older logicians did, that a claim has several different opposites, which are in different kinds of opposition with the claim. Gottlob Frege (8 November 1848 – 26 July 1925)'s Begriffsschrift also presents a square of oppositions, organised in an almost identical manner to the classical square, showing the contradictories, subalternates and contraries between four formulae constructed from universal quantification, negation and implication. Algirdas Julien Greimas (9 March 1917 – 27 February 1992)' semiotic square was derived from Aristotle's work. The traditional square of opposition is now often compared with squares based on inner- and outer-negation. Logical hexagons and other bi-simplexes The square of opposition has been extended to a logical hexagon which includes the relationships of six statements. It was discovered independently by both Augustin Sesmat (April 7, 1885 – December 12, 1957) and Robert Blanché (1898–1975). It has been proven that both the square and the hexagon, followed by a "logical cube", belong to a regular series of n-dimensional objects called "logical bi-simplexes of dimension ." The pattern also goes even beyond this. Square of opposition (or logical square) and modal logic The logical square, also called square of opposition or square of Apuleius, has its origin in the four marked sentences to be employed in syllogistic reasoning: "Every man is bad," the universal affirmative - The negation of the universal affirmative "Not every man is bad" (or "Some men are not bad") - "Some men are bad," the particular affirmative - and finally, the negation of the particular affirmative "No man is bad". Robert Blanché published with Vrin his Structures intellectuelles in 1966 and since then many scholars think that the logical square or square of opposition representing four values should be replaced by the logical hexagon which by representing six values is a more potent figure because it has the power to explain more things about logic and natural language. Set-theoretical interpretation of categorical statements In modern mathematical logic, statements containing words "all", "some" and "no", can be stated in terms of set theory if we assume a set-like domain of discourse. If the set of all 's is labeled as and the set of all 's as , then: "All is " (AaB) is equivalent to " is a subset of ", or . "No is " (AeB) is equivalent to "The intersection of and is empty", or . "Some is " (AiB) is equivalent to