Based on the formal treatment of semantic presupposition in the previous chapter, we are ready to formalize two crucial notions that we have introduced informally so far: the notions of truth-value gap and presuppositional language.
As we have observed repeatedly in chapters 5, 6, and 7, a communication breakdown between two P-language communities is often signified by the occurrence of a truth-value gap between the two languages. How can we explain semantically the occurrence of such a truth-value gap? Presumably, we need some sound semantic theory on truth-value conditions. Although I. Hacking’s styles of reasoning and N. Rescher’s factual commitments are constructed to provide the explanation, they do not work out a satisfactory semantic theory to back up their ideas. Kuhn seems to try to construct a truth-value condition based on possible facts to explain truth-value gaps. But his theory is apparently piggybacking on a dubitable correspondence theory of truth and the notion of possible world and possible fact. Besides, semantically, the linguistic formation of the notions of fact and possible fact is based on the notion of truth and truth-value status: A possible fact is the state of affairs described by a sentence with a truth-value and a fact is what is described by a true sentence. A potential circularity lurks in the background when Kuhn tries to define truth-value status based on the notion of possible fact.
The notion of a P-language based on the notion of semantic presupposition introduced in chapter 6 provides us with a basic conceptual framework for the explanation of a truth-value gap between two P-languages. If some absolute M-presuppositions underlying one P-language PL1 (such as the traditional Chinese medical theory) cannot be grasped or are not accepted by the speakers from the other P-language PL2 (such as contemporary Western medical theory), then many core sentences of PL1 are truth-valueless when considered within the context of PL2. Hence, a truth-value gap between PL1 and PL2 occurs. Now what we need is to work out a truth-value condition based on the notion of semantic presupposition.
The usual theories of truth are supposed to answer the question, ‘What is truth?’ However, this question is ambiguous and misleading. The question is ambiguous since it can be taken as a question about either the nature of truth (‘What is the nature of truth?’) or the conditions of truth (‘Under what conditions is a sentence true?’). M. Devitt calls the former a constitutive issue of truth, which concerns the semantic dimension (more precisely, the metaphysical dimension) of truth, and the latter is an evidential issue concerning the epistemic dimension of truth (Devitt, 1984).
As far as the metaphysical dimension of truth is concerned, there are two opposite views concerning the nature of truth: inflationary versus deflationary accounts of truth. The two accounts disagree on whether Tarski’s Convention T suffices to account for the nature of truth. According to the inflationary account of truth, Tarski’s Convention T is incomplete at explaining the entire conceptual and theoretical roles of truth, for it does not reveal the essential nature of truth. So Convention T has to be inflated with other semantic, epistemic, or pragmatic substantial properties, such as the metaphysical property of corresponding with reality, the epistemic property of being verified in ideal situations, the pragmatic property of facilitating successful activity, or even the logical property of belonging to a harmonious system of beliefs. As P. Horwich (1990) points out, there is a common doctrine shared by all the inflationists: The predicate ‘is true’ seems to attribute a substantial property such as the property of corresponding with reality to assertions. Truth has some hidden structure awaiting our discovery. Such a hidden structure is supposed to be the essential nature of truth. In other words, truth is an ingredient of reality whose underlying essence will, it is hoped, be revealed someday somehow. This essence of truth is by no means revealed by Convention T.
According to the deflationist point of view, such a doctrine about the substantial nature of truth is unjustified, and is actually false. It is caused by a mistaken analogy of the predicate ‘is true’ to some physical predicate, like ‘is magnetic’. But, unlike most other natural properties, ‘being true’ is not used to attribute any sort of property to assertions. There exists no underlying essence of truth to be discovered. This is the reason why Strawson contends that the question, ‘What is truth’, is misleading—it sounds like it only concerns the nature of truth and while doing so it smuggles in an essentialistic notion of truth. In contrast, according to the deflationist account of truth, Tarski’s Convention T is complete. The entire conceptual and theoretical roles of truth are fully captured by the initial triviality of Convention T. Truth is no more than what is presented in Convention T. There is no hidden naturalistic essence underlying the notion of truth. Of course, this does not mean that, after rejecting the idea that the truth predicate designates a naturalistic property, the predicate is no longer philosophically significant. It only means that the function of truth is no longer metaphysical as the inflationists expect. Truth predicates exist mainly for the sake of a certain logical need since we frequently have occasions to endorse one another’s statements, to reaffirm our own statements, and so on. In this sense, truth is still a property of some sort (a sort of logical property). As Kuhn puts it, ‘the essential function of the concept of truth is to require choice between acceptance and rejection of a statement or a theory in the face of evidence shared by all’ (1991, p. 9).
