Talk:Logical hexagon

The article is good because it reads the logical propositions of the logical square as such
(Jean KemperNN (talk) 08:32, 18 November 2010 (UTC)) The article is good. It reads the logical propositions of the square: A E I O as such. For instance A is read "Whatever x may be, if x is a man, then x is white." conformably to the algoritmic expression (x)(M(x) → W(x))of modern logic. Jean-François Monteil, the author of the present remarks, thinks that it is not legitimate to identify A that is (x)(M(x) → W(x)) with the marked universal affirmative of natural language All men are white (or Everyman is white). In his opinion,the logical proposition A Whatever x may be, if x is a man, then x is white exactly represents the common referent of two semantically different universals of natural language: Every man is white on the one hand and Man is white, Men are white on the other. The author of these remarks thinks that if Every man is white and Man is white have the same referent, still they have not the same sense.

They have the same referent in so far as they both make known the same reality: the fact that Whatever x may be, if x is a man, then x is white ,in other words, that the quality 'white'is ascribed to the totality of mankind. It goes without saying that when I examine the expressions: Whatever x may be, if x is a man, then x is white, Men are white, All men are white, I'm acting as logician and linguist and that I don't adhere to their obviously false content.

They have not the same sense in so far as they do not contradict the same antithetic proposition. Man is white, Men are white contradicts Man is not white, Men are are not white whereas All men are white (or Everyman is white)contradicts Not all men are white, Some men are not white. The fact that Some men are not white refers to and the fact that Man is not white refers to are different. What we want to explain is this : the sense of an assertive proposition of natural language is made of two elements: its referent of course but also its power to contradict. References:Tract Eight-8,"knol 000" (Jean KemperNN (talk) 13:09, 19 November 2010 (UTC))

About the representation of the third subcontrary U as A v E
(Jean KemperN (talk) 03:49, 28 December 2010 (UTC)) The article presents U as the disjunction A v E. Hence what one finds in the article to represent U analytically by means of modern algoritmic symbolization: (x)(M(x) → W(x)) v (x)(M(x) → ~W(x)) The statement U may be interpreted as Whatever x may be, if x is a man, then x is white or whatever x may be, if x is a man, then x is non-white. I cannot object to the representation of U as a disjunction A v E that is to say (x)(M(x) → W(x)) v (x)(M(x) → ~W(x)) since it is what we find in Structures intellectuelles of Robert Blanché. However in this talk page I want to explain soon why it would be good to translate U by A w E instead of A v E. A w E or (x)( M(x) → W(x)) w ((x) M(x) → ~W(x)) is to be read something like this: One of two things, either Whatever x may be, if x is a man, then x is white or Whatever x may be, if x is a man, then x is NOT white. (84.100.243.244 (talk) 23:35, 12 January 2011 (UTC)) (84.100.243.244 (talk) 09:01, 28 January 2011 (UTC)) (84.100.243.244 (talk) 02:33, 28 February 2011 (UTC))The great presupposition as far as the logical square and the logical hexagon, the more complete figure, are concerned is this: A and E are necessarily, a priori incompatible. That means that the facts they represent cannot coexist and can be both excluded from reality. The exclusion both of A and E is the conjunction of I (i.e not-E) and O (i.e not-A), which constitutes the third contradictory of Robert Blanché's hexagon symbolized by Y. When U is the case, it means that Y is not the case and that you have not the conjunction: not-A & not-E. On the other hand, you cannot have A & E and that a priori. Therefore, U is equivalent to ~ ( A & E) & ~ (not-A & not-E). Now, p w q (one of two things either p or q) signifies first that one has not both p and q, second that one has not both not-p and not-q. Consequently, if with U you have ~ ( A & E) & ~ (not-A & not-E), you have A W E and not A V E. In my opinion, the drawback of A V E consists in the fact that the form A V E does not exclude explicitly the forbidden conjunction A & E. —Preceding unsigned comment added by 84.100.243.244 (talk) 02:50, 28 February 2011 (UTC) (86.75.111.161 (talk) 22:07, 19 April 2012 (UTC))mindnewcontinent


