Abstract
If to be is to be the value of a bound variable, then the acknowledgment and denial of the existence of chairs amounts to a serious disagreement about the range of a quantifier. However, by resorting to the intrinsic hierarchical structure of hi-world semantics, we find that the varying of domains from worlds to worlds can actually be accommodated within a unified framework. With the introduction of a universal domain D of hi-individuals and an existence predicate E that serves as a realization operator, a new semantics for quantified modal logic is proposed. It allows individual variables to range over individuals in different levels of a hi-world, and is indifferent to the ontological debate about the existence of chairs.
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Notes
Note that while Do may contain things that are not in Dh, Dh may contain things not in Do as well. For example, in daily language, our domain of discourse might not include electrons and quarks.
No binary accessibility relation will be imposed on the set of possible worlds, however.
See Tsai (2012) for more details about hi-world semantics.
Strictly speaking, this is problematic because H is not a predicate for a historical Reagan, i.e., Reagan as a totality through his life. We need to interpret the Hr here charitably as asserting only that the temporal slice of Reagan at time t is happy.
Williamson seems to think differently, see Williamson (1998) p266.
Recall that the tensed theory of time and the tenseless theory of time are often referred to as the A-theory and the B-theory respectively.
I have assumed that we have a well-defined notion of trans-world identities.
Concerning the ontological status of hi-individuals, an anonymous reviewer for this journal raises the question of whether the hi-individuals are derivative or fundamental. Or, in Schaffer (2009)’s terms, is the underlying metaphysical structure Flat, Sorted, or Ordered? I think, while, as mentioned in Section 1, the semantics sketched here is neutral to the ontological debate between the hard ontologist and the easy ontologist, hence with no intention to pin down a universal E (the set of entities, in Schaffer’s terms) for a Flat metaphysical structure, it does not impose a Sorted or an Ordered structure on the domain of hi-individual either. Nevertheless, it is likely that the language L introduced here can serve as a meta-language for discussing ontological debates, such as “What really exists” or even “What grounds what,” through the availability of varying domains Dw (to be defined later), which is closely related to the interpretation of the existence predicate E. However, I have no idea how exactly the notion of “grounding” is to be defined in this scheme as yet.
Alien individuals and un-instantiated properties, insofar as they can be referred to, should be found somewhere in the hierarchy of a hi-world. This is the fundamental conviction of the hi-individual semantics. So, despite that a does not exist in the plain world w0 of a hi-world s0, and P is un-instantiated in w0 as well, there is some plain world w1 of a hi-world s1 in which a can be found and there is some plain world w2 of a hi-world s2 in which P is instantiated, and s1 and s2 are related to s0 through some applications of the sub-hi-world relation. I would like to thank an anonymous reviewer for this journal for reminding me to make this clarification.
An anonymous reviewer for this journal raises the concern that, with the introduction of hi-individuals within a set-theoretic machinery, perhaps we need a hybrid ontology of plain individuals and sets of sets … of plain individuals and a plural quantification over them. Furthermore, the reviewer points out that this might be, metaphysically, quit a controversial ideological commitment. To this, I totally agree. Hence, in this paper, our quantifiers are restricted to range over only hi-individuals—which amount to plain individuals in some plain worlds at some suitable place of a hi-world. In other words, any quantified modal logic developed upon this hi-individual semantics is necessarily first order, and admits no machinery to quantify over properties of any order.
Williamson (1998), p257.
Presumably, adding hints such as, hint 1, some might die, and hint 2, some might come into being, would have made the exercise trivial, so the authors are not bothered with it.
These approaches are criticized in Hayaki (2006) on ontological and other grounds.
It asserts that x exists in the actual (ground) world w0 regardless of whether the predicate is within the scope of a modal operator. For example, ◇ ∃ x(Hx ∧ Ex) asserts that some possible plain world in U1 has a human inhabitant, while \( \diamond \exists x\left( Hx\wedge \overset{\check{} }{\boldsymbol{E}}x\right) \) asserts that some possible plain world in U1 has a human inhabitant that inhabits in the actual world w0 as well.
Recall that he strives to leave “existence” out of the picture in Williamson (1998).
Clearly, if it could have not been one of these, then it is not one of these.
I am much obliged to an anonymous reviewer for this journal for advising me that an example involving iterated modalities is needed here.
References
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Acknowledgments
Many thanks are due to an anonymous reviewer for this journal, whose insightful remarks have led the author to reflect more on the possible metaphysical implications or assumptions that the semantics outlined here might have had. Thanks are also due to all reviewers that have previously commented on earlier versions of this paper.
Funding
This work was supported by the Ministry of Science and Technology, Taiwan (grant numbers MOST 106-2410-H-715-001, MOST 107-2914-I-715-004-A1, and MOST 107-2410-H-715-004-MY3). The author is much obliged.
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Tsai, Cc. Hi-individuals and Where to Find Them—Towards a Hi-world Semantics for Quantified Modal Logic. Acta Anal 35, 165–179 (2020). https://doi.org/10.1007/s12136-019-00401-4
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DOI: https://doi.org/10.1007/s12136-019-00401-4