: We refine the semantic process which reduces predicate logic to propositional tautologies. We then devise a tableau based proof system mirroring the semantic process by purely syntactic (i.e. programmable) means. We obtain a beautifully symmetric set of theorems given in the figure 1. As a byproduct of the refinement we have a simple tableau method of deciding quasi-tautologies and a proof system without the eigen-variable condition. 1 Introduction The idea of reducing predicate logic to (propositional) tautologies can be traced back to the Henkin's proof of completeness of predicate calculus [5] but it was first explicitly formulated by Smullyan in the form of his Fundamental theorem [8]. He calls the idea central to predicate logic. A modern and very readable exposition is by Barwise in [1]. The semantic reduction of predicate logic to tautologies can be expressed as follows: T j= A iff the formula V 1!in A i ! A is a tautology for some n 0 such that A i 2 T[Eq[Q. Here Eq are...