An Introduction to Logic for Computer Science

Total Page:16

File Type:pdf, Size:1020Kb

An Introduction to Logic for Computer Science An introduction to Logic for Computer Science Sujata Ghosh ISI Chennai [email protected] What is logic? What is logic? Reasoning is typically a human activity. What is logic? Reasoning is typically a human activity. There are inference rules which determine whether an argument is correct or not. What is logic? Reasoning is typically a human activity. There are inference rules which determine whether an argument is correct or not. Logic is the study of reasoning, in particular, the study of these inference rules. Example argumentAn example in physics in Physics An example in Linguistics Intelligence without representation* Rodney A. Brooks MIT Artificial Intelligence Laboratory, 545 Technology Square, Rm. 836, Cambridge, MA 02139, USA Received September 1987 Brooks, R.A., Intelligence without representation, Artificial Intelligence 47 (1991), 139–159. * This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the research is provided in part by an IBM Faculty 9 Development Award, in part by a grant from the Systems Development Foundation, in part by the University Research Initiative under Office of Naval Research contract N00014-86-K-0685 and in part by the Advanced Research Projects Agency under Office of Naval Research contract N00014-85-K-0124. Abstract Artificial intelligence research has foundered on the issue of representation. When intelligence is approached in an incremental manner, with strict reliance on interfacing to the real world through perception and action, reliance on representation disappears. In this paper we outline our approach to incrementally building complete intelligent Creatures. The fundamental decomposition of the intelligent system is not into independent information processing units which must interface with each other via representations. Instead, the intelligent system is decomposed into independent and parallel activity producers which all interface directly to the world through perception and action, rather than interface to each other particularly much. The notions of central and peripheral systems evaporateeverything is both central and peripheral. Based on these principles we have built a very successful series of mobile robots which operate without supervision as Creatures in standard office environments. 1. Introduction • We must incrementally build up the capabilities of Artificial intelligence started as a field whose goal intelligent systems, having complete systems at was to replicate human level intelligence in a each step of the way and thus automatically ensure Intelligencemachine. without representation*that the pieces and their interfaces are valid. Early hopes diminished as the magnitude and • At each step we should build complete intelligent Rodneydifficulty A. Brooks of that goal was appreciated. Slow progress systems that we let loose in the real world with real was made over the next 25 years in demonstrating sensing and real action. Anything less provides a MIT Artificial Intelligence Laboratory, 545 Technology Square, Rm. 836, Cambridge, MA 02139, USA isolated aspects of intelligence. Recent work has candidate with which we can delude ourselves. Received tendedSeptember to concentrate1987 on commercializable aspects of Brooks, R.A.,"intelligent IntelligenceAn assistants" without example representation, for human workers.Artificial Intelligence 47in (1991), 139–159.WeArtificial have been following this approach and have built a series of autonomous mobile robots. We have * This reportNo describes one talks research about done replicating at the Artificial the Intelligence full gamut Laboratory of of thereached Massachusetts an unexpected Institute of Technology.conclusion Support(C) and for thehave a research ishuman provided intelligence in part by an any IBM more. FacultyIntelligence Instead 9 Development we see Award,a retreat in part by a grantrather from radical the Systems hypothesis Development (H). Foundation, in part by the University Research Initiative under Office of Naval Research contract N00014-86-K-0685 and in part by the Advanced Research Projects Agencyinto specialized under Office ofsubproblems, Naval Research suchcontract as N00014-85-K-0124. ways to Intelligencerepresent knowledge, without natural languagerepresentation* understanding, (C) When we examine very simple level intelligence Abstract vision or even more specialized areas such as truth we find that explicit representations and models maintenance systems or plan verification. All the of the world simply get in the way. It turns out ArtificialRodney intelligence A. Brooks research has foundered on the issue of representation. When intelligence is approached in an incremental manner, with strict reliancework on in interfacing these subareasto the real isworld benchmarked through perception against and theaction, reliance on representationto be better todisappears. use the worldIn this aspaper its ownwe outline model. our approachMITsorts Artificial to incrementallyofIntelligence tasks Laboratory, humans building 545 Technologycompletedo within Square, intelligent Rm.those 836, Creatures. Cambridge, areas. MAThe 02139, fundamental USA decomposition of the intelligent system is not into independent information processing units which must interface with each other via representations. Instead, the intelligent system