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I'm reading Peter Smith's "An introduction to formal Logic" and I understand that one of the most plausible theories to explain the differences between the material conditional '⊃' and its natural language counterpart "If... then" is the "Robustness theory".

I wonder if there are some serious contenders to this theory and which are they?

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What is the robustness theory (for those who want to understand the reference in the OP's question)?

VERY roughly, that ordinary 'if' stands to '$\supset$' rather as 'but' stands to '$\land$'. In each case the difference is not difference in truth-relevant content but in what more the use of the expression signals about (in a broad sense) the speaker's attitude to the truth.

To explain. We can all agree that 'but' doesn't mean the same as the truth-functional 'and'. But plausibly, the difference between 'P but Q' and 'P and Q' is not a difference in truth-conditions (what has to obtain in the world for those two claims respectively to be true). Rather the difference is that 'but' is reserved for use when, roughly, the speaker takes there to be some kind of contrast/tension between the truth of P and the truth of Q.

Similarly, the story goes, 'if' doesn't mean the same as the truth-functional '$\supset$'. However, again it isn't a difference in truth-conditions. 'If P then Q' and 'P $\supset$ Q' are true in the same circumstances. Rather the difference is that 'if' is reserved for use when, roughly, the speaker is promising to undertake a modus ponens inference and accept Q should it turn out that P.

Contrast: If my ground for asserting P $\supset$ Q is my firm conviction that not-P, then if you persuade me that, after all, P, then I won't conclude Q, but rather I'll take back my claim that P $\supset$ Q! The suggestion is that unlike assertions of P $\supset$ Q, when I assert if P then Q, I'm promising not to take it back should it be discovered that P -- my assertion is robust with respect to that discovery! Which explains why the inference 'not-P hence if P then Q' strikes us as unhappy, even if it is strictly speaking truth-preserving. For although I'll accept the truth-conditional content P $\supset$ Q on the basis of not-P, I won't promise to make the modus ponens inference if I change my mind about whether P!

Such very roughly are the beginnings of a sophisticated story due originally to the philosopher Frank Jackson, and looked on with some favour in the first edition of my IFL which was published in 2003, though mostly written twenty years ago now.

I'm not so enthusiastic about that story now, however, and won't be endorsing it in the second edition of the book (hopefully out by the end of the year).

That's not because there is a positive story about indicative conditionals which I now prefer, but because I think Jackson's story can't cope with some objections. And every other story on the market has problems too! So in fact, in the second edition, I'll take a more Fregean line. Frege, recall, introduced the material conditional into modern logic quite explicitly as a surrogate or replacement for the conditional of ordinary discourse, one that serves some core purposes of mathematicians in particular. But he didn't say that it is, either in whole or in part, an analysis of ordinary 'if'. I think we can live with that!

And so, finally, to answer the question posed by the OP! For an account of various alternative modern theories about the (indicative) conditional, you can't do much better than Dorothy Edgington's encyclopaedia essay here. You'll see that the ins and outs of the debates get pretty complicated!

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