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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
1.7
The Material Conditional
Brief summary:
(1.7.1) The material conditional, symbolized as ⊃, is true when the antecedent is false or the consequent is true. It is thus logically equivalent to ¬A ∨ B. But also on that account, it generates the “paradoxes of material implication,” namely, B ⊨ A ⊃ B and ¬A ⊨ A ⊃ B. In other words, suppose we have some given formula that is true. That means we can make it a consequent in a conditional with any arbitrary antecedent. Or suppose we have a negated formula as true, then we can make the formula’s unnegated form be the antecedent in a conditional with any arbitrary consequent. (1.7.2) The truth conditions for the material conditional allow for technically true but intuitively false sentences that fulfill the conditions for the material conditional but seem false on account of the irrelevance of the antecedent to the consequent, as for example, “If New York is in New Zealand then 2 + 2 = 4.” This seems to contradict the intuitive sense we ascribe to the English conditional, which involves relevance. (1.7.3) The counter-intuitive example conditional sentences are odd because they break certain rules of communication, namely to assert the strongest information.
[The Material Conditional and the Paradoxes of Material Implication]
[Irrelevance and the Material Conditional]
[Conversational Implicature as an Explanation for the Counter-Intuitive Conditionals]
Summary
[The Material Conditional and the Paradoxes of Material Implication]
[The material conditional, symbolized as ⊃, is true when the antecedent is false or the consequent is true. It is thus logically equivalent to ¬A ∨ B. But also on that account, it generates the “paradoxes of material implication,” namely, B ⊨ A ⊃ B and ¬A ⊨ A ⊃ B. In other words, suppose we have some given formula that is true. That means we can make it a consequent in a conditional with any arbitrary antecedent. Or suppose we have a negated formula as true, then we can make the formula’s unnegated form be the antecedent in a conditional with any arbitrary consequent.]
Priest now discusses the material conditional, symbolized as ⊃. [Recall from section 1.3.2 the semantic evaluation for the conditional:
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
(p.5, section 1.3.2)
and here are the ones for negation and conjunction:
v(¬A) = 1 if v(A) = 0, and 0 otherwise.
v(A ∨ B) = 1 if v(A) = 1 or v(B) = 1, and 0 otherwise.(p.5, section 1.3.2)
Using the above two conditions, what would be the evaluation for: ¬A ∨ B? For the whole disjunction to be true, then either disjunct needs to be true. That means we need either v(¬A) = 1 or v(B) = 1 for the disjunction to be true, otherwise it is false. And for v(¬A) = 1, we need v(A) = 0. Thus we see when we combine the conditions for the disjunction with one negated term, we have the same as for the conditional.
v(¬A ∨ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
Priest next shows the paradoxes of material implication, which follow from that equivalence, it seems, but I am not entirely sure I know how it works. For, we start by affirming either the consequent term or the negation of the antecedent term, and we infer the whole conditional from that. One is
B ⊨ A ⊃ B
And the other is:
¬A ⊨ A ⊃ B
Semantic validity is defined in the following way in section 1.3.3:
Let Σ be any set of formulas (the premises); then A (the conclusion) is a semantic consequence of Σ (Σ ⊨ A) iff there is no interpretation that makes all the members of Σ true and A false, that is, every interpretation that makes all the members of Σ true makes A true. ‘Σ ⊭ A’ means that it is not the case that Σ ⊨ A.
(p.5, section 1.3.3)
So suppose that B is true. That means, were it to happen to be the consequent of any conditional whatsoever, that conditional would have to be true. For, it meets the second condition for the conditional to be true:
v(A ⊃ B) = 1 if v(A) = 0 or v(B) = 1, and 0 otherwise.
In other words, for any true formula we can derive a conditional with any arbitrary antecedent. I am not sure, but perhaps it results in counter-intuitive derivations, like, on the basis of knowing that water is wet, we can derive, “if the moon is made of green cheese, then water is wet.” I am just guessing, as I am not sure if the problem here is with relevance. The second one would seem to work similarly, that from any true negated formulation we can derive a conditional with its unnegated form as the antecedent and choose any arbitrary consequent. So on the basis of “it is not the case that pigs can fly,” we might derive “if pigs can fly, than the moon is made of green cheese”. Again, I am not sure.]
The connective ⊃ is usually called the material conditional (or material implication). As its truth conditions show, A ⊃ B is logically equivalent to ¬A ∨ B. It is true iff A is false or B is true. Thus, we have:
B ⊨ A ⊃ B
¬A ⊨ A ⊃ B
These are sometimes called the ‘paradoxes of material implication’.
(12)
[Irrelevance and the Material Conditional]
[The truth conditions for the material conditional allow for technically true but intuitively false sentences that fulfill the conditions for the material conditional but seem false on account of the irrelevance of the antecedent to the consequent, as for example, “If New York is in New Zealand then 2 + 2 = 4.” This seems to contradict the intuitive sense we ascribe to the English conditional, which involves relevance.]
