A-LOGIC is a full-length book (600+ pg). It functions as a system of logic designed to: 1) solve the standard paradoxes and major problems of standard mathematical logic; 2) minimize that logic's anomalies with respect to ordinary language, yet; 3) prove that all theorems in mathematical logic are tautologies.
It covers lst order logic the logic of the words "and", "or", "not", "all" and "some". But it also has a non truth functional "if...then" and differs in its definition of validity, its semantics and its theorems. In the book A-logic is contrasted step by step with standard mathematical logic as presented and defended by Quine.
All of standard logic's theorems are proven tautologies in A-logic. But some argument-forms called "valid" in standard logic are not valid in A-logic -- notably non-sequiturs like "(P and not-P), therefore Q". In addition A-logic has many tautologies with its non-truthfunctional "if ... then" that standard logic can not derive -- e.g., "Not-(if P&Q then not-P)."
A-logic's semantics is based on syntactically defined concepts of logical synonymy and containment of meanings rather than on truth-values and truth-functions. Its "if...then" sentences (called "C-conditionals") are valid if and only if (i) the meaning of the consequent is logically contained in that of the antecedent, and (ii) the antecedent and consequent are jointly consistent. The predicate "valid" holds only of C-conditionals and arguments. No valid C-conditionals are translatable into standard logic though all of them imply tautologies of standard logic.
Logic and foundations of mathematics | Philosophy
Angell, R. B. (2002). A-logic. Lanham, MD: University Press of America.