Paraconsistent Logic - Tradeoff

Tradeoff

Paraconsistency does not come for free: it involves a tradeoff. In particular, abandoning the principle of explosion requires one to abandon at least one of the following three very intuitive principles:

Though each of these principles has been challenged, the most popular approach among logicians is to reject disjunctive syllogism. If one is a dialetheist, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ¬ A, then A is excluded, so the only way A ∨ B could be true would be if B were true. However, if A and ¬ A can both be true at the same time, then this reasoning fails.

Another approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. The disjunction (A ∨ B) is defined as ¬ (¬A ∧ ¬B). In this approach all of the rules of natural deduction hold, except for proof by contradiction and disjunction introduction; moreover, does not mean necessarily that, which is also a difference from natural deduction. Also, the following usual Boolean properties hold: excluded middle and (for conjunction and disjunction) associativity, commutativity, distributivity, De Morgan's laws, and idempotence. Furthermore, by defining the implication (A → B) as ¬ (A ∧ ¬B), there is a Two-Way Deduction Theorem allowing implications to be easily proved. Carl Hewitt favours this approach, claiming that having the usual Boolean properties, Natural Deduction, and Deduction Theorem are huge advantages in software engineering.

Yet another approach is to do both simultaneously. In many systems of relevant logic, as well as linear logic, there are two separate disjunctive connectives. One allows disjunction introduction, and one allows disjunctive syllogism. Of course, this has the disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them.

The three principles below, when taken together, also entail explosion, so at least one must be abandoned:

Both reductio ad absurdum and the rule of weakening have been challenged in this respect, but without much success. Double negation elimination is challenged, but for unrelated reasons. By removing it alone, while upholding the other two one may still be able to prove all negative propositions from a contradiction.