Abstract
There are two ideas underlying this talk of mine. The first one is concerned with the interpretation of a class of first order formulas (called SH-formulas, since they are a class of HORN formulas in SKOLEM open form) in a category satisfying a very weak requirement, namely the sole existence of finite products. This idea was developed some years ago and later I will devote some of my time to discuss it. Of course, it is fashionable to interpret every formula (even of higher order and multi-sorted) in a topos, but to-poi have a very rich structure and satisfy many more properties than just having finite products. I might mention that C. MARCHINI started working on the comparison between the two definitions of truth and will soon publish a paper on the subject (see [1, 2]).
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REFERENCES
COSTE, Logique du 1er ordre dans lès topos élé-mentaires.(Mimeographed notes).
C.MARCHINI, Funtori che conservano e riflettono le SH, Atti Sem.Mat.e Pis.Univ.Modena 22 (1973).
C. MAROHINI, Alcune questioni di semantica categoriale (to appear).
M.SERVT, Una questione di teoria dei modelli nelle categorie con prodotti finiti,Matematiche(Catania)26.
M.SERVT Su alcuni funtori che conservano le SH, Riv.Mat.Univ.Parma(3)3(1974).
M.SERVT A generalization of the exponential functor in connection with the SH-formulas(to appear).
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Servi, M. (2010). SH-Formulas and Generalized Exponential. In: Mangani, P. (eds) Model Theory and Applications. C.I.M.E. Summer Schools, vol 69. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11121-1_3
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