Abstract
In philosophy we are sometimes interested in the invariants of intelligibility. What do all good (adequate, successful, fair) interpretations of a language, or of a person, have in common? What common reality is projected by our understanding of each other?2 What do persons share that makes communication possible? Some voices in the tradition coach us to look for answers in subject matter, or ontology (physical objects, numbers, universais, propositions), in law governed relational systems such as causality, in deep logical structures such as predication and quantification, in universally applied concepts such as identity, truth, order, and in language-wide principles of interchange, such as the principle of extensionality. One may think here also of attitudes, such as belief, that every person takes toward “propositions”, of laws of deductive logic and, after Ramsey and de Finetti, of the calculus of probability.
Never ask for the meaning of a word in isolation, but only in the context in the context of a sentence.
G. Frege
Most of the material in this paper has been presented in classes and seminars or roughed out in short workpapers over the past four years. I am deeply indebted to students and friends who have suggested novel strategies, spotted errors and, more important, helped clarify the point of this work on truth theory. I cannot mention everyone who has helped in these ways but I must mention Nuel Belnap, Richard Grandy, Gilbert Harman, Richard Jeffrey, Sue Larson, David Lewis, Thomas Scanlon, Marc Temin, and Samuel Wheeler. To Donald Davidson I have a deeper debt: when I was a graduate student, his vision of the place of theory of meaning in philosophy and of the relation between a theory of truth and a theory of meaning set me to work on truth, meaning, and philosophy. I have had the benefit of countless discussions with him while the work reported in this paper was in process.
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References
Most of the material in this paper has been presented in classes and seminars or roughed out in short workpapers over the past four years. I am deeply indebted to students and friends who have suggested novel strategies, spotted errors and, more important, helped clarify the point of this work on truth theory. I cannot mention everyone who has helped in these ways but I must mention Nuel Belnap, Richard Grandy, Gilbert Harman, Richard Jeffrey, Sue Larson, David Lewis, Thomas Scanlon, Marc Temin, and Samuel Wheeler. To Donald Davidson I have a deeper debt: when I was a graduate student, his vision of the place of theory of meaning in philosophy and of the relation between a theory of truth and a theory of meaning set me to work on truth, meaning, and philosophy. I have had the benefit of countless discussions with him while the work reported in this paper was in process.
This phrasing comes from C. I. Lewis, Mind and the World Order, New York 1965, p.111.
The singleness of the structure and its philosophical importance have been elucidated by W. V. Quine; see especially the middle chapters of Word and Object, Cambridge, Mass. 1960.
A. Tarski, ‘The Semantic Conception of Truth and the Foundations of Semantics’ in Readings in Philosophical Analysis (ed. by H. Feigl and W. Sellars), New York 1949, p. 63. (This paper originally appeared in Philosophy and Phenomenological Research 4 (1944) 341–375.)
This way of presenting the paradigms is borrowed from W. V. Quine, ‘Notes on the Theory of Reference’ in W. V. Quine, From a Logical Point of View, Cambridge, Mass. 1953, p. 134.
A. Tarski, ‘Der Wahrheitsbegriff in den formalisierten Sprachen’, Studia Philosophica 1 (1936). This paper is translated under the title ‘The Concept of Truth in Formalized Languages’ in A. Tarski, Logic, Semantics, Metamathematics, Oxford 1956, pp. 152–278.
I refer here to the precise condition of adequacy laid down by Tarski in the Wahrheitsbegriff; see especially Sections III and VI.
The theory to be set out here is very close to one used by W. Craig and R. L. Vaught, ‘Finite Axiomatizability Using Additional Predicates’, Journal of Symbolic Logic 23 (1958) 289–308.
D. Hilbert and P. Bernays, Grundlagen der Mathematik I, Berlin 1934, pp. 381ff. The Hilbert-Bernays elimination theory is very clearly explained by W. V. Quine, Set Theory and Its Logic, Cambridge, Mass. 1963, pp. 9–15.
The substitution interpretation is discussed also in the following papers. B. Mates, ‘Synonymity’, University of California Publications in Philosophy 25 (1950) 201–226, esp. p. 223. This paper is reprinted, with original pagination, in Semantics and the Philosophy of Language (ed. by L. Linsky). P. T. Geach, Reference and Generality, Ithaca 1962, pp. 144–167.
R. B. Marcus, ‘Interpreting quantification’, Inquiry 5 (1962) 252–259.
R. B. Marcus, ‘Modalities and Iktensional Languages’ in Boston Studies in the Philosophy of Science (ed. by M. W. Wartofsky), Dordrecht 1963, pp. 77–96.
D. Føllesdal, ‘Interpretation of Quantifiers’ in Logic, Methodology, and Philosophy of Science III (ed. by B. van Rootselaar and J. F. Staal), Amsterdam 1968, pp. 271–281.
W. V. Quine, ‘Reply to Professor Marcus’ in W. V. Quine, The Ways of Paradox, New York 1966, pp. 175–182.
W. V. Quine, ‘Ontological Relativity’, Journal of Philosophy 65 (1968) 85–212.
‘Existence and Quantification’ in Fact and Existence (ed. by J. Margolis), Oxford 1969, pp. 1–17.
