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Comments on a variant form of natural deduction

Published online by Cambridge University Press:  12 March 2014

William Tuthill Parry
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Extract

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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 30 , Issue 2 , June 1965 , pp. 119 - 121
Copyright
Copyright © Association for Symbolic Logic 1965

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References

1 SL 100; italics and bracketed a mine. We retain the ν (üpsilon), though ‘ν’ (nu) was perhaps intended and is sometimes used.

2 SL 54 104, 106; bracketed letters mine.

3 Cf. Methods of Logic (1959) 164Google Scholar; ibid. (1950) 160f. We have worded the italicized rule so that it also excludes such an erroneous inference from (or to) an instance which is not a “conservative instance” [ibid. (1959) 165] as: “*(1) Fyy, *(2) (x)Fxy (1) y(y).” Quine's restriction (C) (ibid. 164), which replaces alphabetic restriction by a possible ordering of the flagged variables, may become: (C′) It must be possible to assign distinct numbers to the distinct flagged variables of a deduction so that the flagged variable of a line never has a greater number than any variable semi-flagged in that line. The erroneous deduction on p. 163 (ibid.) would read (in part):

The “crossing” of z and w manifests the impossibility of satisfying restriction C or C′.

4 The 4-line fallacy above and the proposal to correct the error in UG by strengthening Restriction a were communicated by the present writer to Professor Copi in February 1957. A parallel fallacy is used in SL 58 to show that we may go by UG “from Φν to (μ)Φμ only if ν is a variable that occurs free in Φν at all places where μ occurs free in Φμ” (p. 105).

Examining a sixth printing of SL (1961), I find it is identical with the third printing in all passages relevant to this discussion.

The 4-line fallacy above does not show the incorrectness of the system of Another variant, for it fails to satisfy the restriction b 56 on UG56 (sec. 2 above).