行政院國家科學委員會補助專題研究計畫成果報告
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※ A Quantificational Analysis of
Dthat
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指示詞
D
that 的量限分析
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計畫類別：ˇ個別型計畫
□整合型計畫
計畫編號：NSC 89 – 2411 – H – 002 – 049
執行期間：2000 年 8 月 1 日 至 2001 年 7 月 31 日
計畫主持人：楊金穆
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91
年
3 月
31
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行政院國家科學委員會專題研究計畫成果報告
國科會專題研究計畫成果報告撰寫格式說明
Pr epar ation of NSC Pr oject Repor ts
計畫編號：NSC 89 – 2411 – H – 002 – 049
執行期限：2000 年 8 月 1 日 至 2001 年 7 月 31 日
主持人：楊金穆
A Quantificational Analysis of
D
that
指示詞
D
that 的量限分析
中文摘要
此報告的目的在於呈現指示詞 ‘that’ (以下簡稱Dthat)的量化分析。弗烈格(Frege)對指示詞的
處理、卡普藍(Kaplan)與派瑞(Perry)對弗烈格的批評，以及伊凡斯(Evans)與尤葛若(Yourgrau) 對弗烈格的辯護，將在本文中重新審視。並討論曲那(Künne)、哈寇特(Harcourt)及卡爾(Carl)
的處理方式。筆者將提出對Dthat 的量化分析，視 Dthat 為一種量詞，當述詞應用在 Dthat 所
指涉對象時，Dthat 提供一具體的量限功能。最後我將指出根據一階語言中所謂的索引量詞來
形式化Dthat 的兩種不同方式，及分別適當的語意。基於第一種進路的邏輯系統將可被建立， 而基於第二種進路的邏輯系統之難處則有詳盡的討論。
關鍵字：
Dthat，指示詞，明示，意思/指涉的區別，專名，完整原則，量化
Abstr act
The main burden of this report is to present a quantificational analysis of the demonstrative ‘that’, ‘Dthat’ for short. Frege treatment of demonstratives, Kaplan and Perry's objection to the Fregean analysis, and Evans and Yourgrau's defence of Frege's treatment will be reviewed. Künne, Harcourt, and Carl’s treatments will be discussed. Then I propose a quantificational analysis of ‘Dthat’ by taking an occurrence of ‘Dthat’ as a quantifier, which puts forth a specified quantification over the application of the associated predicate to the object that the very occurrence of ‘Dthat’ is supposed to refer to. Finally, I show two different way of formulating occurrences of ‘Dthat’ in terms of the so-called indexed quantifier(s) in a first-order language, together with appropriate semantics respectively. A logical system on the basis of the first approach will be established. The difficulties with the establishment of a logical system on the second one will be discussed.
Keywor ds: Dthat, demonstratives, demonstration, sense/reference distinction, proper names, pr inciple of completion, quantification
I. The background
Demonstratives (such as ‘that’, ‘I’, ‘now’, ‘here’, ‘actual’, and alike) are referring expressions of a special type with several peculiar characteristics which referring expressions of other kinds do not have, including noticeably, context-sensitivity (or context-dependency, or indexicality), lack of descriptive content, and cognitive-situational immediacy. In view of these characteristics, Frege
seemed to fail to offer an account of the sense of demonstratives coherent with his semantic theory. According to Frege (1984/[1892]; [1918/9]), a referring expression in a sentence used to express a though has a sense, which is supposed to be the cognitive value of the given expression. Moreover,
the reference of a referring expression, if there is any, is supposed to be determined via the sense of that expression, taken as the mode of presentation. Now, it seems perfectly sensible to claim that a sentence with a demonstrative as its grammatical subject, say ‘That is a book’, can be used to express a thought. And in ordinary discourse, it seems beyond reasonable doubt to claim that an occurrence of ‘That’ in a sentence of this kind would has an object as its reference. But, it is also agreed that the occurrence of ‘That’ in this case, as a demonstrative, which is in essence an expression lack of descriptive content, would have no alleged Fregean’s sense. Be that as it may, the cognitive-situational immediacy of demonstratives indicates that the reference of a
demonstrative would not be determined via its sense.
