142. Truth-preserving and consequence-preserving deduction rules. Bulletin of Symbolic Logic. 20 (2014) 130–1. ► JOHN CORCORAN, Truth-preserving and consequence-preserving deduction rules. Philosophy, University at Buffalo, Buffalo, NY 14260-4150, USA E-mail: corcoran@buffalo.edu Following Tarski's truth-definition and consequence-definition papers [3, pp. 152–278, 409– 420], we assume an interpreted formalized language: a first-order language interpreted number-theoretically. We use ordinary variable-enhanced English: for example, the English sentence schema 'every number x is such that P(x)' translates the first-order schema 'x P(x)'. As usual, a deduction is a rule-governed list of sentences beginning with premises and ending with a conclusion. A system of deductions is truth-preserving if each of its deductions having true premises has a true conclusion [3, p. 167]-and consequence-preserving if, for any given set of sentences, each deduction having premises that are consequences of that set has a conclusion that is a consequence of that set [2, p.15]. Consequence-preserving amounts to: in each of its deductions the conclusion is a consequence of the premises. The same definitions apply to deduction rules considered as systems of deductions. Every consequence-preserving system is truth-preserving. It is not as well-known that the converse fails: not every truth-preserving system is consequence-preserving [2, Appendix]. In ordinary first-order Peano-Arithmetic, the induction rule yields the conclusion 'every number x is such that: x is zero or x is a successor'-which is not a consequence of the null set-from two tautological premises, which are consequences of the null set, of course. Truth-preserving rules not consequence-preserving are non-logical or extra-logical rules [1, §4.1]. Such rules are unacceptable to persons espousing traditional truth-and-consequence conceptions of demonstration [2, p.16]: a demonstration shows its conclusion is true by showing that its conclusion is a consequence of premises already known to be true. [1] JOHN CORCORAN, Gaps between logical theory and mathematical practice, Methodological Unity of Science (Mario Bunge, editor), Kluwer, 1973. [2] JOHN CORCORAN, Founding of Logic, Ancient Philosophy, vol. 14 (1994), pp. 9–24. [3] ALFRED TARSKI, Logic, semantics, metamathematics, Hackett, 1983.