Abstract
We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.
Highlights
One of the tasks of formal logic is to provide adequate tools for the formal analysis of certain fragments of natural language, as well as for the languages of particular fields of science
We focus on deduction systems for the logic SCI
The paper consists of five sections: in Section 2, we present the basics of the non-Fregean propositional logic SCI, that is, its language, semantics, and axiomatization
Summary
One of the tasks of formal logic is to provide adequate tools for the formal analysis of certain fragments of natural language, as well as for the languages of particular fields of science. It is commonly accepted that the theory of interpretation of a language is semantics. The choice of semantics determines how we think about a given language and what meaning we assign to its components. It is often acknowledged that the first precisely formulated semantic principles—that serves as a foundation for contemporary formal logic and have determined its development—were presented by Frege in his Begriffsschrift. According to Frege, a correct and adequate formal system of a given language should meet the following conditions: All names and all sentences have meaning and denotation. Meaning is not the same as denotation
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