Editing Paraconsistent logic (section)

=== Strategy ===
Suppose we are faced with a contradictory set of premises &Gamma; and wish to avoid being reduced to triviality. In classical logic, the only method one can use is to reject one or more of the premises in &Gamma;. In paraconsistent logic, we may try to compartmentalize the contradiction. That is, weaken the logic so that &Gamma;→''X'' is no longer a tautology provided the propositional variable ''X'' does not appear in &Gamma;. However, we do not want to weaken the logic any more than is necessary for that purpose. So we wish to retain modus ponens and the deduction theorem as well as the axioms which are the introduction and elimination rules for the logical connectives (where possible).

To this end, we add a third truth-value ''b'' which will be employed within the compartment containing the contradiction. We make ''b'' a fixed point of all the logical connectives.
:<math> b = \lnot b = (b \to b) = (b \lor b) = (b \land b) </math>

We must make ''b'' a kind of truth (in addition to ''t'') because otherwise there would be no tautologies at all.

To ensure that modus ponens works, we must have
:<math> (b \to f) = f ,</math>
that is, to ensure that a true hypothesis and a true implication lead to a true conclusion, we must have that a not-true (''f'') conclusion and a true (''t'' or ''b'') hypothesis yield a not-true implication.

If all the propositional variables in &Gamma; are assigned the value ''b'', then &Gamma; itself will have the value ''b''. If we give ''X'' the value ''f'', then
:<math> (\Gamma \to X) = (b \to f) = f </math>.
So &Gamma;→''X'' will not be a tautology.

Limitations:
(1) There must not be constants for the truth values because that would defeat the purpose of paraconsistent logic. Having ''b'' would change the language from that of classical logic. Having ''t'' or ''f'' would allow the explosion again because
:<math> \lnot t \to X </math> or <math> f \to X </math>
would be tautologies. Note that ''b'' is not a fixed point of those constants since ''b'' ≠ ''t'' and ''b'' ≠ ''f''.

(2) This logic's ability to contain contradictions applies only to contradictions among particularized premises, not to contradictions among axiom schemas.

(3) The loss of disjunctive syllogism may result in insufficient commitment to developing the 'correct' alternative, possibly crippling mathematics.

(4) To establish that a formula &Gamma; is equivalent to &Delta; in the sense that either can be substituted for the other wherever they appear as a subformula, one must show
:<math>(\Gamma \to \Delta) \land (\Delta \to \Gamma) \land (\lnot \Gamma \to \lnot \Delta) \land (\lnot \Delta \to \lnot \Gamma)</math>.
This is more difficult than in classical logic because the contrapositives do not necessarily follow.