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Logic Machines a brief history of the idea George Matthews CC 2018

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Aristotle’s insight ! Valid reasoning is a matter of form not content. ! Cognition is categorization. ! The formal structure of categorical reasoning can be exhaustively mapped out. 384 – 322 BCE

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Aristotle’s insight ! Valid reasoning is a matter of form not content. ! Cognition is categorization. ! The formal structure of categorical reasoning can be exhaustively mapped out. 384 – 322 BCE

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Aristotle’s insight ! Valid reasoning is a matter of form not content. ! Cognition is categorization. ! The formal structure of categorical reasoning can be exhaustively mapped out. 384 – 322 BCE

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Aristotle’s insight ! Valid reasoning is a matter of form not content. ! Cognition is categorization. ! The formal structure of categorical reasoning can be exhaustively mapped out. 384 – 322 BCE

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the Aristotelian Square of opposition A E I O subalternation contraries c o n t r a d i c t o r i e s subalternation c o n t r a d i c t o r i e s sub-contraries All S are P. Some S are P. No S are P. Some S are not P.

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Lull’s fantasy ! All thoughts are combinations of a finite set of concepts. ! We can mechanically determine all possible combinations. ! The true nature of reality, God, morality can be shown for all to see. 1232 – 1316

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Lull’s fantasy ! All thoughts are combinations of a finite set of concepts. ! We can mechanically determine all possible combinations. ! The true nature of reality, God, morality can be shown for all to see. 1232 – 1316

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Lull’s fantasy ! All thoughts are combinations of a finite set of concepts. ! We can mechanically determine all possible combinations. ! The true nature of reality, God, morality can be shown for all to see. 1232 – 1316

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Lull’s fantasy ! All thoughts are combinations of a finite set of concepts. ! We can mechanically determine all possible combinations. ! The true nature of reality, God, morality can be shown for all to see. 1232 – 1316

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Lull’s Art of Finding Truth

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Hobbes’ analysis ! Society is made of individuals. ! Individuals are made of mechanical components. ! Thoughts are computations carried out by these components. 1588 – 1679

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Hobbes’ analysis ! Society is made of individuals. ! Individuals are made of mechanical components. ! Thoughts are computations carried out by these components. 1588 – 1679

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Hobbes’ analysis ! Society is made of individuals. ! Individuals are made of mechanical components. ! Thoughts are computations carried out by these components. 1588 – 1679

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Hobbes’ analysis ! Society is made of individuals. ! Individuals are made of mechanical components. ! Thoughts are computations carried out by these components. 1588 – 1679

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Hobbes’ Leviathan

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Leibniz’ dream ! A perfect language expressing all truths is possible. ! This language could be encoded in an algebra of thought enabling only true conclusions to be derived from other thoughts. ! All questions could be resolved by a machine whose workings mirror the rules of correct thinking. 1646 – 1716

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Leibniz’ dream ! A perfect language expressing all truths is possible. ! This language could be encoded in an algebra of thought enabling only true conclusions to be derived from other thoughts. ! All questions could be resolved by a machine whose workings mirror the rules of correct thinking. 1646 – 1716

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Leibniz’ dream ! A perfect language expressing all truths is possible. ! This language could be encoded in an algebra of thought enabling only true conclusions to be derived from other thoughts. ! All questions could be resolved by a machine whose workings mirror the rules of correct thinking. 1646 – 1716

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Leibniz’ dream ! A perfect language expressing all truths is possible. ! This language could be encoded in an algebra of thought enabling only true conclusions to be derived from other thoughts. ! All questions could be resolved by a machine whose workings mirror the rules of correct thinking. 1646 – 1716

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Leibniz’ calculator

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Boole’s laws ! Logic is a branch of mathematics. ! Logical reasoning can be built up from the elementary operations OR, AND, NOT and the laws governing them. ! Thinking is the mechanical application of these laws to express complex ideas and reasoning. 1816 – 1864

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Boole’s laws ! Logic is a branch of mathematics. ! Logical reasoning can be built up from the elementary operations OR, AND, NOT and the laws governing them. ! Thinking is the mechanical application of these laws to express complex ideas and reasoning. 1816 – 1864

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Boole’s laws ! Logic is a branch of mathematics. ! Logical reasoning can be built up from the elementary operations OR, AND, NOT and the laws governing them. ! Thinking is the mechanical application of these laws to express complex ideas and reasoning. 1816 – 1864

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Boole’s laws ! Logic is a branch of mathematics. ! Logical reasoning can be built up from the elementary operations OR, AND, NOT and the laws governing them. ! Thinking is the mechanical application of these laws to express complex ideas and reasoning. 1816 – 1864

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A Boolean circuit

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Babbage’s ambitions ! Complex mathematical operations can be broken down into simple mindless steps. ! A general purpose machine can be built to carry out complex calculations. ! This machine can be programmed to perform different operations on different inputs. 1814 – 1871

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Babbage’s ambitions ! Complex mathematical operations can be broken down into simple mindless steps. ! A general purpose machine can be built to carry out complex calculations. ! This machine can be programmed to perform different operations on different inputs. 1814 – 1871

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Babbage’s ambitions ! Complex mathematical operations can be broken down into simple mindless steps. ! A general purpose machine can be built to carry out complex calculations. ! This machine can be programmed to perform different operations on different inputs. 1814 – 1871

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Babbage’s ambitions ! Complex mathematical operations can be broken down into simple mindless steps. ! A general purpose machine can be built to carry out complex calculations. ! This machine can be programmed to perform different operations on different inputs. 1814 – 1871

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Babbage’s Difference Engine

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Turing’s proof ! All computations require only the ability to read and write symbols in some location in response to instructions. ! It is possible to construct a machine that can imitate all other machines. ! Thus a universal machine is possible. 1912 – 1954

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Turing’s proof ! All computations require only the ability to read and write symbols in some location in response to instructions. ! It is possible to construct a machine that can imitate all other machines. ! Thus a universal machine is possible. 1912 – 1954

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Turing’s proof ! All computations require only the ability to read and write symbols in some location in response to instructions. ! It is possible to construct a machine that can imitate all other machines. ! Thus a universal machine is possible. 1912 – 1954

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Turing’s proof ! All computations require only the ability to read and write symbols in some location in response to instructions. ! It is possible to construct a machine that can imitate all other machines. ! Thus a universal machine is possible. 1912 – 1954

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Turing’s Automatic Computing Engine