Author: tomr Format: TextI prepared this question in Math-Stackexchange:
http://math.stackexchange.com/questions/1992156/how-to-express-predicate-logic-in-the-categorical-monoidal-logics
Maybe someone knows the asnwer?
The question is:
Rosetta stone in the book "New Structures in Physics" (http://www.springer.com/la/book/9783642128202) is the correspondence between the propositional (linear) logic from the one side and the monoidal categories from the other side. Each proposition can be represented as the object, each deduction (proof) among two literals can be represented as the morphism between repsective objects and each connective can be attributed to the tensor product among objects in category. That works for propositional logic.
The similar correspondence can be extended to the modal logics and modal operators, e.g. in paper http://www.cs.nott.ac.uk/~psznza/papers/Alechina++:01a.pdf
The question is: can this correspondence between logic (deductive system) and the category theory be extended to the predicate logic? There is article where it has been done (it is said so) http://amcm.pcz.pl/get.php?article=2015_1/art_03.pdf This extension is done in chapter 4 of this article. The problem is - I don't understand this chapter from the very basic things. The authors say that predicate can be expresses by the subset of the object - but there is no such notion in the category theory. Maybe someone can explain chapter 4 or give alternative correspondence between predicate logic and category theory?
This correspondence is very exciting subject because it can lead to the universal logic and universal reasoning system - very much welcome in the applied AI!
I prepared this question in Math-Stackexchange: http://math.stackexchange.com/questions/1992156/how-to-express-predicate-logic-in-the-categorical-monoidal-logics Maybe someone knows the asnwer?
The question is: Rosetta stone in the book "New Structures in Physics" (http://www.springer.com/la/book/9783642128202) is the correspondence between the propositional (linear) logic from the one side and the monoidal categories from the other side. Each proposition can be represented as the object, each deduction (proof) among two literals can be represented as the morphism between repsective objects and each connective can be attributed to the tensor product among objects in category. That works for propositional logic.
The similar correspondence can be extended to the modal logics and modal operators, e.g. in paper http://www.cs.nott.ac.uk/~psznza/papers/Alechina++:01a.pdf
The question is: can this correspondence between logic (deductive system) and the category theory be extended to the predicate logic? There is article where it has been done (it is said so) http://amcm.pcz.pl/get.php?article=2015_1/art_03.pdf This extension is done in chapter 4 of this article. The problem is - I don't understand this chapter from the very basic things. The authors say that predicate can be expresses by the subset of the object - but there is no such notion in the category theory. Maybe someone can explain chapter 4 or give alternative correspondence between predicate logic and category theory?
This correspondence is very exciting subject because it can lead to the universal logic and universal reasoning system - very much welcome in the applied AI!