Proof of Ł-decidability of Lewis system S5

Słupecki, Jerzy; Bryll, Grzegorz

doi:10.1007/bf02123824

Cited by 8 publications

(9 citation statements)

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“…Slupecki, Bryll, Wybraniec-Skardowska and others from Slupecki's school developed a general theory, in Tarski's style, of rejected propositions and investigated in detail the properties of the corresponding consequence relation (see [17] , [18]) . Also, Slupecki and Bryll [16] constructed a complete refutation system for the propositional modal logic S5, and Bryll and Maduch [2] proposed refutation axioms for Lukasiewicz's many-valued logics.…”

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confidence: 99%

Refutation systems in modal logic

Goranko¹

1994

Stud Logica

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Complete deductive systems are constructed for the non-valid (refutable) formulae and sequents of some propositional modal logics. Thus, complete syntactic characterizations in the sense of Lukasiewicz are established for these logics and, in particular, purely syntactic decision procedures for them are obtained. The paper also contains some historical remarks and a general discussion on refutation systems. Introduction: historical remarksFormal logic traditionally deals with deductive systems for inference of statements valid according to certain semantics, and is not involved in the inference of the non-valid ones. As Lukasiewicz points out in [9], out of the two intellectual acts -acceptance and rejection of a statement, the latter one has been neglected in the modern formal logic. This neglect seems strange, moreover because the father of the formal logic, Aristotle, already noticed the importance of the systematic rejection of non-valid arguments. He realized that, in order to show that a syllogism was not universally valid, it was not necessary to construct a "refuting model", i.e. an example where the syllogism produces an obviously false conclusion from true premises. Instead, it was enough to infer that syllogism from others, the validity or non-validity of which had already been established, applying certain rules of inference. The typical rule used by Aristotle for that purpose was the so called "modus tollens": If A implies B, and B is rejected, then A is rejected too. Thus, he established a sort of deductive system for rejection of non-valid syllogisms. We shall call deductive systems which infer refutable statements instead of valid ones refutation systems. The statements which are inferred, Le. provably refutable in such a system will be called rejected statements.The history of refutation systems, unlike the history of the "orthodox" ones, according to the author's knowledge, is rather short and scanty. Lukasiewicz in [9], raising the general problem of the formal deduction of the nonvalid statements of a given theory, suggested a complete refutation system for the non-valid classical propositions. The system is a very natural one: the only axiom is "p is rejected" where p is a fixed propositional variable,

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confidence: 99%

Refutation systems in modal logic

Goranko¹

1994

Stud Logica

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“…and the set 3 T − S of all rejected propositions of S with respect to the set T + S and the basis (B S ), on the analogy to D(Sł), can be defined as follows: [43], Słupecki and Bryll [44], Wybraniec-Skardowska [63,64]).…”

Section: Three Different Notions Of Rejectionmentioning

confidence: 99%

“…Afterwards, Słupecki initiated research on Ł-decidability of Lewis system S5. In his and Bryll's paper [44], the proof of Ł-decidability was achieved with an assumption of one rejected axiom (the prepositional variable p) and, apart from Łukasiewicz's rules, a class of rejection rules of the common scheme.…”

Section: Modal Logicmentioning

confidence: 99%

Rejection in Łukasiewicz’s and Słupecki’s Sense

Wybraniec-Skardowska

2018

Studies in Universal Logic

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The idea of rejection originated by Aristotle. The notion of rejection was introduced into formal logic by Łukasiewicz [20]. He applied it to complete syntactic characterization of deductive systems using an axiomatic method of rejection of propositions [22, 23]. The paper gives not only genesis, but also development and generalization of the notion of rejection. It also emphasizes the methodological approach to biaspectual axiomatic method of characterization of deductive systems as acceptance (asserted) systems and rejection (refutation) systems, introduced by Łukasiewicz and developed by his student Słupecki, the pioneers of the method, which becomes relevant in modern approaches to logic.

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“…Perhaps the easiest route to a generalization is to draw on the work of Słupecki and Bryll [15]. They pursue the old idea of the Polish school that a logic should have in addition to its axioms and rules of the ordinary kind, its axioms and rules of acceptance, indicating that certain formulas are acceptable as general laws, some axioms and rules of an opposite kind, axioms and rules of rejection, indicating that certain formulas are unacceptable as general laws.…”

Section: Against Mckmentioning

confidence: 99%

Which Modal Logic Is the Right One?

Burgess¹

1999

Notre Dame J. Formal Logic

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The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Słupecki and Bryll, is presented for the claim that the right demonstrability logic must be contained in S5, and a more speculative argument for the claim that it does not include S4.2 is also presented.

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Proof of Ł-decidability of Lewis system S5

Cited by 8 publications

References 2 publications

Refutation systems in modal logic

Refutation systems in modal logic

Rejection in Łukasiewicz’s and Słupecki’s Sense

Which Modal Logic Is the Right One?

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