 * If the equivalent formulas listed on the page are correct, then A and E have no existential import and so both can be true. For example, the statements ″All unicorns are white" and "All unicorns are not white" are both true, so long as there are no unicorns. If the hexagon is supposed to represent contradiction between U and Y then U must mean an inclusive "or" so as to include the case where A and E are both true. Dezaxa (talk) 12:43, 4 June 2013 (UTC)

From the square of opposition to the logical hexagon of Robert Blanché. The two triads
(84.101.36.19 (talk) 22:15, 2 July 2013 (UTC)) The logical hexagon of Robert Blanché adds two values to the four values A, E, I, O represented in the square of opposition or square of Apuleius. The value Y represents a third contrary constituting with A the universal affirmative and E the universal negative a triad of contraries. Y represents the conjunction of the subcontrary I called particular affirmative and the subcontrary O called particular negative. Between I particular affirmative and E universal negative, there is a relation of contradictoriness due to the fact that they differ in quantity and quality. For the same reason, there is a relation of contradictoriness between A the universal affirmative and O the particular negative. It follows that Y, conjunction of I and O, excludes both the content of A and that of E. As to the other value U added by Robert Blanché, it is to Y what I is to E and what O is to A. This third subcontrary constitutes with the two traditional subcontraries I and O a triad of subcontraries. The subcontrary U can be represented as the exclusive disjunction of A and E. If U contradicts Y, if, in other words, it rejects the fact that  the universals A and E might be both false, that means that one of two things, either A or E is true. Hence the fact that U can be represented by the exclusive disjunction: A w E. (84.101.36.131 (talk) 15:13, 10 July 2013 (UTC))

Some information about the six logical propositions represented in the logical hexagon.
(84.101.36.28 (talk) 13:34, 26 July 2013 (UTC)) The logical hexagon of Robert Blanché adds two propositions Y and U to the four logical propositions A, E, I, O represented in the square of opposition of Aristotle (or Apuleius). Y is a third contrary constituting with A, the universal affirmative and E, the universal negative a triad of contraries A E Y. Y is the conjunction of the subcontrary I called particular affirmative and the subcontrary O called particular negative. Between I particular affirmative and E universal negative, there is a relation of contradiction (in the sense of contradictoriness). Due to the fact that I and E differ both in quality (affirmative versus negative) and in quantity (particular versus universal). The same relation of contradiction (or contradictoriness) exists, and for the same reasons, between A the universal affirmative and O the particular negative. It follows that Y, conjunction of I and O, excludes both the content of A and that of E. As to the other proposition U added by Robert Blanché, it is to Y what I is to E and what O is to A. With the two traditional subcontraries: I and O, the third subcontrary U constitutes a triad of subcontraries U I O. The subcontrary U can be represented as the exclusive disjunction of A and E. If U contradicts Y, in other words, if U rejects the fact that the universals A and E are both false, that means that either A or E is true. Hence the fact that U can be represented by the exclusive disjunction: A w E to be read One of two things, either A or E.

Somewhat detailed information about contradictoriness, contrariety, subcontrariety in the logical hexagon of Robert Blanché
(84.101.36.28 (talk) 13:35, 26 July 2013 (UTC)) In the traditional square of opposition, two logical propositions differing in quantity (universal versus particular) and in quality (affirmative versus negative) are said to be contradictory. That means that because they cannot be both true nor both false, each of them is the negation of the other. The square presents two pairs of contradictories: A versus O, I versus E. For illustration, let us choose the matter of the judgment, utilized by Aristotle in De Interpretatione (or On Interpretation) Chapter 7, which chapter is at the origin of the square of opposition. On the one hand as subject a set, the set of men, on the other as predicate, the quality white. Let us give to the four propositions the form that modern logic of predicates gives to them. One has

