A (Leibnizian) Theory of Concepts

objects are simply different in kind from ordinary objects: the latter are not the kind of thing that could encode properties; the former are not the kind of thing that could be concrete. Moreover, abstract objects necessarily fail to exemplify the properties of ordinary objects— they necessarily fail to have a shape, they necessarily fail to have a texture, they necessarily fail to reflect light, etc. Consequently, by the classical laws of complex properties, abstract objects necessarily exemplify the negations of these properties. But the properties abstract objects encode are more important than the properties they necessarily exemplify, since the former are the ones by which we individuate them. The six principles listed above are cast within the framework of a classical modal (S5 with Barcan formulas) second-order predicate logic. Moreover, this logic is extended by the logical axiom for encoding and the axioms that govern the two kinds of complex terms: (rigid) definite descriptions of the form ıxφ and λ-predicates of the form [λy1 . . . yn φ]. The logical axiom for encoding is: Logic of Encoding : xF → xF Intuitively, this says that if an object encodes a property at any possible world, it encodes that property at every world; thus facts about encoded 76In this principle, F̄ is still defined as [λy ¬Fy]. This λ-expression is governed by the usual principle (see below in the text): [λy ¬Fy]x ≡ ¬Fx Note that encoding satisfies classical bivalence: ∀F∀x(xF ∨ ¬xF ). But the incompleteness of abstract objects is captured by the fact that the following is not in general true: xF ∨ xF̄ . 77By including the Barcan formulas, this quantified modal logic is the simplest one available—it may interpreted in such a way that the quantifiers ∀x and ∀F range over fixed domains, respectively. See Linsky and Zalta [1994], in which it is shown that this simplest quantified modal logic, with the first and second order Barcan formulas, is consistent with actualism. 41 A (Leibnizian) Theory of Concepts properties are not relativized to any circumstance. This axiom and the definition of identity for abstract objects jointly ensure that the properties encoded by an abstract object are in some sense intrinsic to it. We conclude this summary of the system by describing the axioms governing the complex terms and some simple consequences of the foregoing. A standard free logic governs the definite descriptions, along with a single axiom that captures the Russellian analysis: an atomic or (defined) identity formula ψ containing ıxφ is true iff there is something y such that: (a) y satisfies φ, (b) anything satisfying φ is identical to y, and (c) y satisfies ψ. In formal terms, this becomes: Descriptions : ψ y ≡ ∃x(φ & ∀z(φx → z=x) & ψ y ), for any atomic or identity formula ψ(y) in which y is free. To keep the system simple, these definite descriptions are construed as rigid designators, and so this axiom is a classic example of a logical truth that is contingent . Thus, the Rule of Necessitation may not be applied to any line of a proof that depends on this axiom. The final element of the logic concerns complex relations. They are denoted in the system by terms of the form [λy1 . . . ynφ] meeting the condition that φ have no encoding subformulas. These λ-predicates behave classically: λ-Equivalence: [λy1 . . . yn φ]x1 . . . xn ≡ φ1n y1,...,yn In less formal terms: objects x1, . . . , xn exemplify the relation [λy1 . . . ynφ] iff x1, . . . , xn satisfy φ. A comprehension schema for relations is a simple consequence of λ-Equivalence. In what follows, it proves useful to appeal to some simple consequences of our system. Let us define ‘there is a unique x such that φ’ (‘∃!xφ’) in the usual way: ∃!xφ =df ∃x∀y(φx ≡ y=x) 78Further examples and discussion of such logical truths that are not necessary may be found in Zalta [1988b]. 79I should mention here that this logic of relations is supplemented by precise identity principles that permit necessarily equivalent properties, relations, and propositions to be distinct. These principles are expressed in terms of a basic definition of property identity: F =G ≡ ∀x(xF ≡ xG). For a more detailed explanation of this principle and the definitions of relation identity and proposition identity, the reader may consult the cited works on the theory of abstract objects. Edward N. Zalta 42 Then we may appeal to Principles 3 and 5 to prove the following more exact comprehension principle for A-objects: Principle 3 ′: ∃!x(A!x & ∀F (xF ≡ φ)), where φ has no free xs The proof of this principle is simply this: by Principle 3 , we know there is an A-object that encodes just the properties satisfying φ; but there couldn’t be two distinct A-objects that encode exactly the properties satisfying φ, since distinct A-objects have to differ by at least one encoded property. Consequently, for each formula φ that can be used to produce an instance of Principle 3 ′, the following is true: Corollary to Principle 3 ′: ∃y y= ıx(A!x& ∀F (xF ≡ φ)) We are therefore assured that the following description is always welldefined: ıx(A!x& ∀F (xF ≡ φ)) Such descriptions will be used frequently in what follows to define various Leibnizian concepts. Indeed, they are governed by a simple theorem that plays a role in the proof of most of the theorems which follow: A-Descriptions : ıx(A!x & ∀F (xF ≡ φ))G ≡ φF In other words, the A-object that encodes just the properties satisfying φ encodes property G iff G satisfies φ. This theorem is easily derivable from the Descriptions axiom described above. Appendix II: Proofs of Selected Theorems The proofs in what follows often appeal to the Principles described in Appendix I and to the theorems of world theory discussed in Appendix III. • Proof of Theorem 4: We prove the concepts in question encode the same properties. (←) Assume cG⊕cHP . We need to show: ıx∀F (xF ≡ G⇒F ∨ H⇒F )P So by A-Descriptions , we must show: G⇒P ∨ H⇒P . By hypothesis, we know: 43 A (Leibnizian) Theory of Concepts ıx∀F (xF ≡ cGF ∨ cHF )P But by A-Descriptions , it follows that cGP ∨ cHP . Now, for disjunctive syllogism, suppose cGP . Then by definition of of cG, it follows that: ıx∀F (xF ≡ G⇒F )P So by A-Descriptions , we know G⇒P . And by similar reasoning, if cHP , then H⇒P . So by disjunctive syllogism, it follows that G⇒P ∨ H⇒P , which is what we had to show. (→) Assume: ıx∀F (xF ≡ G⇒F ∨ H⇒F )P We want to show: cG⊕cHP . By the definition of real sum, we have to show: ıx∀F (xF ≡ cGF ∨ cHF )P By A-Descriptions , we therefore have to show that cGP ∨ cHP . By applying A-Descriptions to our hypothesis, though, we know: G⇒P ∨ H⇒P So, for disjunctive syllogism, suppose G⇒P . Then, by A-Descriptions: ıx∀F (xF ≡ G⇒F )P That is, by definition of the concept G, we know: cGP . By similar reasoning, if H⇒P , then cHP . So by our disjunctive syllogism, it follows that cGP ∨ cHP , which is what we had to show. • Proof of Theorem 7: (→) Assume (x⊕y)⊕zP . Then, by definition of ⊕, we have: ıw∀F (wF ≡ x⊕yF ∨ zF )P This, by A-Descriptions , entails: x⊕yP ∨ zP Expanding the left disjunct by the definition of ⊕, we have: ıw∀F (wF ≡ xF ∨ yF )P ∨ zP And reducing the left disjunct by applying A-Descriptions , we have: Edward N. Zalta 44