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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
1.9
More Counter-Examples
Brief summary:
(1.9.1) There are three other counter-examples to the material conditional, and they present damning objections to the claim that the English conditional is material. {1} (A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C); for example, “If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on.” {2} (A ⊃ B) ∧ (C ⊃ D) ⊢ (A ⊃ D) ∨ (C ⊃ B); for example, “If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.” And {3} ¬(A ⊃ B) ⊢ A; for example, “It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god” (14-15). (1.9.2) We cannot dismiss these counter-examples on grammatical grounds, because they are all in the indicative mood. And we cannot dismiss them on conversational implicature grounds, because none break the rule of assert the strongest. (1.9.3) We cannot object that in fact the above counter-examples really are valid, provided we stipulate that the English conditional is material in those cases. For, by making that stipulation, we are admitting that naturally the English conditional is not material and is only artificially so. But the whole point of these objections is to show that the English conditional is naturally material.
[Three Potent Counter-Examples to the Claim that the English Conditional is Material]
[The Counter-Examples Evade Objections over Grammatical Mood or Conversational Implicature]
[The Ineffectuality of Objecting that the Counter-Examples are Material Under an Additional Stipulation]
Summary
[Three Potent Counter-Examples to the Claim that the English Conditional is Material]
[There are three other counter-examples to the material conditional, and they present damning objections to the claim that the English conditional is material. {1} (A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C); for example, “If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on.” {2} (A ⊃ B) ∧ (C ⊃ D) ⊢ (A ⊃ D) ∨ (C ⊃ B); for example, “If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.” And {3} ¬(A ⊃ B) ⊢ A; for example, “It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god” (14-15).]
[Previously in section 1.7 and section 1.8, Priest showed how the material conditional is not perfectly suited for understanding the logic of the conditional in the English language. He now discusses even more arguments for this. (They are not the simplest structures, so let us go through them slowly starting sometimes with an intuitive rendition and working to the more formal ones. First, we imagine an electrical circuit in which there are two switches in a series where both need to be closed in order for the circuit to close and the light to go on. Think of the first switch as A and the second switch as B. And think of the light as C. Since both switches need to be turned on for the light to go on, and since that is sufficient to bring about the light to turn on, we can think of the logic here as:
(A ∧ B) ⊃ C
In other words, if switch A and B are both closed, then light C will be on. Now, intuitively, we would say that it is not enough for just one switch to be on for the light to be on. But under the logic of the material conditional, it is valid. So the following formula is valid, and I will try to see if we can make a tableau to show that.
(A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C)
[This table is not in the text and probably is mistaken. Please trust your own proofs over my attempt below.]
| (A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C) | ||
| 1. .
2. .
3. .
4. .
5. .
6. .
7. . 8. . 9. . | (A ∧ B) ⊃ C . ¬((A ⊃ C) ∨ (B ⊃ C)) ↓ ¬(A ⊃ C) ↓ ¬(B ⊃ C) ↓ A ↓ ¬C ↓ B ↙ ↘ ¬(A ∧ B) C ↙ ↘ × ¬A ¬B × ×
| P .
P .
2¬∨D .
2¬∨D .
3¬⊃D .
3/4¬⊃D .
4¬⊃D . 1⊃D (8×6) 8¬∧D (9×5) (9×7) Valid |
(not in Priest’s text)
In other words, in our electrical circuit example where the light turning on relies conditionally on both the switches being closed, were that conditionally material, we would validly conclude that only one of the two is needed for the light to go on, which is in real fact not what would happen. For the next example, we set up a conjunction of two distinct conditionals. But on account of the semantics for the material conditional, it allows us to derive a disjunction of two conditionals with the same antecedents from before but with switched consequents. The problem with this becomes evident with Priest’s illustration, where the conjunction of conditionals is true, but the derived disjunction of conditionals is not, even though it is a proof-theoretic consequence. Here is Priest’s example:
If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.
(15)
So here is the formulation again, which I will try to prove as best I can with my limited time and skills:
(A ⊃ B) ∧ (C ⊃ D) ⊢ (A ⊃ D) ∨ (C ⊃ B)
[This table is not in the text and probably is mistaken. Please trust
your own proofs over my attempt below.]
| (A ⊃ B) ∧ (C ⊃ D) ⊢ (A ⊃ D) ∨ (C ⊃ B) | ||
| 1. .
2. .
3. .
4. .
5. .
6. .
7. . 8. . 9. . 10. . 11. . . . | (A ⊃ B) ∧ (C ⊃ D) . ¬((A ⊃ D) ∨ (C ⊃ B)) ↓ (A ⊃ B) ↓ (C ⊃ D) ↓ ¬(A ⊃ D) ↓ ¬(C ⊃ B) ↓ A ↓ ¬D ↓ C ↓ ¬B ↙ ↘ ¬A B × ×
| P .