W. Sellars, ‘Grammar and Existence: A Preface to Ontology’, Mind n.s. 69 (1960), reprinted in W. Sellars, Science, Perception, Reality, New York and London 1963, pp. 247–281. Some of these writers see in the substitution interpretation an ontological neutrality (Mates, Marcus, Quine, Sellars), some a congenialness to intensionality and modality (Marcus, Geach, Sellars) that distinguish it from the satisfaction interpretation. If my claim is correct, these supposed differences are illusory. But the reader must decide these matters for himself
The point about parentheses is Lesniewski’s; see Quine, ‘Existence and Quantification’, p. 12.
Geach, Reference and Generality; the first quotation is from p. 156, the second from p. 158:
D. Hubert and P. Bernays, Grundlagen der Mathematik II, Berlin 1939, pp. 329ff.
J. C. C. McKinsey, ‘A New Definition of Truth’, Synthese 7 (1948–1949) 428–433. The quotation is from p. 428.
Tarski, Logic, Semantics, Metamathematics, p. 246 (footnote).
This rule is stated explicitly by R. M. Smullyan, ‘Languages in which Self Reference is Possible’, Journal of Symbolic Logic 22 (1957) 55–67.
M. Schoenfinkel, ‘Über die Bausteine der mathematischen Logik’, Mathematische Annalen 92 (1942) 305–316. This paper is reprinted (with introductory comments by W. V. Quine) in From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (ed. by J. van Heijenoort), Cambridge, Mass. 1967.
W. V. Quine, ‘Variables Explained Away’, Proceedings of the American Philosophical Society 104 (1960) 343–347. This paper is reprinted in W. V. Quine, Selected Logic Papers, New York 1966, pp. 227–235. Similar work has been done by P. Bernays, ‘Über eine natürliche Erweiterung des Relationenkalküls’, in Constructivity in Mathematics (ed. by A. Heyting), Amsterdam 1959, pp. 1–14.
The standard reference for possible world model theory is S. A. Kripke, ‘A Completeness Theorem in Modal Logic’, Journal of Symbolic Logic 24 (1959) 1–15. I do not know a paper in which possible world truth theory has been explicitly discussed.
The counterpart idea comes from David K. Lewis, ‘Counterpart Theory and Quantified Modal Logic’, Journal of Philosophy 65 (1968) 113–126.
David Lewis mentions this incompleteness, ibid., p. 117. I do not know whether he would agree with the present point made on the basis of it.
Dana Scott, ‘Advice on Modal Logic’ in Philosophical Problems in Logic: Recent Developments (ed. by Karel Lambert), Reidel, Dordrecht 1970, pp. 143–173.
H. Wang, ‘Truth Definitions and Consistency Proofs’, Transactions of the American Mathematical Society 73 (1952) 243–275, esp. pp. 254–255. In fact, Wang appears to make the mistaken claim that a truth definition exists for the whole language. See Richard Montague’s review, in Journal of Symbolic Logic 22 (1957) 365–367.
B. Russell, ‘On Denoting’, Mind n.s. 14 (1905), 479–493. Reprinted in Russell, Logic and Knowledge, London 1958, and in Feigl and Sellars, Readings in Philosophical Analysis, and in Linsky, Semantics and the Philosophy of Language. For a recent treatment see Quine, Word and Object, sections 37 and 38.
It will be well to note circumstances under which (1) does separate from (2) and (3). It separates when the object language has only a finite number of sentences: one can simply list their truth conditions. It separates when the object language has no quantifiers. It separates when the object language has quantifiers but only one-place predicates: its sentences fall into a finite number of truth-functional equivalence classes. It separates when each sentence is equivalent to a quantifier free sentence; i.e., when it submits to elimination of quantifiers (see G. Kreisel and J. Krivine, Elements of Mathematical Logic, Amsterdam 1967, Ch. 4). It separates when truth in the object language is decidable. It seems likely that the language of a creature that knows and acts intentionally must include quantifications and relations essentially; Gödel showed that truth in a language that contains arithmetic is undecidable. Thus it seems likely that cases in which (1) separates from (2) and (3) are of limited philosophical interest.
W. V. Quine, ‘Two Dogmas of Empiricism’ in Quine, From a Logical Point of View, pp. 20–46; the quotation is from p. 42.
G. Frege, The Foundations of Arithmetic, Oxford 1959, p. x. This is a translation of Die Grundlagen der Arithmetik made by John L. Austin.
D. Davidson, ‘Truth and Meaning’, Synthese 17 (1967) 304–323.
See Quine, Word and Object, Ch. 2; and Quine, ‘Ontological Relativity’.
L. Wittgenstein, Philosophical Investigations, New York 1953, p. 48. This is a translation of Philosophische Untersuchungen made by G. E. M. Anscombe.
My research was supported by Princeton University and the National Science Foundation.
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Wallace, J. (1972). On the Frame of Reference. In: Davidson, D., Harman, G. (eds) Semantics of Natural Language. Synthese Library, vol 40. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2557-7_8
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