Axiomatization and Algebraic Semantics of the Logic of Ordinary Discourse

Core Concepts

The Logic of Ordinary Discourse (OL) is a three-valued propositional logic that challenges classical logic by rejecting some theses and accepting non-classically valid principles. The authors present modular and analytic Hilbert-style multiple-conclusion and single-conclusion axiomatizations for OL and its structural companion sOL, and investigate their algebraic semantics.

Abstract

The content discusses the Logic of Ordinary Discourse (OL), a three-valued propositional logic proposed by W.S. Cooper as a more adequate formalization of everyday reasoning in natural language. The key highlights are:

OL is a notable exception among non-classical logics, as it is not subclassical - it accepts non-classically valid principles such as Aristotle's and Boethius' theses, while rejecting some classical tautologies.

The authors present modular and analytic Hilbert-style multiple-conclusion and single-conclusion axiomatizations for OL and its structural companion sOL. The calculi are obtained via the methods developed in Shoesmith and Smiley [1978] and Caleiro and Marcelino [2019].

The authors prove that sOL is algebraizable, with the quasi-variety OL as its equivalent semantics. OL turns out to be a discriminator variety, making sOL a nearly functionally complete logic.

It is shown that sOL is definitionally equivalent to an expansion of Da Costa and D'Ottaviano's three-valued logic J3, which is an axiomatic extension of paraconsistent Nelson logic.

The authors discuss potential future developments, including a more extensive study of the algebraic counterpart of sOL and the extension of the present approach to other logics definable from the truth tables of OL.

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Axiomatizing the Logic of Ordinary Discourse

by Vito... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2405.03543.pdf

Deeper Inquiries

What are the potential applications of the Logic of Ordinary Discourse (OL) beyond the formalization of everyday reasoning in natural language

The Logic of Ordinary Discourse (OL) has potential applications beyond formalizing everyday reasoning in natural language. One significant application is in the field of artificial intelligence and natural language processing. OL can be utilized to enhance the understanding of natural language statements and improve the reasoning capabilities of AI systems. By incorporating the principles of OL, AI algorithms can better interpret conditional statements with false antecedents, leading to more accurate and contextually relevant responses. This can be particularly useful in chatbots, virtual assistants, and automated reasoning systems.
Furthermore, OL can also find applications in cognitive science and psychology. By modeling the logic of ordinary discourse, researchers can gain insights into how humans process and reason with everyday language. Understanding the underlying logic of human discourse can help in developing cognitive models and theories that better reflect human thought processes. Additionally, OL can be applied in linguistics research to analyze the structure and semantics of natural language expressions, leading to advancements in computational linguistics and language understanding.
In the realm of philosophy, OL can contribute to the study of logic and language. Philosophers can use OL to investigate the validity of arguments, analyze the structure of reasoning in ordinary discourse, and explore the implications of non-classical logic systems. By applying OL in philosophical inquiries, scholars can deepen their understanding of logical principles and their applications in various philosophical contexts.

How do the algebraic properties of OL and sOL, such as being a discriminator variety, relate to their logical features and behavior

The algebraic properties of OL and its structural companion sOL, such as being a discriminator variety, are closely related to their logical features and behavior. The fact that sOL is algebraizable and forms a discriminator variety indicates that it exhibits specific algebraic structures and properties that align with its logical characteristics.
The discriminator variety status of sOL implies that it can discriminate between different elements based on their logical properties. This feature reflects the non-classical nature of sOL, where it can handle contradictions and paraconsistent reasoning in a unique way. The algebraic properties of sOL provide a formal framework for understanding its logical behavior, allowing for the analysis of connectives, truth values, and inference rules within the system.
Furthermore, the algebraic properties of OL and sOL contribute to their expressiveness and computational complexity. By studying the algebraic structures of these logics, researchers can gain insights into their computational properties, decidability, and expressivity. The algebraic semantics of OL and sOL offer a mathematical foundation for analyzing their logical systems, enabling a deeper understanding of their formal properties and relationships.

What insights can be gained by exploring the connections between OL/sOL and other non-classical logics, such as paraconsistent Nelson logic and Da Costa and D'Ottaviano's logic J3

Exploring the connections between OL/sOL and other non-classical logics, such as paraconsistent Nelson logic and Da Costa and D'Ottaviano's logic J3, can provide valuable insights into the relationships and distinctions between these systems. By comparing and contrasting the features of these logics, researchers can uncover commonalities, differences, and potential applications in various domains.
The connection between OL/sOL and paraconsistent Nelson logic highlights the paraconsistent nature of these systems, where contradictions are allowed without leading to triviality. Understanding how OL/sOL relates to paraconsistent logics can shed light on the principles of non-contradiction and consistency in reasoning, offering new perspectives on handling contradictory information in logical systems.
Similarly, exploring the relationship between OL/sOL and Da Costa and D'Ottaviano's logic J3 can reveal the connections between connexive logics and three-valued logics. By identifying the definitional equivalences and extensions between these systems, researchers can uncover the underlying principles that govern their logical structures and semantics. This comparative analysis can lead to a deeper understanding of the expressive power and computational properties of these non-classical logics, paving the way for further research and applications in diverse fields such as AI, linguistics, and philosophy.