P .
1∧D .
1∧D .
2¬∨D .
2¬∨D .
5¬⊃D . 5¬⊃D . 6¬⊃D . 6¬⊃D . 3⊃D (11×7) (11×10) Valid |
(not in Priest’s text)
The next one asserts a negated conditional and then derives the antecedent. It seems simple but is not so intuitive. The example is:
It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god.
(15)
(I am not sure how to grasp the problem with this, but I wonder if it has something to do with the fact that this antecedent asserts the existence of something, which would not seem to be derivable from the negated conditional it is the antecedent in. At any rate, the formula and its possible proof are the following:)
¬(A ⊃ B) ⊢ A
[This table is not in the text and probably is mistaken. Please trust
your own proofs over my attempt below.]
| ¬(A ⊃ B) ⊢ A | ||
| 1. .
2. .
3. .
4. | ¬(A ⊃ B) . ¬A ↓ ¬B ↓ A × | P .
P .
1¬⊃D .
1¬⊃D (4×2) Valid |
(not in Priest’s text)
Here is Priest’s text:]
There are more fundamental objections against the claim that the indicative English conditional (even if it is distinct from the subjunctive) is material. It is easy to check that the following inferences are valid.
(A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C)
(A ⊃ B) ∧ (C ⊃ D) ⊢ (A ⊃ D) ∨ (C ⊃ B)
¬(A ⊃ B) ⊢ A
If the English indicative conditional were material, the following inferences would, respectively, be instances of the above, and therefore valid, which they are clearly not.
(1) If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on. (Imagine an electrical circuit where switches x and y are in series, so that both are required for the light to go on, and both switches are open.) |
(2) If John is in Paris he is in France, and if John is in London he is in England. Hence, it is the case either that if John is in Paris he is in England, or that if he is in London he is in France.
(3) It is not the case that if there is a good god the prayers of evil people will be answered. Hence, there is a god.
(14-15)
[The Counter-Examples Evade Objections over Grammatical Mood or Conversational Implicature]
[We cannot dismiss these counter-examples on grammatical grounds, because they are all in the indicative mood. And we cannot dismiss them on conversational implicature grounds, because none break the rule of assert the strongest.]
[Recall from section 1.8 how objections to some counter-examples for the claim that the English conditional is material involve making a distinction between subjunctive uses of the conditional clause in English from indicative uses, and to argue that indicative uses are material. The examples above in section 1.9.1 are all indicative, so that objection will not apply. Also recall from section 1.7.2 and section 1.7.3 how one might also distinguish certain odd but technically valid cases of the English conditional understood as material as resulting from incorrect uses of conversational rules, namely, to assert the strongest information. (Consider the example, “If New York is in New Zealand then 2 + 2 = 4.” Here, the rule of conversation that can apply is that we should always assert the strongest information. We know the consequent to be true but the antecedent false. So the strongest information is simply that “2 + 2 = 4,” and thus to follow the rule of assert the strongest, we would normally just assert that and leave out “If New York is in New Zealand,” which weakens the assertion with information that we know to be false.) But consider the first problematic counter-example from section 1.9.1 above.
(A ∧ B) ⊃ C ⊢ (A ⊃ C) ∨ (B ⊃ C)
(1) If you close switch x and switch y the light will go on. Hence, it is the case either that if you close switch x the light will go on, or that if you close switch y the light will go on. (Imagine an electrical circuit where switches x and y are in series, so that both are required for the light to go on, and both switches are open.)
(p.14, section 1.9.1)
Here, Priest notes, we cannot say that either of the disjuncts in the conclusion should be asserted instead of the other, because they both appear to be false.]
Notice that all these conditionals are indicative. Note, also, that appealing to conversational rules cannot explain why the conclusions appear odd, as in 1.7.3. For example, in the first, it is not the case that we already know which disjunct of the conclusion is true: both appear to be false.
(15)
[The Ineffectuality of Objecting that the Counter-Examples are Material Under an Additional Stipulation]
[We cannot object that in fact the above counter-examples really are valid, provided we stipulate that the English conditional is material in those cases. For, by making that stipulation, we are admitting that naturally the English conditional is not material and is only artificially so. But the whole point of these objections is to show that the English conditional is naturally material.]
[I do not entirely grasp the next point, but is seems to be the following. There is another objection, which is that someone might say that normally the above counter-examples will seem invalid, but they will have to be valid when we understand the English conditional as material. I am not sure why one would argue that, because it establishes the fact that it can only be valid by an artificial stipulation rather than by an analysis of how natural language works. That might be Priest’s point, because he says that by making this claim, we are admitting that it is not naturally the case that the English conditional is material.]
It might be pointed out that the above arguments are valid if ‘if’ is understood as ⊃. However, this just concedes the point: ‘if’ in English is not understood as ⊃.
(15)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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