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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 Ref
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]
| 14207 | If cats equal cherries, model theory allows reinterpretation of the whole language preserving truth [Putnam] |
| 14212 | A consistent theory just needs one model; isomorphic versions will do too, and large domains provide those [Lewis] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 13636 | An axiomatization is 'categorical' if its models are isomorphic, so there is really only one interpretation [Shapiro] |
| 13670 | Categoricity can't be reached in a first-order language [Shapiro] |
| 10214 | Theory ontology is never complete, but is only determined 'up to isomorphism' [Shapiro] |
| 10238 | The set-theoretical hierarchy contains as many isomorphism types as possible [Shapiro] |
| 10105 | Differences between isomorphic structures seem unimportant [George/Velleman] |
| 13537 | An 'isomorphism' is a bijection that preserves all structural components [Wolf,RS] |
| 10884 | A theory is 'categorical' if it has just one model up to isomorphism [Horsten] |
| 10758 | If models of a mathematical theory are all isomorphic, it is 'categorical', with essentially one model [Rossberg] |