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.This is because the logic of predicate quantifiers in constructive conceptualismis like the logic of first-order quantifiers in free logic in that the logic is freeof existential presuppositions regarding predicate constants, which means thata predicate constant must stand for a  predicative concept in order to be asubstituend of the bound predicate variables of the logic.The predicate quanti-fiers in nominalism, on the other hand, function like the objectual quantifiers ofstandard first-order logic; and that is because, as the paradigms of predicationin nominalism, predicate constants do not differ from one another in their pred-icative role, which is why, under a substitutional interpretation, all predicateconstants are substituends of the bound predicate variables.Consider, for example, a language L containing  " as a primitive two-placepredicate constant, and suppose we formulate a theory of membership in L withthe following as a second-order axiom:("F)("y)("x)[x " y ”! F(x)].(C)Now, in nominalism, where predicate quantifiers are interpreted substitutionally,this axiom seems quite plausible as a thesis, stipulating in effect that everypredicate expression has an extension.But as plausible a thesis as that mightbe, it leads directly to Russell s paradox.For, by the nominalist comprehensionprinciple, (CP!),("F)("x)[F(x) ”! x " x], (D)/is provable under such an interpretation; and, because no predicate quantifieroccurs in x " x, then, by (UI!2), x " x represents a predicate expression that/ /can be properly substituted for F in a universal instantiation of (C).In constructive conceptualism, on the other hand, (D) is not an instanceof the conceptualist principle, (CCP!), and all that follows by Russell s argu-ment from (C) is the fact that  " cannot stand for a  predicative (relational)concept.That is, instead of the contradiction that results when predicate quan-tifiers are interpreted substitutionally, (C), when taken as an axiom of a theoryof membership in constructive conceptualism, leads only to the result that themembership predicate does not stand for a  predicative (relational) concept:¬("R)("x)("y)[R(x, y) ”! x " y].In other words, as a plausible thesis to the effect that every  predicative concepthas an extension, (C) is consistent, not inconsistent, in constructive conceptu-alism.90 CHAPTER 4.FORMAL THEORIES OF PREDICATIONOn the nominalist strategy, the notion of a  predicative context is purelygrammatical in terms of logical syntax; that is, an open formula is  predica-tive in nominalism just in case it contains no bound predicate variables.Inconstructive conceptualism, the notion of a  predicative context is semanti-cal, which means that in addition to being  predicative in nominalism s purelygrammatical sense, it must also stand for a  predicative concept.It is for thisreason that the second-order logic of constructive conceptualism must be free ofexistential presuppositions regarding predicate constants, which is why the bi-nary predicate,  " , in a theory of membership having (C) as an axiom, cannotstand for a value of the bound predicate variables.In general, how we determine which, if any, of the primitive predicate con-stants of an applied language and theory stand for a  predicative conceptdepends on the domain of discourse of that language and theory and how thatdomain is to be conceptually represented.In particular, those primitive predi-cates that are to be taken as standing for a  predicative concept will be stipu-lated as doing so in terms of the  meaning postulates of that theory, whereasthose that are not will usually occur in axioms that determine that fact.Identity and its role in a logical theory marks another important differencebetween nominalism and constructive conceptualism.In nominalism, identity isdefinable in any applied language with finitely many predicate constants.Thisis because such a definition can be given in terms of a formula representingindiscernibility with respect to those predicate constants.Suppose, for exampleL is a language with two just two predicate constants, a one-place predicateconstant P, and a two-place predicate constant R.Then, identity can be definedin theories formulated in terms of L as follows:a = b ”! [P(a) ”! P(b)] '" [R(a, a) ”! R(b, a)] '" [R(a, b) ”! R(b,b)]'"[R(a, a) ”! R(a, b)] '" [R(b,a) ”! R(b, b)]In other words, in any given application based on finitely many predicateconstants, which we may assume to be the standard situation, identity, in nom-inalism, is reducible to a first-order formula, which is why the identity sign isallowed to occur in instances of (CP!) under its nominalistic, substitutionalinterpretation.Such a definition will not suffice in constructive conceptualism,on the other hand, because the first-order formula in question, even were it tostand for a  predicative concept, cannot justify the substitutivity of identicalsin nonpredicative contexts.The identity sign is not eliminable, or otherwise re-ducible, in constructive conceptualism, in other words, because, on the basis ofLeibniz s law, identity must allow for full substitutivity even in nonpredicativecontexts.Thus, whereas,x = y ”! ("F)[F(x) ”! F(y)],is provable in nominalism s second-order logic, as based on its substitutionalinterpretation, the right-to-left direction of this same formula is not provable inthe logic of constructive conceptualism.Finally, note that although  impredicative definitions are not allowed innominalism, they are not precluded in the logic of constructive conceptualism.4.4.RAMIFICATION AND HOLISTIC CONCEPTUALISM 91The difference is determined by the role free predicate variables have in eachof these frameworks.In nominalism, free predicate variables must be construedas dummy schema letters, which in an applied language stand for arbitraryfirst-order formulas of that theory.This means that the substitution rule,if È, then È[Õ/G(x1, [ Pobierz caÅ‚ość w formacie PDF ]