If we get rid of Ex Falso Quadlibet and Modus Ponens, what insight do we gain into logic, maths and language? — Banno
"...the objects which sit on the end of those symbols' interpretations..." It seems to me erroneous to think of numbers as things; rather, they are ways of doing things with words. A number is not an object so much as what we do when we count; a way of using words. — Banno
Why should we assume that there is only one true way to talk about truth - what it really means? — Banno
...a generic end point that a string gets mapped to — fdrake
Perhaps mathematical Platonism will give way to a recognition of maths as another language in which we should look not to meaning but to use. — Banno
If the Russell Contradiction does not spread, then there is no obvious reason why one should not take the view that naive set theory provides an adequate foundation for mathematics, and that naive set theory is deducible from logic via the naive comprehension schema.
But now it seems that there might be an alternative. Rather than an incomplete yet consistent account of mathematics and language, we might construct an inconsistent yet complete account... — Banno
Still wondering if it would be much practical in the real world. — Corvus
the contradictions themselves as they appear in such systems are at the end of the day empty words. — TheMadFool
In other words, if we allow that anything goes, then we are able to do anything, — Metaphysician Undercover
If the case of A and not A, then B. There is some way in which A and not A implies B. But, A and ~A is a contradiction. So, some As are contradictory. Why not? If language is objectively inspired there's no reason the world should have to follow our rules for it. On occasion the rules we have don't match the world.I suspect you have missed something. The supposition is that we might drop (A & ~A)⊃B and yet with suitable alterations retain the "making sense" part. — Banno
(My bolding).Contemporary logical orthodoxy has it that, from contradictory premises, anything follows. A logical consequence relation is explosive if according to it any arbitrary conclusion B is entailed by any arbitrary contradiction A, ¬A (ex contradictione quodlibet (ECQ)). Classical logic, and most standard ‘non-classical’ logics too such as intuitionist logic, are explosive. Inconsistency, according to received wisdom, cannot be coherently reasoned about.
Paraconsistent logic challenges this orthodoxy. A logical consequence relation is said to be paraconsistent if it is not explosive. Thus, if a consequence relation is paraconsistent, then even in circumstances where the available information is inconsistent, the consequence relation does not explode into triviality. Thus, paraconsistent logic accommodates inconsistency in a controlled way that treats inconsistent information as potentially informative.
Get involved in philosophical discussions about knowledge, truth, language, consciousness, science, politics, religion, logic and mathematics, art, history, and lots more. No ads, no clutter, and very little agreement — just fascinating conversations.