Allard Tamminga gave a good talk titled `Lost in Translation: the Logic of Paradox’ at the recent Beyond the Possible conference.
Making use of Greg Restall’s semantics for LP, he provided a nice translation of LP into the modal logic S5. This got me thinking about converting a K3-to-modal logic translation I once came across into an LP-to-modal logic translation. Here is what I have so far: A translation of LP into modal logic.
Some of the philosophical conclusions of Allard’s presentation were interesting. Take the following:
LP is the logic of a fragment of the modal logic S5. As a consequence, Priest is wrong, as far as LP‘s “theoretical account of negation” is concerned, when he states: “Dialethic logic, unlike modal logic, does […] provide a genuine rival theory to that provided by classical logic”.
During question time Graham Priest raised a point which I think went something like this. Many logics can be translated into other logics. In this case, although there is a formal translation of LP into S5, he couldn’t see why this should detract from the significance of LP‘s many-valued semantics basis.
This seems a fair point. I think that the difference between modal logic and LP relative to classical logic is marked. Obviously, classical logic and modal logic are both bivalent whereas LP is trivalent. But more than this, the standard translation of modal logic into first-order logic involves changing boxes into universal quantifiers and diamonds into existential quantifiers; the syntax changes but the meaning or gist of things is pretty much preserved. The translation of LP into modal logic does not have this same degree of meaning preservation.
Furthermore, although it is shown that “LP is the logic of a fragment of the modal logic S5“, the legitimacy of treating some logic as a fragment of another logic is something to bear in mind. Most logics going around can be translated into first-order logic; that doesn’t mean that the meaning significance of those logics can be dismissed.
It is reasonable to claim that the Logic of Paradox can do perfectly well without true contradictions, since it has a classical, two-valued semantics and therefore need not be interpreted under the assumption of dialetheism. That the formal system of LP can be reduced to a classical system is nothing new.
But going the other way, dialetheism needs LP, or at least has no reason to abandon it in favour of a classical translation. Besides the obvious resonance of the truth value b, meaning `true and false’, another thing that comes to mind is how the many-valued semantics for LP provide the basis for paraconsistent probability functions.
In closing, I wonder how a paraconsistent, LP-based modal logic would fare under this translation.