Proof Analysis in Intermediate Logics
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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. -
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). -
Introduction to Formal Logic II (3 Credit
PHILOSOPHY 115 INTRODUCTION TO FORMAL LOGIC II BULLETIN INFORMATION PHIL 115 – Introduction to Formal Logic II (3 credit hrs) Course Description: Intermediate topics in predicate logic, including second-order predicate logic; meta-theory, including soundness and completeness; introduction to non-classical logic. SAMPLE COURSE OVERVIEW Building on the material from PHIL 114, this course introduces new topics in logic and applies them to both propositional and first-order predicate logic. These topics include argument by induction, duality, compactness, and others. We will carefully prove important meta-logical results, including soundness and completeness for both propositional logic and first-order predicate logic. We will also learn a new system of proof, the Gentzen-calculus, and explore first-order predicate logic with functions and identity. Finally, we will discuss potential motivations for alternatives to classical logic, and examine how those motivations lead to the development of intuitionistic logic, including a formal system for intuitionistic logic, comparing the features of that formal system to those of classical first-order predicate logic. ITEMIZED LEARNING OUTCOMES Upon successful completion of PHIL 115, students will be able to: 1. Apply, as appropriate, principles of analytical reasoning, using as a foundation the knowledge of mathematical, logical, and algorithmic principles 2. Recognize and use connections among mathematical, logical, and algorithmic methods across disciplines 3. Identify and describe problems using formal symbolic methods and assess the appropriateness of these methods for the available data 4. Effectively communicate the results of such analytical reasoning and problem solving 5. Explain and apply important concepts from first-order logic, including duality, compactness, soundness, and completeness 6. -
Intuitionistic Logic
Intuitionistic Logic Nick Bezhanishvili and Dick de Jongh Institute for Logic, Language and Computation Universiteit van Amsterdam Contents 1 Introduction 2 2 Intuitionism 3 3 Kripke models, Proof systems and Metatheorems 8 3.1 Other proof systems . 8 3.2 Arithmetic and analysis . 10 3.3 Kripke models . 15 3.4 The Disjunction Property, Admissible Rules . 20 3.5 Translations . 22 3.6 The Rieger-Nishimura Lattice and Ladder . 24 3.7 Complexity of IPC . 25 3.8 Mezhirov's game for IPC . 28 4 Heyting algebras 30 4.1 Lattices, distributive lattices and Heyting algebras . 30 4.2 The connection of Heyting algebras with Kripke frames and topologies . 33 4.3 Algebraic completeness of IPC and its extensions . 38 4.4 The connection of Heyting and closure algebras . 40 5 Jankov formulas and intermediate logics 42 5.1 n-universal models . 42 5.2 Formulas characterizing point generated subsets . 44 5.3 The Jankov formulas . 46 5.4 Applications of Jankov formulas . 47 5.5 The logic of the Rieger-Nishimura ladder . 52 1 1 Introduction In this course we give an introduction to intuitionistic logic. We concentrate on the propositional calculus mostly, make some minor excursions to the predicate calculus and to the use of intuitionistic logic in intuitionistic formal systems, in particular Heyting Arithmetic. We have chosen a selection of topics that show various sides of intuitionistic logic. In no way we strive for a complete overview in this short course. Even though we approach the subject for the most part only formally, it is good to have a general introduction to intuitionism. -
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. -
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. -
On the Intermediate Logic of Open Subsets of Metric Spaces Timofei Shatrov
On the intermediate logic of open subsets of metric spaces Timofei Shatrov abstract. In this paper we study the intermediate logic MLO( ) of open subsets of a metric space . This logic is closely related to Medvedev’sX logic X of finite problems ML. We prove several facts about this logic: its inclusion in ML, impossibility of its finite axiomatization and indistinguishability from ML within some large class of propositional formulas. Keywords: intermediate logic, Medvedev’s logic, axiomatization, indistin- guishability 1 Introduction In this paper we introduce and study a new intermediate logic MLO( ), which we first define using Kripke semantics and then we will try to ratio-X nalize this definition using the concepts from prior research in this field. An (intuitionistic) Kripke frame is a partially ordered set (F, ). A Kripke model is a Kripke frame with a valuation θ (a function which≤ maps propositional variables to upward-closed subsets of the Kripke frame). The notion of a propositional formula being true at some point w F of a model M = (F,θ) is defined recursively as follows: ∈ For propositional letter p , M, w p iff w θ(p ) i i ∈ i M, w ψ χ iff M, w ψ and M, w χ ∧ M, w ψ χ iff M, w ψ or M, w χ ∨ M, w ψ χ iff for any w′ w, M, w ψ implies M, w′ χ → ≥ M, w 6 ⊥ A formula is true in a model M if it is true in each point of M. A formula is valid in a frame F if it is true in all models based on that frame. -
