A Study of Subminimal Logics of Negation and Their Modal Companions
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Classifying Material Implications Over Minimal Logic
Classifying Material Implications over Minimal Logic Hannes Diener and Maarten McKubre-Jordens March 28, 2018 Abstract The so-called paradoxes of material implication have motivated the development of many non- classical logics over the years [2–5, 11]. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. Keywords: reverse mathematics; minimal logic; ex falso quodlibet; implication; paraconsistent logic; Peirce’s principle. 1 Introduction The project of constructive reverse mathematics [6] has given rise to a wide literature where various the- orems of mathematics and principles of logic have been classified over intuitionistic logic. What is less well-known is that the subtle difference that arises when the principle of explosion, ex falso quodlibet, is dropped from intuitionistic logic (thus giving (Johansson’s) minimal logic) enables the distinction of many more principles. The focus of the present paper are a range of principles known collectively (but not exhaustively) as the paradoxes of material implication; paradoxes because they illustrate that the usual interpretation of formal statements of the form “. → . .” as informal statements of the form “if. then. ” produces counter-intuitive results. Some of these principles were hinted at in [9]. Here we present a carefully worked-out chart, classifying a number of such principles over minimal logic. -
Mathematical Logic. Introduction. by Vilnis Detlovs And
1 Version released: May 24, 2017 Introduction to Mathematical Logic Hyper-textbook for students by Vilnis Detlovs, Dr. math., and Karlis Podnieks, Dr. math. University of Latvia This work is licensed under a Creative Commons License and is copyrighted © 2000-2017 by us, Vilnis Detlovs and Karlis Podnieks. Sections 1, 2, 3 of this book represent an extended translation of the corresponding chapters of the book: V. Detlovs, Elements of Mathematical Logic, Riga, University of Latvia, 1964, 252 pp. (in Latvian). With kind permission of Dr. Detlovs. Vilnis Detlovs. Memorial Page In preparation – forever (however, since 2000, used successfully in a real logic course for computer science students). This hyper-textbook contains links to: Wikipedia, the free encyclopedia; MacTutor History of Mathematics archive of the University of St Andrews; MathWorld of Wolfram Research. 2 Table of Contents References..........................................................................................................3 1. Introduction. What Is Logic, Really?.............................................................4 1.1. Total Formalization is Possible!..............................................................5 1.2. Predicate Languages.............................................................................10 1.3. Axioms of Logic: Minimal System, Constructive System and Classical System..........................................................................................................27 1.4. The Flavor of Proving Directly.............................................................40 -
Minimal Logic and Automated Proof Verification
Minimal Logic and Automated Proof Verification Louis Warren A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematics and Statistics University of Canterbury New Zealand 2019 Abstract We implement natural deduction for first order minimal logic in Agda, and verify minimal logic proofs and natural deduction properties in the resulting proof system. We study the implications of adding the drinker paradox and other formula schemata to minimal logic. We show first that these principles are independent of the law of excluded middle and of each other, and second how these schemata relate to other well-known principles, such as Markov’s Principle of unbounded search, providing proofs and semantic models where appropriate. We show that Bishop’s constructive analysis can be adapted to minimal logic. Acknowledgements Many thanks to Hannes Diener and Maarten McKubre-Jordens for their wonderful supervision. Their encouragement and guidance was the primary reason why my studies have been an enjoyable and relatively non-stressful experience. I am thankful for their suggestions and advice, and that they encouraged me to also pursue the questions I found interesting. Thanks also to Douglas Bridges, whose lectures inspired my interest in logic. I cannot imagine a better foundation for a constructivist. I am grateful to the University of Canterbury for funding my research, to CORCON for funding my secondments, and to my hosts Helmut Schwichtenberg in Munich, Ulrich Berger and Monika Seisenberger in Swansea, and Milly Maietti and Giovanni Sambin in Padua. This thesis is dedicated to Bertrand and Hester Warren, to whom I express my deepest gratitude. -
