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On natural deduction

Published online by Cambridge University Press:  12 March 2014

W. V. Quine
W. V. Quine*
Affiliation:
Harvard University
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Extract

For Gentzen's natural deduction, a formalized method of deduction in quantification theory dating from 1934, these important advantages may be claimed: it corresponds more closely than other methods of formalized quantification theory to habitual unformalized modes of reasoning, and it consequently tends to minimize the false moves involved in seeking to construct proofs. The object of this paper is to present and justify a simplification of Gentzen's method, to the end of enhancing the advantages just claimed. No acquaintance with Gentzen's work will be presupposed.

A further advantage of Gentzen's method, also somewhat enhanced in my revision of the method, is relative brevity of proofs. In the more usual systematizations of quantification theory, theorems are derived from axiom schemata by proofs which, if rendered in full, would quickly run to unwieldy lengths. Consequently an abbreviative expedient is usually adopted which consists in preserving and numbering theorems for reference in proofs of subsequent theorems. Further brevity is commonly gained by establishing metatheorems, or derived rules, for reference in proving subsequent theorems. In natural deduction, on the other hand, proofs tend to be so short that the abbreviative expedients just now mentioned may conveniently be dispensed with—at least until theorems of extraordinary complexity are embarked upon. In natural deduction accordingly it is customary to start each argument from scratch, without benefit of accumulated theorems or derived rules.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 15 , Issue 2 , June 1950 , pp. 93 - 102
Copyright
Copyright © Association for Symbolic Logic 1950

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References

1 Gentzen, Gerhard, Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39 (1934), pp. 176–210, 405–431CrossRefGoogle Scholar.

2 This notation, due to Whitehead and Russell, rests on a neat arithmetical analogy: (mk)(n/m) = nk. Confusingly, ‘ϕ(α/β)’ is often written nowadays instead of ‘ϕ(β/α).’ This nversion probably rests on a pun between ‘durch’ in the fractional sense and ‘α durch β ersetzen.’

3 Rule EI, my most conspicuous departure from Gentzen, is not altogether new. Several systems of natural deduction have been presented which differ from the present system and from one another in no essential way except in the restrictions adopted in connection with UG and EI. (They differ also in that in some of them TF is resolved into a bundle of more elementary rules; but this is a trivial matter.) Such a system was set forth by Cooley, John C., A primer of formal logic (Macmillan, 1942), pp. 126–140Google Scholar, but without exact formulation of restrictions. Another such system, exactly formulated, appeared in my mimeographed Short course in logic (Cambridge, Mass., 1946)Google Scholar, but the restrictions here were insufficient. (For a fallacious result which can be obtained in that system, see the ten-line example in the next section.) A revised version appeared in my mimeographed Theory of deduction (Cambridge, 1948)Google Scholar, but here, as J. W. Oliver has pointed out, UG was restricted beyond necessity and convenience. The present system is superior both practically and aesthetically to that of Theory of deduction. I have lately learned that Barkley Rosser has had, since 1940, an exactly formulated system of natural deduction which perhaps resembles the present system more closely than any of the others cited above. He set it forth in some mimeographed lecture notes in 1946–47. Having learned of this during travels in which I am still engaged, I do not yet know the details of his system. I was influenced in latter-day revisions of my present system, however, by information that Rosser's UG and EI were symmetrical to each other.

4 Jaśkowski, Stanisław, On the rules of suppositions in formal logic, Studia logica, no. 1 (Warsaw, 1934; 32 pp.)Google Scholar.

5 Readers accustomed to dot conventions at variance with Whitehead and Russell's should be warned that this means ‘Fxz ▪ (Gz ⊃ Gx).’

6 The definition of implication as validity of the conditional must be borne in mind. To deny that ‘Fy’ implies ‘(x)Fx’ is not to deny that ‘(y)Fy’ implies ‘(x)Fx,’ but it is to deny rather that ‘Fy ⊃ (x)Fx’ is valid, or, in other words, that ‘(y)(Fy ⊃ (x)Fx)’ is valid.

7 Gödel, Kurt, Die Vollständigkeit der Axiome des logischen Funktionenkalküls, Monatshefte für Mathematik und Physik, vol. 37 (1930), pp. 349–360CrossRefGoogle Scholar.