Intermediate Logic
Intermediate Logic Richard Zach Philosophy 310 Winter Term 2015 McGill University Intermediate Logic by Richard Zach is licensed under a Creative Commons Attribution 4.0 International License. It is based on The Open Logic Text by the Open Logic Project, used under a Creative Commons Attribution 4.0 In- ternational License. Contents Prefacev I Sets, Relations, Functions1 1 Sets2 1.1 Basics ................................. 2 1.2 Some Important Sets......................... 3 1.3 Subsets ................................ 4 1.4 Unions and Intersections...................... 5 1.5 Pairs, Tuples, Cartesian Products ................. 7 1.6 Russell’s Paradox .......................... 9 2 Relations 10 2.1 Relations as Sets........................... 10 2.2 Special Properties of Relations................... 11 2.3 Orders................................. 12 2.4 Graphs ................................ 14 2.5 Operations on Relations....................... 15 3 Functions 17 3.1 Basics ................................. 17 3.2 Kinds of Functions.......................... 19 3.3 Inverses of Functions ........................ 20 3.4 Composition of Functions ..................... 21 3.5 Isomorphism............................. 22 3.6 Partial Functions........................... 22 3.7 Functions and Relations....................... 23 4 The Size of Sets 24 4.1 Introduction ............................. 24 4.2 Enumerable Sets........................... 24 4.3 Non-enumerable Sets........................ 28 i CONTENTS 4.4 Reduction.............................. -
Admissible Rules in Logic and Algebra
Admissible Rules in Logic and Algebra George Metcalfe Mathematics Institute University of Bern Pisa Summer Workshop on Proof Theory 12-15 June 2012, Pisa George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 1/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))). However, this rule is only admissible: Nat(s(x)) ⊲ Nat(x). But what if we add to the system: Nat(s(−1)) ??? George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 2/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))). However, this rule is only admissible: Nat(s(x)) ⊲ Nat(x). But what if we add to the system: Nat(s(−1)) ??? George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 2/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))). However, this rule is only admissible: Nat(s(x)) ⊲ Nat(x). But what if we add to the system: Nat(s(−1)) ??? George Metcalfe (University of Bern) Admissible Rules in Logic and Algebra June2012,Pisa 2/107 Derivability vs Admissibility Consider a system defined by two rules: Nat(0) and Nat(x) ⊲ Nat(s(x)). The following rule is derivable: Nat(x) ⊲ Nat(s(s(x))). -
Teach Yourself Logic 2016: a Study Guide
Teach Yourself Logic 2016: A Study Guide Peter Smith University of Cambridge Version of January 1, 2016 Pass it on, . That's the game I want you to learn. Pass it on. Alan Bennett, The History Boys c Peter Smith 2016 The latest version of this Guide can always be downloaded from logicmatters.net/tyl/ URLs in blue are live links to external web-pages or PDF documents. Internal cross-references to sections etc. are also live. But to avoid the text being covered in ugly red rashes internal links are not coloured { if you are reading onscreen, your cursor should change as you hover over such live links: try here. The Guide's layout is optimized for reading on-screen: try, for example, reading in iBooks on an iPad, or in double-page mode in Adobe Reader on a laptop. Contents A very quick introduction iii 1 Preliminaries1 1.1 Why this Guide for philosophers? ..................... 1 1.2 Why this Guide for mathematicians too?................. 2 1.3 A strategy for reading logic books (and why this Guide is so long)... 3 1.4 On the question of exercises ........................ 3 1.5 Assumed background ............................ 4 1.6 Do you really need more logic?....................... 6 1.7 How to prove it ............................... 7 2 How the Guide is structured9 2.1 Logic, in the broad sense .......................... 9 2.2 Mapping the field .............................. 10 2.3 Three comments on the Guide's structure................. 12 2.4 Choices, choices ............................... 13 3 First order logic 15 3.1 FOL: the basic topics............................ 15 3.2 The main recommendations on FOL................... -
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.