The Gödel and the Splitting Translations
The Godel¨ and the Splitting Translations David Pearce1 Universidad Politecnica´ de Madrid, Spain, [email protected] Abstract. When the new research area of logic programming and non-monotonic reasoning emerged at the end of the 1980s, it focused notably on the study of mathematical relations between different non-monotonic formalisms, especially between the semantics of stable models and various non-monotonic modal logics. Given the many and varied embeddings of stable models into systems of modal logic, the modal interpretation of logic programming connectives and rules be- came the dominant view until well into the new century. Recently, modal inter- pretations are once again receiving attention in the context of hybrid theories that combine reasoning with non-monotonic rules and ontologies or external knowl- edge bases. In this talk I explain how familiar embeddings of stable models into modal log- ics can be seen as special cases of two translations that are very well-known in non-classical logic. They are, first, the translation used by Godel¨ in 1933 to em- bed Heyting’s intuitionistic logic H into a modal provability logic equivalent to Lewis’s S4; second, the splitting translation, known since the mid-1970s, that al- lows one to embed extensions of S4 into extensions of the non-reflexive logic, K4. By composing the two translations one can obtain (Goldblatt, 1978) an ade- quate provability interpretation of H within the Goedel-Loeb logic GL, the sys- tem shown by Solovay (1976) to capture precisely the provability predicate of Peano Arithmetic. These two translations and their composition not only apply to monotonic logics extending H and S4, they also apply in several relevant cases to non-monotonic logics built upon such extensions, including equilibrium logic, non-monotonic S4F and autoepistemic logic. -
The Broadest Necessity
The Broadest Necessity Andrew Bacon∗ July 25, 2017 Abstract In this paper we explore the logic of broad necessity. Definitions of what it means for one modality to be broader than another are formulated, and we prove, in the context of higher-order logic, that there is a broadest necessity, settling one of the central questions of this investigation. We show, moreover, that it is possible to give a reductive analysis of this necessity in extensional language (using truth functional connectives and quantifiers). This relates more generally to a conjecture that it is not possible to define intensional connectives from extensional notions. We formulate this conjecture precisely in higher-order logic, and examine concrete cases in which it fails. We end by investigating the logic of broad necessity. It is shown that the logic of broad necessity is a normal modal logic between S4 and Triv, and that it is consistent with a natural axiomatic system of higher-order logic that it is exactly S4. We give some philosophical reasons to think that the logic of broad necessity does not include the S5 principle. 1 Say that a necessity operator, 21, is as broad as another, 22, if and only if: (*) 23(21P ! 22P ) whenever 23 is a necessity operator, and P is a proposition. Here I am quantifying both into the position that an operator occupies, and the position that a sentence occupies. We shall give details of the exact syntax of the language we are theorizing in shortly: at this point, we note that it is a language that contains operator variables, sentential variables, and that one can accordingly quantify into sentence and operator position. -
Modal Logics That Bound the Circumference of Transitive Frames
Modal Logics that Bound the Circumference of Transitive Frames. Robert Goldblatt∗ Abstract For each natural number n we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than n and no strictly ascending chains. The case n =0 is the G¨odel-L¨ob provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When n = 1, this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to in- tuitionistic propositional logic. A topological semantic analysis shows that the n-th member of the sequence of extensions of S4 is the logic of hereditarily n +1-irresolvable spaces when the modality ♦ is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of ♦ as the derived set (of limit points) operation. The variety of modal algebras validating the n-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by n. Moreovereach algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them. 1 Algebraic Logic and Logical Algebra The field of algebraic logic has been described as having two main aspects (see the introductions to Daigneault 1974 and Andr´eka, N´emeti, and Sain 2001). -
