A few days ago I briefly introduced the idea of paraconsistency, that certain statements can be both true and false, and that accepting the possibility of such contradictory statements exist doesn’t necessarily imply the truth of all statements, as it would in classical logic. A consequence of this, however, is that implication, as it is normally understood, doesn’t allow us to draw the conclusions we are accustomed to. For example if we assert A and A -> B we can’t conclude, in a paraconsistent system, that B.

Obviously the inability to deduce conclusions from premises would be a major problem for a paraconsistent logic system, since it would make it effectively useless. Most systems deal with this by redefining implication. For example, an implicative connective may imply that if A is provable then so is B, or that in all possible worlds where A is true so is B. In any case these definitions end up not being equivalent to the classical definition of implication (A -> B = ~A or B), and so allow paraconsistent systems to deduce conclusions from premises, and without the possibility of a contradiction implying that every statement is true. There are some downsides however, specifically that not every valid conclusion in classical logic can be deduced in such a paraconsistent system.

And, perhaps more significantly, the classical reasoning used in everyday life may not be valid in paraconsistent systems. For example we may reason: Bob is at the store or at home, Bob is not at home, therefore he is at the store. This is valid classically but invalid in many paraconsistent systems; from those premises we could conclude only Bob is not at the store or that Bob is at home and not at home. Such failures to capture the reasoning we employ successfully on a daily basis may seem like a good reason to reject paraconsistency.

Paraconsistency can overcome these difficulties, but we must introduce two new ideas. The first is the theorem (although obviously I won’t be giving a proof here) that, with the appropriate paraconsistent implicational connective, if classically a set of sentences implies some conclusion, S, then that set of sentences paraconsistently implies S or a classical “contradiction” (such as b or ~b). The second idea that must be introduced is that of rational acceptability. Rational acceptability is an attempt to capture that idea that some statements are warranted by the evidence and some are not. Although often rationally acceptable statements are often true, and vice versa, it is not always the case. Although I could say a lot about rational acceptability all I need to mention here is that if a statement in the form “A or B” is rationally acceptable, and A is not rationally acceptable. then we can conclude that B is rationally acceptable.

Together these two ideas resolve the apparent difficulty posed by our continuing usage of classical reasoning in most situations. Our acceptance of paraconsistency forces us to conclude that all our classically valid conclusions are really conclusions in the form “S or X”, where X is a classical contradiction and S the conclusion we normally accept. Our premises then make “S or X” rationally acceptable to us. Additionally, we cannot rationally accept X, since in our experience we have never come across a classical contradiction. Thus, from the principle about rational acceptability mentioned above, we should rationally accept S, the conclusion that we make when reasoning classically. And thus paraconsistency is shown to be compatible with classical reasoning, so long as we accept that classical contradictions in the domain in question are impossible or extremely unlikely. And of course we can’t do away with paraconsistency by this kind of argument, since, according to paraconsistent logicians, there are some domains, such as set theory, where certain classical contradictions are rationally acceptable.