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.