A Semantic Approach to wa/ga Distinction

The esoteric distinction between Japanese postpositions -wa and -ga has caused so much rancour all around the galaxy that there exist several planets where the inhabitants pick up sundry exegeses on this topic from the street and throw them at each other in greeting. What we will say below covers a small semantic difference of the two particles. It is hoped that this is going to be a short essay lest it should cause serious damage when used as a projectile by an enraged reader.

We begin by comparing how -wa and -ga have different implications for a statement that English expresses with; "R2-D2 sang Rigoletto."

1. R2-D2はリゴレットを歌った。

• R2-D2-wa rigoretto-o utatta.
• R2-D2-wa Rigoletto-Accusative singPast
  
• Rigoletto is likely to be the only piece that the robot sang.
  

2. R2-D2がリゴレットを歌った。

• R2-D2-ga rigoretto-o utatta.
• R2-D2-ga Rigoletto-Accusative singPast
• The robot is likely to
  have been the only performer who sang Rigoletto.
  

Using logic symbols, we tentatively understand the two sentences as conveying the following basic scheme;

[wa]
S ⊂ P (the subject, S, is part of the set that meets the predicate, P)

[ga]
S ⊃ P (the subject, S, includes as its component the set that meets the predicate, P)


This explanation for wa seems justifiable but ga poses a problem, if we consider the following pair.

3. OK自然数は整数である（ℕ ⊂ ℤ）。

• šizensū-wa sēsū dearu.
• Natural number-wa integer Copula.
• Natural numbers are integers.
  

4. *整数が自然数である（ℤ ⊃ ℕ）。

• *sēsū-ga šizensū dearu.

The sentence is well-formed but hard to interpret. A forced interpretation is, "Integers are natural numbers." This does not convey the mathematical relationship between integers and natural numbers, ℤ ⊃ ℕ. The ga interpretation needs renovating with another method of analysis.

We now consider a set of opera performances in order to analyse the semantic implications of -wa and -ga. The table below lists opera performances at Carnegie Hall in the summer of 2006.

Performance at Carnegie Hall

PERFORMER	PERFORM_DATE	ITEM
Cheshire Cat	2006/07/01	Othelo
Moo	2006/07/03	The Pirates of Pensance
Tom	2006/08/12	Lohengrin
Cheshire Cat	2006/08/23	Lohengrin
Babar	2006/08/23	Mikado
R2-D2	2006/09/01	Rigoletto
Cheshire Cat	2006/09/03	Othelo

Any useful list of such performances should tell vital information such as performer, date and the performed musical pieces. The table captures them in the three columns, for which we now postulate an important relationship; that each column has values independent of those in other columns. The name of a performer poses no constraints on the date he can perform or the piece he can sing, for example.

Looking at performances on 07/01 and 09/03, we realise that Cheshire Cat was the only performer who sang Othelo during this season. Also, the record for 09/01 shows that R2-D2 was the only one to sing Rigoletto. Without contradicting other performance records, wa/ga postpositions can be used to express these facts.

5. チェシャ猫がオテロを歌った。

• Češa Neko-ga otero-o utatta.
• It was Cheshire Cat that sang Othelo.
  

6. R2-D2はリゴレットを歌った。

• R2-D2-wa rigoretto-o utatta.
• R2-D2 sang Rigoletto only.
  

Sentences 5 and 6 are equivalent to the following statements about the table PERFORMANCE
(where p is an instance of performance);

7. ∃p (performer = Cheshire Cat, item = Othelo)
AND ∄ p (performer ≠ Cheshire Cat, item = Othelo)

The first clause states that there is indeed a record of Cheshire Cat singing Othelo at the prestigious venue, whereas the second one says there was no other performer who sang Othelo. Note that this formulation is irrelevant of other performances of the grinning feline. In fact Cheshire Cat sang Lohengrin too. This, however, does not prohibit ga-statement from use here.

8. ∃p (performer = R2-D2, item = Rigoletto)
AND ∄ p (performer = R2-D2, item ≠ Rigoletto)
(optionally) AND ∄ p (performer ≠ R2-D2, item = Rigoletto)

The first clause for wa-statement has the same effect as that for ga-statement. Then second clause reads that there exists no record such that, while having performer = R2-D2, item is not Rigoletto. The third one is equivalent to the second for 7.


We might be able to build a similar table for the sentences about natural numbers and integers (3, 4). A table like below, however, violates the column independence condition being postulated above. If a record has natural_no = Y, then always int = Y. Statements 7 and 8, having foundation on this postulate for the table, cannot be applied where the postulate does not stand.

