A Construction of Heyting Algebra on Categorical Syllogisms

Senturk, Ibrahim

doi:10.37560/matbil1170005s

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“…Recently, the topic is studied extensively and investigated under different treatments. For instance, Stanley Burris examined traditional syllogistic logic by using Boolean Algebras [3], Senturk and Oner constructed Heyting Algebras on categorical syllogisms [18], and Esko Turunen represented Peterson Intermediate Syllogisms by means of MV-Algebras [20]. In addition to these, we recently witness the use of syllogisms in different areas such as computer science [13,16], engineering [11,15], artificial intelligence [10,21], etc.…”

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confidence: 99%

An algebraic analysis of categorical syllogisms by using Carroll’s diagrams

Senturk

Oner

2019

Filomat

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In this paper, we analyze the algebraic properties of categorical syllogisms by constructing a logical calculus system called Syllogistic Logic with Carroll Diagrams (SLCD). We prove that any categorical syllogism is valid if and only if it is provable in this system. For this purpose, we explain firstly the quantitative relation between two terms by means of bilateral diagrams and we clarify premises via bilateral diagrams. Afterwards, we input the data taken from bilateral diagrams, on the trilateral diagram. With the help of the elimination method, we obtain a conclusion that is transformed from trilateral diagram to bilateral diagram. Subsequently, we study a syllogistic conclusion mapping which gives us a conclusion obtained from premises. Finally, we allege valid forms of syllogisms using algebraic methods, and we examine their algebraic properties, and also by using syllogisms, we construct algebraic structures, such as lattices, Boolean algebras, Boolean rings, and many-valued algebras (MV-algebras).

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