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A Brief History of Logic Steffen H¨ olldobler International Center for Computational Logic Technische Universit¨ at Dresden Germany ◮ History ◮ A Simple Example ◮ Literature ◮ Module Foundations Steffen H¨ olldobler A Brief History of Logic 1

History: Basic Ideas Aristotle ( † 322 B.C.) Formalization syllogisms SeP PeQ SeQ Herodot ( † 430 B.C.) Calculization Egyptian stones, abacus Herodot ( † 430 B.C.) Mechanization mechanai Steffen H¨ olldobler A Brief History of Logic 2

History: Combining the Ideas (1) Descartes (1596-1650) Hobbes (1588-1679) Leibnitz (1646-1719) geometry thinking lingua characteristica = calculus ratiocinator calculating universal encyclopedia Lullus (1232-1315) Pascal (1623-1662) Leibnitz (1646-1719) ars magna Steffen H¨ olldobler A Brief History of Logic 3

History: Combining the Ideas (2) DeMorgan (1806-1871) Boole (1815-1864) Frege(1882) Whitehead, Russell (1910-1913) propositional logic Principia Mathematica first order logic “Begriffsschrift” Javins(1869) Babbage (1792-1871) analytical engine evaluating boolean expressions Steffen H¨ olldobler A Brief History of Logic 4

History: Combining the Ideas (3) civil servant’s logic: F | = G iff G ∈ F higher civil servant’s logic: F | = G iff F = { G } Skolem, Herbrand, G¨ odel (1930) completeness of first order logic Zuse (1936-1941) Z1, Z3 Turing (1936) Turing machine Steffen H¨ olldobler A Brief History of Logic 5

History: Finally, Computers Arrive von Neumann (1946) Zuse (1949) Turing (1950) computer Turing test Plankalk¨ ul Can machines think? Steffen H¨ olldobler A Brief History of Logic 6

History: Deduction Systems ◮ early 1950s: Davis: Preßburger arithmetic. ◮ 1955/6: Beth, Sch¨ utte, Hintikka: semantic tableaus. ◮ 1956: Simon, Newell: first heuristic theorem prover. ◮ late 1950s: Gilmore, Davis, Putnam: theorem prover based on Herbrand’s “Eigenschaft B Methode”. ◮ 1960: Prawitz: unification. ◮ 1965: J.A. Robinson: resolution principle. ◮ thereafter: improved resolution rules vs. intelligent heuristics. ◮ 1996: McCune’s OTTER proves Robbin’s conjecture. ◮ today: TPTP library, yearly CASC competition. Steffen H¨ olldobler A Brief History of Logic 7

History: Logic Programming ◮ 1971: A. Colmerauer: System Q � Prolog. brother-of ( X , Y ) ← father-of ( Z , X ) ∧ father-of ( Z , Y ) ∧ male ( X ) . ◮ 1979: R.A. Kowalski: algorithm = logic + control. ◮ late-70s to mid-80s: theoretical foundations. ◮ 1977: D.H.D. Warren: first Prolog compiler. ◮ 1982: A. Colmerauer: Prolog II � constraints. ⊲ Constraint logic programming. Steffen H¨ olldobler A Brief History of Logic 8

A Simple Example ◮ Socrates is a human. All humans are mortal. Hence, Socrates is mortal. human ( socrates ) ( forall X ) ( if human ( X ) then mortal ( X )) mortal ( socrates ) h ( s ) ( ∀ X ) ( h ( X ) → m ( X )) m ( s ) ◮ 5 is a natural number. All natural numbers are integers. Hence, 5 is an integer. Steffen H¨ olldobler A Brief History of Logic 9

Deduction ◮ A world without deduction would be a world without science, technology, laws, social conventions and culture (Johnson–Laird, Byrne: 1991). ◮ Think about it! Steffen H¨ olldobler A Brief History of Logic 10

The Addition of Natural Numbers ◮ The sum of zero and the number Y is Y. The sum of the successor of the number X and the number Y is the successor of the sum of X and Y. ⊲ Are you willing to conclude from these statements that the sum of one and one is two? 0 + Y = Y s ( X ) + Y = s ( X + Y ) s ( 0 ) + s ( 0 ) = s ( s ( 0 )) ⊲ Are you willing to conclude that addition is commutative? 0 + Y = Y s ( X ) + Y = s ( X + Y ) X + Y = Y + X Steffen H¨ olldobler A Brief History of Logic 11

Applications ◮ Functional equivalence of two chips ◮ Verification of hard- and software ◮ Year 2000 problem ◮ Eliminating redundancies in group communication systems ◮ Designing the layout of yellow pages ◮ Managing a tunnel project ◮ Natural language processing ◮ Cognitive Robotics ◮ Semantic web (description logics) ◮ Law ◮ Optimization Problems Logic is Everywhere Steffen H¨ olldobler A Brief History of Logic 12

Some Background Literature ◮ L. Chang and R.C.T. Lee: Symbolic Logic and Mechanical Theorem Proving. Academic Press, New York (1973). ◮ M. Fitting: First–Order Logic and Automated Theorem Proving. Springer Verlag. Berlin, second edition (1996). ◮ J. Gallier: Logic for Computer Science: Foundations of Automated Theorem Proving. Harper and Row. New York (1986). ◮ S. H¨ olldobler: Logik und Logikprogrammierung. Synchron Publishers GmbH, Heidelberg (2009). ◮ D. Poole and A. Mackworth and R. Goebel: Computational Intelligence: A Logical Approach. Oxford University Press, New York, Oxford (1998). ◮ S. Russell and P. Norvig: Artificial Intelligence. Prentice Hall, Englewood Cliffs (1995). ◮ U. Sch¨ oning: Logik f¨ ur Informatiker. Spektrum Akademischer Verlag (1995). Steffen H¨ olldobler A Brief History of Logic 13

Module Foundations ◮ Two lectures ⊲ Logic ⊲ Science of Computational Logic ◮ Logic is offered from now until the end of November. ◮ Science of Computational Logic is offered from beginning of December until end of the lecturing period. ◮ Exact dates will be announced later. ◮ Exams: ⊲ Logic: written exam (Dec 21, 2014) ⊲ Science of Computational Logic: oral exam Steffen H¨ olldobler A Brief History of Logic 14

Logic ◮ Agenda ⊲ Introduction ⊲ Propositional Logic ⊲ First Order Logic ◮ Exercises ⊲ Exercises are announced each week. ⊲ We expect students to discuss their solutions. ◮ Tests ⊲ Their will be two written tests. ⊲ 10% of the final mark will be given based on performance in the tests. ◮ See our web pages for more detail. ◮ Ask questions as soon as they arise, anywhere and at anytime. ◮ Don’t accept a situation, where you do not understand everything. Steffen H¨ olldobler A Brief History of Logic 15