Full Idea
If the number of cats happens to equal the cherries, then it follows from the theory of models that there is a reinterpretation of the entire language that leaves all sentences unchanged in truth value while permuting the extensions of 'cat' and 'cherry'.
Gist of Idea
If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth
Source
Hilary Putnam (Reason, Truth and History [1981], Ch.2)
Book Reference
Putnam,Hilary: 'Reason, Truth and History' [CUP 1998], p.44
A Reaction
This horrifying result seems to come simply from the fact that there is an isomorphism between two models, which in turn seems to rest largely on the cardinality of the models. There seems to be something wrong with model theory here (?).
Related Ideas
Idea 14205 The sentence 'A cat is on a mat' remains always true when 'cat' means cherry and 'mat' means tree [Putnam]
Idea 14206 There are infinitely many interpretations of a sentence which can all seem to be 'correct' [Putnam]
Idea 14212 A consistent theory just needs one model; isomorphic versions will do too, and large domains provide those [Lewis]