As far as the epistemic dimension of truth is concerned, a theory of truth first assumes, based on the principle of bivalence, that a fact-stating sentence has a (classical) truth-value, and then attempts to specify which truth-value it has in terms of some specific truth conditions. For example, according to one version of the correspondence theory of truth, a sentence is true if and only if it corresponds to a fact. According to the coherence theory of truth, a belief is true if and only if it belongs to a coherent system of beliefs.
It is easy to see that the two dimensions of the notion of truth are conceptually connected. The truth conditions specified by a theory of truth are in accordance with the nature of truth identified by the theory. For instance, based on the nature of truth identified by the correspondence theory of truth—namely, that truth is a substantial property of corresponding with reality—the truth conditions of a sentence consist in whether the sentence corresponds to reality. If truth is the essential epistemic property of being verified in ideal situations, then a sentence is true if and only if it can be verified in an ideal situation. However, the above way of thinking seems to encourage a misconception1 that the truth conditions specified by a theory of truth are determined by the very nature of truth identified by the theory. To me, the relation between the nature of truth and the truth conditions is the other way around. It is the truth conditions specified by a theory that indicate the nature of truth to which the theory commits. For example, since the truth conditions of a sentence, according to the correspondence theory, consist in whether the sentence corresponds to reality, the essential property of truth consists in corresponding with reality. In this sense, we can say that the truth conditions are more basic than the nature of truth. Therefore, I shall take the notion of truth as primarily an epistemic (in Devitt’s sense) notion concerning the truth conditions.
For this reason, I prefer to read the original question of truth, ‘What is truth?’ as the question, ‘By virtue of what is a sentence true if it has a truth value?’ The conditional clause of the question leaves room for the notion of truth-value status to figure in. According to this reading, the usual theories of truth—such as the correspondence theory, the coherence theory, or the pragmatic theory, etc.—are theories about the truth conditions. As to whether Convention T reveals all the essential natures of truth (if any), I prefer to leave this issue open considering our limited purpose here.
Since a usual theory of truth concerns the truth conditions of a sentence, it can only be used to determine the truth-value of the sentence by assuming that it has a truth-value. But what we need to know in the case of incommensurability under discussion is whether or not a sentence of concern has a truth-value. A usual theory of truth does not help us with this. For this reason, I would like to introduce a theory of truth-value to specify the truth-value conditions.
According to Tarski’s semantic theory of truth, a theory of truth for language L is a set of axioms that entail, for any sentence in L, a statement of conditions under which that sentence is true. If we have a definition of the truth predicate ‘is true in L’ satisfying Tarski’s Convention T,
(Con-T) s is true in L iff p
we have a theory of truth for L. When ‘s’ is replaced by a canonical description of a sentence S in an object language Lo and ‘p’ by a sentence P of a metalanguage LM, the corresponding T-sentence,
(T) S is true in L iff P,
gives us the truth conditions of sentence S in Lo.
Accordingly, we can define a semantic theory of truth-value as a theory that specifies, for any presupposing sentence in language L, the conditions under which it has a truth-value. I call such conditions the truth-value conditions. A satisfactory theory of truth-value should meet the following requirements: (a) Giving truth-value conditions to every presupposing sentence (i.e., the sentences that carry semantic presuppositions) within language L; (b) Not depending on any specific theory of truth so we can effectively separate our theory of truth-value from a theory of truth; (c) Validating our partial understanding of the notion of truth-valuelessness; (d) Giving an account of truth-value conditions that makes no use of unexplained semantic concepts (such as meaning, reference, synonymy, translation, interpretation, non-linguistic fact, state of affairs, etc.) and propositional attitudes (intention, belief, etc.) of the speaker. Consequently, the proposed truth-value conditions should be equally available to both the speaker and the interpreter involved in linguistic communication.