David Kaplan in a series of papers on demonstratives (1990/[1978], 1989/[1979], 1989, 1989a, 1990) argues that Frege's semantic analysis of demonstratives is inadequate. According to Kaplan, the demonstration indicated by the use of a demonstrative in a sentence is nothing more than a function which takes a certain possible world w and a fixed time t as its argument so that a certain
object can be taken as its value (Kaplan 1990/[1978]: 28). Following Kaplan's footsteps, John Perry claims that we should not conflate epistemological issues with the pure semantic analysis of demonstratives. For Perry, Frege's original notion of sense is supposed to perform a variety of functions, which are hardly satisfied altogether by a single theoretical device. Consequently, an adequate analysis of the demonstratives must take into account the role of a demonstrative in use, which is in essence a rule ‘taking us from an occasion of utterance to a certain object’. (Perry 1990/[1977]: 55)
In defense of Frege's analysis, Evans (1990/[1981]:87) proposes that the way of thinking of an object to which the general Fregean conception of sense direct us is, in the case of a dynamic Fregean thought, a way of keeping track of an object. Thus the sense of a demonstrative, such as ‘today’ on a special occasion (say on d), can be treated as a function which takes whoever is
thinking of d as the current day as its argument and the very day d as its value. Evans then uses
the method of abstraction to formulate such a required function in terms of the ë-operator, that is, ë(R(x,d)), where R(x,d) is a relation to characterize the required function. For instance, since any
two utterances of the sentence ‘Today is F’ on d expresses the same thought, we might equate the
thought with the triple, namely
〈d, ëxëy(R(x,y)), Sense of ‘( ) is F’〉.
On the basis of this analysis, Evans further points out that Perry's account of the demonstratives is merely a notational variant of Frege's semantic treatment. However, Yourgrau (1990/[1986]) insists that the search for a satisfactory analysis of demonstratives indeed brings to light an ingredient missing from Frege's semantics, namely, a non-descriptive mode of 'cognitive' access (my italic).
But, Yourgrau argues that this missing bit can be integrated into Frege's semantic framework.
Künne (1992) has recently insisted that a sentence with ‘That’ as its grammatical subject does express a complete thought. Sticking to the principle of completion, he proposes that we had better treat demonstratives as a kind of hybrid proper names, that is to say, associated to each occurrence of a demonstrative in a sentence, there is a proper name. Thus, the association of an appropriate proper name to the occurrence of a demonstrative in use would make the whole sentence capable of expressing a complete thought. (See Harcourt (1993, 1999) for some criticism of this approach.)
Along a similar line of thought, Carl (1994) claims that Frege seemed to hold that sentences containing demonstratives are not incomplete sentences, nor do they have an incomplete sense. Nonetheless, Carl (1994) maintains that Frege was not concerned with applying the sense/reference distinction to demonstratives. According to Carl, Frege would not consider the question of how to refer to objects by using demonstratives, or what kind of sense we can attribute to them. For Carl, a sentence containing demonstratives is an incomplete expression of a thought. Therefore, the key question with regard to the use of demonstratives is this: How do we manage to express a thought by sentences containing demonstratives and to communicate them to others?
The main burden of this project is to provide a satisfactory analysis of demonstratives. Since there are a variety of demonstratives, it would be implausible to produce an overall account of demonstratives of all kinds in a short-term project. This report therefore merely focuses upon a satisfactory analysis of ‘that’ used as a demonstrative. Following Kaplan's terminology, let us use the term ‘Dthat’ to denote occurrences of ‘that’ used as demonstratives.
II. The under lying thought: Towar d a quantificational analysis of Dthat
It is striking that grammatically a demonstrative can be also used as a subject (or logical subject) of a sentence and that a sentence containing a demonstrative as its subject can be used to express a complete thought. Moreover, whenever a sentence containing a demonstrative as its subject is uttered, the speaker indeed intends to refer, via an use of the very demonstrative, to a particular thing in the desired domain of discourse, which is supposed to be the reference of the very occurrence of the demonstrative in use, and which is supposed to be the object that the thought expressed by the given sentence is said to be about. Therefore, it seems perfectly sensible to maintain that a sentence containing a demonstrative is not merely a complete expression but also used to express a complete thought. Moreover, an occurrence of ‘Dthat’ used in this way does have a sense in addition to its reference. In view of these aspects of the use of demonstratives, it seems to me that the difference between demonstratives and referring expressions of other types (such as proper names and definite descriptions) may not be so sharp in essence as that philosophers have usually recognized.
If our foregoing observation is on the right track, then it seems patent that an analysis of the use of
Dthat must meet the following conditions:
(i) An analysis of Dthat must be able to explain the principle of completion, by showing that a sentence with an occurrence of Dthat as its grammatical subject not only can be used to express a complete thought, but also must be itself a complete expression.
(ii) An analysis of Dthat must be able to explain what is the sense of Dthat in a sentence in which it occurs and how the supposed reference of the very occurrence of ‘Dthat’ is determined via its sense. In particular, such an analysis must be able to explain a non-descriptive mode of ‘cognitive’ access to its reference.
