Refutation Systems: An Overview and Some Applications to Philosophical Logics

“…In our use of words, Inoué [11] observed that the idea of Stahl is nothing but the Stahlization of classical propositional logic in Gentzen-style. His observation is based on the inversion principle after P. Lorenzen with respect to negation in Gentzen's sequent calculus LK 10 . He could extend Stahl's logic to the first-order predicate level on the basis of the Stahlization (see [11] ).…”

Section: Take the Inversion Of What You Knowmentioning

confidence: 99%

“…For axiomatic rejection, see for example Inoué [12] and Inoué, Ishimoto and Kobayashi [14]. Also refer to Carnielli and Pulcini [6], and Goranko, Pulcini and Skura [10].…”

Section: Take the Inversion Of What You Knowmentioning

confidence: 99%

How to Make a New Logic

Inoué

2021

Preprint

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We discuss about how to make a new logic with considering a slogan: take the inversion of what you know. Suppose that we have a Gentzen-style logical system S. The operation to make a new logic from S is the following: for all the axioms and rules of S, change the direction of all the arrows occurred in the sequents of the axiom or the rule to the opposite side. We call this operation Stahlization. We consider certain logics in this respect.

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“…A substantial theory concerning the logic of rejected propositions is presented in [42]. Recently the achievements in the field were recapitulated in [43] and the Łukasiewicz-Słupecki approach to the issue was discussed in [44].…”

Section: (26b)mentioning

confidence: 99%

Aristotle’s Syllogistic as a Deductive System

Kulicki

2020

Axioms

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Aristotle’s syllogistic is the first ever deductive system. After centuries, Aristotle’s ideas are still interesting for logicians who develop Aristotle’s work and draw inspiration from his results and even more from his methods. In the paper we discuss the essential elements of the Aristotelian system of syllogistic and Łukasiewicz’s reconstruction of it based on the tools of modern formal logic. We pay special attention to the notion of completeness of a deductive system as discussed by both authors. We describe in detail how completeness can be defined and proved with the use of an axiomatic refutation system. Finally, we apply this methodology to different axiomatizations of syllogistic presented by Łukasiewicz, Lemmon and Shepherdson.

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Refutation Systems: An Overview and Some Applications to Philosophical Logics

Cited by 22 publications

References 75 publications

Hybrid Deduction-Refutation Systems for FDE-Based Logics

Hybrid Deduction-Refutation Systems for FDE-Based Logics

How to Make a New Logic

Aristotle’s Syllogistic as a Deductive System

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