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A consistent combinatory logic with an inverse to equality

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
Frederic B. Fitch*
Affiliation:
Yale University, New Haven, Connecticut 06520
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Extract

In a previous paper [9] a demonstrably consistent type-free system of combinatory logic CΔ was presented. This system contained a moderately strong exten-sionality principle, the usual equations of combinatory logic, Boolean propositional connectives, unrestricted quantifiers, an unrestricted abstraction principle (“comprehension axiom”), and a limited principle of excluded middle. It also contained elementary arithmetic in its entirety, avoiding arithmetical Gödel incompleteness [14] by being ω-complete. CΔ probably can be shown to contain certain important parts of mathematical analysis, such as the theory of continuous functions, which are contained by the somewhat similar (but nonextensional) system K′ [6], [7]. A stronger system CΓ, having these same properties and more, will be formulated in the present paper. Elsewhere [13] it will be shown that the system Q of my book, Elements of combinatory logic (Yale University Press, New Haven and London, 1974), referred to below as ECL, is essentially a subsystem of CΓ, so that the consistency of CΓ guarantees that of Q.

CΓ will be stronger than both CΔ and Q in having an operator ‘ɿ’ (inverted Greek iota) which denotes the inverse of equality in the sense that ‘ɿ(= a) = a’ is a theorem of the system. Thus ‘ɿ’ denotes a function or operator which operates on a unit class to give the only member of that class. (Here ‘=a’ denotes the unit class whose only member is denoted by ‘a’.) If uninverted Greek iota, ‘ι’, is used as an abbreviation for ‘=’, the above equation could be written as ‘ɿ(ιa) = a’. Here ‘ιa’ is analogous to Bertrand Russell's well-known notation ‘ιa’, denoting a unit class. This same operator ‘ɿ’ makes it possible to transform one-many or many-one relations into the corresponding functions or operators, a kind of transformation not usually available in systems of combinatory logic [1], [2]. It also provides a method for restricting functions by restricting the corresponding relations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 45 , Issue 3 , September 1980 , pp. 529 - 543
Copyright
Copyright © Association for Symbolic Logic 1980

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References

REFERENCES

[1]Curry, H. B. and Feys, R., Combinatory logic, vol. 1, North-Holland, Amsterdam, 1958.Google Scholar
[2]Curry, H. B., Hindley, J. R. and Seldin, J. P., Combinatory logic, vol. 2, North-Holland, Amsterdam, 1972.Google Scholar
[3]Feferman, S., Non-extensional type-free theories of partial operations and classifications. I, Lecture Notes in Mathematics, vol. 500, Kiel Proof Theory Symposium, Springer, Berlin, 1975, pp. 73–118.Google Scholar
[4]Fitch, F. B., Self-reference in philosophy, Mind, vol. 55 (1946), pp. 64–73.CrossRefGoogle Scholar
[5]Fitch, F. B., An extension of basic logic, this Journal, vol. 13 (1948), pp. 95–106.Google Scholar
[6]Fitch, F. B., The Heine-Borel theorem in extended basic logic, this Journal, vol. 14 (1949), pp. 9–15.Google Scholar
[7]Fitch, F. B., A demonstrably consistent mathematics, this Journal, vol. 15 (1950), pp. 17–24; vol. 16 (1951), pp. 121–124.Google Scholar
[8]Fitch, F. B., A definition of negation in extended basic logic, this Journal, vol. 19 (1954), pp. 29–36.Google Scholar
[9]Fitch, F. B., The system CΔ of combinatory logic, this Journal, vol. 28 (1963), pp. 87–97.Google Scholar
[10]Fitch, F. B., Universal metalanguages for philosophy, Review of Metaphysics, vol. 17 (1964), pp. 396–402.Google Scholar
[11]Fitch, F. B., Elements of combinatory logic, Yale University Press, New Haven and London, 1974. (Referred to as ECL.)Google Scholar
[12]Fitch, F. B., Excluded middle and the paradoxes, Abstract, this Journal, vol. 42 (1977), pp. 148–149.Google Scholar
[13]Fitch, F. B., The consistency of system Q, this Journal (to appear).Google Scholar
[14]Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme. I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198.CrossRefGoogle Scholar
[15]Tarski, A., Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1 (1936), pp. 261–405.Google Scholar
[16]Whitehead, A. N. and Russell, B., Principia mathematica, Cambridge University Press, Cambridge, vol. 1 (1910), vol. 2 (1912), vol. 3 (1913); 2nd ed. 1925, 1926, and 1927, respectively.Google Scholar