I, Of The Understanding first published 1739, reprinted as: David Hume, A Treatise of Human Nature, Penguin Classics, 1985.

This work was published by Hume in 1758 as his rewrite of his "juvenile" Treatise of Human Nature: Being An attempt to introduce the experimental method of Reasoning into Moral Subjects Vol.

Thus an example of the expression would look like this: (pig): (Flies(pig) ⊕ ~Flies(pig)) (For all instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously) === Logicians versus Intuitionists === From the late 1800s through the 1930s, a bitter, persistent debate raged between Hilbert and his followers versus Hermann Weyl and L.

Brouwer's philosophy, called intuitionism, started in earnest with Leopold Kronecker in the late 1800s. Hilbert intensely disliked Kronecker's ideas: The debate had a profound effect on Hilbert.

Peters, Wellesley, MA, 1997. van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967.

71) Thus Hilbert was saying: "If p and ~p are both shown to be true, then p does not exist", and was thereby invoking the law of excluded middle cast into the form of the law of contradiction. The rancorous debate continued through the early 1900s into the 1920s; in 1927 Brouwer complained about "polemicizing against it [intuitionism] in sneering tones" (Brouwer in van Heijenoort, p.

The William James Lectures for 1940 Delivered at Harvard University. Bertrand Russell, The Problems of Philosophy, With a New Introduction by John Perry, Oxford University Press, New York, 1997 edition (first published 1912).

71) Thus Hilbert was saying: "If p and ~p are both shown to be true, then p does not exist", and was thereby invoking the law of excluded middle cast into the form of the law of contradiction. The rancorous debate continued through the early 1900s into the 1920s; in 1927 Brouwer complained about "polemicizing against it [intuitionism] in sneering tones" (Brouwer in van Heijenoort, p.

Hilbert's example: "the assertion that either there are only finitely many prime numbers or there are infinitely many" (quoted in Davis 2000:97); and Brouwer's: "Every mathematical species is either finite or infinite." (Brouwer 1923 in van Heijenoort 1967:336). In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g.

71) Thus Hilbert was saying: "If p and ~p are both shown to be true, then p does not exist", and was thereby invoking the law of excluded middle cast into the form of the law of contradiction. The rancorous debate continued through the early 1900s into the 1920s; in 1927 Brouwer complained about "polemicizing against it [intuitionism] in sneering tones" (Brouwer in van Heijenoort, p.

Reprinted with corrections, 1975. Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted).

Thus an example of the expression would look like this: (pig): (Flies(pig) ⊕ ~Flies(pig)) (For all instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously) === Logicians versus Intuitionists === From the late 1800s through the 1930s, a bitter, persistent debate raged between Hilbert and his followers versus Hermann Weyl and L.

The William James Lectures for 1940 Delivered at Harvard University. Bertrand Russell, The Problems of Philosophy, With a New Introduction by John Perry, Oxford University Press, New York, 1997 edition (first published 1912).

149)}} In his lecture in 1941 at Yale and the subsequent paper Gödel proposed a solution: "that the negation of a universal proposition was to be understood as asserting the existence ...

Includes a wonderful essay on "The Art of drawing Inferences". Hans Reichenbach, Elements of Symbolic Logic, Dover, New York, 1947, 1975. Tom Mitchell, Machine Learning, WCB McGraw-Hill, 1997. Constance Reid, Hilbert, Copernicus: Springer-Verlag New York, Inc.

Thus intuitionists absolutely disallow the blanket assertion: "For all propositions P concerning infinite sets D: P or ~P" (Kleene 1952:48). For more about the conflict between the intuitionists (e.g.

19–20 in Robert Maynard Hutchins (ed.), Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago, IL, 1952.

8 in Robert Maynard Hutchins (ed.), Great Books of the Western World, Encyclopædia Britannica, Inc., Chicago, IL, 1952.

Kleene 1952 original printing, 1971 6th printing with corrections, 10th printing 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY, . Kneale, W.

But a good introduction to the concepts. David Hume, An Inquiry Concerning Human Understanding, reprinted in Great Books of the Western World Encyclopædia Britannica, Volume 35, 1952, p. 449 ff.

and Kneale, M., The Development of Logic, Oxford University Press, Oxford, UK, 1962.

Reprinted with corrections, 1975. Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted).

Hilbert's example: "the assertion that either there are only finitely many prime numbers or there are infinitely many" (quoted in Davis 2000:97); and Brouwer's: "Every mathematical species is either finite or infinite." (Brouwer 1923 in van Heijenoort 1967:336). In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g.

Peters, Wellesley, MA, 1997. van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967.

Very easy to read: Russell was a wonderful writer. Bertrand Russell, The Art of Philosophizing and Other Essays, Littlefield, Adams & Co., Totowa, NJ, 1974 edition (first published 1968).

1996, first published 1969.

Kleene 1952 original printing, 1971 6th printing with corrections, 10th printing 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY, . Kneale, W.

Very easy to read: Russell was a wonderful writer. Bertrand Russell, The Art of Philosophizing and Other Essays, Littlefield, Adams & Co., Totowa, NJ, 1974 edition (first published 1968).

Reprinted with corrections, 1975. Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted).

Includes a wonderful essay on "The Art of drawing Inferences". Hans Reichenbach, Elements of Symbolic Logic, Dover, New York, 1947, 1975. Tom Mitchell, Machine Learning, WCB McGraw-Hill, 1997. Constance Reid, Hilbert, Copernicus: Springer-Verlag New York, Inc.

I, Of The Understanding first published 1739, reprinted as: David Hume, A Treatise of Human Nature, Penguin Classics, 1985.

Kleene 1952 original printing, 1971 6th printing with corrections, 10th printing 1991, Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY, . Kneale, W.

Contains a wealth of biographical information, much derived from interviews. Bart Kosko, Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, New York, 1993.

Peters, Wellesley, MA, 1997. van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, 1967.

The William James Lectures for 1940 Delivered at Harvard University. Bertrand Russell, The Problems of Philosophy, With a New Introduction by John Perry, Oxford University Press, New York, 1997 edition (first published 1912).

Includes a wonderful essay on "The Art of drawing Inferences". Hans Reichenbach, Elements of Symbolic Logic, Dover, New York, 1947, 1975. Tom Mitchell, Machine Learning, WCB McGraw-Hill, 1997. Constance Reid, Hilbert, Copernicus: Springer-Verlag New York, Inc.

Hilbert's example: "the assertion that either there are only finitely many prime numbers or there are infinitely many" (quoted in Davis 2000:97); and Brouwer's: "Every mathematical species is either finite or infinite." (Brouwer 1923 in van Heijenoort 1967:336). In general, intuitionists allow the use of the law of excluded middle when it is confined to discourse over finite collections (sets), but not when it is used in discourse over infinite sets (e.g.

Ross (trans.), The Works of Aristotle, Oxford University Press, Oxford, UK. Martin Davis 2000, Engines of Logic: Mathematicians and the Origin of the Computer", W.

Also see: David Applebaum, The Vision of Hume, Vega, London, 2001: a reprint of a portion of An Inquiry'' starts on p. 94 ff == External links == "Contradiction" entry in the Stanford Encyclopedia of Philosophy Classical logic Articles containing proofs Theorems in propositional logic

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