(iii) An analysis of Dthat must be able to explain the context-sensitivity of demonstratives. In particular, it must be able to show how different uses of the term Dthat take us from different
occasions of utterances to distinct objects.
Now, it seems clear that to meet the aforementioned conditions, the key question to an analysis of
Dthat is this:
In what sense a sentence containing ‘Dthat’ can be used to express a thought so that the object that the thought is said to be about can be taken as the r efer ence of the ver y occur rence of ‘Dthat’?
I have recently argued (Yang 1999) that proper names should be treated as quantifiers of a special type, a kind of constant quantifiers, which are used to put forth quantification over the application of the associated predicates to the specified objects. It seems perfectly sensible to take ‘Dthat’ as an indexical quantifier which is supposed to quantify the intended application of the associated
predicate so that the associated predicate can be applied only to the value of the variable bound to the given occurrence of Dthat. I believe that if this is a right approach, then we can have a unified explanation of referring expressions— referring expressions of all kinds can be treated as quantifiers, including indexical expressions of other sorts such as ‘today’, ‘here’ or ‘I’, and so on. And I hope this result can enhance one of my belief that predication should play a central role in the theory of meaning.
From a syntactic point of view, a quantificational treatment of Dthat would meet the first requirement for an analysis of Dthat we have just stated, namely it meets the principle of completion. Moreover, from a semantic point of view, the notion of quantification itself should be able to show how the supposed reference of an occurrence of ‘Dthat’, taken as a quantifier, is determined via its sense— in this case, that is, the quantificational power of Dthat. What remains is to show that we can also establish an appropriate semantic treatment to exemplify the context-sensitivity of Dthat, that is, to show how different uses of the term Dthat take us from
different occasions of utterances to distinct objects.
It is worth noting that in ordinary discourse, the use of a sentence containing ‘Dthat’ as an referring expression is different from sentences with other kinds of referring expressions. Particularly, we would not ask whether a statement ‘That is an F’ is true or not. Instead, we can only ask whether or not an utterance of ‘That is an F’ is true with respect to a certain occasion on which the sentence ‘That is an F’ is utterred. In other words, we can ascribe truth to an utterance of ‘That is an F’ only when appealing to a certain occasion on which the speaker who utters the very sentence in fact points to an object in the domain of discourse and that object does have the property F. Accordingly, for an appropriate semantics for the use of ‘Dthat’, specification of the construction of occasions and that of a function from uses of ‘Dthat’ to occasions are called for.
On the basis of these observations, we can now give a quantificational analysis of ‘Dthat’. In particular, I will show how to formulate the occurrences of Dthat in ordinary discourse into quantifiers of a special kind, called indexed quantifiers, in a first-order language, and appropriate semantic treatments for indexed quantifiers will be given on the basis of our quantificational analysis of Dthat.
III. A Quantificational theor y of Dthat
We have already noted that the evaluation of an utterance of a sentence with ‘Dthat’ as its grammatical subject can be done only when a certain occasion is taken into account. Accordingly, different specification of the construction of occasions may render different (interpretation) function of occurrences of ‘Dthat’, i.e., a function from occurrences of ‘Dthat’ to occasions. And this in turn may render different ways of formulating the occurrences of ‘Dthat’ in ordinary discourse into a formal language.
At the moment, there are two plausible ways of specification of the notion of the required occasions. The first one is to appeal to the notion of types of occasions. The idea comes from Austin’s account of truth. According to Austin (1964/[1950]: 22),
A statement is said to be true when the historic state of affairs to which it is correlated by the demonstrative conventions (the one to which it ‘refer’) is of a type with which the sentence used in making it is correlated by the descriptive conventions.
By descriptive conventions is meant a certain relation, whatever they may be, ‘correlating the words (= sentences) with the types of situation, thing, event, etc., to be found in the world’; while by
demonstrative conventions is meant a certain relation, whatever they can be, ‘correlating the words (= statements) with the historic situations, etc., to be found in the world’.
