XII 92
Definition Reduction Sentence/Carnap/Quine: weaker than definition: provides no equivalent sentences without the term in question, but only implications.
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No full explanation but only partial explanation.
Implication here: the reduction sentences name a few sentences that are implied by sentences with this term and imply some other sentences, that imply sentences with this term. - This does not provide a genuine reduction, but a fictional story of language acquisition. ((s) > "
Rylean Ancestors").
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VII (a) 19
Conceptual Scheme/Reduction/Quine: we want to see how far a physicalist scheme can be reduced to a phenomenalist one.
The latter has epistemological priority.
The choice between conceptual schemes is guided by purposes and interests.
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XI 143
Reduction/Ontology/Quine/Lauener: for ontological reduction, it is not extensional equality that is decisive, but the preservation of the relevant structure.
For example Frege's, v. Neumann's and Zermelo's definitions do not produce equivalent predicates, but are nevertheless suitable for reduction, because all three represent a structure-preserving model of arithmetic.
Extensional Equality(s): ensures the uniformity of the quantities considered. The reduction then takes place at the description level. It would not reduce the ontology.
XI 146
Reduction/Theory/Quine/Lauener: by the condition that an n-tuple of arguments applies to a predicate exactly when the open sentence is fulfilled by the corresponding n-tuple of values, we avert an impending trivialization. We can do this by determining the proxy function.
If the truth values of the closed sentences are preserved, we can actually speak of a reduction to the natural numbers. (Ways of Paradox, p. 203).
XI 145
Def Proxy Function/Quine/Lauener: is a function that assigns each object of the original theory a function of the new theory.
Example "The Goedel number of".
This need not be expressed in one theory or another. It is sufficient if we have the necessary means of expression at the meta level.
Reduction: from one theory to another: so we need a special function for this
XI 146
whose arguments are from the old theory and whose values are from the new theory.
Proxy Function/Quine/Lauener: does not need to be unique at all. Example: Characterization of persons on the basis of their income: here different values are assigned to an argument. For this we need a background theory:
We map the universe U in V in such a way that both the objects of U and their proxies are contained in V. If V forms a subset of U, U itself can be defined as
background theory, within which its own ontological reduction is described.
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VsQuine: this is not a reduction at all, because then the objects must exist.
QuineVsVs: this is comparable to a reductio ad absurdum: if we want to show that a part of U is superfluous, we may presuppose this for the duration of argument U (>
Ontology).
Lauener: that brings us to >
ontological relativity.
Löwenheim/Ontology/Reduction/Quine/Lauener: if a theory of its own requires a super-countable range, we can no longer present a proxy function that would allow a reduction to a countable range.
This would require a much stronger framework theory, which could no longer be discussed away absurdly as reductio ad absurdum according to Quine's proposal.
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XII 60
Specification/Reduction/Quine: we cannot find a clear difference between specifying one item area and reducing that area to another.
We have not discovered a clear difference between the clarification of the concept of "expression" and its replacement by that of number. ((s) >
Goedel Numbers).
And now, if we are to say what numbers actually are, we are forced to reveal them and instead assign a new, e.g. set-theoretical model to arithmetic.
XII 73
Reduction/Ontology/Quine: an ontology can always be reduced to another if we know of a reversibly unique deputy function f.
Reason: for each predicate P of the old system, there is a predicate of the new system that takes over the role of P there. We interpret this new predicate in such a way that it applies exactly to the values f(x) of the old objects x to which P applied.
Example: Suppose f(x): is the Goedel number of x,
Old system: is a syntactical system,
Predicate in the old system: "... is a section of___" an x
New system: the corresponding predicate would have the same extension (coextensive) as the words "...is the Gödel number of a section whose Goedel number is___". (Not in this wording but as a purely arithmetic condition.)
XII 74
Reduction/ontological relativity/Quine: it may sound contradictory that the objects discarded in the reduction must exist.
Solution: this has the same form as a reduction ad absurdum: here we assume a wrong sentence to refute it. As we show here, the subject area U is excessively large.
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Löwenheim/Skolem/strong form/selection axiom/ontology/reduction/onthological relativity/Quine: (early form): thesis: If a theory is true and has a supernumerable range of objects, then everything but a countable part is superfluous, in the sense that it can be eliminated from the range of variables without any sentence becoming false.
This means that all acceptable theories can be reduced to countable ontologies. And this in turn can be reduced to a special ontology of natural numbers. For this purpose, the enumeration, as far as it is explicitly known, is used as a proxy function. And even if the enumeration is not known, it exists. Therefore, we can regard all our items as natural numbers, even if the enumeration number ((s) of the name) is not always known.
Ontology: could we not define once and for all a Pythagorean general purpose ontology?
Pythagorean Ontology/Terminology/Quine: consists either of numbers only, or of bodies only, or of quantities only, etc.
Problem: suppose, we have such an ontology and someone would offer us something that would have been presented as an ontological reduction before our decision for Pythagorean ontology, namely a procedure according to which in future theories all things of a certain type A are superfluous, but the remaining range would still be infinite.
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In the new Pythagorean framework, his discovery would nevertheless still retain its essential content, although it could no longer be called a reduction, it would only be a manoeuvre in which some numbers would lose a number property corresponding to A. We do not even know which numbers would lose a number property corresponding to A.
VsPythagoreism: this shows that an all-encompassing Pythagoreanism is not attractive, because it only offers new and opaque versions of old methods and problems. >
Proxy function.
