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.e., as avalue of the bound objectual variables, is important to note because it has beenclaimed that the objective view of plural objects, i.e., the view of them asobjects (such as classes as many), is refuted by Russell s paradox.3 The factthat the Russell class does not exist in the logic of classes as many is statedin the following theorem.(Proofs will be given only as footnotes.)T10: ¬("x)(x = Rus).43See, e.g., Schein 1993, pages 5, 15, and 32-37.4Proof.By axiom 13 and identity logic, ("A)(Rus = A), and by definition 1,Rus " Rus ”! ("A)[Rus = A '" ("xA)(x = Rus)], and therefore by Leibniz s law, a quantifier-confinement law and tautologous transformations, Rus " Rus ”! ("xRus)(x = Rus).Butthen, by definition of Rus and T8, ("xRus)(x = Rus) ”! ("x/("A)(x = A '" x " A))(x =/Rus), and therefore, by T1, ("xRus)(x = Rus) ”! ("x)[("A)(x = A '" x " A) '" x = Rus],/from which, by Leibniz s law, it follows that ("xRus)(x = Rus) ”! ("x)[Rus " Rus '"/x = Rus]; and, accordingly, by quantifier-confinement laws, and tautologous transformations,("x)(x = Rus) ’! (Rus " Rus ”! Rus " Rus), from which we conclude that ¬("x)(x =/Rus). 11.1.THE LOGIC OF CLASSES AS MANY 239What Russell s argument shows is that not every name concept has an ex-tension that can be object -ified in the sense of being the value of a boundobjectual variable.Now the question arises as to whether or not we can specify a necessaryand sufficient condition for when a name concept has an extension that canbe object -ified, i.e., for when the extension of the concept can be provento exist as the value of a bound objectual variable.In fact, the answer isaffirmative.In other words, unlike the situation in set theory, such a conditioncan be specified for the notion of a class as many.An important part of thiscondition is Nelson Goodman s notion of an atom, which, although it wasintended for a strictly nominalistic framework, we can utilize for our purposesand define as follows.5Definition 6 Atom =[x/¬("y)(y ‚" x)].ÆThis notion of an atom has nothing to do with physical atoms, of course.Rather, it corresponds in our present system approximately to the notion of anurelement, or individual, in set theory.We say approximately because inour system atoms are identical with their singletons, and hence each atom will bea member of itself.This means that not only are ordinary physical objects atomsin this sense, but so are the propositions and intensional objects denoted bynominalized sentences and predicates in the fuller system of conceptual realism.Of course, the original meaning of atom in ancient Greek philosophy was that ofbeing indivisible, which is exactly what was meant by individual in medievalLatin.An atom, or individual, in other words, is a single object, which isapropos in that objects in our ontology are either single or plural.We willhenceforth use atom and individual in just this sense.The following axiom (where y does not occur in A) specifies when and onlywhen a name concept A has an extension that can be object -ified (as a valueof the bound objectual variables).Axiom 15: ("y)(y =[xA]) ”! ("xA)(x = x) '" ("xA)("zAtom)(x = z).ÆStated informally, axiom 15 says that the extension of a name concept A canbe object -ified (as a value of the bound objectual variables) if, and only if,something is an A and every A is an atom.6 An immediate consequence of thisaxiom, and of T8 and T1, is the following theorem schema, which stipulatesexactly when an arbitrary condition Õx has an extension that can be object -ified.T11: ("y)(y =[x/Õx]) ”! ("x)Õx '" ("x/Õx)("zAtom)(x = z).ÆNote that where Õx is the impossible condition (x = x), it follows from T11that there can be no empty class, which, as already noted, is our first basic5See Goodman 1956 for Goodman s account of atoms in nominalism.6That something is an A is perspicuously symbolized by ("y)("xA)(y = x).But because("xA)(x = x) ”! ("y)("xA)(y = x) is provable, we will use ("xA)(x = x) as a shorter way ofsaying the same thing. 240 CHAPTER 11.PLURALS AND THE LOGIC OF CLASSES AS MANYfeature of the notion of a class as many.We define the empty-class concept asfollows and then note that its extension, by T11, cannot exist (as a value ofthe bound objectual variables), as well as that no object can belong to it.Definition 7 ›=[x/(x = x)].ÆT12a: ¬("x)(x =›).T12b: ¬("x)(x " ›).Finally, our last axiom concerns the second basic feature of classes as many;namely, that every atom, or individual, is identical with its singleton.In termsof a name concept A, the axiom stipulates that if at most one thing is an Aand that whatever is an A is an atom, then whatever is an A is identical tothe extension of A, which in that case is a singleton if in fact anything is an A.Where y does not occur in A, the axiom is as follows.Axiom 16: ("xA)("yA)(x = y) '" ("xA)("zAtom)(x = z) ’!("yA)(y =[xA]).ÆA more explicit statement of the thesis that an atom is identical with itssingleton is given in the following theorem.T13: ("zAtom)(x = z) ’! x =[w/(y = x)].7By T13, it follows that every atom is identical with the extension of somename concept, e.g., the concept of being that atom.Of course, non-atoms, i.e.,plural objects, are the extensions of name concepts as well (by the definitionsof Atom, ‚", and "), and hence anything whatsoever is the extension of a nameconcept.T14: ("zAtom)(x = z) ’! ("A)(x = A).T15: ¬("zAtom)(x = z) ’! ("A)(x = A).T16: ("A)(x = A).Note that if A is a proper name of an ordinary, physical object (and hencean atom), then, by the meaning postulate for proper names, the antecedent ofaxiom 16 is true, and therefore, by axioms 16 and 14, ("yA)(y = A).Inother words, if A is a proper name of an atom, then F(A) ”! ("yA)F(y) is true,which in our conceptualist framework explains the role proper names have as singular terms (i.e., as substituends of free objectual variables) in free logic
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