It appears to be quite reasonable to assume that for a collection of occurrences of ‘Dthat’ in ordinary discourse, each occurrence may be used on different occasions, but on each occasion, the speaker,
no matter whoever she/he is, points to the same object in the domain of discourse and use an occurrence of ‘Dthat’ to refer to the very object. Thus, we may treat all occasions of this kind, different as they can be, as a type of occasions. And we may then assume that, associated to each group of occurrences of ‘Dthat’ in ordinary discourse, there is a type of occasions, on each of which the speaker, no matter whoever she/he is, always points to the same object. We can further define a function d from types of occasions to the domain of discourse so that its value will serve as the
semantic value of every occurrence of ‘Dthat’. Of course, when two occurrences of Dthat are to be associated with distinct types of occasions, they are used to refer to different object. Intuitively, we may enumerate all occurrences of Dthat in ordinary discourse in accordance with the associated types of occasions. This can be achieved by adding appropriate subscripts to each occurrence of
Dthat. We may call an occurrence of ‘Dthat’ with appropriate subscript an indexed ‘Dthat’. For
example, consider two occurrences of Dthat in a sentence ‘That is a book’ uttered by John and Merry on two distinct occasions. If they are pointing to the different objects, then they are on distinct type of occasions. We may then say that John utters that ‘That1 is a book’, while Merry
states that ‘That2 is book’. If they are pointing to the same object, then they are actually on the
same type of occasions. Hence both are using ‘Dthat’ with the same indexical subscript. Intuitively, indexed ‘Dthat’ can be treated as a constant quantifier in the sense that different occurrences of the same indexed Dthat will be used to refer to the same object on all occasions. On the basis of the foregoing analysis, we can now give a formal language for Dthat. Let L be a standard countable first-order language with identity. L*D is an expansion of L by adding to the
alphabet of L a collection D of (indexical) quantifier D0, D1, D2, . . . , Dn, . . ., equipped with an extra
formation rule for indexical quantifiers: If ϕ is a formula, so is Dixϕ, for any Di ∈ D. That is, L⊆
L*D(= L ∪ D). The notion of the scope of an occurrence of the quantifier ‘Di’ in a formula is
defined in the usual way.
Now let M be an L-structure. Assume that there is a collection O of types of occasions, each of which is informally understood as a type of occasions on which a speaker (in a particular time t at a
particular location l) using an indexed ‘Dthat’ to refer to a thing in some structure. And define a
mapping d from {M}×O to A (the domain of M). Intuitively, a triple 〈M, o, a〉 for some o ∈O, a ∈A, can be understood as stating that the object a in M is precisely the object to which the speaker is pointing when uttering a sentence with ‘Thati’ as its grammatical subject.
Now, let
M*D = 〈M, d , I(D)〉,
Where I(D) is a function from D to the function d ; i.e., I : D →d . Intuitively, I can be construed as an assignment which assigns to each indexical quantifier Dia type of occasions o, for some o ∈O,
in M, so that the value of d (〈M,o〉), d (M,o) for short, will be the object to which the speaker uttering a sentence containing Di intends to refer. Putting this in another way, this can be understood as an
assignment of the variable x bound to the indexical quantifier Di in M, which takes some object, say
a, in M as the semantic value the bound variable x. The semantic rules we need are the followings: M*D ö iff M ö, for ö, any D–free formula.
M*D Dixö(x) iff for some o ∈O, d (M, o) = I(Di) and M ϕ(x) [d (M, o)] .
As usual, a logical system of Dthat can be presented. On the basis of a first-order logical system of natural deduction, the rules of inference for the indexical quantifier ‘Dthat’ are:
(D-Elimination) Dixö(x) (D-Introduction) ∀xö(x)
∃ xö(x) Di xö(x)
(¬D-Elimination) ¬Dixö(x) (¬D-Introduction) Dix¬ö(x)
Dix¬ö(x) ¬ Dixö(x)
and (ii) for an analysis of Dthat. It also shows how different uses of Dthat take us from different occasions of utterances to distinct objects. Moreover, we can easily construct a required logic of
Dthat. However, this approach apparently violates the required context-sensitivity of demonstratives
in the strict sense. Apparently, in ordinary discourse we by and large would not use the word ‘That’ with subscripts. The loss of the context-sensitivity of Dthat in the strict sense would make the foregoing formal language and its semantics inappropriate, let alone the establishment of such a logical system of Dthat. A different specification of occasions and the required function is then called for, to which we turn our attention now.