Strawson’s notion of semantic presupposition provides us with a basic theoretical framework for such a theory of truth-value. As I have argued in chapter 8, for any presupposing sentence, when considered within language L, the sentence would be truth-valueless if its presupposition failed to be true in L. Take the contrapositive of the same thesis: A presupposing sentence is true or false only when its presupposition is true in L. That means that the truth of a presupposition is necessary for truth-or-falsity of the sentence carrying that presupposition. For example, sentence (16) (‘The present king of France is bald’) presupposes sentence (16a) (‘The present king of France exists’). (16) is true or false only when (16a) is true; otherwise (16) is neither true nor false.
A presupposing sentence may have many different presuppositions. For example, sentence (16) has at least three different presuppositions, i.e., (16a), (16b) (‘There is a country named France’), and (16c) (‘A person can have hair’). If we can identify all the presuppositions of a presupposing sentence, then the truth of the conjunction of these presuppositions will be sufficient for truth-or-falsity of the sentence. For example, the conjunction of (16a), (16b), and (16c) (maybe plus other presuppositions), if true, is sufficient for (16) to be true or false. Here is another example. ‘The box is red’ presupposes ‘The box exists’ and ‘The box is colored’. If these two presuppositions are true, then we know that the sentence, ‘The box is red’, is either true or false.
I will call the conjunction of all the presuppositions of a presupposing sentence its sufficient presupposition, which can be defined as follows:
A sentence B is a sufficient presupposition for a sentence A when considered within language L iff ╞L TL(B) → (TL(A) v FL(A)).
It should be clear that if B is a sufficient presupposition of A, then the truth of B is not only sufficient but also necessary for the truth-or-falsity of A.
Let us formalize the above reasoning. Suppose Bi (i = 1, 2, . . . , n), when considered within language L, is a complete list of all the presuppositions of sentence A. Then its sufficient presupposition is B = {B1 & B2 . . . & Bn). Recall that if A presupposes Bi we have ╞L (TL(A) V FL(A)) → TL(Bi). Since B is a sufficient presupposition of A, we have the following formula:
╞L (TL (A) ∨ FL (A)) ↔ TL (B) |
|
Or |
╞L (TL (A) ∨ FL (A)) ↔ TL (B1 & B2 … & Bn). |
If any of the presuppositions Bi is untrue, then A will be neither true nor false; if all the presuppositions are true, then A is true or false.
Unlike Tarski’s Convention T that is used to determine the truth-value of a sentence in a language, we can use the above formula to determine the truth-value status of a presupposing sentence. I call it ‘Convention P’. Putting in a format analogous to Convention T:
(Con-P) A presupposing sentence A is true or false when considered within language L if and only if A’s sufficient presupposition B is true in L. That is, ╞L (TL(A) v FL(A)) ↔ TL(B), or in brief, A ⇒s B.
Here the subscript ‘s’ indicates that B is a sufficient presupposition of A. In other words, A is true or false from the perspective of L if and only if every presupposition of A is true in L.
Here is an example of the application of Convention P. Consider the fate of a Newtonian sentence (29) when viewed from the perspective of Leibniz’s language of space Lz.
(29) An object in an otherwise empty space could have been located at any number of different spatial points.
(29) presupposes (29a),
(29a) There exists a self-existing empty spatial continuum so that all physical and geometrical properties of it exist independently of different configurations of physical bodies within it.
(29a) is false from the perspective of Lz. Therefore, according to Convention P, (29) is neither true nor false when considered within Lz. Putting the inference into our formal formulation:
(29a) ╞ L (TL(29) v FL(29)) ↔ TL(29a).
But,
╞ L ~TL(29a).
Therefore,
╞ L ~(TL(29) v FL(29).
Our Convention P entails the following rules of presupposition:
(CP1) If A ⇒s B, then ~ A ⇒s B.
(CP2) If A ⇒s B1, B1 ⇒s B2 . . . and Bl ⇒s Bl + 1, then A ⇒s (B1 & B2 & ... & Bl, Bl + 1)
(CP3) If A ⇒ B and B ⇒ C, then A ⇒ C.
‘A ⇒s B’ means that B a sufficient presupposition of A; ‘A ⇒ B’ means that B is a necessary and sufficient presupposition of A.
