# Reviews

## Sung: Part II - Their Exibits

### Chapter IV - Logical Exhibits

This chapter is subtitled "A Whole System of the Symbols of Yi King" and presents an anlysis of the syllogism and then of dichotomous categorization.

An good explanation of the syllogism on the web is given by Wikipedia. A good book source is Copi's Introduction to Logic [Cop] but many introductory logic texts will contain adequate material on syllogistic reasoning.

The syllogism analysis begins by considering the figure of a syllogism (pp59-60). There are generally taken to be four figures, determined by the placement of the middle term, but Sung choses to align them with the trigrams, thus requiring an eight way characterization. This is obtained by taking the four figures associated each with one trigram with a yang base line, and then taking them again with converted conclusions, each associated with a trigram with a yin base line.

However, converting the conclusion does not create new figures. For example, taking the first figure with a converted conclusion is equivalent to the forth figure. This would imply that the trigram connected with the first figure with the converted conclusion (110) is logically equivalent to the trigram connected with the forth figure (100). Sung does not mention this logical equivalence and therefore does not discuss its implications for the trigrams.

A more natural association would be between the figures and the bigrams:

• 1st: 10
• 2nd: 00
• 3rd: 11
• 4th: 01
This mapping takes the first line of the bigram as symbolzing the position of the middle term in the first premise (1 when distributed, 0 when not) and the second line as symbolizing the midde term in the second premise in a similar manner. No doubt there are other mappings that are equally appropriate.

Sung next discusses the moods of the syllogisms (pp61-63). There are sixtyfour moods, derived as from the three-place permutation of four basic sentence types. The obvious connection then, is of the sentence types to bigrams, and the resulting moods to the hexagrams. This is indeed the route that the author takes. He makes the following associations:

• A (universal affirmative): 11, greater yang.
• E (universal negative): 00, greater yin.
• I (particular affirmative): 10, smaller yang.
• O (particular negative): 01, smaller yin.
I presume that this mapping stems from the traditional names of these bigrams: universal affirmative is greater yang, particular affirmative is smaller yang, and so on.
 From C. F. Russell, p VIII
Interestingly, this is exactly the same mapping chosen by Russell, [Rus67, p viii]. This is shown in the figure from p. viii of his Prolog shown here. Although this analysis thus appears in two separate unconnected works, it is not unproblematical. Consider that 00 is the complement of 11, so we might expect 00 to represent the negation of 11. But the negation of the universal affirmative is the particular negative, not the universal negative.

What this shows, and what Russell's diagram emphasises, is that the analysis is not compositional. For example, "universal affirmative" is not an atomic concept, it is composed of the components "universal" and "affirmative". One of those components is shared with the compound concept "universal negative" and another is shared with the concept "particual affirmative". However, in the analysis suggested, there is no way of mapping the individual elements of the Yi representation to the individial elements of the concept being mapped, or vice versa.

Given that logic, and indeed the Yi itself, both have a strongly compositional nature, I take this to be a serious draw back.

More could be said on this topic: there are different types of logical negation (see, for example, Copi's description of the square of opposition [Cop72, p155-159]). However, such a view does not give us a consistent picture of the role of the complement operator. When negating A or E it must be seen as a contrary operator and when negating I or O it must be seen as a subcontrary operator.

This lack of a consistent interpretation of the complement operator is not satisfying. Instead, I suggest the following mapping is more appropriate to the syllogistic domain:

• A (universal affirmative): 11
• E (universal negative): 10
• I (particular affirmative): 01
• O (particular negative): 00
That is, let the first line indicate whether the sentence is universal or particular, and the second line whether it is affirmative or negative. Under this mapping, the logical negations work out as Boolean complements on the bigrams: the representation of the universal affirmative is 11 and the complement of that is 00. However, under the revised mapping, 00 represents the particual negative, which is correct as the negation of the universal affirmative. Thus, we can consistently view the complement operator as producing contradictories, which is the usual interpretation of negation.

Whatever mapping is chosen, there is then a clear relationship between moods and hexagrams: AAA is 111111, EIO is 011000 and so on. Sung presents this on p61.

This is interesting, but what the author does not address is that not all of the combinations of moods and figures (totalling 256 distinct syllogism types) represents a valid argument. I'd venture to suggest that without some way of distinguishing valid from invalid syllogisms, on the basis of their resulting hexagram/bigram mappings, that the translation is a pointless exercise. Without such a mechanism, all the author has shown is that there is a systematic way of mapping a given 64-way characterization on to the hexagrams.

It seems to me that the point of mapping any domain onto the symbols of the Yi is either to illuminate some aspect of that domain or to investigate the Yi from a new perspective. Or both. If it does neither of these things then it is an empty intellectual exercise.

The discussion of dichotomy (pp64-65) is straightforward and follows the development of gua by repeated addition of lines (i.e., Shao Yong's Xiantian development).