Many-valued logic or many-valued semantics?

Abstract
There have been, I am afraid, almost as many answers to the question what is logic? as there have been logicians. However, if logic is not to be an obscure "science of everything", we must assume that the majority of the various answers share a common core which does offer a reasonable delimitation of the subject matter of logic. To probe this core, let us start from the answer given by Gottlob Frege (1918/9), the person probably most responsible for modern logic: the subject matter of logic is "truth", and especially its "laws"1. How should we understand the concept of "laws of truth"? The underlying point clearly is that the truth/falsity of our statements is partly a contingent and partly a necessary, lawful matter: that "Paris is in France" is true is a contingent matter, whereas that "Paris is in France or it is not in France" is true is a necessary matter (let us, for the time being, leave aside the Quinean scruples regarding the delimitation of necessarily true statements). Logic,then, should focus on the statements that are true as a matter of law (i.e. necessarily), or, more generally, the truth of which "lawfully depends" on some other statements (i.e. which are true as a matter of law provided these other statements are true). This renders Fregean laws of truth as, in general, a matter of "lawful truth-dependence" - i.e. of entailment or inference (again, let us now disregard any possible difference between these two concepts). This yields a conception of logic as a theory of entailment or inference, a conception which looms behind many other specifications of the subject matter of logic and which, I think, is ultimately correct. However, we can also see the logician – and this is the view we will stick to here – as trying to separate true sentences from false ones; or, equivalently, to map sentences onto truth and falsity. Let us first consider the case of a non-empirical language with a single, definite truth valuation – like the language of Peano arithmetic..
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