VII (g) 131
Def Ideology/Quine: what ideas can be expressed in a theory? I.e. what is definable?
Example: the theory of real numbers has infinite ontology, but finite ideology: addition, division, multiplication, rationality, algebraicity, etc.
Two theories can have the same ontology and different ideologies. Example:
(1) The real number x is an integer
That can be expressed in one theory, but not in another!
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Ideology, >
Ontology, >
Definitions.
VII (g) 132
Due to Goedel's incompleteness theorem for the integers, we know that Tarski's performance would have been impossible if (1) could have been translated into the notation of T.
Ontology/Theory/Quine: the ontology of a theory can even include objects that are indefinable in this theory.
For example it can be shown that the theory T includes the whole real numbers, although (1) cannot be expressed in its notation.
Ideas/Quine: "Ideas of ideas" we can drop them completely together with "ideology" (expressability). We are more interested in definability (in a theory).
VII (g) 132
Definition definable/Quine: a general term t can be defined, in any part of the language that includes a sentence S such that S contains the variable "x" and is met by all and only the values of "x", of which t is true. - E.g. "whole" is not definable in theory T.
XI 76
Analyticity/Synonymy/Necessity/Quine/Lauener: these terms can only be defined among themselves. We have nothing to break out of this circle with.
XI 122
Diversity/Distinguishability/Definability/Mark WilsonVsQuine/VsDifferentiality/Lauener:
LauenerVsWilson: Quine mistakenly assumes that two theories R and RT are different iff their union is logically incompatible.
Wilson: this is unsatisfactory, because T and RT can be considered formalizations of the same theory, and yet they are not logically equivalent, because their languages are interpreted differently.
Interdefinability/Theory/Wilson: two theories are interdefinable if each can be defined within the other, otherwise they are different. For example, one theory with mathematical vocabulary, the other with physical vocabulary. In addition, no superfluous properties may be introduced.
Quine: ditto. In addition, the application of a theory should not be confused with the theory itself.
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Theories.