is decomposedReceivedAmongst into September independent the 1987 dreamers and parallel still activityin the producers field of whichAI (those all interface directly(H) Representationto the world through is perceptionthe wrong and unit action, of abstractionrather than interfaceBrooks,not R.A., dreamingto Intelligenceeach other about without particularly dollars,representation, much.that Artificial is), The Intelligencethere notions is 47 aof (1991),feeling. central 139–159. and peripheral insystems building evaporateeverything the bulkiest is partsboth centralof intelligent and peripheral. Based on these principles we have built a very successful series of mobile robotssystems. which operate without supervision as Creatures in standard officethat one environments. day all these pieces will all fall into place * This report describes research done at the Artificial Intelligence Laboratory of the Massachusetts Institute of Technology. Support for the researchand is we provided will in seepart by"truly" an IBM Faculty intelligent 9 Development systems Award, emerge.in part by a grant from the Systems Development Foundation, in part by the University Research Initiative under Office of Naval Research contract N00014-86-K-0685 andRepresentation in part by the Advanced has beenResearch the central issue in artificial 1. IntroductionProjects Agency under Office of Naval Research contract N00014-85-K-0124. However, I, and others, believe that human level • We mustintelligence incrementally work over build the uplast the 15 capabilitiesyears only becauseof ArtificialAbstractintelligence intelligence is too started complex as anda field little whose understood goal to be intelligentit has provided systems, an havinginterface complete between otherwisesystems isolatedat Artificialcorrectly intelligence decomposed research has foundered into theon the right issue of subpiecesrepresentation. Whenat the intelligence is approachedmodules in an andincrementa conferencel manner, with papers. was tostrict reliancereplicate on interfacing human to the reallevel world throughintelligence perception and in action, a reliance on representationeach step disappears. of the In thisway paper and we outlinethus automatically ensure ourmoment approach to incrementallyand that evenbuilding ifcomplete we knewintelligent the Creatures. subpieces The fundamental we decomposition of the intelligent system is not into machine.independent information processing units which must interface with each other via representations.that the Instead,pieces the and intelligent their systeminterfaces is are valid. decomposedstill wouldn't into independent know and parallel the activity right producers interfaces which all interfacebetween directly to the world2. through The perception evolution and action, ratherof intelligence than interface to each other particularly much. The notions of central and peripheral systems evaporateeverything is both central and Earlyperipheral. them.hopes Based Furthermore, diminishedon these principles we weas willhave the built never amagnitude very understandsuccessful seriesand ofhow mobile to robots• whichAt operateeach withoutstep supervisionwe should as Creatures build in complete intelligent standard office environments. difficultydecompose of that goal human was levelappreciated. intelligence Slow until progress we've had a systemsWe that already we let loosehave inan the existence real world proof with realof, the was made1. lotIntroduction ofover practice the next with 25simpler years level in demonstratingintelligences. sensingpossibility and real of action. intelligent Anything entities: less provideshuman beings.a • We must incrementallyAdditionally, build up the many capabilities animals of are intelligent to some isolatedArtificial aspects intelligence of intelligence. started as a fieldRecent whose goalwork has intelligent systems,candidate having with complete which systemswe can atdelude ourselves. tendedwas to toconcentrateIn replicatethis paper human on commercializable I leveltherefore intelligence argue in aspects fora a ofdifferenteach step of the waydegree. and thus (This automatically
Recommended publications
  • The Corcoran-Smiley Interpretation of Aristotle's Syllogistic As
    November 11, 2013 15:31 History and Philosophy of Logic Aristotelian_syllogisms_HPL_house_style_color HISTORY AND PHILOSOPHY OF LOGIC, 00 (Month 200x), 1{27 Aristotle's Syllogistic and Core Logic by Neil Tennant Department of Philosophy The Ohio State University Columbus, Ohio 43210 email [email protected] Received 00 Month 200x; final version received 00 Month 200x I use the Corcoran-Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen{Prawitz system of natural deduction, using the universal and existential quantifiers of standard first- order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is a fragment of my (constructive and relevant) system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle. 1. Introduction: the Corcoran-Smiley interpretation of Aristotle's syllogistic as concerned with deductions Two influential articles, Corcoran 1972 and Smiley 1973, convincingly argued that Aristotle's syllogistic logic anticipated the twentieth century's systematizations of logic in terms of natural deductions. They also showed how Aristotle in effect advanced a completeness proof for his deductive system.
    [Show full text]
  • Does Reductive Proof Theory Have a Viable Rationale?
