Abstract
An extension of intuitionism to empirical discourse, a project most seriously taken up by Dummett and Tennant, requires an empirical negation whose strength lies somewhere between classical negation (‘It is unwarranted that. . . ’) and intuitionistic negation (‘It is refutable that. . . ’). I put forward one plausible candidate that compares favorably to some others that have been propounded in the literature. A tableau calculus is presented and shown to be strongly complete.
Highlights
In mathematical discourse a uniform treatment of negated and unnegated statements can be given by defining the former in terms of the latter
It is not obvious that, when we extend these conceptions to empirical statements, there exists any class of decidable atomic statements for which a similar presumption holds good; and it is not obvious that we have, for the general case, any similar uniform way of explaining negation for arbitrary statements. It would remain well within the spirit of a theory of meaning of this type that we should regard the meaning of each statement as being given by the simultaneous provision of a means for recognizing a verification of it and a means for recognizing a falsification of it, where the only general requirement is that these should be specified in such a way as to make it impossible for any statement to be both verified and falsified. (Dummett 1996, pp. 71-72)
There are a number of reasons for desiring conservativity having to do primarily with anti-holism, learnability, anti-realism and consistency, but whichever reasons one has in mind, classical negation is going to be problematic since the usual ways of proof-theoretically extending deductive systems for intuitionistic logic to include classical negation yield nonconservative extensions
Summary
In mathematical discourse a uniform treatment of negated and unnegated statements can be given by defining the former in terms of the latter. It is not obvious that, when we extend these conceptions to empirical statements, there exists any class of decidable atomic statements for which a similar presumption holds good; and it is not obvious that we have, for the general case, any similar uniform way of explaining negation for arbitrary statements It would remain well within the spirit of a theory of meaning of this type that we should regard the meaning of each statement as being given by the simultaneous provision of a means for recognizing a verification of it and a means for recognizing a falsification of it, where the only general requirement is that these should be specified in such a way as to make it impossible for any statement to be both verified and falsified. The following section could serve double duty as a survey of the numerous objections to classical-like negations
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