What logics modify conditional detachment in this way?

What logical systems modify conditional detachment such that it is not permitted in cases that would be allowed by 'normal' logics — in particular, such that A⇒B is not equivalent to ~AvB?

• Linear logic is one such system. Apr 22 '15 at 20:51

The example which I know of is relevance logic, in which propositions such as A⇒B mean that B can be inferred more-or-less semantically as a consequence of A; not that either B is true or ~A is true.

For instance: consider a counterfactual exclamation of the form "If the US Dollar was valued at 2 British Pounds yesterday, then I'm the King of France".

• This statement is no mere facetious dismissal, but is in fact a valid proposition in classical logic: because the US dollar was not valued at 2 British Pounds yesterday, I can claim that any sort of ridiculous thing — including my being the monarch of the Fifth French Republic — would be true if the incorrect premise did hold.

• Such a statement would not be a valid proposition under relevance logic, unless we were acting in a logical system with axioms which would clarify how the state of affairs of such a high valuation of the US Dollar would have any causal connection to the ruler and form of government of France, and my relation to that situation.

To elaborate on Atamiri's comment above, Linear Logic is one particular form of relevance logic, in which propositions (from which one may derive consequences) are used largely as non-renewable resources which are consumed, and implication as a procedure to transform one sort of (propositional) resource into another. Thus an implication A⇒B stands in for a sort of procedure, and can be applied to transform an instance of a proposition A into an instance of proposition B; but from this it does not follow that you have either a ~A or a B to begin with.

• A remark, pertinent to your earlier question about contradictions: relevance logic would not suffice to prevent A⇒~A from sometimes being true when ~A is true, because of the example of the standard proof of the irrationality of the square root of 2. There, the proof of A⇒~A is a result of a semantic connection from a premise A to its own negation, where A is the assumption that some ratio a/b is a reduced fraction for sqrt(2). Apr 24 '15 at 6:59

An obvious example is intuitionistic logic, where A --> B is interpreted as, "from A, one may prove B". This is clearly different from, "either ~A is provable, or B is provable".

If you're very interested in non-classical logic, you may be interested to read Priest's very good book, Non-Classical Logic. If you're very interested in conditionals there are several good books, but my favorite is Bennett's A Philosophical Guide to Conditionals.