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A-LOGIC is a system of logic designed to
- solve the standard paradoxes and major problems of standard mathematical logic,
- minimize that logic's anomalies with respect to ordinary language, yet
- 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.
In Part II an extra-logical operator ('T' for "It is true that...") is introduced, producing an extension of A-logic called Truth-logic. This encompasses standard logic's semantics except for the restriction to two truth-values. The truth-tables are three valued, and the meaning of each row in each truth-table is expressed by a valid C-condtional. Due to its definition of validity, the problem of The Liar is not a logical paradox in A-logic. Further, its C-conditionals have no "paradoxes of material or strict implication". And because a C-conditional is neither true nor false when the antecedent is not true, it avoids Hempel's "Paradox of Confirmation", solves Carnap's "Problem of Dispositional Predicates" and Goodman's first "Problem of Counterfactual Conditionals" and makes the probability of a C-conditional the same as conditional probablility in probability theory. In accomplishing these ends A-logic introduces some plausible qualifications for the C-conditional versions of Transposition and Addition.
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