Sticking to the context-sensitivity of Dthat, no occurrences of ‘Dthat’ with subscripts are permitted. This implies that no so-called types of occasions will be associated with occurrences of ‘Dthat’; instead, associated to every occurrence of ‘Dthat’, there is an occasion on which the speaker intends to point to an object in the given domain of discourse when uttering a sentence with ‘Dthat’ as its subject. And the required formal language will include a sole indexical quantifier. Let L be a standard first-order language with identity. LD is an expansion of L by adding to the alphabet of L a
quantifier D, together with an extra formation rule for the indexical quantifier ‘D’: If ϕ is a formula, so is Dxϕ. Now, let M be an L-structure. Assume that there is a collection O of occasions, and a mapping d from {M}×O to A (the domain of M). Intuitively, a triple 〈M, o, a〉 for some o ∈O, a ∈A, can be understood as stating that the object a in M is precisely the object to which the speaker is pointing when uttering a sentence with ‘Dthat’ as its grammatical subject. To be more precise, let us call an occasiono faithful to a particular occurrence of ‘Dthat’ in a sentence containing ‘Dthat’ as a logical subject when the speaker does utter that sentence on the occasiono and intends to point to an
object by using the very occurrence of ‘Dthat’ as a demonstrative. Now, let
MD,o= 〈M, d , I(D)〉,
where I(D) is a function from {D} to d . Intuitively, I assigns a faithful occasion o, for some o ∈O, to an occurrence of the indexical quantifier D, so that the value of d (M,o) will be the object to which the speaker uttering a sentence containing D intends to refer. This can be understood as an assignment of the variable x bound to the indexical quantifier D in M, which takes some object,
namely some d (M,o), in M as the semantic value of the bound variable x. Now, if the assignment of the occurrence of D in a given sentence, say Dxö(x), takes d (M,o) as its value and the object d (M,
o) satisfies ϕ(x), we say that on the faithful occasion o, Dxö(x) is true in M, in symbols, MD,o
Dxö(x). The semantic rules we need are the followings:
(S1)MD,o ö iff M ö, for ö, any D–free formula.
(S2) For any occasion o ∈O, MD,o Dxö(x) iff I(D) = d (M, o), for o, the occasion faithful to the
given occurrence of D, and M ϕ(x) [d (M, o)].
We can further extend the notion of a faithful occasion to a set of occurrences of D. For simplicity, let us consider only a finite number of occurrences of D. Let 〈D〉n for some n ∈N, a finite set of
occurrences of D. An occasion o is said to be faithful to 〈D〉n if o contains a sequence of occasions
o1, o2, . . . , on, each of which is faithful to the corresponding occurrence of D. Thus, we can
evaluate a sentence with more than one occurrence of D on an occasion o in a structure M. More specifically, let
MD,o= 〈M, d , I(〈D〉n〉,
where I(〈D〉n) is a function from 〈D〉n to d . Intuitively, I assigns to each occurrence of the
indexical quantifier D in 〈D〉n a faithful occasion oi, for some oi∈O, so that the value of d (M,o1) will
be the object to which the first occurrence of D is supposed to refer, and so on. This can be understood as an assignment of the variables x’s bound to each occurrence of the indexical
quantifier D amongst 〈D〉n, which takes a sequence of objects, d (M,o1), d (M,o2), . . . , d (M,on), in M
as the semantic values of the bound variable x’s. For example, we may set as a semantic rule for
conjunction:
faithful to the two occurrences of D, respectively such that for the first occurrence of D,
I(D) = d (M, o1) and M ϕ(x) [d (M, o1)], while for the second occurrence of D, I(D) = d
(M, o2) and M ϕ(x) [d (M, o2)].
Finally, we may say that
(S) MD Dxö(x) iff for all faithful occasion o ∈O, MD,o Dxö(x).
This completes the required semantics for a formal language containing D (to stand for the demonstrative ‘Dthat’) as an indexical quantifier. One can see that this semantics indeed meets the three conditions we put forth before for an appropriate analysis of Dthat. Yet, it is somehow questionable to construct a logical system of Dthat in the language LD on the basis of the established
semantics. Admittedly, the classical concept of logical consequence would no longer hold because we can only talk about the logical consequence of utterances, rather than that of sentences/formulae. Similarly, most of theorems/validities in classical logic would no longer hold. For example, the well-known law of excluded middle collapses on the above semantics, unless we put forth a stipulation which asserts that all occurrences of D in a single (compound) sentence will take the same object as the semantic value of the associated variables. For the law of excluded middle states that Dxö(x) ∨ ¬ Dxö(x) holds in every model MD, and this in turn requires that for all o ∈O,
MD,o Dxö(x) ∨¬Dxö(x). But it is clear that the fact that in some MD, Dxö(x) fails to be true on
some occasion o would not imply that ¬ Dxö(x) will be true on the same occasion. For we may assign different objects in M to distinct occurrences of ‘D’ in ‘Dxö(x)’ and ‘¬ Dxö(x)’, which is equivalent to ‘Dx¬ö(x)’. It is then striking that classical logic cannot serve as the required
underlying system for a desired logic of Dthat. To my knowledge, so far no prevalent logical system can be suitable for such a desired logical system. At the moment, the closest one, perhaps, is a certain version of relevant logic. But some more arguments and further discussions are required. It seems to me that for the construction of a logic of demonstrative Dthat, the best we can do is to adopt the first way of formulation and the proposed semantic treatment with the cost of the loss of context-sensitivity in the strict sense.
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