Great-circle distance

29 yr old Orthopaedic Surgeon Grippo from Saint-Paul, spends time with interests including model railways, top property developers in singapore developers in singapore and dolls. Finished a cruise ship experience that included passing by Runic Stones and Church. Library Technician Anton from Strathroy, has many passions that include r/c helicopters, property developers in condo new launch singapore and coin collecting. Finds the beauty in planing a trip to spots around the globe, recently only returning from Old Town of Corfu. In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation.

Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:

$\mathbf {*4\cdot 13} .\ \ \vdash .\ p\ \equiv \ \thicksim (\thicksim p)$

"This is the principle of double negation, i.e. a proposition is equivalent of the falsehood of its negation."

The principium contradictiones of modern logicians (particularly Leibnitz and Kant) in the formula A is not not-A, differs entirely in meaning and application from the Aristotelian proposition [ i.e. Law of Contradiction: not (A and not-A) i.e. ~(A & ~A), or not (( B is A) and (B is not-A))]. This latter refers to the relation between an affirmative and a negative judgment. According to Aristotle, one judgment [B is judged to be an A] contradicts another [B is judged to be a not-A]. The later proposition [ A is not not-A ] refers to the relation between subject and predicate in a single judgment; the predicate contradicts the subject. Aristotle states that one judgment is false when another is true; the later writers [Leibniz and Kant] state that a judgment is in itself and absolutely false, because the predicate contradicts the subject. What the later writers desire is a principle from which it can be known whether certain propositions are in themselves true. From the Aristotelian proposition we cannot immediately infer the truth or falsehood of any particular proposition, but only the impossibility of believing both affirmation and negation at the same time.

Footnotes

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References

• William Hamilton, 1860, Lectures on Metaphysics and Logic, Vol. II. Logic; Edited by Henry Mansel and John Veitch, Boston, Gould and Lincoln. Available online from googlebooks.
• Christoph Sigwart, 1895, Logic: The Judgment, Concept, and Inference; Second Edition, Translated by Helen Dendy, Macmillan & Co. New York. Available online from googlebooks.
• Stephen C. Kleene, 1952, Introduction to Metamathematics, 6th reprinting with corrections 1971, North-Holland Publishing Company, Amsterdam NY, ISBN 0 7204 2103 9.
• Stephen C. Kleene, 1967, Mathematical Logic, Dover edition 2002, Dover Publicastions, Inc, Mineola N.Y. ISBN 0-486-42533-9 (pbk.)
• Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, 2nd edition 1927, reprint 1962, Cambridge at the University Press, London UK, no ISBN or LCCCN.