Three logical theories

Philosophy of Science 36 (2):153-177 (1969)
Abstract
This study concerns logical systems considered as theories. By searching for the problems which the traditionally given systems may reasonably be intended to solve, we clarify the rationales for the adequacy criteria commonly applied to logical systems. From this point of view there appear to be three basic types of logical systems: those concerned with logical truth; those concerned with logical truth and with logical consequence; and those concerned with deduction per se as well as with logical truth and logical consequence. Adequacy criteria for systems of the first two types include: effectiveness, soundness, completeness, Post completeness, "strong soundness" and strong completeness. Consideration of a logical system as a theory of deduction leads us to attempt to formulate two adequacy criteria for systems of proofs. The first deals with the concept of rigor or "gaplessness" in proofs. The second is a completeness condition for a system of proofs. An historical note at the end of the paper suggests a remarkable parallel between the above hierarchy of systems and the actual historical development of this area of logic
Keywords logic  PHILOSOPHY  LOGICAL TRUTH  QUINE  IMPLICATION  TAUTOLOGY-CONSEQUENCE-DEDUCTION  TRUTH-PRESERVATION  TAUTOLOGY-PRESERVATION  CONSEQUENCE-PRESERVATION  KNOWLEDGE-PRESERVATION
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JOHN CORCORAN AND HASSAN MASOUD, Three-logical-theories redux.

  The 1969 paper, “Three logical theories” [1], considers three logical systems all based on the same interpreted language and having the same semantics.

  The first, a logistic system LS, codifies tautologies (logical truths)—using tautological axioms and tautology-preserving rules that are not required to be consequence-preserving.

  The second, a consequence system CS, codifies valid premise-conclusion arguments—using tautological axioms and consequence-preserving rules that are not required to be cogency-preserving [2]. A rule is cogency-preserving if in every application the conclusion is known to follow from its premises if the premises are all known to follow from their premises.

  The third, a deductive system DS, codifies deductions, or cogent argumentations [2]—using cogency-preserving rules. The derivations in a DS represent deduction: the process by which conclusions are deduced from premises, i. e. the way knowl ... (read more)