I have now gone deeper into my reading of The Fifth Corner of Four. Graham Priest has now touched upon the Paradox of Ineffability. For those who don't know, this arises when we attempt to say that some part of reality cannot be talked about and we give reasons why it cannot be talked about, but in the course of giving those reasons we inevitably talk about that which cannot be talked about. In other words, the ineffable is indeed effable.
Anyone familiar with the history of Philosophy should recognize this dilemma from the works of such gigantic figures like Kant, Heidegger, and Wittgenstein. Now the Paradox of Ineffability cannot be adequately dealt with using the system FDEe that I alluded to in the last post. For if a state of affairs is both effable and ineffable, we cannot just give that state of affairs the value e, because even though that does describe one facet of the situation at hand (namely its ineffability), we have to remember that the paradoxical state of affairs is also effable. So perhaps we should just give the state of affairs the value b? That won't work either, for while a state of affairs with such a value is certainly effable, it is not ineffable. Similar reasoning applies to all of the other cotis.
So a natural thought might be to just add a sixth coti, called 'P', that means something like 'both effable and ineffable.' But this solution is inadequate. For suppose that a certain Philosopher was to say something like the following: 'Ultimate Reality is ineffable because it is beyond all conceptual understanding and dualistic thinking.' This obviously leads to the Paradox of Ineffability, and thus it would be correct to say that Ultimate Reality under this picture is both effable and ineffable. But this isn't the end of the story. For what would be the value of the state of affairs described by the following sentence: 'Ultimate Reality is beyond all conceptual understanding?' Obviously, it has the value e (since all states of affairs that include Ultimate Reality are ineffable), but some thought into the matter makes it clear that it also has the value 'b'. The reason for this is that the sentence in question must describe a true state of affairs (since being beyond conceptual understanding is one of the reasons that Ultimate Reality is ineffable), but the state of affairs it describes is also False since we can conceptually understand that Ultimate Reality is beyond all conceptual understanding. So it would appear that the state of affairs in question has both the value e and the value b.
In order to accommodate this insight, Priest makes use of Plurivalent Logic. Plurivalent Logic is unique among Non-Classical Logics because instead of forcing wffs to be assigned only one truth-value, it allows wffs to be assigned more than one truth-value. The main technical trick that allows one to achieve this is making use of a relation between wffs and truth values, rather than the standard functional assignment of wffs to truth values.
When we do this, we can very easily accommodate the fact that sentences describing states of affairs can have more than one truth value, and thus we can solve the Paradox of Ineffability.
As a side note, one quite surprising feature of Plurivalent FDE is that it has exactly the same theorems as the Many-Valued FDE. So this means we can simply bring over all the deductive apparatus of FDE when applying our new semantical modeling.