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The topic of this paper in a broad phrase is “proof theory". It tries to theorize the general
notion of “proving" something using rigorous definitions, inspired by previous less general
theories. The purpose for being this general is to eventually establish a rigorous framework
that can bridge the gap when interrelating different logical systems, particularly ones
that have not been as well defined rigorously, such as sequent calculus. Even as far as
semantics go on more formally defined logic such as classic propositional logic, concepts
like “completeness" and “soundness" between the “semantic" and the “deductive system"
is too arbitrarily defined on the specific system that is applied to for it to carry as an
adequate definition. What we shall do then is come up with an adequate definition for
a characterization of every logic that one has worked with, and show what can be done
with it for a few basic logical systems that include classic propositional logic, intuitionistic
propositional logic and intuitionistic sequent calculus. To make this definition work with
eloquence, we go the category theory route of constructing a category with objects that
correspond to collections of logical formulae and arrows that correspond to deductions
from one such collection to another.
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This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.
Goodlad, Alex Gabriel, "Constructing a Categorical Framework of Metamathematical Comparison Between Deductive Systems of Logic" (2016). Senior Projects Spring 2016. 137.