Abstract
In celebrating Arthur Prior we celebrate what he gave to the world. Much of this is measured by what others have made of his ideas after his death. The focus of this paper is a little different. It looks at what Prior himself thought he was accomplishing. In particular it considers Prior’s attitude to the semantic metatheory of the logics that he was interested in. The paper sets out some characteristics of the metalogical study of intensional languages in terms of an indexical theory of truth conditions stated in the language of set theory. In examining Prior’s work using these characteristics, it emerges that Prior had serious reservations about this way of studying modal and tense logics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In Prior (1952) we find a ‘possible worlds’ account of modality:
It is not so easy to find a sense for ‘p has the modal value x’ as it is to find a sense for ‘p has the same modal value as q’; but we might say that the modal value of a proposition is the set of possible states of affairs in which, and in which only, the proposition in question is true. (Prior 1952, p. 140)
and a few pages later:
Let us again simplify matters, and suppose that there are only two possible states of affairs. This gives rise to four modal values that a proposition may have—it may be true in neither state of affairs (Value 1), true in the first but not the second (value 2), true in the second but not the first (Value 3) and true in both (Value 4), (If true in neither, it will have a modality of impossibility in this ‘modal universe’; if true in both, one of necessity). (Prior 1952, p. 142)
He then shews how to tabulate this in a four valued ‘modal value table’ from which we may determine values for all formulae of modal logic. There is also what seems to be a sympathetic consideration of a possible-worlds semantics in his earlier unpublished The Craft of Formal Logic. (I owe this information to Aneta Markoska-Cubrinovska.) Material from these works suggests that a significant developmental story needs to be told about the aspect of Prior that I am considering.
An anonymous referee for this paper writes: “I believe that although he had been familiar with the essence of the semantics of modal logic at least since the beginnings of the sixties, it was only when Prior came to UCLA in the fall of ’65 that he was confronted with the method ‘head-on’, so to speak, mostly through the model-theoretic work on Tense Logic which had then just culminated in the dissertation of Nino Cocchiarella.”
Certainly Chaps. 2–4 of Prior (1967b) contain many informal statements of independence proofs which could easily be made precise in a standard model-theoretic way, and the models which achieve this also lead to completeness proofs, though mostly Prior contents himself here with referring to the results of others.
Tractatus, 4.461 ends with: “I know nothing about the weather when I know that it rains or does not rain.” Although there are a number of references to the Tractatus in Prior’s works, none of them indicate a reaction to the truth-conditional theory of meaning found there. (Except possibly for a hint on p. 39 of Prior (1971), where he refers to 4.5.)
KCpqCqp is \((p\supset q)\). \((q\supset p)\) in Russellian notation. In this paper I follow Prior’s polish notation when quoting from his works, except when the quotation is from Prior (2003), where I follow the notation of the 2003 edition.
It was in fact the Presidential Address to the ‘Second Philosophical Congress’ of the ‘New Zealand Section of the Australasian Association of Psychology and Philosophy’, held in Wellington on 27–30 August, 1954. Prior’s address was delivered on the opening afternoon of the congress, on 27 August. I am grateful to Dr Rosemary Mercer, who was present, for supplying a copy of the programme for the archives of the former New Zealand Division of the Australasian Association of Philosophy, and to Dr Colin Cheyne for maintaining and digitalising these archives.
Prior attributes the notation Uab to C.A. Meredith. Meredith (1956) is a one-page note from August 1956 in which Prior “recorded and expanded”, what Meredith had provided of a translation of certain modal axioms into what was called the ‘property calculus’.
Quine (1960, p. 216) [Prior’s footnote 1].
In those years a lot of attention was given to restricted fragments of the predicate calculus. Typical restricted fragments include only one-place predicates and can be even more restricted by using only one individual variable, which is then usually omitted. Such fragments are extensively discussed in the text (Hughes and Londey 1965) which replaced Basson and O’Connor in Wellington. In Hughes and Londey, as in Prior (1955), no semantics is provided for the full lower predicate calculus. There is however a discussion of universes of discourse, though validity is only defined for the restricted class of LPC formulae which need no individual variables. As a matter of autobiography, although we studied the completeness theorem in Henkin (1949), as part of the fourth year logic course in Wellington in 1960, I never really understood what notion of validity it presupposed, despite Henkin’s lucid definition on p. 160. In that respect, doing graduate work with Prior in Manchester in 1961–1963, while more exciting in terms of intensional logic than a more standard experience, still left me without a true appreciation of the power of completeness proofs. It was not until Hughes and I were preparing to write our modal logic book (Hughes and Cresswell 1968) that I really understood what semantics was all about.
The discussion in Prior (1957a) is mainly about a formula equivalent to \(\forall x \square \upalpha \supset \square \forall x\upalpha \), first introduced in Barcan (1946, p. 2) (Axiom 11), which was called the ‘Barcan Formula’ in Prior (1956, p. 60), where it is shewn derivable in S5. Its converse is provable in all the standard modal systems. The formula mentioned in the text of the present article is related to BF, but not equivalent to it.
References
Barcan, R. C. (1946). A functional calculus of first order based on strict implication. The Journal of Symbolic Logic, 11, 1–16.
Basson, A. H., & O’Connor, D. J. (1953). Introduction to symbolic logic. London: University Tutorial Press (2nd ed., 1957).
Carnap, R. (1947). Meaning and necessity. Chicago: University of Chicago Press. (2nd ed., 1956, references are to the 2nd ed.)