Table of Numbers

NO	NATURAL_NO	INT
1	Y	Y
2	Y	Y
-3	N	Y

The ga-statement about natural numbers and integers (4), therefore, cannot be accorded with a proper interpretation. Sentence 3 is still interpreted properly since for a wa-statement, interpretation at table level is not necessary. The relationship between the wa-subject and the predicate is that of set theory (S ⊂ P).

The following is the formal definition of a ga-statement.

For a group of records R, with mutually independent elements from e₁ to e_n,
R = (e₁, e₂, e₃, e₄, ...., e_n)

if a ga-statement exists between two of the elements,
e₁-ga e₂ da.

then the relationship between the elements is:
∃r (e₁ = S, e₂ = P)
AND ∄ r (e₁ ≠ S, e₂ = P)
where S is the subject and P the predicate, of a ga-statement.

This is the logical structure of the ga-statement, often called exhaustive ga.


We have yet to establish a logical representation of the implications of wa-statement outside the table framework.

9. 70点は取った。

• nanajutten-wa totta.
  
• seventyPoints-wa takePast.
  
• I got around seventy points.

In this sentence, -wa substitutes the accusative marker and roughly corresponds to the English "around." If a student is talking about how many points she thinks she got in the last ungraded exam, sentence 9 means 70 is the student's assessment of what she has scored. If the results are out and it turns out the student is graded 67 points, her assessment would have been quite a valid one. The same goes for 76, 71, 68 etc. since these figures are close enough to the mark. An overly high or low grade such as 53 and 99 means that her wa-statement did not match the reality.

Even if individuals may differ in assessing what is different enough from the subject, we can safely assume that a wa-statement entails negative relationships between subjects not being
"similar" to the wa-clause and the predicate. A wa-statement contrasts the subject with other objects belonging to the same class. For the example above, points that are not close to 70 cannot be interpreted as the points scored by the student. If we use Sim(S) as a notation for subjects "similar to S," we get an implication;
10. ¬Sim(S) → ¬P
for a wa-statement.

The primary semantics of a wa-statement, S ⊂ P, can be modified as below in the light of similar subjects:
Sim(S) ⊂ P (S and others similar to S are included in P).

This entails,
11. Sim(S) → P
where
S ∼ Sim(S) ∼ ∞ (∞ ≠ ¬S)
default: Sim(S) = ∞

Due to the subjective nature of Sim(S), we allow it to range from S itself to whatever the speaker pleases, unless it is ¬S. The default value Sim(S) = ∞ is postulated in order to minimise the ¬P conclusion unless context is given to justify shrinking the range of Sim(S).

When Sim(S) is S, 11 and the inverse of 10 are;
S → P
P → S,

This means;
12. S ↔ P
that P and S are equivalents. This has been, in fact, the conclusion of the full-fledged version of statement 8.


The consequence of 11 and 12 is that, when context renders to -wa utmost power of contrasting, the postposition is used for a definite object. If an object in an utterance is equivalent to another, then the object is uniquely identifiable among other objects of the same class; thus the object becomes definite. Even if the opposition definite vs. indefinite plays little role in Japanese syntax,
-wa can be a sort of definite marker by virtue of contrasting. In fact, contrastive function is the primary role of this postposition (contrastive wa).

13. 年とった猫が宿屋に住んでいた。猫は鼠捕りが上手かった。

• toshitotta neko-ga yadoya-ni sundeita. neko-wa nezumitori-ga umakatta.
  
• old cat-ga inn-Locative livePerfectProgressive. cat-wa catchingMice-ga goodAtPast.
  
• An old cat lived in an inn. The cat was good at catching mice.

After an old cat living in an inn has been introduced in the discourse, there is little possibility that another cat is discussed in the second sentence. The "neko" in the second sentence is a definite object, thereby requiring -wa.

Also of note here is that "an old cat" is introduced by -ga. Replacing it with -wa causes interpretive dissonance. This may be accounted for by the different levels that -wa and -ga operate. On one hand, -ga, working on table level, requires that such an object indeed exists that meets the criterion denoted by the predicate (∃r (e₁ = S, e₂ = P)). On the other hand, -wa indicates relationships between propositions, without specifying the truth value of any of them. First, -ga introduces the existence of a cat living in an inn. Then, -wa elaborates on the object at
categorical level.


Conclusion
This has been an attempt at analysing exhaustive ga and contrastive wa in the light of semantics. We have found that the two postpositions work at different semantic levels and that some semantic implications are reflected in syntax. Other syntactic characteristics of wa and ga may be also worthy of semantic examination.