## Abstract

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 Frege^{2} 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.

This article is based in part on material in my dissertation *Foundations of Three- Value Logic*, University of Pittsburgh, 1969. I should like to thank Mr. Michael Byrd for reading a preliminary draft and making a number of very helpful comments.

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### References

For example, see Scott’s essay in this volume and Kripke [14]. All references are to the bibliography.

Frege [9], p. 41.

Strawson [21] and [22], Ch. 6.

Van Fraassen [25] and [26], among others. It was in response to Van Fraassen that the central themes of this essay were first developed.

Another response has been made by Kaplan in [12], in which he applies the methods of free quantificational logic to a propositional calculus with quantifiers.

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.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).See Bochvar [4] and Hallden [10].

Church [5], Ch. L

These tables are functionally complete. For we may define Lukasiewicz’s

*C*and*N*and Slupecki’s*T*as follows:*C*AB =*(T*A ⊃ B)&(A ⊃ *B)*N*A = ~ A*T*A =*u*&(A⊃A). By Slupecki [20] the tables for these connectives are functionally complete.Frege [9], p. 27.

Strawson [21].

Cf. Belnap [2], § 3; also Birkhoff [3].

For a good survey, see Moisil [16].

It should be noted that if we adopt the ‘meaninglessness’ interpretation of u suggested in Section II, then the matter is rather different. For then we should want to regard a formula as expressing a proposition only when it has value; this in turn suggests that

*[A]*be defined as the*partial*function which is defined only on those I for which I*(A) ≠*u. If we then define meet and join as before, we obtain a non-trivial algebra, indeed a Boolean algebra. This algebra cannot, however, be conveniently extended to include an operation corresponding to*T.*Fitch [8]. Fitch’s presentation is slightly less rigorous than that adopted here.

The reason for calling

**K**a rule of weakening has to do with a sequenzen-kalküül formulation of the system; the interested reader may consult Chapter II of my dissertation. A good discussion of Peirce's law is found in Curry [6], Ch. V. Cf. especially his Gentzen rule*Px.*The general technique is that of Henkin [11].

Tarski [23], p. 187.

E.g. Sellars in [18], Ch. 6.

A good discussion of this problem is presented in Van Fraassen [25], p. 143.

Dummett [7], p. 144.

It is perhaps worth noting that in view of Montague’s [17], the situation is exactly similar to that in modal logic.

Strawson [22], p. 175.

(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.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].

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Woodruff, P.W. (1970). Logic and Truth Value Gaps. In: Lambert, K. (eds) Philosophical Problems in Logic. Synthese Library, vol 29. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-3272-8_6

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