Logical Studies

Non-classical Comparative Logic I: Standard Categorical Logic – from SLe to IFLe

Document Type : Research

Authors

Amer Amikhteh 1
Seyyed Ahmad Mirsanei 2

1 Tarbiat Modares University

2 Master of Modern Logic (Classical, extended and non-classical logic), Traditional logic, Comparative Logic, Philosophy and Critical Thinking at Mofid university, Sadra University and Payame Noor (Qom and Tehran)

https://doi.org/10.30465/lsj.2021.35927.1349
Abstract
In this paper, a non-classical axiomatic system was introduced to classify all moods of Aristotelian syllogisms, in addition to the axiom "Every a is an a" and the bilateral rules of obversion of E and O propositions. This system consists of only 2 definitions, 2 axioms, 1 rule of a premise, and moods of Barbara and Datisi. By adding first-degree propositional negation to this system, we prove that the square of opposition holds without using many of the other rules of classical logic (including double negation elimination).
We then show that the Propositional Substructural Logic SLe is the best logic to study Aristotelian Syllogisms. Also, based on the IFLe square of opposition, the rules of conversation and the rules of negation are completely proved in Muzaffar's logic. For this purpose, we used the monadic first-order logic with the same standard deductive apparatus of quantifiers in classical logic, plus the axioms of "some a is an a" and "some not-a is a not-a". Finally, to show that there is no existential commitment to general terms in categorical logic, the Strong Four-Valued Relevant-classical Logic KR4 was used. With the same existential interpretation of the quantifiers and the standard translation of the quarter quantified.

Keywords

Categorical logic
Aristotelian syllogism
Non-classical logic
Substructural logic
Axiomatic method
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Logical Studies
Volume 12, Issue 1
April 2021
Pages 1-24

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