<abstract xmlns="http://www.w3.org/1999/xhtml"><p> In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued propositional logic as a propositional logic language with a logical matrix over n truth-values. The algebra of the logical matrix has operations expanding the operations of the classical propositional logic. Therefore we look over the Łukasiewicz, Post, Heyting and Rosser style expansions of the operations negation, conjunction, disjunction and with a special emphasis on implication.</p><p>In the frame of consequence operation, some notions of semantic consequence are examined. Then we continue with the decision problem and the logical calculi. We show that the cause of difficulties with the notions of semantic consequence is the weakness of the reviewed expansions of negation and implication. Finally, we introduce an approach to finding implications that preserve both the modus ponens and the deduction theorem with respect to our definitions of consequence.</p></abstract>

In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued propositional logic as a propositional logic language with a logical matrix over n truth-values. The algebra of the logical matrix has operations expanding the operations of the classical propositional logic. Therefore we look over the Łukasiewicz, Post, Heyting and Rosser style expansions of the operations negation, conjunction, disjunction and with a special emphasis on implication.In the frame of consequence operation, some notions of semantic consequence are examined. Then we continue with the decision problem and the logical calculi. We show that the cause of difficulties with the notions of semantic consequence is the weakness of the reviewed expansions of negation and implication. Finally, we introduce an approach to finding implications that preserve both the modus ponens and the deduction theorem with respect to our definitions of consequence.

Many-valued logics—implications and semantic consequences

Acta Universitatis Sapientiae, Informatica

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

{"article-title":"Many-valued logics\u2014implications and semantic consequences"}

In this paper an application of the well-known matrix method to an extension of the classical logic to many-valued logic is discussed: we consider an n-valued...

Many-valued logics—implications and semantic consequences

Published Online: May 30, 2014

Page range: 145 - 166

Received: Jun 05, 2013

DOI: https://doi.org/10.2478/ausi-2014-0008

Keywords
many-valued logic, extensions of implication, notions of consequence

© 2014

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Many-valued logics—implications and semantic consequences

Published Online: May 30, 2014

Page range: 145 - 166

Received: Jun 05, 2013

DOI: https://doi.org/10.2478/ausi-2014-0008

Keywordsmany-valued logic, extensions of implication, notions of consequence

© 2014

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Keywords
many-valued logic, extensions of implication, notions of consequence