I have been reading Carnielli and Rodrigues' "An epistemic approach to paraconsistency: a logic of evidence and truth" when they discuss the notion of semi-valuation (BLE stands for Basic Logic of Evidence, a paraconsistent + paracomplete logic they invented).
I don't understand what Definition 3 and the quasi-matrix are doing. Specifically:
- Why is the 3rd row of $A\lor (A\to B)$ like that? I get the situation where $A=0$ and $B=1$, $A=1$ and $B=0$, and $A=B=1$ (covered by clause 1 or 2 of Def. 2). But where $A=B=0$ is not covered by any clause. Yet the matrix somehow says that when $A=B=0$, $A\to B$ may either be $1$ or $0$ - why?
- Does Def. 3 have something to do with validity/invalidity? The notion of validity was never discussed until this point, so I don't really understand when is a formula valid (I understand when is a classical, 1st order formula valid of course, but that doesn't seem to explain what's happening here because BLE is supposedly paraconsistent)
- Is Def. 3 just saying that, when all antecendents of a conditional is true but the consequent is false, the conditional is false? But isn't this just a generalisation of clause 1 of Def. 2?
You have to note the peculiar def of semivaluation for $\to$: it does not define the resulting truth value for $A =0$ and $B=0$. Thus in the matrix below there is an additional column $s_1$ that is not allowed in classical truth table.
Please note authors’ comment: semantics for connectives is not compositional.
Now, having $0$ in semival $s_1$, we have that formula $A \lor (A \to B)$ is no more valid simply because there is a valuation that gives value $0$ to the formula.