-A with the form (x) m(x) ⊃ w(x)) Whatever x may be, if x is man, then  x is white, -E with the form (x) m(x) ⊃ ~w(x)) Whatever x may be, if x is man , then  x is non-white, -I with the form (∃x)(m(x) & w(x)) There exists at least one x both man and white. -0 with the form (∃x)(m(x) & ~w(x)) There exists at least one x both man and non-white. If, for instance, the logical proposition Whatever x may be, ifx is man, thenx is white is true, then the logical proposition 0 There exists at least one x both man and non-white is false. Conversely, if the logical proposition 0 There exists at least one x both man and non-white is true, then the logical proposition A Whatever x may be, ifx is man, thenx is white is false. The logical proposition A, universal in quantity, affirmative in quality, and 0, particular in quantity, negative in quality, are contradictory. Each can be thought as negating the other. The same relation of contradictoriness exists between the logical proposition I, particular in quantity, affirmative in quality and the logical proposition E, universal in quantity, negative in quality. Whatever x may be, if x is man, then x is non-white contradicts There exists at least one x both man and white. Conversely, There exists at least one x both man and white contradicts Whatever x may be, if x is man, then x is non-white. Traditionally, one says that the two logical universals A and E are mutually contrary because they cannot be both true but can be both false. The content of Whatever x may be, if  x is man, thenx is white and that of Whatever x may be, if  x is man , then  x is non-white are obviously incompatible. But as the content of A and that ofE can be both excluded, when one is in the position to say that only some men are white, the two universals A and E can be both false. Traditionally, one says that the two logical particulars I and 0 are subcontraries Thereby, one means that they can be both true but not be both false. The two subcontraries I and O are both true when the two contraries A and E are both false. The conjunction ofI and O, when I and O are both true, may be represented thus : (∃x)(m(x) & w(x)) & (∃x)(m(x) & ~w(x)) On the one hand, there exists at least one x both man and white, on the other, there exists at least one x both man and non-white. I and 0 cannot be both false. In effect, since the universalE is the contradictory ofIand the universal A the contradictory of 0, ifI and O were both false, that would mean that the universals A and E would be both true, which is absolutely impossible. The conjunction of I and O: (∃x)(m(x) & w(x)) & (∃x)(m(x) & ~w(x)) is nothing else than the case where the two contraries of the square of opposition are both false. This case is what Y symbolizes. It is not represented in the square of opposition, it is represented in the hexagon. Therefore, the logical hexagon of Robert Blanché is superior to the square of opposition originating from Aristotle because it is obviously more complete. The proposition Y assertingI & O : On the one hand, there exists at least one x both man and white, on the other, there exists at least one x both and non-white (∃x)(m(x) & w(x)) & (∃x)(m(x) & ~w(x)) constitutes the third contrary. As Y is the third contrary introduced by Robert Blanché’s hexagon, so U is the third subcontrary introduced by the same. The subcontrary U entertains a relation of contradictoriness with the contrary Y. U is to Y what the subcontrary I is to the contrary E and what the subcontrary 0 is to the contrary A. The subcontrary U consists in rejecting the case where A and E might be both false. Consequently, it consists in saying that there are merely two possibities either one has the content of the universal affirmative A or that of the universal negative E. Hence the fact that the third subcontrary U can be advantageously represented in the form of an exclusive disjunction : A w E. Y entertains a relation of contrariety with each of the traditional contraries: A and E. For example, Y and E cannot be both true since the content of Y and that of E are incompatible. Y et E can be both false when it is the case that the contrary A is true. U entertains what one could call a relation of subcontrariety with each of the two traditional subcontraries of the square. For instance, let us consider the relation of subcontrariety between U and I. The two propositions U and I can be both true. That occurs when the universal affirmative A is true. The content of the universal affirmative A : Whatever x may be, if x is man, then x is white excludes the content of the proposition  Y : On the one hand, there exists at least one x both man and white, on the other, there exists at least one x both man and non–white and, of course, excludes the content of the universal negative E: Whatever x may be, if x is man, then x is non–white. The subcontraries U and I cannot be both false for if they were false, that would mean that Y contradicting U and that E contradicting I are both true, which is absolutely impossible since Y and E are mutually contrary.

You are still completely missing the point about existential import. If A means, as you state, (x) m(x) ⊃ w(x))  then this does not assert the existence of any m. Neither does E. On the other hand I and O do assert existence. This means that A and E can both be true, in the event that there are no m, and I and O can both be false. To see this, just substitute unicorns for men in your example. Dezaxa (talk) 16:07, 27 July 2013 (UTC) (86.75.111.153 (talk) 15:38, 28 July 2013 (UTC)) Thanks for the remark. I shall think of it earnestly. Jean-François Monteil PS Type on Google : KNOLmnc 0 Sites and topics