Abstract
This paper develops a metaphysically flexible theory of quantification broad enough to incorporate many distinct theories of objects. Quite different, mutually incompatible conceptions of the nature of objects and of reference find representation within it. Some conceptions yield classical first-order logic; some yield weaker logics. Yet others yield notions of validity that are proper extensions of classical logic.
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Notes
Constructivists generally count an existential sentence true if an object satisfying the corresponding open sentence has been constructed—or if an algorithm for constructing it has been specified. That suggests distinct interpretations of the stage-by-stage constructions we model in this paper; we might classify objects for the construction of which an algorithm has been specified at a given stage as available at that stage or as available only at some future stage.
Some approaches to real number theory yield a hierarchy of stages that fits neither the minervan nor the marsupial conception. The entities that make up the finite stages of this hierarchy are not the real numbers themselves. Each such segment has a transitional status; at the next stage it is replaced by a couple of smaller segments, which it yields by division and which classify the potential reals more finely. The real numbers themselves are not entities at any finite stage, but have the status of limits that can be approximated by sequences of segments that delimit them ever more closely. Constructions of this kind, which produce at each stage classifications of the targeted entities that are subsequently replaced by finer classifications, we call amoebic. The reals are not the only ontological category for which an ‘amoebic’ conception makes sense. Another candidate is a constructivist version of the bundle theory of objects, for example, which might construe objects as infinite limits of finite bundles of properties. Vague boundaries might be understood as infinite limits of precisifications. Quantities might be understood as infinite limits of measurements. In all such cases, we would have to extend the theory we present here to incorporate transfinite limits among its methods of construction.
Our languages are without predicates of -arity 0; that is, they are without propositional constants. This restriction has been imposed merely for convenience: excluding propositional constants simplifies both the formulation of a number of our theorems and often also their proofs. Not surprisingly, our results can be generalized to languages with propositional constants, albeit at the cost of a certain amount of extra work.
The substitution of a term t for all free occurrences of a variable x in a formula φ will be denoted either as (φ)t/x or as [φ]t/x, depending on which notation seems more perspicuous in any given context.
For an example let L be the language {R,S}, where R and S are 2-place predicates, and let \(\mathfrak {M}\) be a model for L with universe A = {a 1,a 2,b 1,b 2} in which \([[R]]_{\mathfrak {M}}\) = {<a 1,b 1>} and \([[S]]_{\mathfrak {M}}\) = {<a 2,b 2>}. In \(\mathfrak {M}\) the sentence ∃x(∃y R x y∧∃z S x z) is clearly false. But in \(M^{**}(\mathfrak {M})\) this sentence is true. For let B 0, B 1, B 2 be the sets {a 1}, {a 1,b 1}, {a 2,b 2}, and let f 0, f 1, f 2 be the functions with domains B 0, B 1, B 2, respectively, and defined by: f 0(a 1) = c 1; f 1(a 1) = c 1, f 1(b 1) = c 2; f 2(a 2) = c 1, f 2(b 2) = c 2. Then the pairs (B 0,f 0), (B 1,f 1) and (B 2,f 2) determine the diagrams D 0, D 1, D 2 of \(M^{**}(\mathfrak {M})\) given by: (i) D i (R c 1 c 1) = D i (S c 1 c 1) = 0 for i = 0, 1, 2; (ii) D 1(R c 1 c 2) = 1; D 1(R c 2 c 1) = D 1(R c 2 c 2) = D 1(S c 1 c 2) = D 1(S c 2 c 2) = D 1(S c 2 c 1) = 0; (iii) D 2(R c 1 c 2) = 1; D 2(R c 2 c 1) = D 2(R c 2 c 2) = D 2(S c 1 c 2) = D 2(S c 2 c 2) = D 2(S c 2 c 1) = 0. Since D 0⊆D 1 and D 0⊆D 2, the sentence ∃y R c 1 y∧∃z R c 1 z is true in \(M^{**}(\mathfrak {M})\) at diagram D 0. So, the sentence ∃x(∃y R x y∧∃z R x z) is true in \(M^{**}(\mathfrak {M})\) at its empty core. This example also confirms the informal observation we made earlier that the demonstrative conception of reference poorly fits the minervan conception of objects if demonstrative designators for these objects are used in a substitution-based definition of truth.
Not all parametric models reflecting both the proper-name and minervan conceptions are strong nets. The process of stage development may be nondeterministic; the emergence of new objects of one kind—that is, satisfying one set of predicates—may prevent the emergence of individuals of some other kind. Thus, among the diagrams of a parametric model there may be some including objects of the first kind and some including objects of the second kind, but none including objects of both kinds.
Note that the simplicity of the axioms that can be used to axiomatize T h(O) does not by itself entail the decidability of T h(O), since the only sets of such simple axioms for T h(O) might not themselves be decidable. We return to this point below when proving T h(O)’s decidability. We thank an anonymous reviewer for suggesting that we bring forward to its present location the point that the class of ∀∃-formulas of predicate logic should not be confused with the decidability of ∀∃ theories.
Note that the condition ‘ c ′≠c’ means that c ′ and c are distinct symbols. It does not mean that they have distinct denotations.
In fact this is a theorem of ordinary predicate logic. Note that the formula is meaningful, since there are only finitely many extensions D i that are diagrams for the language \(L \bigcup \{c_1,...,c_m\}\). So \(\bigvee _i \bigwedge D_i(\mathbf {x}, y)\) is a finite disjunction.
Note that because of the properties of M the choice of constants here is immaterial.
This switch is strictly one of convenience; we could, at some slight cost, have persisted with languages based on the logical vocabulary consisting of just ¬,→ and ∀.
Note that this differs from the usual definition of tableau closure. Here, what corresponds to a model is not a branch but an entire tableau structure set.
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Acknowledgments
We are grateful to Nicholas Asher and to anonymous referees for their comments on drafts of this paper. We would also like to thank Johan van Benthem, who first suggested to us the affinity between our approach to quantification and modal logic.
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Bonevac, D., Kamp, H. Quantifiers Defined by Parametric Extensions. J Philos Logic 46, 169–213 (2017). https://doi.org/10.1007/s10992-016-9398-6
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DOI: https://doi.org/10.1007/s10992-016-9398-6