Abstract
On the 4th of December 1967, Hans Kamp sent his UCLA seminar notes on the logic of ‘now’ to Arthur N. Prior. Kamp’s two-dimensional analysis stimulated Prior to an intense burst of creativity in which he sought to integrate Kamp’s work into tense logic using a one-dimensional approach. Prior’s search led him through the work of Castañeda, and back to his own work on hybrid logic: the first made temporal reference philosophically respectable, the second made it technically feasible in a modal framework. With the aid of hybrid logic, Prior built a bridge from a two-dimensional UT calculus to a one-dimensional tense logic containing the ‘now’ operator J. Drawing on material from the Prior archive, and the paper “‘Now”’ that detailed Prior’s findings, we retell this story. We focus on Prior’s completeness conjecture for the hybrid system and the role played by temporal reference.
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Notes
A copy of the notes Kamp used for his presentation can be found in the Prior Archive at the Bodleian Library, Oxford (Kamp 1967).
Prior’s paper was originally published as “‘Now”’ in Noûs, 2:101–119, 1968. An addendum, correcting a technical glitch, appeared as “‘Now’, corrected and condensed”, in Noûs, 2:411–412, 1968. A combined version (in which Prior’s Polish notation was converted to standard notation) appeared under the title “‘Now”’ in the new edition of Papers on Time and Tense (Prior 1968/2003). In this paper, all quotations from and page references to “‘Now”’ are to this combined version.
Prior’s own example is: “If I am sitting down, then it will always be the case that I am now sitting down” (Prior 1968/2003, p. 173).
Even more specifically, the problem can be viewed the following way as Kamp (1971, p. 238) later did. As \(\varphi \) and \(J\varphi \) should be semantically equivalent in the sense just mentioned, we arrive at the following problem. Suppose we’re are looking for \(>\)something\(<\) such that \(\mathfrak {M},t\models J\varphi \) iff \(>\)something\(<\), then this \(>\)something\(<\) would have to be equivalent to the condition for \(\mathfrak {M},t\models \varphi \) (as \(\varphi \) and \(J\varphi \) should be equivalent whenever \(\varphi \) is uttered). Bearing this in mind, consider any model with at least two points, \(t_1\) and \(t_2\), such that \(t_1 \prec t_2\), and with p true at \(t_1\) but false at \(t_2\). Such a model falsifies \(p\supset GJp\) at \(t_1\).
To put Prior’s point in contemporary terms—it’s all very well saying that we should just ban such rule applications, but can this be done without sacrificing completeness?
Prior cites their joint paper (Meredith and Prior 1965) which deals with axiomatisation, but notes that Meredith’s work on the subject dates back until at least 1953.
Prior did not speak of hybrid logic; that term only gained currency in the 1990s, long after Prior’s death; the term was first used in passing in (Blackburn 1990), and the publication of Blackburn and Seligman (1995) was the baptismal event. Prior regarded hybrid logic as a part of tense logic, indeed it was the third grade of tense logical involvement, as he explained in his paper “Tense Logic and the Logic of Earlier and Later”, which also can be found in (Prior 1968/2003). As the phrase “too much of a hybrid” seems to suggest, Prior’s views on hybrid logic (third grade tense logic) were somewhat equivocal; see (Blackburn 2006) for further discussion.
That Prior was the inventor of hybrid logic is surprisingly little known given the central role they play in his work on temporal logic; this curious state of affairs is discussed in detail in (Blackburn 2006). In the present paper we have (by and large) adopted contemporary hybrid logical notation and terminology; for example, Prior would have spoken of world-variables rather than nominals. But to make the comparison with “‘Now”’ more transparent we have followed Prior and used \(\Diamond \) and \(\Box \) for the universal modalities.
Prior is insistent here: “For the present, however, let us simply consider the system \(\text {K}_{\text {t}}\) which is in a sense ‘minimal’. It is well to confine ourselves to this because I want to show that it is awkward to introduce into tense-logic an operator with the properties of the idiomatic ‘now’, but if the tense-logic into which I introduce this operator is richer than \(\text {K}_{\text {t}}\) it is too easy to suggest that the trouble arises from my having made rash assumptions about time in the first place.” (Prior 1968/2003, p. 178).
These operators are nowadays often written \(\mathsf {E}\) (there exists some time) and \(\mathsf {A}\) (at all times) and are usually called universal modalities. They have played an important role in the development of hybrid logic (see (Gargov and Goranko 1993) and (Blackburn and Seligman 1995)) and are important in their own right (see Goranko and Passy 1992). Nowadays the notation \(@_i \varphi \) is standardly used to state that \(\varphi \) is true at the time named by the nominal i. This is sometimes introduced as a primitive symbol, and sometimes as shorthand for \(\Diamond (i\wedge \varphi )\) and \(\Box (i\supset \varphi )\).
Recall Prior’s early comment on restricting the generalisation laws: “this is a tall order if we are to think of \(K_t\) being embeddable in an an earlier-later calculus in the usual way” (p. 180). Well, he has successfully filled his tall order, and done so by linking with the UT calculus in an unusual way, namely via hybrid logic.
One remark. Recall the restriction put on RL. From \(\vdash \varphi \) we can conclude \(\vdash \Box \varphi \), given that \(\varphi \) does not contain occurrences of n or J. This restriction does not affect logical completeness. If \(\varphi \) is a logical validity containing occurrences of n or J, then choose a nominal k not occurring in \(\varphi \) and replace all occurrences of n by k. This new formula, \(\varphi [n\leftarrow k]\), is also logically valid and hence provable. But then we prove \(\varphi \) in one more step by substituting n for k.
References
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Acknowledgments
Patrick Blackburn would like to acknowledge the financial support received from the project Hybrid-Logical Proofs at Work in Cognitive Psychology, funded by The Velux Foundation (VELUX 33305), Denmark.
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Appendix: Prior’s system for now
Appendix: Prior’s system for now
Here we present Prior’s axiomatisation of his tense hybrid logic of ‘now’. The primitive symbols in the language are \(\mathord {\sim }, \supset , H\) and G. The other symbols \(\vee ,\wedge , \equiv \) and P, F are defined in the usual way. For propositional logic, Prior takes any complete propositional axiomatisation (so we may assume we have the rule Modus Ponens either as a primitive or derived proof rule). To this he then adds the following 12 axioms and the modified RL rule (in A3, n is the special symbol for ‘now’, and in A4 and A5, i is a nominal):
Included in this axiomatisation is a rule of substitution allowing one to substitute nominals for nominals and arbitrary formulas for propositional symbols. Prior defines his Kamp-style ‘now’-operator \(J\varphi \) to be \(\Box (n\supset \varphi )\). As noted in the text, G and H generalization goes via RL: if \(\varphi \) is provable, then by RL so is \(\Box \varphi \). Thus from axiom L4 we get \(G \varphi \), and from L5 we get \(H \varphi \). That is, both RG and RH are derived rules.
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Blackburn, P., Jørgensen, K.F. Arthur Prior and ‘Now’. Synthese 193, 3665–3676 (2016). https://doi.org/10.1007/s11229-015-0921-z
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DOI: https://doi.org/10.1007/s11229-015-0921-z