Tuesday, July 21, 2009

A.N. Prior on double indexing

``Hans Kamp devised in 1967 a consistent semantic interpretation for `now' which can be presented, with slight modifications, as a new sort of UT-calculus, in which T ties each tense-logical proposition not to one instant but to two, i.e. our basic form is not Tap but Tabp. The proposition p is related to the instants a and b in different ways; the essential difference is that the elimination of complexities from what is put after Tab may take us to other instants than a, but never to other instants than b. And wherever we may have been taken from a by operators like G and H, the one place to which we are always immediately taken by J [`It is now the case that'] is the instant b, i.e. the instant represented by the second argument of T. We might read the form Tabp as `From b it is the case at a that p', and `From b it is that case at a that p--now' = `From b it is the case at b that p'." -- Prior, A. N.: 1968, `"Now"', Nous 2(2), 101–119.

The 1967 work that Prior refers to is Kamp, H.: 1967, "The treatment of `now' as a 1-place sentential operator", multilith circulated to a graduate seminar at the University of California in Los Angeles. The results were reported in A.N. Prior (1968) and full treatment was published in Kamp (1971),`The formal properties of ``Now"', Theoria 37, pp. 227-274.

Prior presents problematic sentences such as,

(A) It will be the case that it is now the case that I am sitting,

and attributes to Kamp the above ``the first solution offered" to such problems. Prior agrees that this works but goes on to present an alternative (singly-indexed) treatment by introducing an instant-constant n for a designated instant (and then doing some other tricks due to Prior & Meredith (1953) -- I don't think he just gives extensional treatment of the operators but it seems very similar).

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