Logical hexagon
The logical hexagon is a conceptual model of the relationships between the truth values of six statements. It is an extension of Aristotle's square of opposition. It was discovered independently by both Augustin Sesmat and Robert Blanché.[1]
In 1966, Blanché published Structures intellectuelles, wherein he presented a hexagonal figure introducing Y and U, a third pair of contrary and subcontrary logical propositions for a total of six, in addition to those of the traditional square, the contraries A and E, and the subcontraries I and O.
The logical hexagon consists in introducing a third contrary Y to be added to the two traditional contraries that is the two universals A and E and in introducing a third subcontrary U to be added to the two traditional subcontraries, that is, the two particulars I and O. So as indicated above this results in introducing a third pair of contradictories Y versus U on the model of the two pairs A versus I, I versus E. The third contrary Y can be represented as the conjunction of the two traditional particulars I and O. Y can be read: On the one hand, there exists at least one x that is both man and white, on the other there exists at least one x that is both man and non-white. The logical hexagon of Robert Blanché also introduces a third subcontrary U to be represented as the alternative A w E. This expression is to be read: One of two things, either A whatever x may be, if x is man, then x is white or E whatever x may be, if x is man, then x is non-white. Between the third contrary Y and the third subcontrary U, there is a relationship of contradictoriness. Y and U contradict each other. Each negates the other. The proposition One of two things, either A whatever x may be, if x is man, then x is white or E whatever x may be, if x is man, then x is non-white obviously rejects the fact apprehended by the proposition On the one hand, there exists at least one x that is both man and white, on the other there exists at least one x that is both man and non-white. Conversely, the latter excludes the content of One of two things, either A Whatever x may be, if x is man, then x is white or E Whatever x may be, if x is man, then x is non-white.
A and I are the first vowels of the Latin verb AFFIRMO, E and O the two vowels of the Latin verb NEGO. A Whatever x may be, if x is man, then x is white and I There exists at least one x that is both man and white are respectively defined as the universal affirmative and the particular affirmative whereas E Whatever x may be, if x is man, then x is non-white and O There exists at least one x that is both man and non-white are respectively defined as the universal negative and the particular negative. One can see that in AFFIRMO and NEGO, the first vowel symbolizes universal quantity whereas the second one symbolizes particular quantity. Between the two universals A and E there is a relationship of contrariety: both cannot be true together but both can be false together. Between the two particulars I and O there is a relationship of subcontrariety: both can be true together but both cannot be false together. One must also evoke the relationship of contradictoriness between A the universal affirmative and O the particular negative. Traditionally, it is said that two propositions are mutually contradictory if on the one hand they cannot be true together and on the other they cannot be false together. Each can be thought as negating the other. Obviously, A Whatever x may be, if x is man, then x is white and O There exists at least one x that is both man and non-white are mutually contradictory. Whatever x may be, if x is man, then x is white means It is not the case that there exists at least one x that is both man and non-white and conversely There exists at least one x that is both man and non-white means It is not the case that whatever x may be, if x is man, then x is white. In the same way, there is a relationship of contradictoriness between I the particular affirmative and E the universal affirmative. If there exists at least one x that is both man and white, it is not the case that whatever x may be, if x is man, then x is non-white and conversely if whatever x may be, if x is man, then x is non-white, then it is not the case that there exists at least one x that is both man and white.
(Jean KemperNN (talk) 14:27, 16 November 2010 (UTC))
Reading of A,E,I,O. In the modern logical expressions to be found below M represents the predicate MAN and W the predicate WHITE
-(x) M(x) → W(x) Whatever x may be, if x is man, then x is white
- (x) M(x) → ~W(x) Whatever x may be, if x is man, then x is non-white
- (∃x) M(x) & W(x) There exists at least one x that is both man and white
- (∃x) M(x) & ~W(x) There exists at least one x that is both man and non-white
Representation of Y that is I & O and U that is A w E
-(Y) (∃x)M(x) & W(x) & (∃x) M(x) & ~W(x)
-(U) (x) M(x) → W(x) w (x) M(x) → ~W(x)
(Jean KemperNN (talk) 15:16, 16 November 2010 (UTC))
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Further extension
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 n.” The pattern also goes even beyond this.[2]
- ^ N-opposition theory logical hexagon
- ^ Moretti, Pellissier