Carnap, R. (1963). Carnap’s intellectual biography. In P. A. Schilpp (Ed.), The philosophy of Rudolf Carnap (pp. 3–84). La Salle, IL: Open Court.
Davidson, D. (1967). Truth and meaning. Synthese, 17, 304–323.
Davidson, D. (1969). On saying that. In D. Davidson & K. J. J. Hintikka (Eds.), Words and objections: Essays on the work of W.V. Quine (pp. 158–174). Dordrecht: Reidel.
Henkin, L. (1949). The completeness of the first-order functional calculus. The Journal of Symbolic Logic, 14, 159–66.
Hughes, G. E., & Cresswell, M. J. (1968). An introduction to modal logic. London: Methuen.
Hughes, G. E., & Londey, D. G. (1965). The elements of formal logic. London: Methuen.
Leblanc, H. (1962). Review of A. N. Prior ‘Formal logic’. The Journal of Symbolic Logic, 27, 218–219.
Lewis, D. K. (1972). General semantics. In D. Davidson & G. Harman (Eds.), Semantics of natural language (pp. 169–218). Dordrecht: Reidel.
Lewis, D. K. (1978). Truth in fiction. American Philosophical Quarterly, 15(1), 37–46.
Meredith, C. A. (1956). Interpretations of different modal logics in the ‘Property calculus’ (Cyclostyled). Christchurch: Philosophy Department, Canterbury University College (recorded and expanded by A. N. Prior).
Montague, R. M. (1974). Formal philosophy. New Haven: Yale University Press.
Prior, A. N. (1952). In what sense is modal logic many-valued? Analysis, 12, 138–143.
Prior, A. N. (1954). Entities. Australasian Journal of Philosophy, 32, 159–168 [Prior 1976, pp. 25–32].
Prior, A. N. (1955). Formal logic. Oxford: Clarendon Press. Second Edition, 1962.
Prior, A. N. (1956). Modality and quantification in S5. The Journal of Symbolic Logic, 21, 60–62.
Prior, A. N. (1957a). Time and modality. Oxford: Clarendon Press.
Prior, A. N. (1957b). Critical notice of Alfred Tarski. Logic, Semantics and Metamathematics, Mind, 66, 401–410.
Prior, A. N. (1958). The syntax of time distinctions. Franciscan Studies, 18(2), 105–120.
Prior, A. N. (1962a). Nonentities. In R. J. Butler (Ed.), Analytical philosophy (pp. 120–132). Oxford: Blackwell [Prior 1976, pp. 109–121].
Prior, A. N. (1962b). What is logic? In Prior (1976, pp. 122–129).
Prior, A. N. (1967a). The correspondence theory of truth. In P. Edwards (Ed.), Encyclopedia of philosophy (Vol. 2, pp. 223–232). New York: MacMillan.
Prior, A. N. (1967b). Past, present and future. Oxford: Clarendon Press.
Prior, A. N. (1968a). Egocentric logic. Noûs, 2, 191–207 [Prior 2003, pp. 223–240].
Prior, A. N. (1968b). Tense logic for non-permanent existents [Prior, 2003, pp. 257–274].
Prior, A. N. (1968c). Modal logic and the logic of applicability. Theoria, 34, 183–202 [Prior 2003, pp. 275–292].
Prior, A. N. (1968d). Papers on time and tense. Oxford: Clarendon Press.
Prior, A. N. (1968e). Quasi-propositions and quasi-individuals’ [Prior 1968d, pp. 135–144; 2003, pp. 213–221].
Prior, A. N. (1969). Tensed propositions as predicates. The American Philosophical Quarterly, 6, 290–297 [Prior 2003, pp. 197–211].
Prior, A. N. (1971). In P. T. Geach & A. J. P. Kenny (Eds.), Objects of thought. Oxford: Clarendon Press.
Prior, A. N. (1976). Papers in logic and ethics. London: Duckworth.
Prior, A. N. (1979). The doctrine of propositions and terms. London: Duckworth.
Prior, A. N. (2003). P. Hasle, P. Øhrstrøm, T. Braüner, & J. Copeland (Eds.), Papers on time and tense (New ed.). Oxford: Oxford University Press.
Quine, W. V. O. (1953). Two dogmas of empiricism. In From a logical point of view (pp. 20-46). Cambridge, MA: Harvard University Press (2nd ed., 1961).
Quine, W. V. O. (1960). Word and object. Cambridge, MA: MIT Press.
Russell, B. A. W. (1918). The philosophy of logical atomism. In R. C. Marsh (Ed.), Logic and knowledge (pp. 177–281). New York: Capricorn Books, 1955.
Tarski, A. (1936). The concept of truth in formalized languages. In Tarski (1956, pp. 152–278).
Tarski, A. (1956). Logic, semantics and metamathematics. Oxford: Clarendon Press.
Whitehead, A. N., & Russell, B. A. W. (1910). Principia Mathematica (3 Vols.). Cambridge: Cambridge University Press, 1st ed., 1910–1913, 2nd ed., 1923–1927.
Wittgenstein, L. (1922). Tractatus Logic-Philosophicus (C. K. Ogden, Trans.). Kegan Paul: London.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Cresswell, M.J. Prior on the semantics of modal and tense logic. Synthese 193, 3607–3623 (2016). https://doi.org/10.1007/s11229-015-0949-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11229-015-0949-0