Philosophical Problems in Logic pp 121-142 | Cite as

# Logic and Truth Value Gaps

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## 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.

## Keywords

Modal Logic Atomic Formula Natural Deduction Proof Theory Valid Inference## Preview

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## Bibliography

- [1]
- [2]N. D. Belnap, Jr., ‘Intensional Models for First-Degree Formulas’,
*Journal of Symbolic Logic***32**(1967) 1–22.CrossRefGoogle Scholar - [3]G. Birkhoff,
*Lattice Theory*, 3rd ed., Providence, R.I., 1967.Google Scholar - [4]D. Bochvar ‘Ob odnom trechznachnom ischislenii i evo primenenii k analyzu paradoksov klassicheskovo rashirennovo funktionalnovo ischislenii’,
*Matemati- cheskij Sbornik***46**(n.s. 4) (1938) 287–308. See also review in*Journal of Symbolic Logic 4*(1939) 98, and correction,*ibid*.**5**(1940) 119.Google Scholar - [5]A. Church,
*Introduction to Mathematical Logic*, Vol. I, Princeton, N.J., 1956.Google Scholar - [6]H. B. Curry, Foundations of Mathematical Logic, New York 1963.Google Scholar
- [8]F. B. Fitch,
*Symbolic Logic*, New York 1952.Google Scholar - [9]G. Frege, ‘Über Sinn und Bedeutung’,
*Zeitschrift für Philosophie und Philosophische Kritik***100**(1892) 25–50.Google Scholar - [10]S. Hallden
*,The Logic of Nonsense*, Uppsala 1949.Google Scholar - [11]L. Henkin, ‘The Completeness of the First-Order Functional Calculus’,
*Journal of Symbolic Logic***14**(1949) 159–66.CrossRefGoogle Scholar - [12]D. Kaplan, ‘The System R: Sentential Logic with Non-Denoting Sentences’, unpublished manuscript, October 1967.Google Scholar
- [13]S. C. Kleene,
*Introduction to Metamathematics*, Princeton, N.J., 1952.Google Scholar - [14]S. Kripke, ‘Semantical Considerations on Modal Logic’,
*ActaPhilosophica Fennica***16**(1963).Google Scholar - [15]J. Lukasiewicz, ‘O Logice Trojwartosciowej’,
*Ruch Filozoficzny*5 (1920) 169–71.Google Scholar - [16]G. C. Moisil, ‘Les logiques non-chrysippiennes et leurs applications’,
*Acta Philosophica Fennica***16**(1963) 137–52.Google Scholar - [17]R. Montague, ‘Syntactical Treatments of Modality, with Corollaries on Reflexion Principles and Finite Axiomatizability’,
*Acta Philosophica Fennica***16**(1963) 153–66.Google Scholar - [18]W. F. Sellars,
*Science, Perception and Reality*, New York 1963.Google Scholar - [19]J. Slupecki, ‘Der volle dreiwertige Aussagenkalkül’, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Cl. iii
**29**(1936) 9–11.Google Scholar - [20]J. Slupecki, ‘Pelny trojwartosciowej rachunek zda’,
*Annales Universitatis Mariae Curie-Sklodowska*(Lublin)**1**(1946) 193–209.Google Scholar - [21]
- [22]
- [23]A. Tarski, Logic, Semantics, Metamathematics, Oxford 1956.Google Scholar
- [24]B. C, van Fraassen, ‘Singular Terms, Truth-Value Gaps, and Free Logic’,
*Journal of Philosophy***63**(1966) 481–95.CrossRefGoogle Scholar - [25]B. C. van Fraassen, ‘Presupposition, Implication and Self-Reference’,
*Journal of Philosophy***65**(1968) 136–52.CrossRefGoogle Scholar - [26]B. C. van Fraassen, ‘Presuppositions, Supervaluations and Free Logic’, in
*The Logical Way of Doing Things*(ed. by K. Lambert), New Haven, Conn., 1969.Google Scholar

## References

- 1.For example, see Scott’s essay in this volume and Kripke [14]. All references are to the bibliography.Google Scholar
- 2.Frege [9], p. 41.Google Scholar
- 3.Strawson [21] and [22], Ch. 6.Google Scholar
- 4.Van Fraassen [25] and [26], among others. It was in response to Van Fraassen that the central themes of this essay were first developed.Google Scholar
- 5.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.Google Scholar
- 6.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 - 7.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 - 8.See Bochvar [4] and Hallden [10].Google Scholar
- 9.Church [5], Ch. LGoogle Scholar
- 10.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.Google Scholar - 11.Frege [9], p. 27.Google Scholar
- 12.Strawson [21].Google Scholar
- 13.Cf. Belnap [2], § 3; also Birkhoff [3].Google Scholar
- 14.For a good survey, see Moisil [16].Google Scholar
- 15.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.*Google Scholar - 16.Fitch [8]. Fitch’s presentation is slightly less rigorous than that adopted here.Google Scholar
- 17.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.*Google Scholar - 18.The general technique is that of Henkin [11].Google Scholar
- 19.Tarski [23], p. 187.Google Scholar
- 20.E.g. Sellars in [18], Ch. 6.Google Scholar
- 22.A good discussion of this problem is presented in Van Fraassen [25], p. 143.Google Scholar
- 23.Dummett [7], p. 144.Google Scholar
- 24.It is perhaps worth noting that in view of Montague’s [17], the situation is exactly similar to that in modal logic.Google Scholar
- 25.Strawson [22], p. 175.Google Scholar
- 26.(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 - 27.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