A few points about Convention P should be kept in mind. First, similar to Davidson’s notion of truth, the notion of truth-value status denotes a primitive irreducible concept, which can be explained and exemplified in a specific linguistic context, but it cannot be given a general definition. Just as Tarski’s Convention T does not define the notion of truth, but defines the truth predicate ‘is true in L’, what we define in Convention P is not the notion of truth-value status, but rather the truth-value predicate ‘is true-or-false in L’.
Second, to avoid extraneous problems about the notion of truth, I use the conditions of holding-true in Davidson’s sense, instead of the conditions of being-true, as the truth conditions in determining the truth-values of the presuppositions of sentences in Convention P. To say that a sentence is held to be true in a language is to say that the language community accepts or believes the sentence to be true based on some available sufficient reasons. In this way, the truth-value of a sentence is attainable for the language community (Davidson, 1984, pp. 152, 173, 224). Truth-values are thereby language dependent and become a language community’s shared beliefs. However, our adoption of the conditions of holding-true does not make the notion of truth subjective. Davidson has argued that the fact that the speaker of a language holds a sentence to be true, under observed circumstances, is prima facie evidence that the sentence is true under those circumstances.2 Moreover, truth-values of sentences are relative to a specific language. Therefore, different languages can assign different truth-values to the same sentence. But the generic primitive notion of truth can still remain the same for upholders of different languages, for example, the notion defined by Tarski’s Convention T. In addition, it is essential to have attainable or recognizable truth conditions in order for truth-value conditions to be available to both the speaker and the interpreter involved in linguistic communication.
Third, as we can see from Convention P, whether a sentence S has a truth-value when considered within the context of language L depends on the truth-value of S’s sufficient presupposition in L. This establishes that the notion of truth-value status is a relational concept in the sense that the truth-value status of a sentence is internal to a specific language within which the sentence is considered. It is the language itself that creates the possibility of truth-or-falsity. A sentence may be a candidate for truth-or-falsity in one language but not in another. For example, sentence (30),
(30) Element a contains more phlogiston than element b,
has a truth-value when considered within the language of phlogiston theory, but has no truth-value when considered within the language of modern chemistry. This is because (30) presupposes (30a),
(30a) Phlogiston exists.
(30a) is held to be true within phlogiston theory, but untrue within modern chemistry. For this reason, (30) is true or false when considered from the perspective of phlogiston theory, but neither true nor false from the perspective of modern chemistry.
Fourth, to say that a sentence is true (or false) in terms of some given truth conditions presupposes that the sentence has a positive truth-value status in terms of some given truth-value conditions. In this sense, a theory of truth-value (the notion of truth-value status and truth-value conditions) is more fundamental than a theory of truth (the notion of truth and truth conditions). The notion of truth presupposes the notion of truth-value status. Therefore, to say of a sentence that it is true (or false) means that it has two properties. The first is that it satisfies some given truth conditions. The second is that it satisfies some given truth-value conditions. This second property is omitted entirely in the usual theories of truth. Classical concepts of truth have succeeded only at the expense of ignoring truth-value conditions by taking the positive truth-value status for granted. By including truth-value conditions as a precondition of truth, our double-truth-condition treatment of truth (the combination of truth conditions and truth-value conditions) is, I believe, a more rigorous development of a philosophical analysis of the notion of truth in general.
Suppose that a sentence B when considered within L1 is the sufficient presupposition of a sentence A. Assume further that B is held to be true in L1
╞ L1 TL1(B) → (TL1(A) v FL1(A)), and ╞L1 TL1(B).
Then we have ╞ L1TL1(A) v FL1(A). This means that A is true or false when considered within L1. By contrast, suppose that B is held to be untrue in another competing language L2:
╞ L2 ~TL2(B) → (TL2(A) v FL2(A)), and ╞L2 ~TL2(B).
Then, ╞ L2 ~(TL2(A) v FL2(A)). This establishes that A is neither true nor false when considered within L2. In this case, we say that there is a truth-value gap occurring between the two languages regarding an individual sentence A.
The occurrence of a truth-value gap regarding individual sentences is not just a theoretical hypothesis, but what has actually occurred in the history of science. Take sentence (30) as an example again. (30) presupposes both (30a) and (30b),
(30b) ‘a’ and ‘b’ are not empty (have proper denotations).
