W. V. Quine. On natural deduction. The journal of symbolic logic, vol. 15 (1950), pp. 93–102. See erratum, ibid., p. iv.

In theCritique of Pure Reason, Kant conceives of general logic as a set of universal and necessary rules for the possibility of thought, or as a set of minimal necessary conditions for ascribing rationality to an agent (exemplified by the principle of non-contradiction). Such a conception, of course, contrasts with contemporary notions of formal, mathematical or symbolic logic. Yet, in so far as Kant seeks to identify those conditions that must hold for the possibility of thought in general, such conditions must holda fortiorifor any specific model of thought, including axiomatic treatments of logic and standard natural deduction models of first-order predicate logic. Kant's general logic seeks to isolate those conditions by thinking through – or better, reflecting on – those conditions that themselves make thought possible.

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A system of implicit quantification

Journal of Symbolic Logic ◽

10.2307/2270176 ◽

1968 ◽

Vol 32 (4) ◽

pp. 480-504 ◽

Cited By ~ 8

Author(s):

J. Jay Zeman

Keyword(s):

Subject Matter ◽

Traditional Method ◽

Natural Deduction ◽

Propositional Calculus ◽

Axiom System ◽

Symbolic Logic ◽

Quantification Theory ◽

The Subject

The “traditional” method of presenting the subject-matter of symbolic logic involves setting down, first of all, a basis for a propositional calculus—which basis might be a system of natural deduction, an axiom system, or a rule concerning tautologous formulas. The next step, ordinarily, consists of the introduction of quantifiers into the symbol-set of the system, and the stating of axioms or rules for quantification. In this paper I shall propose a system somewhat different from the ordinary; this system has rules for quantification and is, indeed, equivalent to classical quantification theory. It departs from the usual, however, in that it has no primitive quantifiers.

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Comments on a variant form of natural deduction

Journal of Symbolic Logic ◽

10.2307/2270129 ◽

1965 ◽

Vol 30 (2) ◽

pp. 119-121 ◽

Cited By ~ 1

Author(s):

William Tuthill Parry

Keyword(s):

Natural Deduction ◽

Journal Article ◽

Symbolic Logic ◽

Variant Form ◽

Propositional Function ◽

The Third ◽

Formal Statement

This note shows that the system of natural deduction proposed by Copi in this Journal (1956), made by varying one restriction on Universal Generalization (UG) of the system of his Symbolic logic (1954), is incorrect. The original Symbolic logic system, also incorrect, was corrected in the third printing (1958) by modification of another restriction on UG; but combining this modification with that of the Journal article does not give a correct system.1. In Symbolic logic — cited as SL (or as SL54 to indicate the first printing) — Copi precedes his formal statement of the quantification rules by a set of conventions applying to all the rules.The expression ‘Φμ’ will denote any propositional function in which there is at least one free occurrence of the variable denoted by ‘μ’. The expression ‘Φν’ will denote the result of replacing all free occurrences of μ in Φμ by ν, with the added provision that [a] when ν is a variable it must occur free in Φν at all places at which μ occurs free in Φμ.The italicized provision we call Restriction a. In the third printing of Symbolic logic — cited as SL58 — Restriction a appears explicitly as part of each of the four quantification rules.

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