Abstract
A Leibnizian semantics proposed by Becker in 1952 for the modal operators has recently been reviewed in Copeland’s paper The Genesis of Possible World Semantics (Copeland in J Philos Logic 31:99–137, 2002), with a remark that “neither the binary relation nor the idea of proving completeness was present in Becker’s work”. In light of Frege’s celebrated Sense-Determines-Reference principle, we find, however, that it is Becker’s semantics, rather than Kripke’s semantics, that has captured the true spirit of Frege’s semantic program. Furthermore, for Kripke’s possible world semantics to fit in Frege’s framework of senses, worlds and referents, it will have to be thoroughly reformulated. By introducing the notion of a hi-world into the picture, we manage to keep the key ingredients of Becker’s semantics intact, while at the same time solve a fatal problem that used to shadow Becker’s original semantics—it had not been able to make sense of inhomogeneous modality. The resulting generalized Beckerian semantics provides, in effect, a Beckerian analysis of the Kripkean possible worlds. It reveals the subtle hierarchical internal structure of a Kripkean world that has not been discovered before.
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Notes
Copeland’s translation is adopted here, see (Copeland 2002).
The truth of ‘Pw’ depends on an interpretation that we have tacitly assumed: the referent of P is the set of all worlds such that p is true, while the referent of w is the actual world w 0.
For simplicity, some notations are abused here and hereafter, in particular, that of P and w
Here we only assume a nodding acquaintance with higher order logic.
Basically, \(\widetilde{{\square \beta }} = A\widetilde{\beta },{\text{ }}\widetilde{{\diamondsuit \beta }} = E\widetilde{\beta }\;{\text{and}}\;\widetilde{P} = P\) for a non-modal sentence P.
Note that cards here resemble worlds.
Strictly speaking, Becker’s semantics cannot deal with inhomogeneous modal sentences to be defined later. However, we shall generalize it to accommodate them in the next section.
There is an alternative way to get around the problem. One can insist that we do not need to consider inhomogeneous modal sentences in the first place, because these sentences are the results of people’s misusage of a language. People are not always clear about what they are actually talking about. Should they always know what their words mean, they would not come up with the conjunction of P and \(\square P,\) as the two are actually about things of different nature, the former is about a world while the latter a set of worlds. As long as people don’t utter inhomogeneous modal sentences, there is no need to provide meaning for them.
It is remarkable that it turns out that the reflexivity, symmetry and transitivity of this relation can be discussed completely in terms of some intrinsic properties of the hi-worlds.
By ‘genuine’ symmetry and ‘genuine’ transitivity, we mean properties that are not accidentally achieved by arbitrarily removing hi-worlds from X. For example, a set left with only one hi-world is necessarily symmetric and transitive, thus imposing no constraint on the internal structure of the hi-world, yet we are ‘not’ interested in these accidental properties.
These are less precise yet definitely more ‘philosophical’ in flavor. To have a vivid grasp of the conditions, it helps also to sum up these conditions metaphorically as (using the me-father-grandfather relationship to resemble the U n − U n+1 − U n+2 relationship, while using uncles to resemble elements of U n+2): it is true of every world that 1) it lies in its father world; 2) it lies in each of its uncle worlds; and 3) each of its uncle worlds is a subset of its father world.
Only in the rare case that there is indeed a binary accessibility relation between the plain-worlds, and the hi-worlds are generated from the plain-worlds via the accessibility relation as suggested in Sect. 2, can we afford not to distinguish between plain-worlds, hi-worlds, and Kripkean possible worlds.
Or, more explicitly, the sense of an expression determines its reference, inasmuch as its reference follows from its sense, taken together with relevant facts about extra-linguistic reality, cf. (Dummett 1991, p. 123.)
Epistemic logic is a typical example where there seems to exist a natural accessibility relation on the set of (plain-) possible worlds. But a careful reader should have noticed by now that in such case, the accessibility relation is simply inseparable from the worlds—it has already been tacitly imposed on the ‘plain-worlds’ to turn them into ‘hi-worlds’.
The modal part is only a first order one.
This time it requires a full two sorted type theory.
The author would like to thank two anonymous referees of this journal for their helpful suggestions. Thanks are also due to all individuals that have commented on earlier versions of this paper.
References
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Tsai, CC. The Genesis of Hi-Worlds: Towards a Principle-Based Possible World Semantics. Erkenn 76, 101–114 (2012). https://doi.org/10.1007/s10670-011-9328-5
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DOI: https://doi.org/10.1007/s10670-011-9328-5