On the Blok-Esakia Theorem
On the Blok-Esakia Theorem Frank Wolter and Michael Zakharyaschev Abstract We discuss the celebrated Blok-Esakia theorem on the isomorphism be- tween the lattices of extensions of intuitionistic propositional logic and the Grze- gorczyk modal system. In particular, we present the original algebraic proof of this theorem found by Blok, and give a brief survey of generalisations of the Blok-Esakia theorem to extensions of intuitionistic logic with modal operators and coimplication. In memory of Leo Esakia 1 Introduction The Blok-Esakia theorem, which was proved independently by the Dutch logician Wim Blok [6] and the Georgian logician Leo Esakia [13] in 1976, is a jewel of non- classical mathematical logic. It can be regarded as a culmination of a long sequence of results, which started in the 1920–30s with attempts to understand the logical as- pects of Brouwer’s intuitionism by means of classical modal logic and involved such big names in mathematics and logic as K. Godel,¨ A.N. Kolmogorov, P.S. Novikov and A. Tarski. Arguably, it was this direction of research that attracted mathemat- ical logicians to modal logic rather than the philosophical analysis of modalities by Lewis and Langford [43]. Moreover, it contributed to establishing close connec- tions between logic, algebra and topology. (It may be of interest to note that Blok and Esakia were rather an algebraist and, respectively, a topologist who applied their results in logic.) Frank Wolter Department of Computer Science, University of Liverpool, U.K. e-mail: wolter@liverpool. ac.uk Michael Zakharyaschev Department of Computer Science and Information Systems, Birkbeck, University of London, U.K. -
Metamathematics in Coq Metamathematica in Coq (Met Een Samenvatting in Het Nederlands)
Metamathematics in Coq Metamathematica in Coq (met een samenvatting in het Nederlands) Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht op gezag van de Rector Magnificus, Prof. dr. W.H. Gispen, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op vrijdag 31 oktober 2003 des middags te 14:30 uur door Roger Dimitri Alexander Hendriks geboren op 6 augustus 1973 te IJsselstein. Promotoren: Prof. dr. J.A. Bergstra (Faculteit der Wijsbegeerte, Universiteit Utrecht) Prof. dr. M.A. Bezem (Institutt for Informatikk, Universitetet i Bergen) Voor Nelleke en Ole. Preface The ultimate arbiter of correctness is formalisability. It is a widespread view amongst mathematicians that correct proofs can be written out completely for- mally. This means that, after ‘unfolding’ the layers of abbreviations and con- ventions on which the presentation of a mathematical proof usually depends, the validity of every inference step should be completely perspicuous by a pre- sentation of this step in an appropriate formal language. When descending from the informal heat to the formal cold,1 we can rely less on intuition and more on formal rules. Once we have convinced ourselves that those rules are sound, we are ready to believe that the derivations they accept are correct. Formalis- ing reduces the reliability of a proof to the reliability of the means of verifying it ([60]). Formalising is more than just filling-in the details, it is a creative and chal- lenging job. It forces one to make decisions that informal presentations often leave unspecified. The search for formal definitions that, on the one hand, con- vincingly represent the concepts involved and, on the other hand, are convenient for formal proof, often elucidates the informal presentation. -
Esakia's Theorem in the Monadic Setting
Esakia’s theorem in the monadic setting Luca Carai joint work with Guram Bezhanishvili BLAST 2021 June 13, 2021 • Translation of intuitionistic logic into modal logic (G¨odel) • Algebraic semantics (Birkhoff, McKinsey-Tarski) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic logic (Glivenko) Important developements in 1930s and 1940s in the study of intuitionistic logic: • Axiomatization (Kolmogorov, Glivenko, Heyting) Intuitionism is an important direction in the mathematics of the 20th century. Intuitionistic logic is the logic of constructive mathematics. It has its origins in Brouwer’s criticism of the use of the principle of the excluded middle. • Translation of intuitionistic logic into modal logic (G¨odel) • Algebraic semantics (Birkhoff, McKinsey-Tarski) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic logic (Glivenko) Intuitionism is an important direction in the mathematics of the 20th century. Intuitionistic logic is the logic of constructive mathematics. It has its origins in Brouwer’s criticism of the use of the principle of the excluded middle. Important developements in 1930s and 1940s in the study of intuitionistic logic: • Axiomatization (Kolmogorov, Glivenko, Heyting) • Translation of intuitionistic logic into modal logic (G¨odel) • Topological semantics (Stone, Tarski) Important translations: • Double negation translation of classical logic into intuitionistic -
