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• W1564305366 abstract "objects are those, that are not possibly concrete; and ordinary objects are those that are possibly concrete. The notion of an ordinary object allows Zalta in other projects to propose a theory of merely possible and also of fictional objects.9 This, however, will be of no concern here. Abstract objects enter OT via a comprehension schema for abstract objects (OC): (OC) ∃x(A!x ∧ ∀F (xF ≡ φ)), where ‘x’ is not free in φ This axiom schema asserts that for any formula φ (minding the restriction on free variables), there exists an abstract object that encodes all and only those properties F that satisfy φ; or, expressed in a more sloppy way, for any collection of properties, there is an abstract object encoding them. OC guarantees that any (abstract) object that is described by an expression of the form ‘ix(A!x ∧ ∀F (xF ≡ φ))’ exists (where there is no free ‘x’ in φ). So, there is, for example, an abstract object that encodes the property of being Zalta (or being identical to Zalta):10 ix(A!x ∧ ∀F (xF ≡ ∀y(Fy ≡ y = Zalta))) OT was originally developed as a formal theory of fictional, abstract, and intensional objects inspired by the work of Meinong’s student Ernst Mally: see (Zalta, 1983). All of the following examples are, of course, dependent on the English names and predicates entering the formal language in some way. How this is done for mathematical terms is described below. Moreover, identity is a defined notion in OT. So, strictly speaking, one would have to specify that the identity relation referred to in our examples is identity between concrete, rather than abstract, objects. What is the Purpose of Neo-Logicism? 41 An abstract object that encodes being either Linsky or Zalta: ix(A!x ∧ ∀F (xF ≡ ∀y(Fy ≡ (y = Linsky ∨ y = Zalta)))) An abstract object that encodes all the properties Zalta has: ix(A!x ∧ ∀F (xF ≡ F (Zalta))) Note that Zalta himself is not identical to any of these objects (since he is concrete and not abstract). He exemplifies, rather than encodes the respective properties. Sherlock Holmes, on the other hand, is an abstract object, viz. the abstract object that encodes all the properties that (the fictional character) Sherlock Holmes has according to the stories by Arthur Conan Doyle. (The devise for fomalising this will be introduced below in the discussion of mathematical theories.) There is also an abstract object that encodes being a square circle: ix(A!x ∧ ∀F (xF ≡ ∀y(Fy ≡ (y is a circle ∧ y is square)))) Moreover, there is an abstract object that encodes being a set that contains all and only those sets that do not contain themselves. In order to avoid inconsistency, the second-order comprehension schema for predicates:11 ∃X∀x(Xx ≡ φ(x)), where X is not free in φ For simplicity’s sake we only give the comprehension schema for monadic second-order variables. The restrictions apply in the same way for the general formulation for polyadic variables. We here use the common formulation of second-order logic introduced in (Church, 1956); the current bible of second-order logic is (Shapiro, 1991). Linsky and Zalta use an equivalent formulation that employs λ-conversion, which requires an analogous restriction. 42 Philip A. Ebert & Marcus Rossberg (and likewise the thirdand higher-order comprehension schemata) has to be restricted. It has to be demanded of the standardly unrestricted second-order comprehension schema that φ does not contain any descriptions or “encoding subformulae”. So, the fully explicit formulation of φ must not contain subformulae of the form pxY q, i.e. subformulae containing the encoding mode of predication.12 Identity between abstracta, ‘=A’, is a defined relation. Two abstract objects are identical if, and only if, they necessarily encode the same properties: x =A y =df A!x ∧A!y ∧2∀F (xF ≡ yF ) With this criterion for identity at hand, we can see that the abstract object introduced above which encodes being Zalta is distinct from the object encoding all of Zalta’s properties: the latter encodes using a Mac while the former does not. So much for the formal background. Linsky and Zalta now suggest that mathematical theories can be identified as those abstract objects, that encode all the mathematical propositions that are true according to them.13 This needs some unpacking. First, encoding was introduced as a mode of predication, i.e. a second-level relation that holds between an object and a property. In order for mathematical theories to be able to encode propositions, they are handled as zero-place properties. Any proposition p thus gives rise to a property being such that p; using the notation of λ-conversion, this can be expressed as: ‘[λy p]’. One might complain that object comprehension, OC, is suspect on the grounds that with the introduction of OC the well established second-order comprehension schema, considered logical by many, needs to be restricted to avoid inconsistency. We will not follow this criticism here. See (Linsky and Zalta, 1995), pp. 538–539, and (Linsky and Zalta, 2006), pp. 89–90. What is the Purpose of Neo-Logicism? 43 Being true according to a mathematical theory t can then be characterised using the resources of OT: it is simply defined as t encoding that particular truth:" @default.
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• W1564305366 title "What is the Purpose of Neo-Logicism ?" @default.
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