    DOES REDUCTIVE PROOF THEORY HAVE A VIABLE RATIONALE? Solomon Feferman Abstract The goals of reduction and reductionism in the natural sciences are mainly explanatory in character, while those in mathematics are primarily foundational. In contrast to global reductionist programs which aim to reduce all of mathematics to one supposedly “univer- sal” system or foundational scheme, reductive proof theory pursues local reductions of one formal system to another which is more jus- tified in some sense. In this direction, two specific rationales have been proposed as aims for reductive proof theory, the constructive consistency-proof rationale and the foundational reduction rationale. However, recent advances in proof theory force one to consider the viability of these rationales. Despite the genuine problems of foun- dational significance raised by that work, the paper concludes with a defense of reductive proof theory at a minimum as one of the principal means to lay out what rests on what in mathematics. In an extensive appendix to the paper, various reduction relations between systems are explained and compared, and arguments against proof-theoretic reduction as a “good” reducibility relation are taken up and rebutted. 1 1 Reduction and reductionism in the natural sciencesandinmathematics. The purposes of reduction in the natural sciences and in mathematics are quite different. In the natural sciences, one main purpose is to explain cer- tain phenomena in terms of more basic phenomena, such as the nature of the chemical bond in terms of quantum mechanics, and of macroscopic ge- netics in terms of molecular biology. In mathematics, the main purpose is foundational. This is not to be understood univocally; as I have argued in (Feferman 1984), there are a number of foundational ways that are pursued in practice.
    [Show full text]
  • Notes on Proof Theory
    Notes on Proof Theory Master 1 “Informatique”, Univ. Paris 13 Master 2 “Logique Mathématique et Fondements de l’Informatique”, Univ. Paris 7 Damiano Mazza November 2016 1Last edit: March 29, 2021 Contents 1 Propositional Classical Logic 5 1.1 Formulas and truth semantics . 5 1.2 Atomic negation . 8 2 Sequent Calculus 10 2.1 Two-sided formulation . 10 2.2 One-sided formulation . 13 3 First-order Quantification 16 3.1 Formulas and truth semantics . 16 3.2 Sequent calculus . 19 3.3 Ultrafilters . 21 4 Completeness 24 4.1 Exhaustive search . 25 4.2 The completeness proof . 30 5 Undecidability and Incompleteness 33 5.1 Informal computability . 33 5.2 Incompleteness: a road map . 35 5.3 Logical theories . 38 5.4 Arithmetical theories . 40 5.5 The incompleteness theorems . 44 6 Cut Elimination 47 7 Intuitionistic Logic 53 7.1 Sequent calculus . 55 7.2 The relationship between intuitionistic and classical logic . 60 7.3 Minimal logic . 65 8 Natural Deduction 67 8.1 Sequent presentation . 68 8.2 Natural deduction and sequent calculus . 70 8.3 Proof tree presentation . 73 8.3.1 Minimal natural deduction . 73 8.3.2 Intuitionistic natural deduction . 75 1 8.3.3 Classical natural deduction . 75 8.4 Normalization (cut-elimination in natural deduction) . 76 9 The Curry-Howard Correspondence 80 9.1 The simply typed l-calculus . 80 9.2 Product and sum types . 81 10 System F 83 10.1 Intuitionistic second-order propositional logic . 83 10.2 Polymorphic types . 84 10.3 Programming in system F ...................... 85 10.3.1 Free structures .
    [Show full text]
  • Proof Theory of Constructive Systems: Inductive Types and Univalence
    Proof Theory of Constructive Systems: Inductive Types and Univalence Michael Rathjen Department of Pure Mathematics University of Leeds Leeds LS2 9JT, England [email protected] Abstract In Feferman’s work, explicit mathematics and theories of generalized inductive definitions play a cen- tral role. One objective of this article is to describe the connections with Martin-L¨of type theory and constructive Zermelo-Fraenkel set theory. Proof theory has contributed to a deeper grasp of the rela- tionship between different frameworks for constructive mathematics. Some of the reductions are known only through ordinal-theoretic characterizations. The paper also addresses the strength of Voevodsky’s univalence axiom. A further goal is to investigate the strength of intuitionistic theories of generalized inductive definitions in the framework of intuitionistic explicit mathematics that lie beyond the reach of Martin-L¨of type theory. Key words: Explicit mathematics, constructive Zermelo-Fraenkel set theory, Martin-L¨of type theory, univalence axiom, proof-theoretic strength MSC 03F30 03F50 03C62 1 Introduction Intuitionistic systems of inductive definitions have figured prominently in Solomon Feferman’s program of reducing classical subsystems of analysis and theories of iterated inductive definitions to constructive theories of various kinds. In the special case of classical theories of finitely as well as transfinitely iterated inductive definitions, where the iteration occurs along a computable well-ordering, the program was mainly completed by Buchholz, Pohlers, and Sieg more than 30 years ago (see [13, 19]). For stronger theories of inductive 1 i definitions such as those based on Feferman’s intutitionic Explicit Mathematics (T0) some answers have been provided in the last 10 years while some questions are still open.
