ B and ¬A ∨ B), that means the logic of the English conditional is adequately expressed by A ⊃ B.)]
[Priest now says that “suppose that ¬A ∨ B is true. Then from this and A we can deduce B, by the disjunctive syllogism: A, ¬A ∨ B ⊢ B. Hence, by (*), ‘If A then B’ is true” (16). Maybe this corresponds to (*) when we take ¬A ∨ B to be the C term in (*), but I am guessing. My problem still is I do not know how to put all the ideas together from all the sections here, even though the ideas in each one by themselves make sense. At least let us note the following for now. In section 1.7.1 we said that A ⊃ B is equivalent to ¬A ∨ B. So my following explanation is wrong, but I cannot think of anything else at the moment. Maybe we are to think of how A, ¬A ∨ B ⊢ B works by disjunctive syllogism and A, A ⊃ B ⊢ B by modus ponens. Both cases also fulfill (*), which confirms “if A then B”. I am incorrectly guessing that the overall idea is that (*) gives us our definition for the English conditional ‘if A then B’, and the inference which gives us the basic “logic” of the conditional (modus ponens or disjunctive syllogism formulated equivalently) fulfills the definition in (*). I am very sorry that at the moment I cannot put all these sections together coherently, so please read this whole section 1.10 for yourself.]
Now, suppose that ¬A ∨ B is true. Then from this and A we can deduce B, by the disjunctive syllogism: A, ¬A ∨ B ⊢ B. Hence, by (*), ‘If A then B’ is true.
(16)
[Noting Disjunctive Syllogism in the Argumentation]
[What later proves important in the above argumentation is the use of disjunctive syllogism.]
[Priest notes finally that:]
We will come back to this argument in a later chapter. For now, just note the fact that it uses the disjunctive syllogism.
(16)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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