Abstract
Appeal to standard set theory in (mainstream) minimalist syntax is shown to be in conflict with the goal of analyzing dependency formation, a.k.a. movement, as involving genuine constituent copies. The underlying tension is due to (the axiom of) extensionality, which—other things being equal—favors a perspective on dependencies in terms of multidominance. The above argument is developed against the backdrop of a recent exposition of minimalist syntax (Chomsky et al. in Catal J Linguist (Special Issue):229–261, 2019), which can be seen as exemplary. The resulting critical assessment should be taken as removing obstacles on the way toward proper minimalist foundations for syntactic theory.
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Notes
I cite selected examplary work from the vast body of literature without in any way implying exhaustiveness.
Scare quotes ("…")—not formally distinguished from direct quotes— are used throughout to flag terminology that would require critical discussion and thorough motivation, sidestepped here in the interest of conciseness.
Langendoen (2003:309) proposes to distinguish the respective cases as Merge(γ1/γ2,γ2) and Merge(γ2/γ1,γ1). If in addition the "Extension Condition" (Chomsky 1993:23) or the "No Tampering Condition" (Chomsky 2007:8) holds, the root node of the containing constituent is a ("global") root in the technical sense of being undominated.
The definition of "sp" is given in (i):
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(i) a
sp(p) = {p} b sp(a) = \(\bigcup\)x∈a sp(x)
This crucially presupposes the existence of "urelements" (Barwise 1975:1.1), a rather "natural" assumption in the context of syntactic structures "projected from" lexical formatives, "roots," or whatever else constitutes "terminals." Variable p ranges over urelements, a over sets, and x over the union of the sets of urelements and sets (Barwise 1975:10).
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(i) a
People who like concrete examples could imagine that a is Artemis and that (1b) describes the current organizational structure of the scientific board for the preservation of Keynesianism in Greece (SBPKG). Imagine further that the statutes of the SBPKG stipulate that it consist of exactly one leader and a two person subcommittee, the advisory committee for the SBPKG, i.e., ACSBPKG. Then (1b) says that currently Artemis is leader of the SBPKG and happens, at the same time, to be member of the ACSBPKG, together with Basilius. At a meeting of the SBPKG, Artemis will be present only once. Noone would expect (or allow) any copy or double of Artemis to show up. Whether she has one or two votes is another matter.
Guimarães (2000: section 4) relies on this fundamental set-theoretic property in his analysis of "Self-Merge," where, applied to example (1), Merge(γ1,γ1) = { a }.
The definition by Troelstra (1992:2) reads: "A multiset over A is a mapping f: A → ℕ, where f(a) = n means that a occurs with multiplicity n. If f(a) = 0, a is not an element of f (that is to say, a occurs with multiplicity 0)." Thus, assuming A = { a, b, c }, the multiset in (4b) is given by: f(4b) = { 〈a,2〉, 〈b,1〉, 〈c,0〉 }.
CGO's discussion of phases, Spell-Out, Transfer, and (in)accessibility of constituents (pp. 240–242) is relevant for determining their perspective here.
"Consider a syntactic object SO = { S1, { S1, S2 } }. We say that S1 occurs twice in SO. The position of an occurrence is given by a "path" from SO to the particular occurrence. A path is a sequence of syntactic objects 〈SO1, SO2, …, SOn〉 where for every adjacent pair SOi, SOi+1 of objects in the path, SOi+1 ∈ SOi (i.e., SOi+1 is immediately contained in SOi)." Collins and Stabler (l.c.) also discuss open questions about and shortcomings of alternative definitions.
As, for example, the system of propositional logic is usually taken to form part of the system of predicate logic.
The syntax of set theory (as relevant here) can be built from individual constants "a", "b", "c" etc., the two-place predicate constant "∈", and the "syncategorematic" expressions "{" (left bracket), "}" (right bracket), and "," (comma). And its (extensional) semantics (as relevant here) can be given in terms of a model ⟨D,I〉 with D the universe of things set theory is about and I an interpretation function such that, in particular, I(a), I(b), I(c) etc. are members of D, and I(∈)⊆D×D. Given that I is a function, any use of the name "a" refers to the same thing, I(a). This is another way of saying that "standard set theory" does not support interpretation of (1b) ({ a, { a, b } }) as involving two "copies" in the sense of two entities. To insist on the latter amounts to an appeal to a particular syntax of set theory with its subtle "excess of notation over subject matter" (Quine, 1995:5).
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Acknowledgements
For reactions to an earlier draft I am grateful to David Adger, Norbert Hornstein, and Thomas Ede Zimmermann. Likewise, I am grateful to the anonymous reviewer. Remaining shortcomings are my own responsibility.
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Gärtner, HM. Copies from "Standard Set Theory"? A Note on the Foundations of Minimalist Syntax in Reaction to Chomsky, Gallego and Ott (2019). J of Log Lang and Inf 31, 129–135 (2022). https://doi.org/10.1007/s10849-021-09342-x
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DOI: https://doi.org/10.1007/s10849-021-09342-x