    [Show full text]
  • Proof Theory for Modal Logic
    Proof theory for modal logic Sara Negri Department of Philosophy 00014 University of Helsinki, Finland e-mail: sara.negri@helsinki.fi Abstract The axiomatic presentation of modal systems and the standard formula- tions of natural deduction and sequent calculus for modal logic are reviewed, together with the difficulties that emerge with these approaches. Generaliza- tions of standard proof systems are then presented. These include, among others, display calculi, hypersequents, and labelled systems, with the latter surveyed from a closer perspective. 1 Introduction In the literature on modal logic, an overall skepticism on the possibility of developing a satisfactory proof theory was widespread until recently. This attitude was often accompanied by a belief in the superiority of model-theoretic methods over proof- theoretic ones. Standard proof systems have been shown insufficient for modal logic: Natural deduction presentations of even the most basic modal logics present difficulties that have been resolved only partially, and the same has happened with sequent calculus. These traditional proof system have failed to meet in a satisfactory way such basic requirements as analyticity and normalizability of formal derivations. Therefore alternative proof systems have been developed in recent years, with a lot of emphasis on their relative merits and on applications. We review first the axiomatic presentations of modal systems and the standard formulations of natural deduction and sequent calculus for modal logic, and the difficulties that emerge with
    [Show full text]
  • Nonclassical First Order Logics: Semantics and Proof Theory
    Nonclassical first order logics: Semantics and proof theory Apostolos Tzimoulis joint work with G. Greco, P. Jipsen, A. Kurz, M. A. Moshier, A. Palmigiano TACL Nice, 20 June 2019 Starting point Question How can we define general relational semantics for arbitrary non-classical first-order logics? I What are the models? I What do quantifiers mean in those models? I What does completeness mean? 2 / 29 Methodology: dual characterizations q Non − classical algebraic Non − classical proof th: semantics first order models Classical algebraic Classical proof th: semantics first order models i 3 / 29 A brief recap on classical first-order logic: Language I Set of relation symbols (Ri)i2I each of finite arity ni. I Set of function symbols ( f j) j2J each of finite arity n j. I Set of constant symbols (ck)k2K (0-ary functions). I Set of variables Var = fv1;:::; vn;:::g. The first-order language L = ((Ri)i2I; ( f j) j2J; (ck)k2K) over Var is built up from terms defined recursively as follows: Trm 3 t ::= vm j ck j f j(t;:::; t): The formulas of first-order logic are defined recursively as follows: L 3 A ::= Ri(t) j t1 = t2 j > j ? j A ^ A j A _ A j :A j 8vmA j 9vmA 4 / 29 A brief recap on classical first-order logic: Meaning The models of a first-order logic L are tuples D D D M = (D; (Ri )i2I; ( f j ) j2J; (ck )k2K) D D D where D is a non-empty set and Ri ; f j ; ck are concrete ni-ary relations over D, n j-ary functions on D and elements of D resp.
    [Show full text]
  • The Relative Consistency of the Axiom of Choice — Mechanized Using Isabelle/ZF (Extended Abstract)
    The Relative Consistency of the Axiom of Choice | Mechanized Using Isabelle/ZF (Extended Abstract) Lawrence C. Paulson Computer Laboratory, University of Cambridge, England [email protected] G¨odel[3] published a monograph in 1940 proving a highly significant the- orem, namely that the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are consistent with respect to the other axioms of set theory. This theorem addresses the first of Hilbert's famous list of unsolved problems in mathematics. I have mechanized this work [8] using Isabelle/ZF [5, 6]. Obvi- ously, the theorem's significance makes it a tempting challenge; the proof also has numerous interesting features. It is not a single formal assertion, as most theorems are. G¨odel[3, p. 33] states it as follows, using Σ to denote the axioms for set theory: What we shall prove is that, if a contradiction from the axiom of choice and the generalized continuum hypothesis were derived in Σ, it could be transformed into a contradiction obtained from the axioms of Σ alone. G¨odelpresents no other statement of this theorem. Neither does he introduce a theory of syntax suitable for reasoning about transformations on proofs, surely because he considers it to be unnecessary. G¨odel'swork consists of several different results which, taken collectively, express the relative consistency of the axiom the choice. The concluding inference takes place at the meta-level and is not formalized. Standard proofs use meta- level reasoning extensively. G¨odelwrites [3, p. 34], However, the only purpose of these general metamathematical consid- erations is to show how the proofs for theorems of a certain kind can be accomplished by a general method.