Take L1 as the language of phlogiston theory, L2 as the language of modern chemistry. In either language, (30a) and (30b) are presuppositions of (30) and the conjunction of (30a) and (30b) is a sufficient presupposition of (30). We know that (30a) is held to be true in phlogiston theory, but untrue in modern chemistry. For simplicity, let us further suppose that ‘a’ and ‘b’ do denote real objects. According to Convention P, from
╞L1 TL1(30a & 30b) → (TL1(30) v FL1(30)) and ╞L1 TL1(30a & 30b),
we have ╞L1TL1(30) v FL1(30). Therefore, (30) is true or false for the advocates of phlogiston theory. On the other hand,
╞L2 ~TL2(30a) → ~(TL2(30) v FL2(30)) and ╞L2 ~TL2(30a).
From these two formulae, we derive that ╞ L2 ~(TL2(30) v FL2(30)). It means that (30) is neither true nor false for the advocates of modern chemistry theory. Consequently, there is a truth-value gap occurring between the two languages regarding (30).
Of course, a sentence of a language could be truth-valueless even when it is considered within the context of the same language. For example, (30) could be neither true nor false even when considered within phlogiston theory if ‘a’ does not denote ((30b) is untrue). But there is an essential difference between the case in which (30) is truth-valueless because (30b) fails to be true and the case in which (30) is truth-valueless because (30a) fails to be true. The falsity of (30b) can be granted in phlogiston theory without endangering the integrity of the theory. However, the rejection of (30a) will endanger the integrity of phlogiston theory, which is not permissible by the theory.
The above analysis of the occurrence of truth-value gaps is restricted to individual sentences. However, the more significant cases are not the cases in which only a few isolated sentences are truth-valueless due to the failure of a few separated assumptions presupposed by these sentences, but rather the cases in which a substantial number of sentences of one language, which have truth-values in this language, lack truth-values when considered within the context of another competing language due to the failure of one or more shared presuppositions. In other words, it is possible that the occurrence of truth-valueless sentences in a language due to the failure of semantic presuppositions may spread throughout the whole language if those semantic presuppositions are absolute presuppositions of the language. In fact, such languages are what we call P-languages. Recall that by a P-language we mean a language whose core sentences share one or more absolute presuppositions.
Based on Convention P, the possibility that many core sentences of one P-language, when considered within the context of another P-language, might be truth-valueless is perfectly conceivable. Suppose that a set of sentences of one language PL1 <Sl>, i = 1, 2, ... n, presupposes the same assumption B when those sentences are considered within another competing language PL2. Suppose further that B is held to be false in PL2:
╞ L2 (TL2(Sl) v FL2(Sl)) → (TL2(B) and ╞L2 ~TL2(B), i = 1, 2, ... n.
Then according to Convention P, all the sentences in <Si>, when considered within the context of PL2, are truth-valueless. That is, ╞ L2 ~ (TL2(Si) v FL2(Si)). In this case, we say that a truth-value gap occurs between languages PL1 and PL2.
For illustration, let us reconsider the debate between the Newtonian language and the Leibnizian language of space. What causes the truth-value gap between the two P-languages of space? Based on our formal treatment of the two languages, the sentences in class SN-SL of the Newtonian language presuppose the existence of absolute space. One can speak of absolute changes of locations and absolute motion. Hence it is factually meaningful to talk about a comparison of the locations of a given body at different times as described in sentences (9) (‘The body b at time t could have located in a different place’) and (10) (‘The spatial location of the body b at time t1 is different from its location at time t2’). The assumption that there exists a self-existing absolute space is a shared universal assumption underlying most sentences in class SN-SL. Therefore, to accept the truth-or-falsity of sentences in SN-SL amounts to accepting Newtonian absolute space. Suspending or rejecting this assumption will change the truth-value status of these sentences from having truth-values to having no truth-values. Therefore, the different attitudes toward the truth-value status of sentences in SN-SL indicate different attitudes toward the assumption of Newtonian absolute space. This is exactly what happened when switching from the Newtonian language to the Leibnizian language. Due to the denial of the assumption of absolute space, the sentences in SN-SL when considered within the context of the Leibnizian language were truth-valueless. Consequently, a truth-value gap occurred between two languages.
A P-language is an interpreted language PL consisting of, but not limit to, at least the following two essential components:
(PL1) An uninterpreted language Lu = <Syn, Val> such that Syn is a syntax and Val is a set of logically possible valuations/interpretations that, by specifying a set of logically possible contexts, maps the sentences in Syn into logically possible truth-values.
