Bennett's Logic of Indicative Conditionals

In Chapter 9, Bennett codifies the logic of indicatives by endorsing some inferences (as probabilistically valid), and rejecting others.

As regards the acceptable inferences, Bennett allows for restricted modus ponens. The restriction deals with what will count as the second premise in a canonical mp-inference. The unrestricted version allows anything that entails the antecedent of the conditional to be part of the inference. The restricted version allows only the antecedent as a premise.

There are four (other) rejected inference forms:
(1) Or-to-if: (P ∨ Q) ∴ (¬P → Q)
(2) Contraposition: (P → Q) ∴ ¬Q → ¬P
(3) Transativity: (P → Q), (Q → R) ∴ (P → R)
(4) Antecedent Strengthening: (P → Q) ∴ ((P ∧ R) → Q)

Bennett derives what he calls the "Security Thesis".

Security Thesis: If X is an argument whose conclusion is an indicative conditional A → C, and if what results from replacing → by ⊃ throughout X is a classically valid inference, then X is probabilistically secure to the extent that P(A) is high. (Bennett, 141)

One rejects or-to-if because accepting it would entail the ⊃-analysis of → (if one accepts the propositional analysis of →). To see this, recall that, at least according to Bennett (p.24) that → is at least as strong as ⊃. This means that (P → Q) ∴ (P ⊃ Q). To get the other direction, that (P ⊃ Q) ∴ (P → Q):

1. (P ⊃ Q)
2. (¬P ∨ Q) [1 Def of ⊃ and ∨]
3. (P → Q) [2 Or-To-If]
So, (P ⊃ Q) ∴ (P → Q)

Since (P → Q) entails (P ⊃ Q), and (P ⊃ Q) entails (P → Q), (P → Q) ≡ (P ⊃ Q).

I leave it to the reader to finish the proofs for the other rejected inferences.

Reading Update

Let's read chapter 9, The Logic of Indicative Conditionals (pp. 127-151), of Bennett's APGTC for the next round of posts. I'd like to see both comments and posts from some of you non-posting, non-commenting fellows.

Chapter 10 starts Bennett's discussion of subjunctive/counterfactual conditiionals. I don't want to start it until we've discussed indicatives a bit more.

No more meetings. The summer session is over, at least for me, so from here on out, let the blog be our meeting place.

--ijd

Bradley's Preservation Condition

Bradley (2000, p.220) offers the following as a "preservation condition" for conditionals:

If Pr(A) > 0, but Pr(B) = 0, then Pr(A → B) = 0.

He goes on to explain that all this means is "that one cannot be certain that B is not the case if one thinks that it is possible that if A then B, unless one rules out the possibility that A as well." (op cit)

Is this condition correct for all real conditionals (by "real" I simply mean to focus attention on conditionals that we find, as opposed to those we make up)? Moreover, is this condition similar to Jackson's robustness-condition for conditional assertion? In reading and rereading his explanation of the preservation condition I've come to wonder how different this is from Jackson's account as both can be explained in terms of modus ponens.

Recall that for Jackson (1987, pp. 28ff.), a conditional is assertible just in case it both has high enough probability on its own, and is robust with respect to its antecedent. This kind of robustness is explained by appealing to whether or not one's subjective probability regarding a conditional's acceptability is affected negatively were one to come to believe the antecedent. This rules out as unacceptable those conditionals whose truth depends solely upon the falsity of their antecedents. We would be wary of empolying such conditionals in modus ponens inferences.

Now look at the preservation condition and its explanation. The reason that you can't both doubt B and accept (even just a little that) A → B, unless you discount A, is that A and (A → B) would force B's truth (by modus ponens).

What is curious is that this preservation condition, unlike robustness wrt antecedent, seems to facilitate modus tollens as well. Accepting (even just a little that) (A → B) and denying that B, forces that you deny A (by modus tollens). If the preservation condition is true of all real conditionals, then Jackson's robustness condition is too narrow for assertibility. Instead, we would have to add two-sided robustness to Jackson's account (which I have advocated for other reasons).

References
Bradley, R. (2000), "A preservation condition for conditionals," Analysis 60.3, pp. 219-222.
Jackson, F. (1987) Conditionals Basil Blackwell.