Free Logics are Cut-Free
TL;DR: In this paper, a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics are presented.
Abstract: The paper presents a uniform proof-theoretic treatment of several kinds of free logic, including the logics of existence and definedness applied in constructive mathematics and computer science, and called here quasi-free logics. All free and quasi-free logics considered are formalised in the framework of sequent calculus, the latter for the first time. It is shown that in all cases remarkable simplifications of the starting systems are possible due to the special rule dealing with identity and existence predicate. Cut elimination is proved in a constructive way for sequent calculi adequate for all logics under consideration.
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TL;DR: In this paper, the authors present rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’.
Abstract: This paper presents rules in sequent calculus for a binary quantifier I to formalise definite descriptions: Ix[F, G] means ‘The F is G’. The rules are suitable to be added to a system of positive free logic. The paper extends the proof of a cut elimination theorem for this system by Indrzejczak by proving the cases for the rules of I. There are also brief comparisons of the present approach to the more common one that formalises definite descriptions with a term forming operator. In the final section rules for I for negative free and classical logic are also mentioned.
3 citations
06 Sep 2021
TL;DR: In this article, the authors provide a tableau approach to definite descriptions, based on several formalizations of minimal free description theory (MFD) usually formulated axiomatically in the setting of free logic.
Abstract: The paper provides a tableau approach to definite descriptions. We focus on several formalizations of the so-called minimal free description theory (MFD) usually formulated axiomatically in the setting of free logic. We consider five analytic tableau systems corresponding to different kinds of free logic, including the logic of definedness applied in computer science and constructive mathematics for dealing with partial functions (here called negative quasi-free logic). The tableau systems formalise MFD based on PFL (positive free logic), NFL (negative free logic), PQFL and NQFL (the quasi-free counterparts of the former ones). Also the logic \(\textsf {NQFL}^{-}\) is taken into account, which is equivalent to NQFL, but whose language does not comprise the existence predicate. It is shown that all tableaux are sound and complete with respect to the semantics of these logics.
1 citations
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TL;DR: In this paper, the authors provide a tableau approach to definite descriptions, based on several formalizations of minimal free description theory (MFD) usually formulated axiomatically in the setting of free logic.
Abstract: The paper provides a tableau approach to definite descriptions. We focus on several formalizations of the so-called minimal free description theory (MFD) usually formulated axiomatically in the setting of free logic. We consider five analytic tableau systems corresponding to different kinds of free logic, including the logic of definedness applied in computer science and constructive mathematics for dealing with partial functions (here called negative quasi-free logic). The tableau systems formalise MFD based on PFL (positive free logic), NFL (negative free logic), PQFL and NQFL (the quasi-free counterparts of the former ones). Also the logic NQFLm is taken into account, which is equivalent to NQFL, but whose language does not comprise the existence predicate. It is shown that all tableaux are sound and complete with respect to the semantics of these logics.
1 citations
06 Sep 2021
TL;DR: In this paper, a dual domain semantics for a theory of definite descriptions with a binary quantifier is presented, where the quantifier I forms a formula from two formulas, i.e. Ix[F, G] means ‘The F is G’.
Abstract: This paper presents a sequent calculus and a dual domain semantics for a theory of definite descriptions in which these expressions are formalised in the context of complete sentences by a binary quantifier I. I forms a formula from two formulas. Ix[F, G] means ‘The F is G’. This approach has the advantage of incorporating scope distinctions directly into the notation. Cut elimination is proved for a system of classical positive free logic with I and it is shown to be sound and complete for the semantics. The system has a number of novel features and is briefly compared to the usual approach of formalising ‘the F’ by a term forming operator. It does not coincide with Hintikka’s and Lambert’s preferred theories, but the divergence is well-motivated and attractive.
1 citations
01 Jan 2022
TL;DR: In this article , a characterization of elementary ontology as a sequent calculus satisfying desiderata usually formulated for rules in well-behaved systems in modern structural proof theory is presented.
Abstract: Abstract The ontology of Leśniewski is commonly regarded as the most comprehensive calculus of names and the theoretical basis of mereology. However, ontology was not examined by means of proof-theoretic methods so far. In the paper we provide a characterization of elementary ontology as a sequent calculus satisfying desiderata usually formulated for rules in well-behaved systems in modern structural proof theory. In particular, the cut elimination theorem is proved and the version of subformula property holds for the cut-free version.
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Book•
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TL;DR: N- systems and H-systems and Gentzen systems, proof theory of arithmetic, second-order logic, and solutions to selected exercises.
Abstract: This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of formalization of first-order logic. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic (intuitionistic as well as classical); the theory of logic programming; category theory; modal logic; linear logic; first-order arithmetic and second-order logic. In each case the aim is to illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. There are numerous exercises throughout the text. In general, the only prerequisite is a standard course in first-order logic, making the book ideal for graduate students and beginning researchers in mathematical logic, theoretical computer science and artificial intelligence. For the new edition, many sections have been rewritten to improve clarity, new sections have been added on cut elimination, and solutions to selected exercises have been included.
808 citations
Book•
01 Jan 2001
TL;DR: In this paper, a historical perspective of classical logic and the material conditional is presented, along with a set of relevant logics, including Intuitionist logic, Basic relevant logic, Mainstream relevant logic and Fuzzy logic.
Abstract: Introduction 1. Classical logic and the material conditional 2. Basic modal logic 3. Normal modal logics 4. Non-normal worlds strict conditionals 5. Conditional logics 6. Intuitionist logic 7. Many-valued logics 8. First degree entailment 9. Basic relevant logic 10. Mainstream relevant logics 11. Fuzzy logic 12. Conclusion: a historical perspective.
368 citations
Book•
01 Jan 2001TL;DR: In this paper, a proof editor for sequent calculus is presented, based on a simple type theory and a categorical grammar, and a proof analysis of axiomatic theories.
Abstract: Introduction 1. From natural deduction to sequent calculus 2. Sequent calculus for institutionistic logic 3. Sequent calculus for classical logic 4. The quantifiers 5. Variants of sequent calculi 6. Structural proof analysis of axiomatic theories 7. Intermediate logical systems 8. Back to natural deduction Conclusion: diversity and unity in structural proof theory Appendix A. Simple type theory and categorical grammar Appendix B. Proof theory and constructive type theory Appendix C. A proof editor for sequent calculus.
349 citations