**Authors:** Yakov A. Iosilevskii

In contrast to Church, who proved in 1936, based on papers by Gödel, that a dual decision problem for the conventional axiomatic first-order predicate calculus is unsolvable, I have solved a trial decision problem algebraically (and hence analytically, not tabularily) for a properly designed axiomatic first-order algebraico-predicate calculus, called briefly the trial logic (TL), and have successfully applied the pertinent algebraic decision procedures to all conceivable logical relations of academic or practical interest, including the 19 categorical syllogisms. The structure of the TL is a synthesis of the structure of a conventional axiomatic first-order predicate calculus (briefly CAPC) and of the structure of an abstract integral domain. Accordingly, the TL contains as its autonomous parts the so-called Predicate-Free Relational Trial Logic (PFRTL), which is parallel to a conventional axiomatic sentential calculus (CASC), and the so-called Binder-Free Predicate Trial Logic (BFPTL), which is parallel to the predicate-free part of a pure CAPC. This treatise, presenting some of my findings, is alternatively called “the Theory of Trial Logic” (“the TTL”) or “the Trial Logic Theory” (“the TLT”). The treatise reopens the entire topic of symbolic logic that is called “decision problem” and that Church actually closed by the fact of synecdochically calling the specific dual decision problem, the insolvability of which he had proved, by the generic name “decision problem”, without the qualifier “dual”. Any additional axiom that is incompatible with the algebraic decision method of the trial logic and that is therefore detrimental for that method is regarded as one belonging to either to another logistic system or to mathematics.

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