Disentangling Conditional Statements from their Corresponding Inference Schemes

Suppose we're looking at two objects:

(i) the conditional statement (A --> B)

(ii) the inference A |- B

In propositional logic, we have that (i) is true if and only if (ii) is valid. My question is if there exist any systems of logic where this does NOT hold? If so, is anyone familiar with arguments as to why these two things could ever disentangle? It seems intuitively true that conditional statements (at least indicative conditional statements) are just arguments in disguise. In other words, there doesn't seem to be any difference whatsoever in someone uttering "If A, then B" and "An argument with premise A entails B". Or is there?

Yes, there are such systems, but they are rare. Most logicians agree with you. They want the deduction theorem to hold.

Among the logicians who disagree are German logician H. Wessel and some of his scholars. In his theory of strict logical consequence A|-B only holds if and only if

``````1.) A->B is a tautology
2.) B contains only variables that A also contains
3.) A is not a contradiction and B not a tautology
``````

In other words, the so-called paradoxes of the material conditional are accepted, as the conditional is only a truth function, but their analogues for logical consequence are avoided. Confusing logical consequence with the material conditional is considered a grave conceptual error in this view. This work is only available in German, see this German Wikipedia link.

Relevance logics have been developed in the Anglo-American tradition to alleviate the problems of the material conditional. Since they change the conditional itself, as far as I know adequate versions of the deduction theorem hold in them, so they do not speak against your point of view.

– Lawl
Commented Apr 17, 2013 at 10:32
• @GeorgeSinger For the sake of completeness, Relevance Logics tend to have a modified version of the deduction theorem called the Relevant Deduction Theorem: If there is a deduction in R of B from A1,...,An, A that is relevant with respect to A, then there is a deduction in R of A -> B from A1,...,An. Furthermore the new deduction will be `as relevant' as the old one, i.e. any Ai that was used in the given deduction will be used in the new deduction. So, although a version of the deduction theorem holds, not every instance of the classical theorem will hold (due to the relevance requirement) Commented Apr 17, 2013 at 13:13
• See Dunn and Restall's paper on Relevance Logic for a comprehensive overview from two of the leading experts. Section 1.4 discusses the deduction theorem. Commented Apr 17, 2013 at 13:14