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Additional info for Commutative algebra [Lecture notes]
Clearly, we do not want to attribute such a triviality to Plato. 4 Yet it is difficult to conceive of it as an interesting metaphysical truth from within the Russellian framework. In object theory, however, we may think of Forms as just a special kind of A-object. When (OMP) is translated into our language, it turns out to be an interesting theorem. x&(F)(xF == F = G). 42 CHAPTER II So a Form of G is any abstract object which encodes just G. So we have: THEOREM 1. (G)(3x)Form(x,G). Proof. By A-OBJECTS.
So we postpone further investigation until the modal theory has been developed. 3. THE PROBLEM OF EXISTENCE The property of existence has puzzled philosophers for years. The assertion that some particular thing fails to exemplify existence (or being) strangely carries with it a commitment to the existence (or being) of the very thing which serves as the subject of the assertion. This is partly a result of trying to keep the theory of language as simple as possible - we try to account for the truth of a simple sentence by supposing that the objects denoted by the object terms are in an extension of the relation denoted by the relation term.
J' function, maps the simple names of the language to elements of the appropriate domain. @. J'(Kn)E ~n' Since "E! @. We ELEMENTARY OBJECT THEORY 23 call this subset of ~ the set of existing objects ("1&""). )))) the set of abstract objects ("d"). B. ff' which assigns to each primitive variable an element of the domain over which the variable ranges. ff on the primitive variables, and (3) assigns denotations to the complex terms on the basis of the denotations of their parts and the way in which they are arranged.