    [Show full text]
  • Combinatorial Proofs and Decomposition Theorems for First
    Combinatorial Proofs and Decomposition Theorems for First-order Logic Dominic J. D. Hughes Lutz Straßburger Jui-Hsuan Wu Logic Group Inria, Equipe Partout Ecole Normale Sup´erieure U.C. Berkeley Ecole Polytechnique, LIX France USA France Abstract—We uncover a close relationship between combinato- rial and syntactic proofs for first-order logic (without equality). ax ⊢ pz, pz Whereas syntactic proofs are formalized in a deductive proof wk system based on inference rules, a combinatorial proof is a ax ⊢ pw, pz, pz ⊢ p,p wk syntax-free presentation of a proof that is independent from any wk ⊢ pw, pz, pz, ∀y.py set of inference rules. We show that the two proof representations ⊢ p,q,p ∨ ∨ ax ⊢ pw, pz, pz ∨ (∀y.py) are related via a deep inference decomposition theorem that ⊢ p ∨ q,p ⊢ p,p ∃ ∧ ⊢ pw, pz, ∃x.(px ∨ (∀y.py)) establishes a new kind of normal form for syntactic proofs. This ⊢ (p ∨ q) ∧ p,p,p ∀ yields (a) a simple proof of soundness and completeness for first- ctr ⊢ pw, ∀y.py, ∃x.(px ∨ (∀y.py)) ⊢ (p ∨ q) ∧ p,p ∨ order combinatorial proofs, and (b) a full completeness theorem: ∨ ⊢ pw ∨ (∀y.py), ∃x.(px ∨ (∀y.py)) ⊢ ((p ∨ q) ∧ p) ∨ p ∃ every combinatorial proof is the image of a syntactic proof. ⊢ ∃x.(px ∨ (∀y.py)), ∃x.(px ∨ (∀y.py)) ctr ⊢ ∃x.(px ∨ (∀y.py)) I. INTRODUCTION First-order predicate logic is a cornerstone of modern logic. ↓ ↓ Since its formalisation by Frege [1] it has seen a growing usage in many fields of mathematics and computer science. Upon the development of proof theory by Hilbert [2], proofs became first-class citizens as mathematical objects that could be studied on their own.
    [Show full text]
  • Model Theory on Finite Structures∗
    Model Theory on Finite Structures∗ Anuj Dawar Department of Computer Science University of Wales Swansea Swansea, SA2 8PP, U.K. e-mail: [email protected] 1 Introduction In mathematical logic, the notions of mathematical structure, language and proof themselves become the subject of mathematical investigation, and are treated as first class mathematical objects in their own right. Model theory is the branch of mathematical logic that is particularly concerned with the relationship between structure and language. It seeks to establish the limits of the expressive power of our formal language (usually the first order predicate calculus) by investigating what can or cannot be expressed in the language. The kinds of questions that are asked are: What properties can or cannot be formulated in first order logic? What structures or relations can or cannot be defined in first order logic? How does the expressive power of different logical languages compare? Model Theory arose in the context of classical logic, which was concerned with resolving the paradoxes of infinity and elucidating the nature of the infinite. The main constructions of model theory yield infinite structures and the methods and results assume that structures are, in general, infinite. As we shall see later, many, if not most, of these methods fail when we confine ourselves to finite structures Many questions that arise in computer science can be seen as being model theoretic in nature, in that they investigate the relationship between a formal language and a structure. However, most structures of interest in computer science are finite. The interest in finite model theory grew out of questions in theoretical computer science (particularly, database theory and complexity theory).