(PL2) A set of postulates of truth-value status <PT> relative to Lu = <Syn, Val> such that <PT>, by specifying a set of conceptually possible contexts, maps the sentences in Syn into conceptually possible truth-values.
Syn is the syntax of PL that contains logical symbols with fixed meanings, descriptive symbols without fixed meanings, formation rules connecting logical symbols with descriptive symbols to form well-formed formulae, and sentences formed in terms of language formation rules. Take the Newtonian language of space LN as an example. In LN, logical symbols are quantifiers and truth-functional connectives. The descriptive symbols in LN include time terms and space terms (constants and variables), the proper names of physical bodies, temporal predicates, equality symbols, and two special function symbols.
The notion of possible truth-value refers to some possible usage of a sentence to say something true or false in a certain possible context. This is because whether a sentence can be used to make a statement depends upon certain contexts. For example, sentence (16) (‘The present king of France is bald’), when uttered by someone in the reign of Louis XV, was used to make a statement since (16) is actually true or false in this context. In contrast, the same sentence (16) might not be used to make a statement in another context. If (16) is uttered by someone today, then the sentence is neither true nor false since its presupposition that (16a) (‘The present king of France exists’) fails in this context.
Obviously, the notion of ‘context’ here does heavy duty. Although the notion as used here is a broad one, it is primarily a linguistic notion. For instance, the above utterances of (16) are two different speech acts occurring in two different linguistic contexts. Specifically, we can treat a language as a whole as a context (the context of a language). As we have pointed out earlier, the truth-value status of a sentence is language dependent. A sentence (say, sentence (30)) might be true or false when considered within the context of one language (say, the language of phlogiston theory), but lacks a truth-value within another (say, the language of modern chemistry). This confirms that the truth-value status of a sentence is (linguistically) context dependent. It is possible that a sentence could be neither true nor false when considered within one fixed context, but true or false when considered within another possible context. Especially, the same sentence could have a different truth-value status when considered within different languages.
The thesis of linguistic-context dependence of truth-value status implies that one sentence that is actually neither true nor false in a current context might be true or false in a different possible context. If (16) is uttered today, then it has no truth-value. Nevertheless, it could have a truth-value in some possible context, say, when uttered in the reign of Louis XV. Sentence (30), when considered within the context of modern chemistry, has no truth-value. However, it does not prevent the same sentence from having a potential truth-value when considered within some other possible contexts, say, within the context of phlogiston theory. For this reason, we need to distinguish the actual truth-value status of a sentence in a fixed context and its potential truth-value status in other possible contexts:
To say that a sentence has a possible truth-value means that it could be used to make a statement in some possible context C because its (sufficient) presupposition could be held to be true in C.
Presumably, whether the above definition of possible truth-value status makes sense depends on how to specify a possible context in which a sentence could be used to make a statement. Whether such a context is possible is relative to a particular language. Otherwise, anything is possible. So our question becomes: What counts as a possible context in which a sentence could be used to make a statement relative to a language? To clarify the notion of possible context, we need to make use of a common distinction between logical possibility and conceptual possibility.
One effective way to define logically possible contexts in which sentences could be used to make statements is to appeal to the notion of interpretation. A possible context in which a sentence could be used to make a statement relative to a language is a state of affairs that can be described by the language. Whether a state of affairs can be described by the sentences of a particular language depends on the interpretations that the language could assign to the descriptive terms of its sentences. Therefore, the possible contexts relative to a language can be specified by the so-called possible interpretations or valuations that could be given to the descriptive symbols of the language. As long as we can identify a set of possible interpretations relative to a language, we can identify the corresponding set of possible contexts.
An interpretation of a sentence S is logically possible relative to a language containing Lu = <Syn, Val> if the meanings assigned to the descriptive symbols in S are consistent with the intended readings of the logical terms in Syn and S is formed in terms of the formation rules of Syn.
