# A formulation of the simple theory of types

@article{Church1940AFO, title={A formulation of the simple theory of types}, author={Alonzo Church}, journal={Journal of Symbolic Logic}, year={1940}, volume={5}, pages={56 - 68} }

The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view…

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## References

SHOWING 1-10 OF 12 REFERENCES

### Grundlagen der Mathematik

- 1934

Der vorliegende Band schliesst die Darstellung der Beweistheorie ab, die ich vor einigen Jahren zusammen mit P. BERNAYS begann. Auf meinen Wunsch hat P. BERNAYS wieder die Abfassung des Textes uber…

### Grundzüge der theoretischen Logik

- Philosophy
- 1928

Die theoretische Logik, auch mathematische oder symbolische Logik genannt, ist eine Ausdehnung der fonnalen Methode der Mathematik auf das Gebiet der Logik. Sie wendet fUr die Logik eine ahnliche…

### Abriss der Logistik: Mit Besonderer Berücksichtigung der Relationstheorie und Ihrer Anwendungen

- Philosophy
- 1929

### Xm 0 -Xn 0 '\/aa(wla

### In certain cases a formula f"'«' may be obtained which does not involve descriptions. In particular, for addition and multiplication of non-negative integers we may use the definitions due to

- J. B. Rosser: !<

### Sa'a'0a' = x tt > |-S"'a-(x a "S a 'a-0 a ') = S a 'a'X a -. Hence by conversion, x a ''<Sa'a'0 a '=z"' |-S a -. a ,.Xa"S a 'a'0a' = Sa'a'X a >. Also, by 23°', N 0 a"Xa" h N, a "{S a

- =0a'. Hence using 22" and pZ3.qZ)pq and 13"' , we have |-(3x