Constructive Embeddings of Intermediate Logics
Constructive embeddings of intermediate logics Sara Negri University of Helsinki Symposium on Constructivity and Computability 9-10 June 2011, Uppsala The G¨odel-Tarski-McKinsey embedding of intuitionistic logic into S4 ∗ 1. G¨odel(1933): `Int A ! `S4 A soundness ∗ 2. McKinsey & Tarski (1948): `S4 A ! `Int A faithfulness ∗ 3. Dummett & Lemmon (1959): `Int+Ax A iff `S4+Ax∗ A A modal logic M is a modal companion of a superintuitionistic ∗ logic L if `L A iff `M A . So S4 is a modal companion of Int, S4+Ax∗ is a modal companion of Int+Ax. 2. and 3. are proved semantically. We look closer at McKinsey & Tarski (1948): Proof by McKinsey & Tarski (1948) uses: I 1. Completeness of intuitionistic logic wrt Heyting algebras (Brouwerian algebras) and of S4 wrt topological Boolean algebras (closure algebras) I 2. Representation of Heyting algebras as the opens of topological Boolean algebras. I 3. The proof is indirect because of 1. and non-constructive because of 2. (Uses Stone representation of distributive lattices, in particular Zorn's lemma) The result was generalized to intermediate logics by Dummett and Lemmon (1959). No syntactic proof of faithfulness in the literature except the complex proof of the embedding of Int into S4 in Troelstra & Schwichtenberg (1996). Goal: give a direct, constructive, syntactic, and uniform proof. Background on labelled sequent systems Method for formulating systems of contraction-free sequent calculus for basic modal logic and its extensions and for systems of non-classical logics in Negri (2005). General ideas for basic modal logic K and other systems of normal modal logics. -
Problems for Temporary Existence in Tense Logic Meghan Sullivan* Notre Dame
Philosophy Compass 7/1 (2012): 43–57, 10.1111/j.1747-9991.2011.00457.x Problems for Temporary Existence in Tense Logic Meghan Sullivan* Notre Dame Abstract A-theorists of time postulate a deep distinction between the present, past and future. Settling on an appropriate logic for such a view is no easy matter. This Philosophy Compass article describes one of the most vexing formal problems facing A-theorists. It is commonly thought that A-theo- ries can only be formally expressed in a tense logic: a logic with operators like P (‘‘it was the case that’’) and F (‘‘it will be the case that’’). And it seems natural to think that we live in a world where objects come to exist and cease to exist as time passes. Indeed, this is typically a key component of the most prominent kind of A-theory, presentism. But the temporary existence assumption cannot be upheld in any tense logic with a standard quantification theory. I will explain the problem and outline the philosophical and logical considerations that generate it. I will then consider two possible solutions to the problem – one that targets our logic of quantification and one that targets our assumptions about change. I survey the costs of each solution. 1. Univocal Existence and Temporary Existence Metaphysicians should be applauded when they make efforts to be sensible. Here are two particular convictions about existence and change that seem worthy of defense: UNIVOCAL EXISTENCE: There is just one way that objects exist; and TEMPORARY EXISTENCE: Some objects came to exist or will cease to exist. -
Tense Logic Over Medvedev Frames
1SFQSJOUPG+VMZ'PSUIDPNJOHJO4UVEJB-PHJDB Wesley H. Holliday On the Modal Logic of Subset and Superset: Tense Logic over Medvedev Frames Abstract. Viewing the language of modal logic as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic logic is the class of Medvedev frames W, R where W is h i the set of nonempty subsets of some nonempty finite set S,andxRy i↵ x y, or more ◆ liberally, where W, R is isomorphic as a directed graph to }(S) , . Prucnal h i h \{;} ◆i [32] proved that the modal logic of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a di↵erence modality (for quantifying over those y such that x = y), or 6 a complement modality (for quantifying over those y such that x y). It follows that either 6◆ the logic of Medvedev frames in one of these tense languages is finitely axiomatizable— which would answer the open question of whether Medvedev’s [31] “logic of finite problems” is decidable—or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics.