A Logical Perspective on Discourse of White Horse

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A Logical Perspective on “Discourse on White-Horse”

Yiu-ming Fung - Division of Humanities, HKUST http://www.ust.hk/~webhuma/publications/A%20Logical%20Perspective%20on%20Discourse%20on%20White%20Horse.pdf

(1) The Issue of Reference in “Discourse on White-Horse”

In “Discourse on Name and Reality” (Ming-Shih Lun名實論) Kung-sun Lung defines the term “name” as “denoting reality” (名者實謂也). Even though his “reality” has different interpretations in the literature, the relation that the name is connected to the reality can be unequivocally identified as referential. We know from the historical description in Ssu-ma Ch’ien’s (司馬遷) Record of History (Shih-chi 史記) that the most important issue of Kung-sun Lung’s thought is the dialectical discussion of “chien-pai” (堅白hard and white) and “tung-yi” (同異 identity and difference). If we put this in the words of contemporary philosophy of language, the issue can be understood as discussion about some specific and general problems of reference.

With respect to the issue of reference in the book of Kung-sun Lung Tzu (公孫龍子), especially in “Discourse on White-Horse” (白馬論Pai-Ma Lun), there has been no consensus in the field about the logical function of the name, what kind of reality denoted by the name, and the mode of reference. However, some of the famous or influential interpretations based on different theoretical considerations for analysing and explaining the issue in the Discourse can be found in the literature. As I know, there are at least four representative interpretations which address different opinions on the issue from different perspectives. The first one is Hu Shih’s (胡適) “Description-Theoretical Interpretation” which seems to be an application of Bertrand Russell’s theory of definite descriptions or, more accurately, is an interpretation which is unconsciously based on the theory; the second one is Fung Yu-lan’s (馮友蘭) “Realistic Interpretation”, an application of Platonic “Idea” to the text in a quite consistent way; the third one is Janusz Chmielewski’s “Set-Theoretical Interpretation” which assigns a logicist status to Kung-sun Lung in terms of a particular notion of set; the last one is Chad Hansen’s “Nominalistic Interpretation” which is based on a bold hypothesis of mass noun and is considered by himself as a replacement of the above three and other kinds of abstract interpretations.

In the next section I will discuss the main points of these four old interpretations and 1demonstrate their theoretical characteristics and difficulties in dealing with the issue inside or outside the Discourse. In the third section I will use the first-order predicate logic to analyse the logical structures of both Kung-sun Lung’s and his opponent’s arguments, and then I will borrow Saul Kripke’s and other theorists’ direct theory of reference to explain the issue of reference in the Discourse. Based on these analysis and explanation, I think, I can provide not only a comprehensive and coherent interpretation for the Discourse, but also a solid base for the interpretation and explanation of the remaining chapters in the book of Kung-Sun Lung Tzu which seems unable to be done by the old interpretations.

Briefly speaking, “nominalism” means a view that all referring terms in a language (for example, our natural language) can be used to refer only to concrete entities, such as physical objects or sensible things; all other (abstract) terms are used non-referentially and are nothing but descriptions of psychological association (i.e. psychological nominalism) or linguistic abstraction of concrete entities (i.e. linguistic nominalism). “Realism” as an opposite term of “nominalism” means a contrary view that all terms of not merely syntactically functioned, either concrete or abstract, are used referentially in the sense that concrete terms refer to concrete entities while abstract terms refer to abstract entities, such as Platonic universals or other kinds of non-sensible objects. We can say that an interpretation based on nominalism is a concrete interpretation while that based on realism an abstract one. As regard to the issue in “Discourse on White-Horse”, the terms “white” and “horse” are interpreted by Fung Yu-lan as names referring to Platonic universals as whiteness and horseness respectively; it is obvious that his interpretation is abstract and can be identified as based on (Platonic) realism. Instead, Chmielewski considers these terms as class names; this is also an abstract interpretation though class or set is not usually understood as Platonic Idea. Hu Shih thinks that what the name refers to is wu-chih (物指attribute of thing), such as shape and colour; that is the attribute or property of physical object. Since Hu does not explain the ontological status of wu-chih, i.e. whether the attributes or properties are ontologically independent of physical objects, it seems that we do not have enough evidence to identify his interpretation as concrete or abstract. Among these four representative interpretations, Hansen’s nominalistic interpretation is the only clear case of the non-abstract characteristic. His mass noun hypothesis helps him to explain away not only all the abstract reference in the text but also all concrete reference of physical objects, because only the names of mass-stuff can be considered as referring terms under his interpretation.

2(2) A Review of Old Interpretations

(2.1) Hu Shih’s Description-Theoretical Interpretation

After analyzing Kung-sun Lung’s “Discourse on Chih (Marks / Fingers) and Wu (Things)” (指物論), Hu Shih concludes that “the word ‘finger’ here means ‘a mark’ or ‘a sign’, that which ‘signifies’….It seems that by ‘mark’ or ‘sign’ is here meant the attribute or quality by which a thing is known. ‘There are no things which are not marks’ [Wu-mo-fei-chih 物莫非指] means that things are what their attributes indicate them to be; that is, what they are perceived to be. This subjectivism is immediately qualified by the realistic statement, ‘But marks are no marks’ [Erh-chih-fei-chih而指非指]; that is, marks are not entities in themselves but marks of things. ‘For without things, can there be marks?’ [Tian-hsia-wu-wu, ke-wai-chih-fu? 天下無物 / 可謂指乎]” So, taking “fingers” to mean “marks” or “attributes of things”, Hu thinks that he can analyze all kinds of the so-called paradoxes mentioned by Kung-sun Lung in a sensible way. For example, he says, “We know a horse by its horse-ness [not as a Platonic universal but as a perceived attribute], a white horse by its white-horse-ness [also not as a Platonic Idea], and a white stone by its white-ness and solidity.” Based on this general thesis, one of the interpretations for the statement “White-horse-not-horse” (白馬非馬) made by Hu Shih can be elaborated as follows: Since the term “white-horse” signifies the attribute of white-horse, the term “horse” signifies the attribute of horse, and the former attribute is not identical with the latter attribute, therefore we can assert that “white-horse is not horse”. (Hu Shih, The Development of the Logical Method in Ancient China (Shanghai: The Oriental Book Company, 1922), p.126. )


According to Hu’s thesis of “peu-de” (表德 attribute), the relation between name and reality can be understood as that between description and the property described. Put in the words of contemporary philosophy of language, it is an indirect relation of reference in the sense that a name or description can fix or identify its referent through its descriptional property, attribute, or condition; it cannot directly refer to the object which has the property or attribute. For example, Hu mentions that, “After obtaining ‘wu-chih’ [attribute of thing], we can have ‘ming’ [名 name]. The name of an object is the symbol which represents all the attributes of the object, such as ‘fire’ representing all the attributes of fire, ‘Mei-lan-fong’ [梅蘭芳] (Hu Shih (胡適) An Outline of History of Chinese Philosophy, vol.1 (中國哲學史大綱, 卷上) (Shanghai: Commercial Press, 1947) (商務印書館) p.248. (胡適說:「有了『物指』,然後有『名』。一物的名 乃是代表這物一切物指的符號。如『火』代表火的一切性質,『梅蘭芳』代表梅蘭芳的一切狀態 性質。」))

This seems to be an application of the decriptional theory of reference or the indirect theory of reference originated from G. Frege, Bertrand Russell, and L. Linski. According to this theory, a term’s meaning can be divided into two parts: its reference (or referent) and its sense (or the Fregean Sinn), and the sense of a term can be used as a criterion of identification of its reference. For Russell and also for Hu, a proper name is nothing but an abbreviation of a series of descriptions (i.e. a conjunction or disjunction of descriptions). In this sense, all referring terms can refer to their referent only indirectly through their descriptional sense; it is quite different from J. Mill’s idea that a proper name has no sense but refers to its referent directly if it’s referent is not empty. For example, the proper name “Plato” can be understood by a theorist of indirect reference as an abbreviation of a series of descriptions such as “Socrates’s student”, “Aristotle’s teacher”, and so on. If an object (person) satisfies the attributes described by a series of descriptions, we can base on this criterion to identify the object (person) as the referent of the name (The theoretical characteristics of the descriptional theory of reference or the direct theory of reference is discussed in detail in N. Salmon, Reference and Essence (Basil Blackwell: Oxford, 1982), pp.9–22. ).


Let’s put aside the problem whether the decriptional theory is right or not. Even though the theory is right, Hu Shih’s (unconscious) application of the theory to the interpretation of the text or his similar idea in the interpretation of the text still has serious problems. With respect to the issue in “Discourse on White-Horse”, according to Hu’s interpretation, “white-horse” represents two attributes while “horse” represents only one attribute, so we can arrive at the conclusion as claimed by Kung-sun Lung that “White-horse is not horse”. Although an object with two attributes is not identical with an object with only one attribute, it is obvious that having two attributes can be said having one of the two attributes. However, we know from the text that Kung-sun Lung does obviously assert that “having white-horse cannot be called having horse” (You-pai-ma-bu-ke-wai-wu-ma 有白馬不可謂有馬), because, as indicated by the text, horse can be separated from white as an independent entity. It is what Kung-sun Lung says about the mere horse without selecting or excluding any colour (Wu-chu-chiu-yeu-tse-chi-ma 無去取于色之馬) and the mere white without fixing into any thing (Bu-ting-shao-pai 不定所白). We know that “Discourse on Hard and White” (Chien-Pai Lun 堅白論) also has the same idea of “non-fixing” for “white” and for “hard”; it says that these non-fixing entities can be separated and self-hided (Li-erh-ji-choung 離而自藏) from the sensible world. In “Discourse on Understanding Change” (Tung-Bin Lun 通變論), if we follow Hu’s idea to treat the concept “one” as meaning one attribute that an object has and “two” as meaning two attributes that another object has, it would be impossible for us to assert that “two does not have one” (Erh-wu-i 二無一). However, if we treat “one” as a non-sensible simple and “two” as a sensible compound which is combined from two separate one and thus changed into a fixing whole in the sense that “the right [one] with a compound partner (the left one) can be called ‘changed’ ” (右有與 / 可謂變), then the above sentence would be intelligible. The simple one is separated from others in the sense that it is without any partner to constitute a compound, and is thus not yet to be changed into an object in the sensible world. In other word, this non-fixing simple is unchanged. When it is combined with others in the sense that it has partner to form a compound object in the sensible world, it is changed. Since “the changed is not the unchanged” (Bin-fei-bu-bin 變非不變), we can follow Kung-sun Lung to say that “the Left [one] combined with the right [one] can be called two” (Cho-yu-you-ke-wai-erh 左與右可謂二) and at the same time to say that “two does not have one”, “two does not have left” (Erh-wu-cho 二無左), and “two does not have right” (Erh-wu-you 二無右). This coherent picture inside or outside “Discourse on White-Horse” is obviously unable to be drawn by Hu Shih’s pen.

In other words, his description-theoretical interpretation is not sound.

(2.2) Fung Yu-lan’s Realistic Interpretation

Fung thinks that, instead of emphasizing, as did Hui Shih, that actual things are relative and changeable, Kung-sun Lung stresses that names are basically absolute and permanent. In this way, as indicated by Fung, Kung-sun Lung arrives at the same concept of Platonic Ideas or universals. Among the main three arguments in “Discourse on White-Horse”, Fung identifies the third argument as the most important one which demonstrates a realistic idea of general terms. He says: “The third argument is: ‘Horses certainly have colour. Therefore there are white horses. Suppose there is a horse without colour, then there is only the horse as such. But how, then do we get a white horse? Therefore a white horse is not a horse. A white horse is horse together with white. Horse with white is not horse.’ In this argument, Kung-sun Lung seems to emphasize the distinction between the universal, ‘horseness’, and the universal, ‘white-horseness’. The universal, horseness, is the essential attribute of all horses. It implies no colour and is just ‘horse as such’. Such ‘horseness’ is distinct from ‘white-horseness’. That is to say, the horse as such is distinct 5from the white horse as such. Therefore a white horse is not a horse.”

Evidence from other places in the Discourse also confirms this Platonic idea through Fung’s interpretation. For example, “White [as such] does not specify what is white. But ‘white horse’ specifies what is white. Specified white is not white.” Specified white interpreted by Fung as the concrete white colour which is seen in this or that particular white object. The word here translated as “specified” is ting (定), which also has the meaning of “determined”. So, he thinks that the concrete white is determined by this or by that object but the universal whiteness is not determined by any particular white object. It is the whiteness unspecified.

Fung’s interpretation seems also applicable to some arguments in “Discourse on Hard and White”, another famous piece of Kung-sun Lung’s writings. Fung’s analysis is that, since both hardness and whiteness, as universals, are unspecified in regard to what particular object it is that is hard or that is white, they can be manifested in any or all white or hard objects. The epistemology that “seeing does not give us what is hard but only what is white” (Sse-bu-de-ch’i-shao-chien-erh-de-ch’i-shao-pai 視不得其所堅而得其所白) and “touching does not give us what is white but only what is hard” (Fu-bu-de-ch’i-shao-pai-erh-de-ch’i-chien 拊不得其所白而得其所堅), for Fung, suggests that such hardness and whiteness are quite independent of the existence of physical stones or other objects that are hard and white. Fung uses a general term to refer to this kind of objective and abstract entities, that is the key term “chih” (meaning “finger”, “pointer”, “idea”, “concept” or “to indicate”) in Kung-sun Lung’s another piece, “Discourse on Chih and Wu” (指物論).

Unlike Hu Shih’s thesis of “attribute”, Fung Yu-lan’s thesis of “universal” has to presuppose two levels of existence or two different but related worlds. Similar to Plato’s two-world theory, he also makes a distinction between the world of particulars and the world of universals. He thinks that “On the one hand, the referent of a name is an individual, this is the so-called ‘a name is denoting reality’. On the other hand, the referent of a name is a universal. For example, in addition to this [particular] horse and that [particular] horse, there is the horse[ness] in the sense of ‘having horse as such’; in addition to this [particular] white thing and that [particular] white thing, there is the

4 Fung Yu-lan, Selected Writings of Fung Yu-lan (Beijing: Foreign Languages Press, 1991), p.289. 5 Op. cit., p.289-290 6 Op. cit., p.291. 6white[ness] in the sense of ‘the white as non-fixing white’. These ‘horse[ness]’ and ‘white[ness]’ are the so-called ‘universals’ or ‘elements’ in modern [Western] philosophy. These are also the referents of names. Kung-sun Lung makes a distinction of chih [指] and wu [物], his chih can be recognized as the universal referred by the name.” 7 It is obvious that Fung’s interpretation is based on Plato’s realism. According to Fung’s view, Kung-sun Lung’s “wu” designates all the concrete individual things in our sensible world, his “chih” means all the abstract universals which are ontologically independent of the objects in the sensible world, and his “wu-chih” (物指) is used to characterize all the concrete properties of sensible objects which are emerged from the chih’s embodying or fixing into particular things. Based on this triple distinction, Fung is able to explain what Hu cannot explain. For example, Fung can explain why Kung-sun Lung says that “having white-horse cannot be called having horse” in the sense that having a particular white-horse (or having the universal white-horse-ness) is not having the universal horseness. In other words, Kung-sun Lung’s specific thesis “white-horse-not-horse” can be understood as an example of the general thesis “wu is not chih”, “the fixing white (which is fixing into a concrete object such as white-horse or white-stone) is not the non-fixing white” can also be considered as an example of “[wu-]chih is not [tu-]chih” (獨指 single, simple, and separate universal). Unfortunately, Fung does not use this triple distinction to interpret the whole book of Kung-sun Lung Tzu in a consistent way. He considers all the referring terms in the book, no matter it is a simple or a compound term, can be used to refer to universals. For example, he asserts that, “the referents of ‘horse’, ‘white’, and ‘white-horse’ are what is called by ‘chih’ in ‘Discourse on Chih and Wu’ of Kung-sun Lung Tzu.” 8

They are all considered by Fung independently separate universals. However, if both white-horse and horse are universals, we would be unable to use the distinction between wu and chih to interpret the difference between these two universals. Furthermore, if Fung accepts that “white-horse” represents something of the characteristic of “an object of fixing into white”, it is definitely not something of the characteristic of non-fixing. Or, if he accepts that “white-horse” refers to something which selects some colour (i.e. white), it is clearly not something which does not select any colour. Put the words more generally, if white-horse is a fixing object, and fixing is not non-fixing, then white-horse is not non-fixing. Since non-fixing is universal, we can say that white-horse in this sense is not universal. Based on Kung-sun Lung’s acceptance

7 Fung Yu-lan (馮友蘭) A History of Chinese Philosophy (中國哲學史) (Hong Kong: Pacific Book Company, 1961)(香港:太平洋圖書公司)p257. 8 Op. cit., p.258. 7of his opponent’s assertion that “there is white-horse in tian-hsia [天下this world]” and his own assertion that “Chih is not something in tian-hsia; but wu is something in tian-hsia” (Chih-yeh-tse-tian-hsia-chi-sho-wu-yeh, wu-yeh-tse-tian-hsia-chi-sho-you-yeh 指也者天下之所無也 / 物也者天下之所有也), it is obvious that his “white-horse” is used as an individual name like “this [white-horse]” or “some [white-horse]” which refers to something in this sensible world, while horse[ness] as chih is not the member of this world but something separate and self-hided from this world.

Fung’s interpretation is also handicap in dealing with “Discourse on Understanding Change”. He interprets “two” as referring to a universal which is different from the universal referred by “one”. But this cannot help in explaining both “two has no left / right [one]” and “two is combined from left [one] and right [one]” (二者左與右) consistently. Because what is referred by “two” is “changed”, and what is named by “one” is “unchanged”. Kung-sun Lung’s thesis on “the changed is not the unchanged” is inevitably committed to two ontological levels for the different residence of the changed and the unchanged. If I am right at this point, to interpret “two” as universal is inaccurate. The accurate interpretation is to understand the term “two” as referring to something combined from fixing the two independent and separate chih (one) into a compound wu. According to this interpretation, the expressions “two has no left / right [one]”, and “two is combined from left [one] and right [one]” can be interpreted as that “the compound wu has no simple and unchanged chih in it, though it is emerged from two unchanged chihs’ in terms of their fixing into something combined which is changeable.” In comparison with this interpretation, I can say that the shortcomings of Fung’s interpretation are coming from his realism not all the way down.

(2.3) Chmielewski’s Set-Theoretical Interpretation

Janusz Chmielewski interprets Kung-sun Lung’s “horse” and “white”, not as general terms, whether referring to abstract entities (i.e. horseness and whiteness) or to individual objects within the scope that the concept characterizes, but as names of class or set. He thinks that, from the premises “horse is what commands shape [and only shape]”, “white is what commands colour [and only colour]”, and “what commands colour [and only colour] is not what commands shape [and only shape]”, we cannot derive the conclusion “White horse is not horse” as commonly thought. However, he continues, if we consider Kung-sun Lung’s argument as based on a special kind of theory of classes, “narrow and incomplete” as not admitting class inclusion and its negation, we can infer the conclusion 8“the class of white objects is not identical with the class of horses”. Since “what commands colour [and only colour] is not what commands shape [and only shape]” indicates the intersection is null, and white horse is an instance of this kind of intersection, we can validly infer the conclusion “White horse is not horse”. (From (B≠A) we can have (B˙A)≠(A˙A) and (B˙A)≠A) if not admitting (A⊂B).) From this and from another Kung-sun Lung’s argument, Chmielewski infers that in “White horse is not horse” Kung-sun Lung uses the negative copula fei (非 not / non) solely to deny identity, and is misunderstood by the attackers who supposes him to be denying class inclusion. Mou Tsung-san (牟宗三) also interprets Kung-sun Lung’s concepts in “Discourse on White-Horse” as names of class; but he does not provide detailed formulation for the Discourse with this idea. Chmielewski’s interpretation is more specific. He thinks that the terms, such as “white” and “horse”, are used in the Discourse to denote the class of white objects and the class of horses respectively, and the term “white horse” to denote the intersection of these two sets which results in an empty class or null set. I think one of the reasons why Kung-sun Lung regards “white horse” as a name of empty class, for Chmielewski, is that if the class is not empty, Kung-sun Lung would have to accept that the class of white horse is included in the class of horse, i.e. the former is a sub-set of the latter. According to Chmielewski’s view, however, Kung-sun Lung accepts only the relation of non-identity but not that of class inclusion between them. 9 If we agree that white horse is an empty class and horse is not, the conclusion “White horse is not horse” would be naturally deduced. Although Chmielewski’s interpretation seems to be coherent in some sense, it cannot escape from the awkwardness or oddity in understanding Kung-sun Lung’s thought. According to the naïve set theory, for example, an empty set can be regarded as a sub-set of any set; it is no evidence to assert that Kung-sun Lung does not accept the class inclusion between white horse and horse because the former is empty. To understand Kung-sun Lung’s thought as rejecting class inclusion and as asserting the empty status of the class of white horse is not in accord with the text. It is obvious that one of the plausible interpretations of Kung-sun Lung’s expression, “Someone who seeks horse will be just as satisfied with yellow-horse or black-horse” (求馬黃黑馬皆可致), is that the class of yellow-horses or the class of black-horses is included in the class of horses.

9 Januz Chmielewski, “Notes on Early Chinese Logic, (I)”, Rocznik Orientalistyczny, Vol.26, No.1 (1962), pp. 12–18. 9“White-horse” is not a name of empty class, because Kung-sun Lung agrees with his opponent that “there is white-horse” (有白馬). He says, “Certainly horse has colour, which is why one has white-horse.” If white-horse is an empty class, so is yellow-horse, how can Kung-sun Lung asserts the difference between white-horse and yellow-horse through the question “If you deem having white-horse having horse, is it admissible to say that having white-horse is to be deemed having yellow-horse”? It is clear from the text that white-horse is not an empty class and Kung-sun Lung does not reject the possibility of having the relation of class inclusion between white-horse and horse if the relevant terms are really used as class names. Furthermore, most importantly, it is evident from the text that Kung-sun Lung makes a distinction between the fixing white and the non-fixing white, and their names can both be simplified as “white”. If we use the term “white” as a class name, their difference would not be identified except that we could provide two different class names for these two kinds of white. Chmielewski’s set-theoretical interpretation of the Discourse is mainly focused on the argument of “naming shape and naming colour” and on that of “rejecting-selecting colour”. He uses the former as the main argument to demonstrate his interpretation. The common translation of this first debate is as follows:

Q: Is it admissible that white-horse is not horse? A: It is admissible. Q: Why? A: “Horse” is that by which we name the shape, “white” is that by which we name the colour. To name the colour is not to name the shape. Therefore I say, “White-horse is not horse”. 「白馬非馬,可乎?」 「可。」 「何哉?」 「馬者,所以命形也;白者,所以命色也。命色者非命形也。故曰:白馬非馬。」

Chmielewski’s translation is quite different from the common one. According to J.W. Hearne’s observarion, Chmielewski’s interpretation is quite different from Graham’s. There are at least three points deserved of notice: (1) In this first round debate, he translates the term “ming” (命) in “ming-hsin” (命形) and “ming-tse” (命色) not as “ming-ming” (命名 to name), but as “ming-ling” (命令 to command); (2) He does not 10treat “pai” (白 white) and “ma” (馬 horse) as names mentioned or as names used to refer to itself, but as names used to refer to the class of white objects and to refer to the class of horses respectively; (3) He considers Kung-sun Lung’s reasoning as a class theory in a special narrow sense: a theory which stresses the relation of non-identity and excludes the relation of class inclusion.

10 His translation can be summarized in the following four sentences:

11

(1) Horse is what commands shape [and only shape]; (2) White is what commands colour [and only colour]; (3) What commands colour [and only colour] is not what commands shape [and only shape]; (4) …White horse is not horse.

It can be formulated into the following forms:

(1’) ΦA (2’) ΨB (3’) (Ŷ) ΨY · (Ŷ)ΦY = 0 (4’) B · A ≠ A

Here “Φ” represents “is what commands shape [and only shape]”, “Ψ” represents “is what commands colour [and only colour]”, and “A” and “B” represent “the class of horses” and “the class of white objects” respectively. In this regard, to treat “ming” not as “to name”, but as “to command”, seems to be able for Chmielewski to explain away the complicate problem of interpretation in meta-language, but his translation still makes no sense though “ming” used as “to command” could be understood as a metaphor.

Furthermore, to add the words “and only shape” into the text after the original sentence is not an acceptable reading; and to treat the original sentence or the common reading of the sentence “to name the colour is not to name the shape” (Ming-tse-che-fei-ming-hsin-yeh 命色者非命形也) as an empty intersection is also not a reasonable interpretation. This interpretation for (3) cannot put the word “fei” (not or non) into a readable place in ancient Chinese as meaning the same as in (4).

10

J. W. Hearne, Classical Chinese as an Instrument of Deduction (Dissertation), University of California 

Riverside, 1980, p.36. 11 Chmielewski, op. cit., pp.12-18. 11If we use the first–order predicate logic to formulate the argument, it can be understood as an argument with three explicit premises together with two implicit premises as follows: (i) “Horse” is used to name something of the shape [i.e. the shape of horse]. (ii) “White” is used to name something of the colour [i.e. the colour of white]. (iii) To name something of colour is not to name something of shape. (iv) Therefore, “white-horse” is not “horse”.

(i) 馬者所以命形也。
(ii) 白者所以命色也。
(iii) 命色者非命形也。
 ∴ (iv)  白馬非馬。

Here we use “a1”, “a2”, and “a3” to stand for the name “horse” mentioned, for “white” mentioned, and for “white-horse” mentioned, respectively, use “b1”, “b2”, and “b3” to represent “the entity shape” (or “the shape of horse”), “the entity colour” (or “the colour of white”), and “the entity of both shape and colour”, respectively, and use “R” to represent the relation “to name”. The argument form can be elaborated as below:

   (i’) (a1Rb1) 
   (ii’) (a2Rb2) 
(iii’) (∀x)(∀y){[(xRb2).(yRb1)]→~(x=y)}
 ∴ (iv’) ~( a3=a1) 

According to this formulation, (iii) says that, for all objects used to name colour and for all objects used to name shape, they are not identical. Based on these three premises, we cannot validly infer the conclusion “white-horse-not-horse”. However, if we accept (i) and (ii), it is quite natural to think that Kung-sun Lung implicitly takes for grant that “the name ‘white-horse’ is used to name the entity of both colour and shape (or of both the colour of white and the shape of horse)” (Pai-ma-ji-ming-tse-yi-ming-hsin 白馬既命色 亦命形). For the reason of economy of writing and printing in the ancient time of China, if something is understood, it is no need to say it explicitly. To make it explicit, we can infer the conclusion from the premise (i) together with the following implicit premises which are consistent with and shown by the three explicit premises. These implicit sentences are:

12 (iia’) (a3Rb3)

and

iiia’) (∀x)(∀y){[(xRb3).(yRb1)]→~(x=y)}

The conclusion “white-horse-not-horse” can be validly deduced from (i’), (iia’), and (iiia’). The function of (ii’) is to reflect the implicit premise (iia’) through its comparison with (i’), and explicit sentence (iii’) is significant in suggesting the implicit sentence (iiia’). If the above analysis is right, I think it is not necessary to interpret the argument as based on a class theory in a special narrow and peculiar sense.

(2.4) Hansen’s Nominalistic Interpretation

Fung Yu-lan tries to make the “White-Horse” thesis more interesting by taking it to deny the identity of the universals “white-horse” and “horse”. But A. C. Graham rejects this interpretation because he thinks that there is no evidence of a realist doctrine of universals in ancient China and that discussions of the common name, in the Mohist Canons (墨經) and Hsun Tzu (荀子), take what we would call nominalism for granted, though it is well perceived that nominalism as an anti-thesis which is normally occurred after realism in the history of Western philosophy.

Sharing with Graham’s dissatisfaction with Fung’s and Chmielewski’s abstract interpretations, Chad Hansen proposes a radical shift of viewpoint, as following from his questionable hypothesis that Chinese nouns resemble the mass nouns rather than the count nouns of Indo-European languages. This would mean that ma (horse) functions like the English mass noun “sand”, and different ma will be conceived as like scattered grains of sand; the relation of abstraction between (abstract) class and (concrete) member now becomes the relation of aggregation between (concrete) whole and (concrete) part, distinguished from others by the similarity of parts. It is notable that the Mohist Canons use the same terms, chien (兼collection) and t’i (體unit), to cover both relations, with the latter defined as “a part in a chien.” Although Graham does not agree with the detail of Hansen’s interpretation, he recognizes Hansen’s shift of angle as opening up a new perspective from which the previously unrelated fall into place. Without being committed to the mass-noun hypothesis, Graham thinks that one can be awakened to the recognition that Kung-sun Lung does present the white-horse as a whole of which white and horse are parts. Of the other two essays of Kung-sun Lung, the “Left and Right” (i.e. “Understanding Change”) and the “Pointings and Things” (i.e. “Chih and Wu”), which he 13believes to be the only other genuine writings of Kung-sun Lung, are explicitly or implicitly about whole and part. 12

Hansen believes that there are two kinds of compound terms in the Mohist Canons: (1) “ox-horse” (“ox ’n’ horse”) refers to some kind of compound objects (chien) the two components (t’i) of which cannot interpenetrate with each other; (2) “hard-white” refers to some other kind of compound objects the two components of which cannot be separated. In the former case, we can say that “ox-horse is non-ox-non-horse”, because the non-ox is the same as horse while the non-horse is the same as ox, and thus the sentence means the same as “ox-horse is horse-ox”. In the latter case, we cannot make the same kind of proposition, because hard-white is an intersection of the two components, and thus we cannot find something hard which is not white or something white which is not hard in the compound, for example, in the hard-white stone. Under the interpretation based on his mass noun hypothesis, however, Hansen thinks that what these two kinds of compound terms refer to are mass stuffs, not physical objects or abstract entities. He calls the former compound “mass sum” or “union compound” in contrast with the latter “mass product” or “intersection compound”.

According to Hansen’s analysis, Kung-sun Lung uses compound terms not in full compliance with the later Mohist usages. He thinks, “Gongsun Long [i.e. Kung-sun Lung] regards ‘intersection’ or ‘interpenetrating’ compounds as contrary to the one-name-one-thing principle. If white horse consists of two names, each should consistently name (scattered) things. Used in combination, their ‘naming’ should remain consistent. Thus, they should name the sum of the two stuffs and, as in the case of ‘ox-horse’, ‘not-horse’ would be assertable of it.” 13

In other words, according to Hansen’s view, Kung-sun Lung accepts the Mohist Cannon’s “mass sum” as the only one mode of compound in accord with the so-called “one-name-one-thing” principle, and thus the compound terms “hard-white” and “white-horse” cannot be understood as the Mohist thinks that their units are intersected or interpenetrated with each other respectively but as mereolgical sum in the same way as “ox-horse”. In this regard, we can say that “white-horse is not / non horse” because “not / non horse” means the “white” part of the compound; we can also say that “white-horse is horse” because horse is one of the unit of the compound. However, the ontological implication of this linguistic arrangement is

12 A. C. Graham, The Disputer of Tao (La Salle, Illinois: Open Court 1989), p.83. 13 Chad Hansen, “Logic in China”, in Routledge Encyclopedia of Philosophy Online, §.7, p.2. 14 startling, if not interesting. It makes the ancient Chinese language committed to an ontology which has no physical objects and abstract entities; and this is quite awkward a view in the sense that the well received concept of language-hood in the philosophy of language today is necessarily having the concepts of physical object and abstract entity. Furthermore, It is a fact that mereological sum can be interpreted not only as a nominalistic idea as Hansen thinks, but also as a realistic one as I demonstrate elsewhere. 14 Hansen’s idea that ancient Chinese language has no abstract concepts but notion of mereological stuff is too bold to be acceptable, at least, that is not in accord with the ancient Chinese texts. For example, if we follow Hansen’s hypothesis, we should interpret the name “circle” in the sentence, “The circle of a small circle is the same as that of a large circle” (小圜之圜與大圜之圜同) in the Mohist Cannons as meaning “mereological round stuff” and the sentence as “The mereological round stuff of a small mereological round stuff is the same as that of a large mereological round stuff”. Nevertheless, this interpretation is not acceptable because they are not the same. I think the most plausible, if not the only correct, interpretation is “The round (or circular) property of a small circle is the same as that of a large circle”. If this interpretation is acceptable, I don’t think it is possible to regard that there are no abstract but mereological concepts in the Mohist Cannons, no matter what kind of abstract entities referred by these concepts. Another example in the Mohist Cannons is the following argument based on the distributive use of a term: “One horse is a horse….That horses have four leys means that each horse has four leys” (一馬馬也‥‥馬四足者 / 一馬而四足也). It can be formulated as below: (1) That one [horse] is a horse. [Ha] (2) All horses have four leys. [(∀x)(Hx→Fx)] (3)∴ That one[horse] has four leys. [Fa] This form together with our experience clearly demonstrates that the argument is not only valid but also sound. However, we would have invalid argument if we follow Hansen’s mereological idea and formulate the argument in mereology as follows:

14

Fung Yiu-ming, Kung-sun Lung Tzu: A Perspective of Analytic Philosophy (Taipei: Tung-tai Book

Company, 2001). 15(1') (a< h) (2') (h’< f) (3') ∴ (a< f) Here, “<” means “is a part of”, “a” represents the mass-stuff of one horse, “h” the mass-stuff of all horses, “h’” the mass-stuff of each horse, and “f” the mass-stuff of all the things with four-leys. If we change the “h’” in (2’) into “h”, the above argument form would become valid. Nevertheless, this modification is not acceptable, because what (2’) says is not all the horses collectively have four leys but each one of them has (i.e. in a distributive sense, all the horses have) four leys. It is obvious that these two examples demonstrate that the mereological interpretation is not adequate for the ancient Chinese texts, or, at least, not accurate for the Mohist Cannons. Hansen’s belief that the linguistic characteristic of the so-called “mass-like” usage in ancient Chinese language suggests the “mass-stuff” ontology is also groundless both in theory and in text. In theory, even if this may not commit to linguistic determinism, this idea of suggestion is at most based on speculation, which is not well grounded on empirical evidence. In text, according to the Mohist Cannons’ usage, even though “t’i” together with “chien” means the part-whole relation, it does not necessarily commit to the mass-stuff ontology. For example, one of the popular instances of Mohist’s usage of “t’i” and “chien” is about the geometrical notion of “point” (tuan 端) and “line” (che尺). These notions, however, cannot be interpreted as “a corporate stuff unit” and “a corporate stuff whole”. The ideas of “t’i-ai” (體愛partial love) and “chien-ai” (兼愛universal love), sometimes named “bei-ai” (別愛) and “jo-ai” (周愛) respectively, is another famous example, which also demonstrate that there is no such commitment as to the so-called stuff-ontology. Hansen’s interpretation is not taking care of all the paragraphs in “Discourse on White-Horse”, on the one hand, and other chapters of the book, on the other. Even in some of his selective paragraphs, he has to interpret the text in a peculiar way in order to fix into his idea. For example, he thinks that Kung-sun Lung asserts both “White-horse is not / non horse” and “White-horse is horse” because white-horse as a mass-sum as ox-horse in the Mohist Cannons can be understood as including both non-horse (i.e. white) and horse. In order to fit into this speculation, Hansen has to find out some evidence in the text to demonstrate that Kung-sun Lung does assert that, “White-horse is horse”. However, the only evidence he can “find out” is the sentences 16“Pai-ma-tse-ma-yu-pai-ye. Ma-yu-pai-ma-ye” (白馬者馬與白也 / 馬與白馬也). His translation for these two sentences is that “A white horse is horse and white combined. Horse and white combined is horse”. However, most of the translations in the literature are not treating the last structural word “yeh” (也) as a sign of the function of making assertion, but as equivalent to another “yeh” (耶), a sign of the function of questioning. Graham’s translation, for example, follows this common philological arrangement: “A white horse is horse and white combined. Is that horse and white combined horse?” Suppose Graham’s and other popular translations were all wrong and we thus accept Hansen’s awkward treatment, it still has a serious problem of making sense of his reading in the context. We know that the following sentence is “Therefore I say, ‘White-horse is not horse’.” If we follow Hansen to regard Kung-sun Lung as asserting both “White-horse is horse” and “White-horse is not / non horse” in the “ox-horse” model, it would be very odd that he puts the word “therefore” between these two sentences. It is as odd as saying that “A cup of oily water has oil, therefore it has non-oil (i.e. water)”, instead of saying that “A cup of oily water has oil, and it also has non-oil”. In addition to using the “ox-horse” model to interpret Kung-sun Lung’s “white-horse” as a sum-compound term in his thesis as asserting that “White-horse is both horse and non-horse”, Hansen also thinks that Kung-sun Lung follows Confucius’s principle of one-name-one-thing in arguing that “white-horse”, if regarded as a sui generis non-compound name, could tie to anything for it has different scope from the term “horse”. So, he thinks Kung-sun Lung’s argument then becomes a dilemma. 15

However,

it seems to me that it is not a dilemma in its proper sense, because the consequents of the two horns are not meaning the same thing. It is clear that, the “not / non horse” in “White-horse is not / non horse” means “the complement of horse” (i.e. “white”) if “white-horse” understood as a sum-compound term, while that in “White-horse is not horse” means “not identical with horse” if “white-horse” regarded as a sui generis non-compound name. In other words, even though we, for the sake of argument, accept Hansen’s arrangement to treat the term “not / non” (fei) in the main contexts of the dialogue as meaning both “complementary negation” and “non-identity”, he still cannot interpret Kung-sun Lung’s argument as a dilemma as motivated by the principle of interpretation that Hansen calls the “principle of humanity” (Richard Grandy’s principle?) It is also clear that the term “fei” in almost all of Kung-sun Lung’s arguments are uttered together with the term “yi” (異 differ or different); and the latter is unambiguously

15

   Ibid.

17meaning “non-identity”. Treating Kung-sun Lung’s term “fei” in the above contexts as having two kinds of sense without providing evidence from the text seems not too “humane” according to Hansen’s own principle of humanity. Besides the above sentences selected and interpreted by Hansen as demonstrating the so-called dilemma, the only other selected paragraph interpreted by him is also not too “humane”. This is the paragraph about seeking horse discussed by Kung-sun Lung as below: A: Someone who seeks horse will be just as satisfied with yellow-horse or black-horse; someone who seeks white-horse will not be satisfied with yellow-horse or black-horse. Supposing that white-horse were after all horse, what they seek would be one and the same; that what they seek would be one and the same is because the white-thing would not be different from the horse. If what they seek is not different, why is it that yellow-horse or black-horse is admissible in the former case but not in the latter? Admissible and inadmissible are plainly contradictory. Therefore that yellow-horse and black-horse are one and the same in that they may answer to “having horse” but not to “having white-horse” is conclusive proof that white-horse is not horse. In regard to this dialogue, unfortunately, what Hansen provides for us is an over-simplified interpretation of the original argument as follows: (i) All yellow horses are horses. [A is B] (ii) No yellow horse is a white horse. [A is ~C] (iii) Therefore, a white horse is not a horse. [B is ~C] This argument is clearly invalid according toWestern logic. Hansen thinks, “No culture that routinely accepted such inferences could have had sagacity enough to rule that empire for two thousand years!” So, he continues, this paragraph should be understood as discriminating the boundaries of the substances or stuffs referred by names, instead of representing the individual objects in the world. In this regard, he thinks, it is not appropriate to use Western logic to interpret the argument’s form. However, his accusation of inappropriateness is unfortunately based on his inappropriate treatment of the logical form of the argument. Let’s put aside his inappropriate treatment of the original statements which include quantification as statements without quantifiers, 18Hansen’s translation together with his formulation is totally wrong. If we look more carefully into the inner structure of Kung-sun Lung’s sentences, we can know that the argumentative structure is a Modus Tollens including an application of the Leibniz’s Law which is not the ridiculous form as Hansen elaborates above. The argument embedded in the paragraph can naturally elaborated as follows: (1) If white-horse is /were horse, it would be no difference between yellow-horse (or black-horse) in response to seeking white-horse and seeking horse. (2) There is a difference. (3) Therefore, white-horse is not horse. Its argument form is:

 (1’) [(a=b)→(Fa↔Fb)] 

(2’) ~(Fa↔Fb)

  ∴(3’) ~(a=b) 

This argument is not only valid, but also sound if the two sides of the debate agree that (2) is empirically true. This pretty good argument indicates that Hansen’s interpretation not only violates the principle of humanity, but also violates Davidson’s principle of charity. (3) A New Realistic Interpretation After examining the four kinds of old interpretation discussed above, I think only Fung Yu-lan’s interpretation can be survived if we make necessary adjustment or revision for it. Although some theorists emphasize that Platonic realism had been never occurred in the history of Chinese philosophy and Kung-sun Lung should not be an exception. However, I think, on the first hand, committing to abstract entities or entertaining one-many relationship between abstract and concrete entities in a thinker’s theory or in the natural language of ancient China or other non-Western cultures is not tantamount to holding Platonism; on the other hand, it is implausible, if not impossible, for any mature language, whether ancient or modern, to be a language without abstract reference and one-many relationship in its linguistic structure or conceptual scheme. Appealing to the speculation that something never happened in non-Western cultures is not finding evidence that there is really no instance or exception. In order to explain away all the possible instances of 19abstract reference, some theorists make too much effort to interpret ancient Chinese texts in an unnatural way. For example, they want to interpret Lao Tzu’s “Tao” as “guiding principle” and to explain away all the literal meaning of abstract entity by adding words into the original text or by reading the text into an artificial way. Nevertheless, Lao Tzu’s idea of “cosmological emergence” from Tao to the myriad things in the natural world is not easy to re-interpret as merely meaning “natural law” or “spiritual guidance” based on Tao. Furthermore, some new versions of Lao Tzu found recently, together with the piece on “the Great One Emerges Water” (太一生水) inside one of the versions, strengthen the evidence of abstract reference which is unlikely to be explained away. The one-many relationship is not Plato’s prestige; it is embedded in ancient Chinese language. For example, yin and yan (陰陽) used by ancient Chinese people as a pair of abstract ideas can be applied to many concrete phenomena as the pair’s instances, including female and male, night and day, death and life, and soft and strong (things). “Wu-hsin” (五行) is another abstract scheme which can be exemplified into different things, including the five organs of human body, the five colour scheme, and the five taste classification. With respect to Kung-sun Lung’s case, based on the two reasons mentioned above, I think it is hard and unnatural to explain away the abstract reference in his essays. Kung-sun Lung’s idea of “chih” is similar to but not exactly the same as Platonic “Idea”. Generally speaking, chih (or specifically speaking, horse) for Kung-sun Lung is like Plato’s universal (or horseness) in the sense that they both exist not in the phenomenal world and both do not have the sense of concreteness defined by physical characteristics of phenomena; but they are different from each other in the sense that Kung-sun Lung’s chih can emerge into phenomenal thing through its chien (combining or joining) with other chih or through its ting (specifying or fixing) into a concrete thing while Plato’s universal can be exemplified into each of its instances without any meaning of emergence. In Plato’s words, his Idea is absolutely transcendent and unchanged as an ontological ground which can be exemplified but cannot participate into the phenomenal world in a cosmological sense; Kung-sun Lung’s chih can emerge into phenomena though before emergence it is transcendent and unchanged as a separate simple. Just like the Taoist metaphysical Tao, Kung-sun Lung’s chih plays a double role of ontological ground and cosmological origin; it is quite different from Plato’s and other Western philosophers’ metaphysics in terms of the separate role assigned to their key terms of ontological and cosmological entities respectively. In this sense, we may say that Kung-sun Lung has a realistic commitment in his language which is not necessarily in the Platonic sense. In this section, I will provide a similar interpretation to Fung Yu-lan’s, but mine is new in 20the sense that I don’t regard all terms (simple and compound) as referring to abstract entity and that my argument and philosophical explanation is quite different from Fung’s. As regard to the problem of abstract reference, I think only simple terms, such as “white” (白), “horse” (馬), “hard” (堅), and “stone” (石), are used by Kung-sun Lung as referring to abstract entities; this is quite different from the view of Fung Yu-lan who treats all the terms in the Discourse as names referring to Platonic universals. I regard the compound (referring) terms, such as “white-horse”(白馬), “yellow-horse” (黃馬), and “hard-stone” (堅石), used by Kung-sun Lung as implicit expressions which have the same function as the phrases explicitly expressed in English which denote definite or indefinite individual object. In ancient Chinese language, for example, “pai-ma” (“white-horse”) can be used as having the same function as “some [white-horse]”, “huang-ma” as “this [yellow-horse]”, and “chien-shi” as “that [hard-stone]”. These compound terms function like demonstratives which directly refer to some kind of concrete object in the world. If we consider them as Saul Kripke’s rigid designators, they should not be understood as descriptions and their logical forms should be formulated as individual terms. As I know, all formulations in descriptional form by predicate construction for the compound terms used by Kung-sun Lung cannot provide a coherent interpretation for the whole dialogue. But, if we interpret the debate between Kung-sun Lung and his opponent as the conflict between the use of language based on a direct theory of reference and that based on a descriptional theory of reference, the arguments provided by both sides would be understandable and can be judged as reasonable. According to my interpretation which is based on the principle of charity, the opponent is also reasonable to say that, “White horse is horse” because it is an analytic truth of the form as follows: (∀x)[(Wx.Hx)→Hx] In other words, “white” and “horse” in the opponent’s statement should be represented by the predicates “W” and “H” respectively. However, if we formulate Kung-sun Lung’s key words in his statement “White-horse is not horse” by the same predicates, his statement would become wrong, i.e. analytically false. Based on the direct theory of reference, I think, Kung-sun Lung basically uses a referring term to refer to the same referent in all possible worlds if it does exist in each world. A simple term (non-fixing) “white”, for example, can be understood as referring to the same simple and unchanged chih in all possible worlds and cannot be regarded as referring to a class because a class of some entities in this world could be different from a class of the same kind of entities in other possible world. Based on this consideration, I think his key words should be represented 21by individual terms “a”, “b”, “c”, etc. or understood as the value of individual variables “x”, “y”, “z”, etc. So, instead of formulating Kung-sun Lung’s statement as (∀x)[(Wx.Hx)→~Hx] we can have the form ~(a=b) which can be proved validly based on his premises. Of course, Kung-sun Lung does not notice very well the distinction between use and mention and thus sometimes uses these words such as “horse”, “white”, and “white-horse” not to refer to entity but to refer to itself, i.e. to mention the name itself. However, his thesis can be proved to be valid based on his premises whether “White-horse is not horse” means “White-horse is not the same [entity] as horse” or “The name ‘white-horse’ is not [referring to] the same [thing] as the name ‘horse’.” The consequence of this arrangement is that the arguments provided by both sides can be formulated in an intelligible way and nothing has to be explained away. In the following, I will use the first-order predicate logic to formulate and to prove each of the arguments provided by both sides. 16 (3.1) The Argument of “Shape and Colour” Q: Is it admissible that white-horse is not horse? A: It is admissible. Q: Why? A: “Horse” is that by which we name the shape, “white” is that by which we name the colour. To name the colour is not to name the shape. Therefore I say, “White-horse is not horse”. 「白馬非馬,可乎?」 「可。」 「何哉?」

16 The translation of “Discourse on White-Horse” is based on Graham’s with minor revision. Please see Graham’s translation in his “Three Studies of Kung-Sun Lung”, in Studies in Chinese Philosophy and Philosophical Literature (Singapore: The Institute of East Asian Philosophies, 1986), pp.125-215. 22「馬者,所以命形也;白者,所以命色也。命色者非命形也。故曰:白馬非馬。」 In this first round discussion, the opponent asks what is the reason for Kung-sun Lung to claim that, “White-horse is not horse”. Kung-sun Lung provides an argument which is related to the difference between naming shape and naming colour as follows:

  • (1) “Horse” is used to name something of shape [i.e. the shape of horse].
  • (2) “White” is used to name something of colour [i.e. the colour of white].
  • (3) To name something of colour is not to name something of shape.
    • (4a) Therefore, [the name] “white-horse” is not [the name] “horse”.
    • (4b) Therefore, white-horse (what is named by “white-horse”) is not horse (what is

named by “horse”).

This complicate argument can be formulated into two parts. In the first part, we can have

  • (4a) based on some of the explicit premises together with two implicit premises which are

suggested by the explicit premises: (Here “a1”, “a2 ”, and “a3” stand for the names “horse”, “white”, and “white-horse”, respectively, “b1”, “b2”, and “b3 ” for “the entity shape” (or “the shape of horse”), “the entity colour” (or “the colour of white”), and “the entity of both colour (white) and shape (horse)”, respectively, and “R” for the relation “to name”.)

  • (1’) (a1Rb1)
  • (2’) (a2Rb2)
  • (3’) (∀x)(∀y){[(xRb2).(yRb1)]→~(x=y)}
  • ∴ (4a’) ~( a3=a1)

The first implicit premise is suggested by the comparison between (1) and (2); it says that,

“ ‘White-horse’ is used to name both shape and colour”. Its form can be like this: (We can only have “~(a2=a1)” as a conclusion of the argument if without adding the implicit premises.)

  • (2a’) (a3Rb3)

The second implicit premise is suggested by (3); it says that, “To name something of both shape (horse) and colour (white) is not to name something of shape (white) only”. Its form is:

  • (3a’) (∀x)(∀y){[(xRb3).(yRb1)]→~(x=y)}

For the second part of the argument, we need one implicit premise as additional premise to derive (4b). This premise says that, “What is named by ‘white-horse’ is not what is named by ‘horse’ ” which is suggested by the sentence with the form (3a’) in a reverse way of speaking. Its form can be like this:

  • (3b’) (∀x)(∀y){[(a3Rx).(a1Ry)]→~(x=y)}

Based on the premises (1’), (2a’), and (3b’), we can derive the conclusion:

  • (4b’) ~(b3=b1)

The conclusion, now, does not say that, “the name ‘white-horse’ is not [referring to the same thing as] the name ‘horse’,” but that, “the entity white-horse is not the entity horse”.

  • (3.2) The argument of “Seeking Horse”

Q: Having white-horse cannot be called “lacking horse”. What cannot be called “lacking horse” is not having horse? If having white-horse is deemed having horse, why if judged to be white is it not horse?

有白馬,不可謂無馬也。不可謂無馬者,非馬也〔耶〕?有白馬為有馬,白之非 馬,何也?」

Here the first of the opponent’s arguments is:

  • (∃x)(Wx.Hx)/∴ ~(∀x)~Hx

The opponent’s second argument can be obtained from the first one based on the definition “~(∀x)~Hx =d f

  • (∃x)Hx” as follows:
  • (∃x)(Wx.Hx)/∴ (∃x)Hx

Based on the above two arguments, the opponent provides his or her third argument to reject “White-horse is not horse”:

  • (∃x)(Wx.Hx)/∴ ~(∀x)[(Wx.Hx)→~Hx]

All these arguments are valid and very easy to prove. In order to persuade the opponent to accept his thesis, Kung-sung Lung gives the following answer:

A: Someone who seeks horse will be just as satisfied with yellow-horse or black-horse; someone who seeks white-horse will not be satisfied with yellow-horse or black-horse. Supposing that white-horse were after all horse, what they seek would be one and the same; that what they seek would be one and the same is because the white-thing would not be different from the horse. If what they seek is not different, why is it that yellow-horse or black-horse is admissible in the former case but not in the latter? Admissible and inadmissible are plainly contradictory. Therefore that yellow-horse and black-horse are one and the same in that they may answer to “having horse” but not to “having white-horse” is conclusive proof that white-horse is not horse.

求馬,黃、黑馬皆可致,求白馬,黃、黑馬不可致。使白馬乃馬也,是所求一也; 所求一者,白者不異馬也。所求不異,如〔而〕黃、黑馬有可有不可,何也?⋯⋯ 白馬之非馬,審矣。」

The main premise of Kung-sun Lung’s argument here is based on an instance of the Leibniz’s Law, i.e. the Principle of the Indiscernability of Identicals; and the argument is obviously of the form of Modus Tollens.

  • (1) If white-horse is /were horse, it would be no difference between yellow-horse (or

black-horse) in response to seeking white-horse and seeking horse.

  • (2) There is a difference.
  • (3) Therefore, white-horse is not horse.
  • (1) 使白馬乃馬也,則所求一也。
  • (2) 所求不一。
  • ∴ (3) 白馬非馬。

The argument’s form can be elaborated as follows:

  • (1’) [(a=b)→(Fa↔Fb)]
  • (2’) ~(Fa↔Fb)
  • ∴(3’) ~(a=b)

Here “Fa” and “Fb” are abbreviations of “Rac1c ”2 and “Rbc1c ”2 respectively, which represent “Yellow-horse or black-horse is able to satisfy the request of seeking horse”.

This argument is not only valid but also sound if both sides agree it is the case that there is a difference between “Fa” and “Fb” in seeking horse.

  • (3.3) The Argument of “Horse and White Combined”

Q: You deem horse which has colour not horse. It is not the case that the world has colourless horse; is it admissible that the world has no horse?

「以馬之有色為非馬,天下非有無色之馬,天下無馬,可乎?」

It is clear that the opponent here appeals to Reductio Ad Absurdum in his or her fourth argument against Kung-sun Lung’s thesis. From a perspective of common sense, it is obvious that there is horse in the world. If we deem horse which has colour not horse and agree that the world has no colourless horse, then we can infer that there is no horse in the world, a conclusion which is contradictory to our common sense view. Finally, we can reject the supposition that, “You deem horse which has colour not horse”. The argument runs like this:

  • (1) (∃x)Hx
  • (2) (∀x)[(Hx.Cx)→~Hx]
  • (3) ~(∃x)(Hx.~Cx)
  • ∴ (4) ~ (∃x) Hx
  • ∴ (5) ~(∀x)[(Hx.Cx)→~Hx]

In response to the opponent’s argument, Kung-sun Lung stresses the problem of combination. He says:

A: Certainly horse has colour, which is why one has white-horse. If horse is colourless, there would be only simple horse, how would one obtain white-horse? Therefore the white-thing is not the horse. White-horse is horse and white combined. Is that horse and white combined horse? Therefore I say, “White-horse is not horse”.

「馬固有色,故有白馬。使馬無色,有馬如〔而〕已耳,安取白馬?故白者非馬也。 白馬者,馬與白也。馬與白,馬也〔耶〕?故曰:白馬非馬也。」

Facing the opponent’s challenge, Kung-sun Lung argues back on the distinction between the combined and the uncombined. What he means by “the combined” is the individual combined with two units and what he means by “the uncombined” is the individual of the simplest kind. If we use the words in “Discourse on Understanding Change”, “the combined” is “two” which is combined with the “left (one)” and the “right (one)” and “the uncombined” is “one” (either left or right) which has not yet been combined into “two”. As indicated in the other Discourses, “the fixing white” can be meant either “white-thing”, such as “white-horse” and “white-stone”, or the “white property (in the white thing)” while the latter is different from “the non-fixing white” which is “the uncombined simple”. In this regard, “the non-fixing white” can be understood as an instance of “the uncombined one” while “the fixing white” in the latter sense is an instance of “the combined one”. Since “the fixing white” is a property of white-thing which can be changed in temporal space from one point to another and “the non-fixing white” is unchanged when it is in the state of isolated simple from the changing world, “the changed is not unchanged” can be hold. If we put in the words of the “Discourse on Chih and Wu”, the statement “two is combined with left (one) and right (one)” can be understood as “There is no wu which is not [emerged from] chih” [and] the statement “the changed [one] is not the unchanged [one]” means “[Wu-]Chih is not [tu-]chih”. To formulate this argument, let’s use “Ca” to stand for “White-horse has colour”, “~Cb” for “Horse has no colour”, and ~(a=b) for “White-horse is not horse”. Based on the above distinction, we can elaborate Kung-sung Lung argument into the following form:

  • (Ca.~Cb) /∴ ~(a=b)

or

  • ~(Ca→Cb) /∴ ~(a=b)

The argument is also valid and is not hard to prove.

  • (3.4) The Argument of “Differentiating Yellow-Horse from Horse”

Q: If horse not yet combined with white is deemed horse, and white not yet combined with horse is deemed white, and you bring horse and white together under the compound name “white-horse”, this is applying to those which are combined a name for those which are uncombined, which is inadmissible. Therefore I say, “It is inadmissible that white-horse is not horse”.

「馬未與白為馬,白未與馬為白。合馬與白,復名白馬。是相與以不相與為名,未 可。故曰:白馬非馬未可。」

After the debate through the above three argumentative dialogues, the opponent begins to know that there is a problem of “combination” in Kung-sun Lung’s arguments. For the sake of argument, the opponent thinks that even we agree with Kung-sun Lung’s assertion that “White-horse is not horse”, it would force Kung-sun Lung to reject “horse” in the combined name “white-horse” to be the same name as the uncombined name “horse”. However, from the view of common sense, they are the same name. Hence, the supposition that “White-horse is not horse” is false. The opponent’s fifth argument is like this:

  • (1) (p→~q)
  • (2) q
  • ∴(3) ~~p

As a matter of fact, I don’t think there is any thinker in ancient China who is so stupid to use a name in the way as based on Hansen’s “one-name-one-thing” principle. Most of the words of all the languages in the world, whether ancient or modern, have to be used by their language users in an equivocal way: in order to use a limited numbers of words to express unlimited concepts and thoughts and to refer to a huge of objects and events, they have to use the same word to express different meanings or to refer to different things ranging over from one context to another. It is a necessity for any expressible language. In the case of Kung-sun Lung’s language use, it is definitely not an exception. His distinction between “white” as “the fixing white” and “white” as “the non-fixing white” clearly illustrates that the same word can be used in different context to mean something differently or to refer to different things. His emphasis on the difference between the uncombined “one” (the simple “left” or “right”) and the combined “one” (the “left” or “right” in “two”) also illustrates this common usage. So, to use the same word in different contexts to mean different things is not a theoretical difficulty for Kung-sun Lung. In order to persuade the opponent, he also uses the same argumentative strategy to suppose the opponent’s statement “To deem having white-horse having horse” together with the statement “It is inadmissible to say that having white-horse is to be deemed having yellow-horse” which is agreed by both sides and then forms an argument of the kind of Reductio Ad Absurdum.

A: If you deem having white-horse having horse, is it admissible to say that having white-horse is to be deemed having yellow-horse?

Q: Inadmissible.

A: To deem having horse different from having yellow-horse is to differentiate yellow-horse from horse. To differentiate yellow horse from horse is deem yellow-horse not horse. To deem yellow-horse not horse, yet in the case of white-horse deem it having horse, this is “Flying things go underwater, inner and outer coffins are in different place”, it is the world’s worst fallacy and inconsistency.

以有白馬為有馬,謂有白馬為有黃馬,可乎?」 未可。」 以有馬為異有黃馬,是異黃馬于馬也;異黃馬于馬,是以黃馬為非馬也。以黃馬 為非馬,而以白馬為有馬,此飛者入池,而棺槨異處,此天下之悖言亂辭也。」

Kung-sun Lung here provides a series of arguments to persuade the opponent. The first part of the series is:

  • (I) [For the argument’s sake] Having white-horse is to deem having horse.
  • (II) It is inadmissible to say (or it is not the case) that having white-horse is to be deemed having yellow-horse.
  • (III) ∴ Having horse is different from having yellow-horse.

The argument form can be elaborated as follows:

  • (I’) (∀x)[(x=a)→(x=b)]
  • (II’) ~(∃x)[(x=a)→(x=c)]
  • (III’) ∴ (∀x){[(x=b)→~(x=c)].[(x=c)→~(x=b)]}

The validity of this argument form is very easy to prove. Based on this valid argument (form), Kung-sun Lung continues to address the second part of this series of arguments:

  • (IV) To deem having horse different from having yellow-horse is to differentiate yellow-horse from horse.
  • (V) To differentiate yellow-horse from horse is to deem yellow-horse not horse.
  • (VI) ∴ Yellow-horse is not horse.

Its form can be formulated in the following:

  • (IV’) (∀x){[(x=b)→~(x=c)].[(x=c)→~(x=b)]}→~(∃x)[(x=c)↔(x=b)]
  • (V’) ~(∃x)[(x=c)↔(x=b)]→~(c=b)
  • (VI’) ∴ ~(c=b)

The conclusion of the argument is that “Yellow-horse is not horse” [~(c=b)]. In this regard, Kung-sun Lung seems to appeal to our linguistic intuition that “white-horse” and “yellow-horse” can be put into the same argument place and thinks that to assert “Yellow-horse is not horse” but to reject “White-horse is not horse” is absolutely absurd or ridiculous. According to the spirit of argument, if we assign black-horse, instead of white-horse, as the value of “a” and assign white-horse, instead of yellow-horse, as the value of “c”, the substituted instance of the conclusion would be “White-horse is not horse”.

In this series of complicate arguments, on the first hand, we cannot use the so-called principle of “one-name-one-thing” or “one-name-one-reality”, as claimed by Hansen, to interpret the key concepts. If we agree that it were the case, the supposition “having white-horse is to deem having horse” would be unnecessary or irrelevant, if not nonsense. Based on this principle, it is redundant or superfluous to make the supposition for the argument’s sake, because we can use the principle to prove “Yellow-horse is not horse” directly without the necessity of making any other supposition. On the other hand, it also makes no sense for the argument if we appeal to Hansen’s mass-sum analysis. If we follow Hansen’s idea, the sentence, “To differentiate yellow-horse from horse is to deem yellow-horse not horse” (異黃馬于馬 / 是以黃馬為非馬也), should be interpreted as “To differentiate yellow-horse from horse is to deem yellow-horse (compound) non-horse (unit)”. It doesn’t make sense because it is not necessary to mention the non-horse unit (i.e. the white unit) which is irrelevant and also unhelpful to the argument. So, it is 30obvious that, either to treat “white-horse” or “yellow-horse” as a sui generis non-compound name or to regard it as a sum-compound term, the interpretation based on Hansen’s dilemmatic elaboration doesn’t make any sense to the text.

(3.5) The Argument of “Non-Fixing White”

A: In “having white-horse cannot be called ‘lacking horse’,” the point is that one separates off the white. Without separating off the white, having white-horse cannot be called “having horse”. Therefore the reason why it is deemed having horse is the horse alone, it is not having white-horse which is deemed having horse. Therefore with respect to the [combined] white-horse’s being deemed having horse, the horse cannot be called “horse” [uncombined].

The view regarding the white as white not fixing into anything may be left out of account at this moment. “White-horse” mentions the white as white fixing into something; the white as white fixing into something is not the [non-fixing] white. Horse selects and excludes none of the colours, therefore one may answer it with either yellow[-horse] or black[-horse]. White-horse selects some colour and excludes others, and yellow[-horse] and black[-horse] are both excluded on grounds of colour; therefore one may answer to it only with white-horse. To select and to exclude none is not to select and to exclude some. Therefore I say, “White-horse is not horse”.

「以有白馬不可謂無馬者,離白之謂也;不離者,有白馬不可謂有馬也。故所以為 有馬者,獨以馬為有馬耳,非有白馬為有馬。故其為有馬也,不可以謂馬馬也。以 白者不定所白,忘之而可也。白馬者,言白定所白也,定所白者非白也。馬者,無 去取于色,故黃、黑〔馬〕皆所以應;白馬者,有去取于色,黃、黑馬皆所以色去, 故唯白馬獨可以應耳。無去者非有去也,故曰白馬非馬。」

In this last part of the dialogue, Kung-sun Lung makes a concluding remark with more emphasis on his idea of “separation” and “fixing” which is discussed more detailed in “Discourse on Hard and White”. In this remark, he provides a series of arguments which includes three parts. The first part is:

  • (1) If the [fixing] white can be separated from white-horse, then having white-horse cannot be called “lacking horse”.
  • (2) If the [fixing] white cannot be separated from white-horse, then having white-horse cannot be called “having horse”
  • (3) It is the case that the [fixing] white cannot be separated from white-horse.
  • ∴ (4) Having white-horse cannot be called “having horse”.
  • (1) 如離白(即白馬中之白可分離出來),則有白馬不可謂無馬。
  • (2) 如不離白,則有白馬不可謂有馬。
  • (3) 不離白(即白馬中之白不可分離出來)
  • ∴ (4) 有白馬不可謂有馬。

Its logical form is as follows:

  • (1’) {P→(∀x)[(x=a)→(x=b)]}
  • (2’) {~P→(∀x)~[(x=a)→(x=b)]}
  • (3’) ~P
  • ∴ (4’) (∀x)~[(x=a)→(x=b)]

This argument includes a rhetorical comparison of the conditional sentences between separation and inseparation together with a Modus Ponens. Even though the consequent of (1), i.e. “(∀x)[(x=a)→(x=b)]” and the consequent of (2), i.e. “(∀x) ~[(x=a)→(x=b) ]” are not contradictory; they are contrary to each other. If we can prove (4) (Having white-horse cannot be called “having horse”), the consequent of (1) (Having white-horse cannot be called “lacking horse”) must be false. Here, I think, Kung-sun Lung’s idea is that the fixing white in white-horse is not the non-fixing white which is not fixing into any concrete thing. So, even though “horse” in the term “white-horse” and “horse” as a single term share the same type of word, they refer to different things.

The second part of the series is an argument based on the identification of white-horse as fixing white [on horse] into something combined; and the third part is a similar argument based on the identification of white horse as selecting colour [white and excluding other

It seems inadequate that some interpretations regard the first part of this paragraph as the opponent’s saying, because one of the main points of this part is to refute that, “Having white-horse is deemed having horse”. As regard to the sentence “Therefore with respect to the [combined] white-horse’s being deemed having horse, the horse cannot be called ‘horse’ [uncombined]”, I think the two “ma” in the original Chinese sentence “Bu-ke-ye-wai-ma-ma-yeh” (不可以謂馬馬也) should not be interpreted as Pong Pu’s (龐樸) explanation that since “ma” in “pai-ma” means ma, if we regard the whole “pai-ma” as meaning ma, “pai-ma” would become “ma-ma”. I don’t think it makes any sense of Kung-sun Lung’s argument. Instead, I think that the first “ma” is used to refer to horse and the second “ma” is mentioned; and the sentence can be translated as “the horse cannot be called ‘horse’ (不可以謂馬『馬』也). Pong Pu’s view can be found in his A Study of Kung-sun Lung Tzu (公孫龍子研究), pp.16-17. colours] in something combined.

The former argument is:

  • (1) White-horse means a fixing white-thing.
  • (2) A fixing white thing is not [non-fixing] white.
  • (3) Therefore, white-horse is not [non-fixing] white
  • (1) 白馬者,言白定所白也。
  • (2) 定所白者非白也。
  • ∴ (3) 白馬非白。

Its form is:

  • (1’) (∀x)[(x=a)→(x=b)]
  • (2’) (∀x)[(x=b)→~(x=c)]
  • ∴(3’) (∀x)[(x=a)→~(x=c)]

The latter argument is:

  • (1) Horse is something not selecting and excluding colour and yellow-horse and

black-horse can meet the request of seeking horse on this ground.

  • (2) White-horse is something selecting and excluding colour and yellow-horse and

black-horse cannot meet the request of seeking horse on this ground.

  • (3) Something not excluding [and selecting] colour is not something excluding [and

selecting] colour.

  • (4) Therefore, white-horse is not horse.
  • (1) 馬者,無去取于色,故黃、黑馬皆所以應。
  • (2) 白馬者,有去取于色,黃、黑馬皆所以色去。
  • (3) 無去(取于色)者非有去(取于色)也。
  • ∴ (4) 白馬非馬。

The form of the argument is:

  • 1’) (∀x){[(x=a)→~Sx].[~Sx→P]}
  • (2’) (∀x){[(x=b)→Sx].[Sx→~P]}
  • (3’) (∀x)~(~Sx↔Sx)
  • ∴ (4’) ~(a=b)

The validity of all the arguments formulated above can be proved easily as demonstrated in the Appendix below.

(4) Appendix: Proofs for the Arguments in “Discourse on White-Horse”

(3.1B) The Proof for the Argument of “Shape and Colour”:

The first part of Kung-sun Lung’s argument tries to prove that the name “white-horse” is different from the name “horse” as follows:

  • [1] 1. (a1Rb1) Assumption
  • [2] 2. (a3Rb3) Assumption
  • [3] 3. (∀x)(∀y){[(xRb3).(yRb1)]→~(x=y)} Assumption
  • [3] 4. (∀y){[(a3Rb3).(yRb1)]→~(a3=y)} 3, U. E.
  • [3] 5. {[(a3Rb3).(a1Rb1)]→~(a3=a1)} 4, U. E.
  • [1,2] 6. [(a3Rb3).(a1Rb1)] 1,2, Conjunction
  • [1,2,3] 7. ~(a3=a1) 5,6, M.P.

The step [7] shows that the conclusion “The name ‘white-horse’ is not the name ‘horse’ ” can be validly deduced from [1], [2], and [3].

The conclusion of the second part of the argument is “The entity white-horse is not the entity horse”; its proof is:

  • [1] 1. (a1Rb1) Assumption
  • [2] 2. (a3Rb3) Assumption
  • [3] 3. (∀x)(∀y){[(a3Rx).(a1Ry)]→~(x=y)} Assumption
  • [3] 4. (∀y){[(a3Rb3.(a1Ry))→~( b3=y)] 3, U. E.
  • [3] 5. [(a3Rb3).(a1R b1)→~( b3=b1)] 4, U. E.
  • [1,2] 6. [(a3Rb3).(a1Rb1)] 1,2, Conjunction
  • [1,2,3] 7. ~(b3=b1) 5,6, M.P.

(3.2A1) The Proof for the Argument (1) of the Opponent:

  • [1] 1. (∃x)(Wx.Hx) Assumption
  • [2] 2. (∀x)~Hx Assumption
  • [2] 3. ~Ha 2, U.E.
  • [1] 4. (Wa.Ha) 1, E.E.
  • [1] 5. Ha 4, simplification
  • [1,2] 6. (Ha.~Ha) 3,5, Conjunction
  • [1] 7. ~(∀x)~Hx 6, R.A.A.

From [1], “There is white-horse [in the world]”, we can infer [7], “It is not the case that there is no horse”.

(3.2A2) The Proof for the Argument (2) of the Opponent:

  • [1] 1. (∃x)(Wx.Hx) Assumption
  • [1] 2. (Wa.Ha) 1, E.E.
  • [1] 3. Ha 2, Simplification
  • [1] 4. (∃x)Hx 3, E.I.

From the same premise as (3.2A1), we can have [4], “There is horse [in the world]”, as conclusion.

(3.2A3) The Proof for the Argument (3) of the Opponent:

  • [1] 1. (∃x)(Wx.Hx) Assumption
  • [2] 2. (∀x)[(Wx.Hx)→~Hx] Assumption
  • [2] 3. [(Wa.Ha)→~Ha] 2, U.E.
  • [1] 4. (Wa.Ha) 1, E.E.
  • [1,2] 5. ~Ha 3,4, M.P.
  • [1] 6. Ha 4, Simplification
  • [1,2] 7. (Ha.~Ha) 5,6, Conjunction
  • [1] 8. ~(∀x)[(Wx.Hx)→~Hx] 7, R.A.A.

From the same premise as (3.2A1), we can infer that “It is not the case that white-horse is not horse”, i.e. [8], based on Reductio Ad Absurdum.

(3.2B) The Proof for the Argument of “Seeking Horse”:

  • [1] 1. [(a=b)→(Fa↔Fb)] Instance of Leibniz’s Law
  • [2] 2. ~(Fa↔Fb) Assumption
  • [1,2] 3. ~(a=b) 1,2, M.T.

(3.3A) The Proof for the Argument (4) of the Opponent:

  • [1] 1. (∃x)Hx Assumption
  • [2] 2. (∀x)[(Hx.Cx)→~Hx] Assumption
  • [3] 3. ~(∃x)(Hx.~Cx) Assumption
  • [3] 4. (∀x)~(Hx.~Cx) 3, Definition
  • [3] 5. (∀x)(Hx→Cx) 4, Definition
  • [3] 6. (Ha→Ca) 5, U.E.
  • [7] 7. Ha Assumption
  • [3,7] 8. Ca 6,7, M.P.
  • [3,7] 9. (Ha.Ca) 7,8, Conjunction
  • [2] 10. [(Ha.Ca)→~Ha] 2, U.E.
  • [2,3,7] 11. ~Ha 9,10, M.P.
  • [2,3,7] 12. (Ha.~Ha) 7,11, Conjunction
  • [2,3] 13. ~Ha 12, R.A.A
  • [2,3] 14. (∀x)~Hx 13, U.I.
  • [2,3] 15. ~(∃x)Hx 14, Definition
  • [1,2,3] 16. [(∃x)Hx.~(∃x)Hx] 1,15, Conjunction
  • [1,3] 17. ~(∀x)[(Hx.Cx)→~Hx] 16, R.A.A.

(3.3B) The Proof for the Argument of “Horse and White Combined”:

This argument can be proved as based on an instance of the Leibniz’s Law, “[( a=b)→ (Ca↔Cb)]”, as one of the premises as follows:

  • 1. [(a=b)→(Ca↔Cb)] Theorem
  • 2. {(a=b)→[(Ca→Cb).(Cb→Ca)]} 1, Definition
  • [3] 3. (a=b) Assumption
  • [3] 4. [(Ca→Cb).(Cb→Ca)] 2,3, M.P.
  • [3] 5. (Ca→Cb) 4, Simplification
  • 6. [(a=b)→(Ca→Cb)] 5, R.C.P.
  • [7] 7. ~(Ca→Cb) Assumption
  • 36[7] 8. ~(a=b) 6,7, M.T.

(3.4A) The Proof for the Argument (5) of the Opponent:

Here, “p” stands for “White-horse is not horse” and “~q” means “It is not the case that the two names of ‘horse’ in the uncombined ‘horse’ and the combined ‘white-horse’ are the same.” The proof of the argument is reflected in the following Modus Tollens:

  • [1] 1. (p→~q) Assumption
  • [2] 2. q Assumption
  • [1,2] 5. ~~ p 1,2, M.T.

(3.4B) The Proof for the Argument of “Differentiating Yellow-Horse from Horse”:

If we use “(∀x)[(x=a)→(x=b)]” (Having white-horse is to be deemed having horse有白 馬為有馬) and "~(∃x)[(x=a)→(x=c)]” (It is inadmissible to say that having white-horse is to be deemed having yellow-horse不以有白馬為有黃馬) as premises, we can infer “~(∃x)[(x=b)→(x=c)]” or “(∀x)~[(x=b)→(x=c)]” (It is inadmissible to say that having horse is to be deemed having yellow-horse不以有馬為有黃馬) and then “(∀x){[(x=b)→ ~(x=c)].[(x=c)→~(x=b)]} (Having horse is considered as different from having yellow-horse以有馬為異有黃馬) and “~ (c = b)” (Yellow-horse is not horse黃馬非馬). If “Yellow-horse is not horse” can be proved, “White-horse is not horse” can be proved on the same ground. The proof can be elaborated as follows:

  • [1] 1. (∀x)[(x=a)→(x=b)] Assumption
  • [2] 2. ~(∃x)[(x=a)→(x=c)] Assumption
  • [1] 3. [(c’=a)→(c’=b)] 1, U.E.
  • [2] 4. (∀x)~[(x=a)→(x=c)] 2, Definition
  • [2] 5. ~[(c’=a)→(c’=c)] 4, U.E.
  • [2] 6. [(c’=a).~(c’=c)] 5, Definition
  • [2] 7. (c’=a) 6, Simplification
  • [1,2] 8. (c’=b) 3,7, M. P.
  • [2] 9. ~(c’=c) 6, Simplification
  • [1,2] 10. [(c’=b).~(c’=c)] 8,9, Conjunction
  • [1,2] 11. ~[(c’=b)→(c’=c)] 10, Definition
  • [1,2] 12. (∀x)~[(x=b)→(x=c)] 11, U.I.
  • 37[13] 13. ~(∀x){[(x=b)→~(x=c)].[(x=c)→~(x=b)]} Assumption
  • [13] 14. (∃x) ~{[(x=b)→~(x=c)].[(x=c)→~(x=b)]} 13, Definition
  • [13] 15. ~{[(c’=b)→~(c’=c)].[(c’=c)→~(c’=b)]} 14, E.E.
  • [13] 16. {~[(c’=b)→~(c’=c)]V~[(c’=c)→~(c’=b)]} 15, Definition
  • [13] 17. {[(c’=b).(c’=c)]V[(c’=b).(c’=c)]} 16, Definition
  • 18. {[(c’=b).(c’=c)]V[(c’=b).(c’=c)]}→[(c’=b).(c’=c)]
                     Theorem
  • [13] 19. [(c’=b).(c’=c)] 17,18, M.P.
  • [13] 20. (c’=c) 19, Simplification
  • [2,13] 21. [(c’=c).~(c’=c)] 9,20, Conjunction
  • [2] 22. (∀x){[(x=b)→~(x=c)].[(x=c)→~(x=b)]} 21, R.A.A
  • [2] 23. {[(b=b)→~(b=c)].[(b=c)→~(b=b)]} 22, U.E.
  • [2] 24. [(b=b)→~(b=c)] 23, Simplification
  • 25. (b=b) Theorem
  • [2] 26. ~(b=c) 24,25, M.P.

(3.5B1) The Proof for the Argument (1) of “Non-Fixing White”:

  • [1] 1. {~P→(∀x)~[(x=a)→(x=b)]} Assumption
  • [2] 2. ~P Assumption
  • [1,2] 3. (∀x)~[(x=a)→(x=b)] 1,2, M. P.

(3.5B2) The Proof for the Argument (2) of “Non-Fixing White”:

  • [1] 1. (∀x)[(x=a)→(x=b)] Assumption
  • [2] 2. (∀x)[(x=b)→~(x=c)] Assumption
  • [3] 3. (a’=a) Assumption
  • [1] 4. [(a’=a)→(a’=b)] 1, U.E.
  • [1,3] 5. (a’=b) 3,4, M.P.
  • [2] 6. [(a’=b)→~(a’=c)] 2, U.E.
  • [1,2,3] 7. ~(a’=c) 5,6, M.P.
  • [1,2] 8. [(a’=a)→~(a’=c)] 7, R.C.P.
    • [1,2] 9. (∀x)[(x=a)→~(x=c)] 8, U.I.

(3.5B3) The Proof for the Argument (3) of “Non-Fixing White”:

  • [1] 1. (∀x){[(x=a)→~Sx].[~Sx→P]} Assumption
  • [2] 2. (∀x){[(x=b)→Sx].[Sx→~P]} Assumption
  • [1] 3. {[(a=a)→~Sa].[~Sa→P]} 1, U.E.
  • [1] 4. [(a=a)→~Sa] 3, Simplification

5. (a=a) Theorem

  • [1] 6. ~Sa 4,5, M.P.
  • [2] 7. {[(a=b)→Sa].[Sa→~ P]} 2, U.E.
  • [2] 8. [(a=b)→Sa] 7, Simplification
  • [1,2] 9. ~(a=b) 6,8,M.T.
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