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A simplification of basic logic

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
Frederic B. Fitch*
Affiliation:
Yale University
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Extract

The program of “basic logic” can be summarized as follows:

I. To treat every syntactical system as a subclass of a certain fixed infinite classUof “U-expressions.” This can always be done by modifying in trivial ways the notation of each syntactical system which is not already such a subclass. As a result all syntactical systems become comparable with each other in the sense that they are merely different subclasses of a single class of expressions. The class U can be chosen in such a way as to be inductively definable thus in terms of a fixed symbol ‘σ’: (1) The symbol ‘σ’ is a U-expression. (2) The result of placing two U-expressions (or two occurrences of the same U-expression) next to each other, and enclosing this total expression within a pair of parentheses, is a U-expression.

II. To formulate a particular syntactical systemKwithin which every syntactical system (and indeedKitself) is “represented.” Such a system is here said to be a “basic system,” and an appropriate interpretation of it is said to be a “basic logic.” Within such a logic every finitary logic is definable, as well as the basic logic itself. Such a logic should be of fundamental importance, especially if it is so constructed as to be the weakest such logic and so contain no theorems that are not essential to its being basic.

With the above considerations in view, the system K has been defined in such a way that it is a subclass of U and is a basic syntactical system. A simpler definition of K will be given than heretofore in previous papers, and the minimum character of K will be made more clear.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 18 , Issue 4 , December 1953 , pp. 317 - 325
Copyright
Copyright © Association for Symbolic Logic 1953

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References

1 See Fitch, F. B., A basic logic, this Journal, vol. 7 (1942), pp. 105–114Google Scholar. This paper will be referred to as BL. In BL the class of U-expressions is built up from an infinite class Y of atomic symbols. In the present treatment we regard Y as having only one member, the atomic symbol ‘σ’. In another paper, Fitch, Representations of calculi, ibid., vol. 9 (1944), pp. 57–62, the class Y is treated as finite, though not necessarily a unit class. This latter paper will be referred to as RC. The class Y is treated as a unit class by John R. Myhill in his paper, A finitary metalanguage for extended basic logic, ibid., vol. 17 (1952), pp. 164–178. I am indebted to Dr. Myhill for several helpful criticisms of an earlier version of the present paper.

2 A class C of U-expressions is said to be represented in a class D of U-expressions by a U-expression ‘c’ if for every U-expression ‘a’, the U-expression ‘(ca)’ is in D if and only if the U-expression ‘a’ is in C. See 5.2 of BL.

3 See Fitch, A minimum calculus for logic, ibid., vol. 9 (1944). This paper will be referred to as MCL. In MCL we consider systems which are basic with respect to various operations. In the present paper attention is restricted to the single operation of 1(2) above.

4 By “constructively definable” we mean “recursively enumerable” in the sense of the first paragraph of RC. A class of U-expressions will be said to be recursively enumerable if the class of Gödel numbers of its members is recursively enumerable, and it will be said to be recursive if the class of Gödel numbers of its members is recursive. This terminology will, in a similar way, be applied to relations among U-expressions. An assignment of Gödel numbers is of course presupposed.

5 For example the system K′ in Fitch, An extension of basic logic, ibid., vol. 13 (1948), pp. 95–106. This system can be shown to be basic with respect to a large class of non-constructive systems, of which it is itself one. It can be simplified in very much the same way as K. Not only are the rules for the ancestral and the dual of the ancestral superfluous as part of the definition of K′, but so too are the rules involving the symbol for negation, since negation can be shown to be definable in terms of the other undefined concepts of K′. The details of this will be presented in a forthcoming paper.

6 In general, we will say that a calculus is a “minimum” of a certain class of calculi if it can be defined by a set of rules such that its belonging to the class in question requires it to have a derived or underived rule of the form of each of the underived rules that define it.

7 In referring to the “form of a rule” we do not here mean to refer also to the form of the representing U-expression. The basicness of the system cannot entail that the representing U-expression be of a certain form, but only that there be a U-expression that does the required representing. In the case of each rule there is exactly one U-expression which we are to think of as the representing U-expression that corresponds to that rule. For example, in the case of the rule for ‘έ’ we regard ‘έ’ as the representing U-expression, rather than ‘έb’ or ‘έba’ or any expression that might be treated as a part of ‘έ’. The “form of the rule” refers to material external to this representing U-expression.

8 This rule has the effect of satisfying requirement (2) on page 91 of MCL, since we may choose ‘(a 0rb)’ and ‘(a 0pb)’ as both the same as ‘(ab)’ of the present paper.

9 Fitch, , Self-referential relations, to be presented at the Eleventh International Congress of Philosophy, Brussels, 1953Google Scholar.

10 Details will be presented in the same paper as that mentioned at the end of footnote 5.