2.0 Knowledge Generation 2.0 Logic as a T ool First appearance - PDF document

07.04.2009 2. Logics for Knowledge Bases 2.0 Introduction to Logics 2.1 Syntax of First Order Logic 2.2 Semantics of First Order Logic Knowledge-Based Systems and Deductive Databases Wolf-Tilo Balke Christoph Lofi Institut für Informationssysteme Technische Universität Braunschweig http://www.ifis.cs.tu-bs.de Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 2 2.0 Knowledge Generation 2.0 Knowledge Generation • Basic question: How can we generate new • Inference comes in two major flavors knowledge? – Inductive inference : Perform multiple observations and draw a conclusion – Start with some knowledge that is (generally?) considered true ( axioms ) – Deductive inference : Provide some true facts (axioms) and rules and then – Derive new knowledge in a combine them to generate consistent and understandable conclusions (theorems) fashion… ( inference ) – Hmmm, …seems far from trivial 3 4 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 2.0 Knowledge Generation 2.0 Logic as a T ool • First appearance of formal logics was • Let‟s do some time travel… sophism (5 th century BC) around 330 BC in „Prior Analytics‟ and „On Interpretation‟ appearing in – Pre-socratic philosophy Aristotle’s Organon – Only fragments survive • Logic was intended as a tool for Protagoras – Known through the writings of valid philosophical arguments opponents like Plato or Aristotle • Aimed at formal and safe inference • Rhetoric as a (paid) skill – Describing the process of deriving new knowledge – Used for persuasion of others Gorgias from old knowledge or observations – Use ambiguities of language in order to – Discovers many sophistic tricks and fallacies in „On deceive or to support fallacious reasoning Sophistical Refutations„ in the Organon Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 5 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 6 1

07.04.2009 2.0 Propositional Logic 2.0 Propositional Logic • Propositional Logic deals with atomic logical • First approaches date back to Aristotle who statements and logical connectives in a merely discussed some basic principles in the collection „Metaphysics‟ (around 4 th century BC) structural sense – Atomic statements cannot be further divided – „A statement and its contradiction cannot be true at the same time‟ • Examples are „The earth is flat‟ or „Socrates is dead‟ – Connectives are „AND‟, „OR‟ and the implication „ ⟹ ‟ – „Every statement or its contradiction has to be true‟ – Basically all connectives are truth functions that – The technique of indirect proofs evaluate to „true‟ or „false‟ in bivalent logic • There are also multi-valued logics , think for instance • Propositional logic then has been about NULL values in relational databases heavily refined during medieval times Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 7 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 8 2.0 Propositional Logic 2.0 Propositional Logic • The first real calculus with implications was • A first sound and complete then formalized by Gottlob Frege (1879) and formalization for truth subsequently refined by Bertrand Russell (1910) values was given by George Boole in 1847 with his • But propositional logic is the simplest kind of algebraic calculus logical calculus… – Boolean Algebra – It does not investigate the statements themselves • Graphic representation by Venn diagrams – For instance quantifiers or predicates are not used, which limits the applications – Sometimes referred to as zero th -order logic x AND y x OR y NOT x 9 10 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 2.0 T erm Logic 2.0 T erm Logic • For the special application in deductive inference • T erms from statements Aristotle introduced term logic – A term per se is neither true nor false – T erm logics remained the dominating logical paradigm until – Examples: Aristotle, man, mortal, blue, … the advent of predicate logics in the late 19 th century • Propositions • Consists of three basics constructs – Provide a statement which is either true or false – T erm : A word representing „something‟ – Propositions have a quantity and a quality – Proposition : A combination of two terms (the subject • Universal and affirmative: ' All men are mortal' and the predicate) • Existential and affirmative: ' Some men are philosophers' – Syllogism : An inference where some proposition • Universal and negative: ' No man is immortal' ( conclusion ) directly follows from two others ( premises ) • Existential and negative: ' Some men are not philosophers' Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 11 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 12 2

07.04.2009 2.0 T erm Logic 2.0 T erm Logic • The syllogism is the actual device of inference • The square of opposition defines the allowed logical conversions proposition & proposition ⟹ proposition minor premise major premise conclusion – The minor premise contains a minor term (subject) and a middle term (predicate) – The major premise contains the same middle term (subject) and a major term (predicate) – The conclusion contains the minor term as subject and the major term as predicate Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 13 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 14 2.0 T erm Logic 2.0 T erm Logic • In syllogisms of the four terms in the premises, • Later, singulars have been introduced predicating one has to make the connection only one thing and treated as universals – Thus, one term has to appear twice and work as – All Socrates are men. & All men are mortals. subject and predicate ⟹ All Socrates are mortals. – All Greeks are men . & All men are mortal. • Introduced in the Port-Royal-Logic ⟹ All Greeks are mortal. by Antoine Arnauld and • For Aristotele in propositions and syllogisms only Pierre Nicole (1662) plurals (universal terms) are possible • Obviously, this is a little – Term logic largely ignores singular terms awkward… – Can you say „Every Socrates is a philosopher‟? 15 16 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 2.0 Early Criticism 2.0 Early Criticism • Eubulides of Miletus (4 th century BC) • But some fallacies are not Aristotle’s fault – Philosopher of the Megarian School – For instance „ Quaternio Terminorum ‟ • Heavily criticized Aristotle‟s – All adults love children . & All children love chocolate . ⟹ All adults love chocolate . syllogisms – A grain of sand is no heap. Adding a single grain does not make a heap. • Where is the fallacy? ⟹ There is no heap of sand! – All adults are children-lovers . All children are – I still have, what I have not lost. chocolate-lovers ⟹ …Nothing!!! I have not lost horns. – Because only three terms are allowed in syllogisms ⟹ I have horns! Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 17 Knowledge-Based Systems and Deductive Databases – Wolf-Tilo Balke – IfIS – TU Braunschweig 18 3