# 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. Commented Apr 22, 2015 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). Commented Apr 24, 2015 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.