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Reduction to a symmetric predicate

Published online by Cambridge University Press:  12 March 2014

Alan Cobham*
Affiliation:
Cambridge, Mass

Extract

It has been shown by Quine that an interpreted theory Θ, formulated in the notation of quantification theory, is translatable into a theory Θ′ in which the only primitive predicate is a dyadic F. In establishing this result Quine takes for the universe of Θ′ a set which comprehends the universe of Θ and has the property that if x and y are members then so is {x, y}. For this fragmentary set theory the distinctness of x from {x, y} and {{x}} is assumed. It will be shown here that, if a pair of somewhat more stringent restrictions are assumed for the set theory, then there is a symmetric dyadic predicate G definable within Θ′, i.e., in terms of F alone, in terms of which F is in turn definable. It follows from this extended result that the theory Θ is translatable into a theory in which the only primitive predicate is symmetric and dyadic.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1956

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References

1 Quine, W. V., Reduction to a dyadic predicate, this Journal, vol. 19 (1954), pp 180182Google Scholar,