    [Show full text]
  • Topics in the Theory of Recursive Functions
    SETS, MODELS, AND PROOFS: TOPICS IN THE THEORY OF RECURSIVE FUNCTIONS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by David R. Belanger August 2015 This document is in the public domain. SETS, MODELS, AND PROOFS: TOPICS IN THE THEORY OF RECURSIVE FUNCTIONS David R. Belanger, Ph.D. Cornell University We prove results in various areas of recursion theory. First, in joint work with Richard Shore, we prove a new jump-inversion result for ideals of recursively enu- merable (r.e.) degrees; this defeats what had seemed to be a promising tack on the automorphism problem for the semilattice R of r.e. degrees. Second, in work spanning two chapters, we calibrate the reverse-mathematical strength of a number of theorems of basic model theory, such as the Ryll-Nardzewski atomic-model theorem, Vaught's no-two-model theorem, Ehrenfeucht's three-model theorem, and the existence theorems for homogeneous and saturated models. Whereas most of these are equivalent over RCA0 to one of RCA0, WKL0, ACA0, as usual, we also uncover model-theoretic statements with exotic complexities such as :WKL0 _ 0 ACA0 and WKL0 _ IΣ2. Third, we examine the possible weak truth table (wtt) degree spectra of count- able first-order structures. We find several points at which the wtt- and Turing- degree cases differ, notably that the most direct wtt analogue of Knight's di- chotomy theorem does not hold. Yet we find weaker analogies between the two, including a new trichotomy theorem for wtt degree spectra in the spirit of Knight's.
    [Show full text]
  • Reflections on Finite Model Theory
    Reflections on Finite Model Theory Phokion G. Kolaitis∗ IBM Almaden Research Center San Jose, CA 95120, USA [email protected] Abstract model theory, to examine some of the obstacles that were encountered, and to discuss some open problems that have Advances in finite model theory have appeared in LICS stubbornly resisted solution. proceedings since the very beginning of the LICS Sympo- sium. The goal of this paper is to reflect on finite model 2 Early Beginnings theory by highlighting some of its successes, examining ob- stacles that were encountered, and discussing some open In the first half of the 20th Century, finite models were problems that have stubbornly resisted solution. used as a tool in the study of Hilbert’s Entscheidungsprob- lem, also known as the classical decision problem, which is the satisfiability problem for first-order logic: given a 1 Introduction first-order sentence, does it have a model? Indeed, even before this problem was shown to be undecidable by Turing During the past thirty years, finite model theory has devel- and Church, logicians had identified decidable fragments of oped from a collection of sporadic, but influential, early re- first-order logic, such as the Bernays-Schonfinkel¨ Class of sults to a mature research area characterized by technical all ∃∗∀∗ sentences and the Ackermann Class of all ∃∗∀∃∗ depth and mathematical sophistication. In this period, fi- sentences. The decidability of these two classes was estab- nite model theory has been explored not only for its con- lished by proving that the finite model property holds for nections to other areas of computer science (most notably, them: if a sentence in these classes has a model, then it has computational complexity and database theory), but also in a finite model (see [14, Chapter 6] for a modern exposition its own right as a distinct area of logic in computer science.
    [Show full text]
  • What Is Mathematical Logic? a Survey
    What is mathematical logic? A survey John N. Crossley1 School of Computer Science and Software Engineering, Monash University, Clayton, Victoria, Australia 3800 [email protected] 1 Introduction What is mathematical logic? Mathematical logic is the application of mathemat- ical techniques to logic. What is logic? I believe I am following the ancient Greek philosopher Aristotle when I say that logic is the (correct) rearranging of facts to find the information that we want. Logic has two aspects: formal and informal. In a sense logic belongs to ev- eryone although we often accuse others of being illogical. Informal logic exists whenever we have a language. In particular Indian Logic has been known for a very long time. Formal (often called, ‘mathematical’) logic has its origins in ancient Greece in the West with Aristotle. Mathematical logic has two sides: syntax and semantics. Syntax is how we say things; semantics is what we mean. By looking at the way that we behave and the way the world behaves, Aris- totle was able to elicit some basic laws. His style of categorizing logic led to the notion of the syllogism. The most famous example of a syllogism is All men are mortal Socrates is a man [Therefore] Socrates is mortal Nowadays we mathematicians would write this as ∀x(Man(x) → Mortal (x)) Man(S) (1) Mortal (S) One very general form of the above rule is A (A → B) (2) B otherwise know as modus ponens or detachment: the A is ‘detached’ from the formula (A → B) leaving B. This is just one example of a logical rule.
    [Show full text]