The logic of paradox (LP) is the simplest, and one of the oldest, of the paraconsistent logics. Instead of assigning truths to statements A, it instead uses relationships binary relationships v(A,1) and v(A,0). "A is true" is encoded by v(A,1); "A is false" is similarly encoded by v(A,0). Each statement A is required to be true or false, but it can be both (in which case we could say that "A is undetermined"). "A is strictly true" means v(A,1) but not v(A,0); strict falsity is the converse.
The usual symbols →, ∨, ∧ and ¬, retain their standard meanings, and a compound statement takes all possible values it could take, seeing all the possible values its components could take. So, for instance if A is (strictly) true, then ¬A is (strictly) false. If A is undetermined, then ¬A is undetermined. If A is undetermined and B is strictly false, then A ∨ B is undetermined and A ∧ B is strictly false - though if B were strictly true, then A ∨ B would be strictly true and A ∧ B undetermined.
These properties make LP quite easy to work with, and one can determine the truths of many statements using truth tables. In fact, it can be seen that every tautology of classical logic is a tautology of LP. This derives from the fact that tautologies are true regardless of the truth values of their components; hence they remain true in LP whether we take undertermined statements to be true or false. Consequently, all of the following are true in LP:
- A → (B → A)
- (A ∧ ¬A) → B
- ((A ∨ B) ∧ ¬A) → B
- A ∧ (A → B) → B
But wait a second. Isn't the second line a statement of the principle of explosion - the fact that we can derive anything from a contradiction? Indeed it is. LP can state the principle of explosion as a (true) theorem - but it can't actually use it as a rule of deduction. Similarly, the third line is a statement of the disjunctive syllogism - a true theorem, but not a valid rule of deduction. That is easy to see: let A be undetermined, and B strictly false. Then A ∨ B is true, and so is ¬A - and yet we cannot deduce that B is true from this information.
So LP can accept contradictions without blowing up, has all the tautologies of classical logic, but lacks some of the rules of inference. "Some" of the rules of inference? LP even lacks modus ponens! As before, let A be undetermined and B strictly false; then A and (A → B) are both true, but B is not.
So while LP is a pleasant logic to play with, it isn't particularly useful. Another weakness is that is still defines the material conditional (A → B) as (¬A ∨ B): false statements still imply anything, and we haven't solved the Löbian problem for UDT. In the next post, I'll look at relevance logics, which have a more restricted use of →, and do allow modus ponens.