Logic and Truth Value Gaps

  • Peter W. Woodruff
Part of the Synthese Library book series (SYLI, volume 29)


A salient feature of contemporary philosophical logic is the great interest in so-called ‘free logics’, logics admitting non-denoting terms without paraphrase. Proponents of such logics have generally followed one of two approaches, each of which was considered and rejected by Russell in ‘On Denoting’. The first, suggested by Meinong, requires the introduction of possible but non-existent objects as ‘references’ for non-denoting terms. This approach has been by far the more popular among contemporary free logicians, perhaps because many of them came to free logic by way of modal logic.1 The second approach was first suggested by Frege2 and later developed at length by Strawson.3 Roughly put, it characterizes sentences containing non-denoting terms as truth-valueless, i.e. as neither true nor false, while at the same time insisting that such sentences are meaningful and express (truth-valueless) propositions.


Modal Logic Atomic Formula Natural Deduction Proof Theory Valid Inference 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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    The tables for conjunction, disjunction and negation are of Lukasiewicz [15]. The implication operator is Kleene’s [13], and the truth operator is found in Bochvar [4], Hallden [10] and Åqvist [1]. The present system U is perhaps closest to Åqvist’s calculus A; the latter lacks only the constant u (see below). The interpretation of U is however different from that of any of the systems mentioned above.Google Scholar
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    Our T should not be confused with the connective T of Slupecki [19], which does for him what u does for us; i.e., TA is always undefined (see note 10).Google Scholar
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    These tables are functionally complete. For we may define Lukasiewicz’s C and N and Slupecki’s T as follows: CAB =(TA ⊃ B)&(A ⊃ *B) NA = ~ A TA =u&(A⊃A). By Slupecki [20] the tables for these connectives are functionally complete.Google Scholar
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    (added in proof) The remark on p. 138 is false, with the embarrassing result that the rule E E, wich reflects this remark, leads to inconsistency when added to our previous rules. For u has no free terms, hence by u I and E E it is a theorem. But then by u E every wff is a theorem.Google Scholar
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    The problem is that a constant like u has no place in a system with the ‘Strawsonian’ motivation of UE, and hence I was wrong to think of UE as, strictly speaking, an extension of U. A better course would be to give up functional completeness for the time being and take ~ as primitive instead of u. Then when an appropriate description theory had been added, together with the requisite predicate constants, we could reproduce formally the informal definition of u given in Section I, or something like it. Kaplan suggests something like this in [12].Google Scholar

Copyright information

© D. Reidel Publishing Company, Dordrecht, Holland 1970

Authors and Affiliations

  • Peter W. Woodruff
    • 1
  1. 1.University of CaliforniaIrvineUSA

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