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The use of dots as brackets in Church's system

Published online by Cambridge University Press:  12 March 2014

A. M. Turing
A. M. Turing*
Affiliation:
King's College, Cambridge
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Extract

Any logical system, if its use is to be carried beyond a rather elementary stage, needs powerful conventions about abbreviations: in particular one usually wants to modify the bracketing so as to make the formulae more readable, and also possibly shorter. The present note has been written in the belief that Church's formulation of the simple theory of types is particularly suitable as a basis for work on that theory, and that it is therefore worth while introducing special conventions which take into account the needs of this particular system. The conventions which I shall describe are ones which I have used a good deal myself, and have always found adequate. I intend to make use of them in forthcoming papers. They may be regarded as an extension of Curry's conventions.

I shall begin with a general discussion of punctuation by means of groups of dots. This general theory is applicable, with some modifications, to Russell's, Quine's, and Curry's bracketing systems as well as to the present one.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 7 , Issue 4 , December 1942 , pp. 146 - 156
Copyright
Copyright © Association for Symbolic Logic 1943

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References

1 Church, Alonzo, A formulation of the simple theory of types, this Journal, vol. 5 (1940), pp. 56–68.Google Scholar

2 A. M. Turing, Some theorems about Church's system, and The theory of virtual types, forthcoming.

3 Curry, H. B., On the use of dots as brackets in logical expressions, this Journal, vol. 2 (1937), pp. 26–28.Google Scholar

4 Whitehead, and Russell, , Principia mathematica, vol. 1, pp. 9–11.Google Scholar

5 Quine, W. V., Mathematical logic (New York 1940), pp. 37–42.Google Scholar