A Brief History of Logic ¨ Steffen Holldobler International Center for Computational Logic ¨ Dresden Technische Universitat Germany

I History I A Simple Example I Literature I Module Foundations

¨ Steffen Holldobler A Brief History of Logic

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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

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History: Combining the Ideas (1)

Descartes (1596-1650)

Hobbes (1588-1679)

Leibnitz (1646-1719)

geometry

thinking = calculating

lingua characteristica calculus ratiocinator universal encyclopedia

Lullus (1232-1315)

Pascal (1623-1662)

Leibnitz (1646-1719)

ars magna

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History: Combining the Ideas (2)

DeMorgan (1806-1871) Boole (1815-1864) propositional logic

Frege(1882)

Whitehead, Russell (1910-1913)

first order logic “Begriffsschrift”

Principia Mathematica

Javins(1869)

Babbage (1792-1871)

evaluating boolean expressions

analytical engine

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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, Godel (1930) completeness of first order logic Zuse (1936-1941) Z1, Z3 Turing (1936) Turing machine

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History: Finally, Computers Arrive

von Neumann (1946)

Zuse (1949)

Turing (1950)

computer

Plankalkul ¨

Turing test Can machines think?

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History: Deduction Systems

I early 1950s: Davis: Preßburger arithmetic. I 1955/6: Beth, Schutte, ¨ Hintikka: semantic tableaus. I 1956: Simon, Newell: first heuristic theorem prover. I late 1950s: Gilmore, Davis, Putnam:

theorem prover based on Herbrand’s “Eigenschaft B Methode”. I 1960: Prawitz: unification. I 1965: J.A. Robinson: resolution principle. I thereafter: improved resolution rules vs. intelligent heuristics. I 1996: McCune’s OTTER proves Robbin’s conjecture. I today: TPTP library, yearly CASC competition.

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History: Logic Programming

I 1971: A. Colmerauer: System Q

Prolog.

brother-of (X , Y ) ← father-of (Z , X ) ∧ father-of (Z , Y ) ∧ male(X ). I 1979: R.A. Kowalski: algorithm = logic + control. I late-70s to mid-80s: theoretical foundations. I 1977: D.H.D. Warren: first Prolog compiler. I 1982: A. Colmerauer: Prolog II

constraints.

. Constraint logic programming.

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A Simple Example

I 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)

I 5 is a natural number. All natural numbers are integers. Hence, 5 is an integer.

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Deduction

I A world without deduction would be a world without science, technology, laws,

social conventions and culture (Johnson–Laird, Byrne: 1991). I Think about it!

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The Addition of Natural Numbers I 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

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Applications I Functional equivalence of two chips I Verification of hard- and software I Year 2000 problem I Eliminating redundancies in group communication systems I Designing the layout of yellow pages I Managing a tunnel project I Natural language processing I Cognitive Robotics I Semantic web (description logics) I Law I Optimization Problems

Logic is Everywhere

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Some Background Literature I L. Chang and R.C.T. Lee: Symbolic Logic and Mechanical Theorem Proving.

Academic Press, New York (1973). I M. Fitting: First–Order Logic and Automated Theorem Proving.

Springer Verlag. Berlin, second edition (1996). I J. Gallier:

Logic for Computer Science: Foundations of Automated Theorem Proving. Harper and Row. New York (1986). ¨ I S. Holldobler: Logik und Logikprogrammierung. Synchron Publishers GmbH, Heidelberg (2009). I D. Poole and A. Mackworth and R. Goebel:

Computational Intelligence: A Logical Approach. Oxford University Press, New York, Oxford (1998). I S. Russell and P. Norvig: Artificial Intelligence.

Prentice Hall, Englewood Cliffs (1995). ¨ I U. Schoning: Logik fur ¨ Informatiker. Spektrum Akademischer Verlag (1995).

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Module Foundations

I Two lectures

. Logic . Science of Computational Logic I Logic is offered from now until the end of November. I Science of Computational Logic is offered from beginning of December

until end of the lecturing period. I Exact dates will be announced later. I Exams:

. Logic: written exam (Dec 21, 2014) . Science of Computational Logic: oral exam

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Logic I Agenda

. Introduction . Propositional Logic . First Order Logic I Exercises

. Exercises are announced each week. . We expect students to discuss their solutions. I Tests

. Their will be two written tests. . 10% of the final mark will be given based on performance in the tests. I See our web pages for more detail. I Ask questions as soon as they arise, anywhere and at anytime. I Don’t accept a situation, where you do not understand everything.

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