Priest then says that there is reason to think that the conditional in English may not be represented as the material conditional ⊃. He then gives some example sentences to demonstrate this. [I am not sure how the English element factors in here. It seems more to illustrate the paradoxes of material implication. So maybe the objection is to the semantics of material implication, here illustrated in English. Or maybe the idea is that the English conditional form has a certain sense to it that is contradicted by the sorts of counter-intuitive sentences that the material conditional should allow us to make. The first example has an obviously false antecedent and an obviously true consequent, with there being no relation between them. The next one has both an obviously true antecedent and an obviously true consequent, but they again seem unrelated. And the final one has an obviously true antecedent but an obviously false consequent, but again they are unrelated. So the issue seems to be relevance as far as I can tell so far.]
People taking a first course in logic are often told that English conditionals may be represented as ⊃. There is an obvious objection to this claim, though. If it were correct, then the truth conditions of ⊃ would ensure the | truth of the following, which appear to be false:
If New York is in New Zealand then 2 + 2 = 4.
If New York is in the United States then World War II ended in 1945.
If World War II ended in 1941 then gold is an acid.
(12-13)
[Conversational Implicature as an Explanation for the Counter-Intuitive Conditionals]
[The counter-intuitive example conditional sentences are odd because they break certain rules of communication, namely to assert the strongest information.]
[Priest next will explain one way to understand why these formulations from 1.7.2 are counter-intuitive even though they are technically true in terms of the semantics of the material condition. I will not summarize this well, so it is best to skip to the quotation below. But let us first look at the ideas here. In actual communication in real situations, we can draw inferences not from the content of what is said but rather from the fact that some particular thing is said. Priest names two such inferences. One is relevance. We suppose that someone asks “How do you use this drill?” That creates a certain context. Then someone says. “There’s a book over there.” We suppose that the person is obeying the rule of conversation that what we say is relevant as a reply. So supposing that “There is a book over there” is relevant to “How do you use this drill?” we can infer that the book is a drill manual of sorts. For otherwise, the reply would be irrelevant. The other rule is “assert the strongest claim you are in a position to make.” This is the important one for the example sentences, but I do not quite get it. What qualifies as the “strongest claim” or “the strongest information”? From the example, my guess is the following. Suppose someone asks, what day is your birthday? and you, knowing the correct answer, say, it is either January 1st or May 2nd. This is true, but maybe the answer is weakened by adding the false alternative. I am wildly guessing. So in Priest’s example, we suppose one person asks, “Who won the 3.30 at Ascot?” then the other person replies, “It was a horse named either Blue Grass or Red Grass,” and from this we can infer that the speaker does not know which. I am guessing that had the person known the correct answer, they would have said which, with that being the “strongest information,” but since both options were given, that means the speaker’s strongest information is not either disjunct, and hence we can infer that they do not know which one is true. Now we need to apply this to the counter-intuitive conditional sentences above. Priest says that they are odd, because the person asserting them is breaking the rule of assert the strongest, on account of them being in a position to assert either the consequent or the negation of the antecedent (or both). So let us look at them, as I am not following so well yet. Suppose someone says:
If New York is in New Zealand then 2 + 2 = 4.
Here the person is in a position to assert the consequent. By also asserting the conditional, they are not asserting the strongest information. Why is that? I am still guessing that by adding the false antecedent, we have “weakened” the statement somehow. But I am not following, because the person is also in a position to assert the negation of the antecedent.
If New York is in the United States then World War II ended in 1945.
Here the person is in a position to assert the consequent.
If World War II ended in 1941 then gold is an acid.
Here the person is in a position to assert the negation of the antecedent or the consequent. I am still missing the point. My best guess at the moment is that on the basis of the rule assert the strongest, when we assert a conditional, it should be that we assert it with the meaning the suggestion that the whole conditional itself is true and not just instead one of the following other options: that in the conditional we asserted only the consequent is true, that only the antecedent is false, or that both. My confusion is still with this example:
If New York is in the United States then World War II ended in 1945.
We supposedly are not objecting on the basis of relevance. But here both the antecedent and consequent are true. So what is the strongest information here? It cannot be the relevance of the antecedent to the consequent. Sorry, let me quote:]
It is possible to reply to this objection as follows. These examples are, indeed, true. They strike us as counterintuitive, though, for the following reason. Communication between people is governed by many pragmatic rules of conversation, for example ‘be relevant’, ‘assert the strongest claim you are in a position to make’. We often use the fact that these rules are in place to draw conclusions. Consider, for example, what you would infer from the following questions and replies: ‘How do you use this drill?’, ‘There’s a book over there.’ (It is a drill manual. Relevance.) ‘Who won the 3.30 at Ascot?’, ‘It was a horse named either Blue Grass or Red Grass.’ (The speaker does not know which. Assert the strongest information.) These inferences are inferences, not from the content of what has been said, but from the fact that it has been said. The process is often dubbed ‘conversational implicature’. Now, the claim goes, the examples of 1.7.2 strike us as odd since anyone who asserted them would be violating the rule assert the strongest, since, in each case, we are in a position to assert either the consequent or the negation of the antecedent (or both).
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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