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Generalized Quantifiers and Computation: 9th European Summer School in Logic, Language, and Information ESSLLI’97Workshop Aix-en-Provence, France, August 11–22, 1997 Revised Lectures Author: Jouko Väänänen Published by Springer Berlin Heidelberg ISBN: 978-3-540-66993-7 DOI: 10.1007/3-540-46583-9 Table of Contents: Generalized Quantifiers, an Introduction Counting and Locality over Finite Structures A Survey A Perspective on Lindström Quantifiers and Oracles Generalized Quantifiers in Logic Programs A Generalized Quantifier Concept in Computational Complexity Theory Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers

Here's a summary of logic: What is Logic? Logic is the systematic study of the principles of valid inference and reasoning. It involves the use of language, symbols, and rules to evaluate arguments and arrive at conclusions. Key Concepts: 1. Propositions: Statements that can be true or false. 2. Arguments: Sets of propositions used to support a conclusion. 3. Inference: The process of deriving a conclusion from premises. 4. Validity: An argument is valid if its conclusion follows logically from its premises. 5. Soundness: An argument is sound if it is both valid and has true premises. Types of Logic: 1. Deductive Logic: Concerned with certain and absolute conclusions. 2. Inductive Logic: Concerned with probable conclusions based on evidence. 3. Abductive Logic: Concerned with making educated guesses or hypotheses. 4. Modal Logic: Concerned with possibility, necessity, and obligation. 5. Non-Classical Logic: Includes fuzzy logic, intuitionistic logic, and others. Logical Operators: 1. Conjunction (AND) 2. Disjunction (OR) 3. Negation (NOT) 4. Implication (IF-THEN) 5. Equivalence (IF AND ONLY IF) Rules of Inference: 1. Modus Ponens 2. Modus Tollens 3. Hypothetical Syllogism 4. Disjunctive Syllogism 5. Conjunctive Syllogism Fallacies: 1. Ad Hominem 2. Straw Man 3. False Dilemma 4. Slippery Slope 5. Begging the Question   Early 19th Century (1800-1840)   - George Boole (1815-1864):     - Published "The Mathematical Analysis of Logic" (1847)     - Developed symbolic logic, using algebraic methods to represent logical operations     - Introduced the concept of a "logical algebra" - Augustus De Morgan (1806-1871):     - Published "Formal Logic" (1847)     - Developed relational logic, emphasizing the importance of relations between terms     - Introduced the concept of "De Morgan's laws"   Mid-19th Century (1840-1870)   - William Hamilton (1788-1856):     - Published "Lectures on Logic" (1833)     - Developed the concept of "quantification of the predicate"     - Influenced the development of mathematical logic - Charles Babbage (1791-1871):     - Proposed the idea of a mechanical computer, the Analytical Engine     - Recognized the importance of logic in computer design   Late 19th Century (1870-1900)   - Gottlob Frege (1848-1925):     - Published "Begriffsschrift" (1879)     - Developed modern predicate logic, using quantifiers and variables     - Introduced the concept of "sense and reference" - Ernst Schröder (1841-1902):     - Published "Vorlesungen über die Algebra der Logik" (1890-1905)     - Developed the algebra of logic, using lattice theory     - Influenced the development of mathematical logic - Giuseppe Peano (1858-1932):     - Published "Arithmetices principia, nova methodo exposita" (1889)     - Developed the Peano axioms for arithmetic     - Introduced the concept of "formal language"