Regular Logic and Regular Categories
Regular logic is the fragment of first-order logic that can express statements of the form
where and are regular formulae i.e. formulae built up from atomic formulae, the truth constant, binary meets and existential quantification. Such formulae can be interpreted in a regular category, and the interpretation is a model of a sequent
if the interpretation of factors through the interpretation of . This gives for each theory (set of sequences) and for each regular category C a category Mod(T,C) of models of T in C. This construction gives a functor Mod(T,-):RegCat→Cat from the category RegCat of small regular categories and regular functors to small categories. It is an important result that for each theory T and for each category C, there is a category R(T) and an equivalence
which is natural in C. Up to equivalence any small regular category C arises this way as the classifying category, of a regular theory.
Read more about this topic: Regular Category
Famous quotes containing the words regular, logic and/or categories:
“A regular council was held with the Indians, who had come in on their ponies, and speeches were made on both sides through an interpreter, quite in the described mode,the Indians, as usual, having the advantage in point of truth and earnestness, and therefore of eloquence. The most prominent chief was named Little Crow. They were quite dissatisfied with the white mans treatment of them, and probably have reason to be so.”
—Henry David Thoreau (18171862)
“The logic of worldly success rests on a fallacy: the strange error that our perfection depends on the thoughts and opinions and applause of other men! A weird life it is, indeed, to be living always in somebody elses imagination, as if that were the only place in which one could at last become real!”
—Thomas Merton (19151968)
“Kitsch ... is one of the major categories of the modern object. Knick-knacks, rustic odds-and-ends, souvenirs, lampshades, and African masks: the kitsch-object is collectively this whole plethora of trashy, sham or faked objects, this whole museum of junk which proliferates everywhere.... Kitsch is the equivalent to the cliché in discourse.”
—Jean Baudrillard (b. 1929)