Doing zeroth-order logic

We now have a cursory acquaintance with the semantics of $\mathcal{L}_0$, so we know how the meaning of formulas are specified, and in particular, how truth-values are assigned to formulas, both atomic and complex. Thus, we are in the position to start talking about logic. Now what logic is exactly is hard to say, since by now, the field is vast and intertwined with many other disciplines. At any rate, one of the fundamental aspects of logic is examining how the truth-values of certain sentences relate to the truth-values of certain other sentences. This sounds like a semantic question, since in this formulation, it is about truth-values. On the other hand, over the years, logicians have come up with a myriad of ways to examine these relations without talking about truth-values, or anything else in semantics. To do this, they introduced syntactic deductive systems, which rely purely on some syntactic rules of transformation to tell you whether certain sentences entailed other sentences, or in other words, whether those sentences were logical consequences of the initial set (among other things). Now since this book likes to be unorthodox, we will be using a deductive system that is technically syntactic in its nature, since it relies on purely syntactic rules for transforming the formulas of the language into other formulas of the language, but it is also highly tied to the semantics we previously examined. The approach is variously called analytic tableau, semantic tableau, or the truth tree method. In fact, in its modern form, it was invented by the logician-mathematician-philosopher-magician-pianist Raymond Smullyan, who was once a CUNY professor!