Conditionals IV: Is Modus Ponens Valid?

Conditionals IV: Is Modus Ponens Valid? UC Berkeley, Philosophy 142, Spring 2016 John MacFarlane 1 The intuitive counterexamples McGee [2] offers these intuitive counterexamples to Modus Ponens: 1. (a) If a Republican will win the election, then if Reagan will not win, Anderson will win. (b) A Republican will win the election. (c) So, if Reagan will not win, Anderson will win. 2. (a) If that creature is a fish, then if it has lungs, it is a lungfish. (b) That creature is a fish. (c) So, if it has lungs, it is a lungfish. 3. (a) If Uncle Otto doesn t find gold, then if he strikes it rich, he will strike it rich by finding silver. (b) Uncle Otto won t find gold. (c) So, if Uncle Otto strikes it rich, he will strike it rich by finding silver. 2 Some important distinctions We have seen from Stalnaker the importance of distinguishing between two questions: 1. If the premises are true at a world, does the conclusion have to be true at that world? 2. If the premises are accepted (true throughout the context set), does the conclusion have to be accepted? Showing (2) doesn t establish (1). We have already seen several examples of inferences that are acceptance-preserving but not truth-preserving. For example, B, therefore if A, then B. Has McGee given us counterexamples to the claim that Modus Ponens is acceptancepreserving? It seems not. Surely, if it is believed with certainty (or accepted or presupposed) that a Republican will win, we must conclude that if Reagan does not win, Anderson will. But, even if MP is acceptance-preserving, it might not be truth-preserving. Suppose that we have good but not conclusive grounds for accepting the premises; that is, grounds for assigning a high probability. Might we still have conclusive grounds for rejecting the conclusion? We can map our credences as follows (bigger area = larger probability): April 14, 2016 1

2. Some important distinctions Carter Reagan Anderson 1. If a Republican will win the election, then if Reagan will not win, Anderson will win. 2. A Republican will win the election. 3. So, if Reagan will not win, Anderson will win. Not a fish Lungs Fish 1. If that creature is a fish, then if it has lungs, it is a lungfish. 2. That creature is a fish. 3. So, if it has lungs, it is a lungfish. Otto finds gold Otto gets rich Otto finds silver 1. If Uncle Otto doesn t find gold, then if he strikes it rich, he will strike it rich by finding silver. 2. Uncle Otto won t find gold. 3. So, if Uncle Otto strikes it rich, he will strike it rich by finding silver. If we apply Edgington s test for the acceptability of an indicative conditional, this will give us cases where the premises are acceptable to a high degree, but the conclusion is not. This answers the question: 3. If we assign high credence to the premises, must we assign high credence to the conclusion? But note that a No answer to (3) doesn t imply a No answer to (1). Compare (for a 6-sided die): The die will not land 1. The die will not land 2. The die will not land 3. The die will not land 4. The die will not land 5. So, the die will land 6. Here we can have a high credence to each premise but a low credence in their conjunction, and hence a low credence in their conclusion. We might ask instead: April 14, 2016 2

