On Friday, January 18, 2019 at 7:30:14 AM UTC-6, Bruno Marchal wrote: > > > On 18 Jan 2019, at 09:49, Philip Thrift <cloud...@gmail.com <javascript:>> > wrote: > > > *The semantic view of theories and higher-order languages* > Laurenz Hudetz > https://link.springer.com/article/10.1007/s11229-017-1502-0 > > "every family of set-theoretic structures has an associated language of > higher-order logic and an up to signature isomorphism unique > model-theoretic counterpart" > > > *Several philosophers of science construe models of scientific theories as > set-theoretic structures. Some of them moreover claim that models should > not be construed as structures in the sense of model theory because the > latter are language-dependent. I argue that if we are ready to construe > models as set-theoretic structures (strict semantic view), we could equally > well construe them as model-theoretic structures of higher-order logic > (liberal semantic view). I show that every family of set-theoretic > structures has an associated language of higher-order logic and an up to > signature isomorphism unique model-theoretic counterpart, which is able to > serve the same purposes. This allows to carry over every syntactic > criterion of equivalence for theories in the sense of the liberal semantic > view to theories in the sense of the strict semantic view. Taken together, > these results suggest that the recent dispute about the semantic view and > its relation to the syntactic view can be resolved.* > > > It cannot do that in the Mechanist Frame, where we cannot assume, neither > a physical universe, nor analysis or set theory. Since recently, I have > realised that we cannot even assume the induction axioms, but we can use it > in the definition of the Löbian entities, whose existence is then a > consequence of the theories without induction. Of course, we need induction > at the meta level, and even the whole of informal mathematics, like in any > science pointing toward some reality independent of us. > > Bruno > > > > The study of the dances between languages and their semantics is what both philosophy and science are all about - from the view at least of 'language-oriented' philosophers.
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