There can be a pluralism of logics because where is classical negation. This can be justified because “1 2″ and “1 is not 2” are basically the same statement. However “not 2” has different meanings in different logics, so 1 2 means something different in paraconsistent logic (We’ll use DeCosta’s C1), we mark this difference with 1 2, and claim that the difference between classical negation () and paraconsistent negation () is marked . Hence .
I mentioned in a previous post that we cannot generalize this statement into because it is a contradiction. Nevertheless, a sense in which is true is the sense .
Now, vagueness is the situation where it is not clear if 1 2. To illustrate, take this curve:
here is vagueness on whether we have two or one “heaps”. This thing could be 2 or it could be 1 and so in a sense 2=1. How is this handled by logical pluralism? Paraconsistent logic would allow that A: “1 is not 2” and ~A: “1 is 2” has a sense in which it is true, while classical logic would explode. The reason for this is entirely based on the difference in negation. C1 creates a new sense each time a true possibility is negated, making the negation of a possibly true row in the truth-table have two senses, one in which the negation is true, another in which the negation is false.
(A ~A) (A ~A)
The point is that how you handle negation changes how vagueness is handled (or not handled). A difference in negation also gives rise to an entirely different logic. Vagueness can be described completely as a failed distinction/negation, so that even though we want “1 is not 2” vagueness makes this distinction fail. A different negation yields a different way distinction fails, but no logic “solves” vagueness completely. This is the mathematical relationship between vagueness and logical pluralism.
This may be made clearer with another example. Vagueness is the situation when a distinction fails, which can be described by the failure to distinguish = from , so .
Now for classical logic if vagueness renders 1 = 2 we can prove , likewise in paraconsistent logic there is no problem having as another non-explosive contradiction. this means in particular that and . Substituting, we get that things can get so vague we can’t tell the difference between and , in other words = . and from our previous statement we have and vagueness is now mathematically related to logical pluralism.