3. Direct assessments 4. If we assign high credence to the conjunction of the premises, must we assign high credence to the conclusion? In McGee s cases, do we assign a high credence to the conjunction of the premises? Well, here there are only two premises. Assume that A and A B are probabilistically independent. Then the probability of their conjunction is the product of their probabilities. If we have 0.9 credence in both, then, our credence in their conjunction should be 0.81, which is still fairly high. Note also that the (a) premise in each of McGee s examples seems to be something we might believe with credence 1. 3 Direct assessments Instead of asking about whether it would be okay to infer the conclusions from the premises in McGee s examples, we might try assessing the truth values of the premises and conclusions in different scenarios. It is important to keep in mind here that, on most semantics for the indicative conditional, it will be highly contextually sensitive. For example, on Stalnaker s view, the truth of if A then B can depend on (a) what respects of similarity are contextually relevant and (b) on what has been presupposed (what the context set is). For example, as we have seen, if B is presupposed and A is compatible with what is presupposed, then A B is guaranteed to be true. (The closest A-world will have to be in the context set, and so it will be a B world.) But if B is not presupposed, A B can be false, even if B is true. Now let s suppose that, shortly before the 1980 election, Sarah uttered (1a) and (1b). Suppose that the context set governing her conversation at this time includes worlds where Reagan, worlds where Carter, and worlds where Anderson. As far as she is presupposing, any of the candidates could win. Did she speak truly in uttering (1a)? Plausibly, yes. She knew that Anderson and Reagan were the only Republican candidates. Did she speak truly in uttering (1b)? We know that she did, since Reagan, a Republican, did win the election. Did she speak truly in uttering (1c)? That is harder to answer, since the truth conditions of indicative conditionals are disputed. But let s think about it from the standpoint of Stalnaker s view. What was the closest world to the actual world where Reagan did not win? Presumably it was a world where Carter won, not a world where Anderson did. So (1c) was false, in Sarah s context. Granted, if it had been accepted in Sarah s context that a Republican would win, then (1c) would have been true in her context, because there would be no worlds in the context set where Carter. But let s imagine a case where this is not accepted. (Imagine either that Sarah is not making an assertion in uttering (1b), or that she is making an assertion, but her conversational partners don t accept it.) April 14, 2016 3

4. Modus ponens vs exportation But hold on! We know that on Stalnaker s semantics, modus ponens is valid. If A and A B are both true at a world, B has to be true at that world. And now we re saying that Stalnaker s semantics agrees with McGee that (1c) is false, in the imagined context. So what s going on? Stalnaker will have to deny that either (1b) or (1a) expresses a truth. (1b) is undeniably true, so he will have to reject (1a). If we think about Stalnaker s semantics, it s easy to see that it doesn t vindicate our intuition that Sarah s utterance of (1a) is true. Consider the actual world. It s a world where a Republican will win the election. So (1a) is true at that world if (1c) is true at that world. We have just seen that (1c) is not true at that world, since the closest world where Reagan does not win is a world where Carter. So (1a) is not true either. McGee would say: This doesn t show that we don t have a counterexample to modus ponens. It just shows that Stalnaker s semantics is not right. For clearly (1a) is true in the imagined context. 4 Modus ponens vs exportation McGee argues that if we want a counditional that is stronger than the material conditional and weaker than strict implication, we need to choose between modus ponens and Exportation (A B) C A (B C ) He gives the argument with a great deal of rigor [2, pp. 465 6]. The basic idea, though, is very simple (it is similar to an idea from Allan Gibbard s [1]): 1. Suppose A B. 2. (A B) A logically implies B. (fact) 3. So, ((A B) A) B. (StrImp: If φ logically implies ψ, then φ ψ is true) 4. So, (A B) (A B). (Exportation) 5. So, A B. (Modus ponens for from 1 and 4) So, if we have exportation and StrImp, the material conditional implies the indicative! We can fix this by: (a) giving up the assumption that is weaker than strict implication and stronger than. (b) giving up exportation. (c) giving up modus ponens. McGee argues that we should choose (c). Exportation is plausible across the board, he thinks arguing inductively from some examples! Can you think of counterexamples? McGee notes that Stalnaker s semantics does not seem to endores exportation. What would Stalnaker s semantics say about the following two conditionals? April 14, 2016 4

REFERENCES REFERENCES (1) If Reagan hadn t won the election and a Republican had won, it would have been Anderson. (2) If Reagan hadn t won the election, then if a Republican had won, it would have been Anderson. References [1] Allan Gibbard. Two Recent Theories of Conditionals. In: IFS: Conditionals, Belief, Decision, Chance, and Time. Ed. by William L. Harper, Robert Stalnaker, and Glenn Pearce. 1981, pp. 211 247. [2] Vann McGee. A Counterexample to Modus Ponens. In: Journal of Philosophy 82 (1985), pp. 462 471. April 14, 2016 5