Sentence examples for conservativeness criterion from inspiring English sources

Now, the Conservativeness criterion can be made precise as follows.

\(\mathbf{H}\) meets the Conservativeness criterion, but not that of Eliminability.

Conservativeness criterion (syntactic formulation): Any formula of \(L\) that is provable in \(L^\) is provable in \(L\).

(So, ostensive definitions can fail to meet the Eliminability criterion explained below; they can fail to meet also the Conservativeness criterion, also explained below).

It is a plausible requirement on any answer to these questions that two criteria be respected.[4] First, a stipulative definition should not enable us to establish essentially new claims call this the Conservativeness criterion.

The criteria of Conservativeness and Eliminability can now be made precise thus: Conservativeness criterion (semantic formulation): For all formulas \(A\) of \(L\) and all interpretations \(M\), if \(A\) is valid in \(L^\) in \(M\) then \(A\) is also valid in \(L\) in \(M\).

The Conservativeness and Eliminability criteria are, it appears, satisfied.

The following two definitions are also not in normal form: But both should count as legitimate under the traditional account, since they meet the Conservativeness and Eliminability criteria.

(If \(T^*\) is infinite then a stipulation of the above form will be needed for each sentence \ \psi\) in \(T^*\).)[11] The definition is legitimate, according to the traditional account, so long as it meets the Conservativeness and Eliminability criteria.

Observe that the satisfaction of Conservativeness and Eliminability criteria, whether in their semantic or their syntactic formulation, is not an absolute property of a definition; the satisfaction is relative to the ground language.

But there is bound to be a violation of one or both of the two criteria, Conservativeness and Eliminability.[13] A final example: We know by a theorem of Tarski that no theory can be an admissible definition of the truth predicate, \(Tr\), for the language of Peano Arithmetic considered above.