This is exactly the function of Val in our P-language PL. Val does not assign any specific meanings to the descriptive symbols in Syn. Instead, it represents a range of all the logically possible interpretations consistent with the intended senses of the logical symbols and the formation rules in Syn. By assigning these logically possible meanings to the descriptive symbols in Syn and by following the formation rules, the interpreted language describes a set of logically possible contexts in which the sentences could be used to make statements. In this sense, we say that Val is a set of functions that, by specifying a set of logically possible contexts, map the sentences in Syn into logically possible truth-values. Suppose I ∍ Val and S ∍ Syn. Then, I(S) = a logically possible truth-value of S. Roughly speaking, as long as a sentence in Syn is in a good syntactic and semantic order according to Val, it could have a logically possible truth-value.
However, to say that a sentence has a logically possible truth-value for a language does not mean that such a possible truth-value is conceptually available for the speaker of the language. This is because to say that a context specified by a logically possible interpretation of a language is logically possible for a language does not mean that the context is conceptually accessible or recognizable to the speaker of the language. For illustration, let us consider an example given by M. Schlick. Schlick (1991) invites us to imagine an opponent who holds that ‘within every electron there is a nucleus which is always present, but which produces absolutely no effect outside’. When the sentence,
(31) Electrons have eternally hidden nuclei,
is considered within the context of modern atomic physics, it is logically possible for there to exist a context in which (31) could be used to say something true or false; since the sentence, not like the expression, ‘hidden UFO eternally electrons’, is in a good syntactic and semantic order. However, an atomic physicist cannot conceptually identify and comprehend a possible context in which the truth-value status of (31) can be verified (in terms of conceptual verifiability or testability). In other words, the truth-value conditions of (31), although they might be logically possible, are not conceptually accessible to the physicist. A similar analysis applies to sentence (4) (‘The association of the yin and rain makes people sleepy’). Western physicians cannot identify and comprehend the pre-modern Chinese mode of reasoning underlying (4). Therefore, they are unable to figure out a possible context in which the sentence could be used to make a statement. In this sense, the truth-value conditions of the sentence, although they are logically possible, are not conceptually accessible to them.
On the other hand, some contexts in which a sentence is put forward to make a statement are both logically and conceptually accessible to the interpreter. Take the Leibnizian language of space as an example. When considered within the Leibnizian language of space LZ, the sentences in SN-SZ, such as sentences (9) and (10) in chapter 5, have logically possible truth-values in a Newtonian space since the meanings of the terms in (9) and (10) are consistent with the intended meanings of the logical symbols, and these terms are connected according to the formation rules in Syn of LZ. In addition, the truth-value conditions of the sentences in SN-SZ are conceptually accessible to a Leibnizian. A Leibnizian is able to identify and comprehend the underlying presupposition of the sentences, i.e., the existence of Newtonian absolute space.
Presumably, to know the conceptually possible contexts in which a sentence could be used to make a statement is to know conceptually its possible truth-value conditions. As long as one can identify and comprehend conceptually the truth-value conditions of a sentence, one knows the possible contexts in which a sentence could be used to make a statement. Thus, an effective way to define (conceptually) possible contexts is to appeal to one’s knowledge of the truth-value conditions.
A context in which a sentence could be used to make a statement is conceptually possible for the interpreter of a language if he or she is able to identify and comprehend its truth-value conditions.
According to Convention P, the truth-value status of a presupposing sentence S is determined by its presuppositions. To know those underlying presuppositions of S is to know its truth-value conditions. For a P-language, the most significant presuppositions are shared absolute presuppositions, which we call metaphysical presuppositions (M-presuppositions). Thus, to be able to identify and comprehend M-presuppositions of a P-language is essential for the interpreter to identify and comprehend the truth-value conditions of its core sentences. Thus, we can define conceptually possible contexts for a P-language as follows:
A context in which a core sentence S of a P-language PL1 could be used to make a statement is conceptually possible to the interpreter of another P-language PL2 only if he or she is able to identify and comprehend (not necessarily accept) the M-presuppositions of PL1.
It should be clear now that M-presuppositions are what we have specified as the postulates of truth-value status, i.e., <PT> of a P-language PL. <PT> singles out a specific set of conceptually possible contexts from all the logically possible contexts described by Val of Lu = <Syn, Val>. By specifying a set of conceptually possible contexts, <PT> maps the sentences in Syn into conceptually possible truth-values: <PT> (S) = a conceptually possible truth-value of S.
One might be able to identify and comprehend the truth-value conditions of a sentence, i.e., its underlying presuppositions, without accepting them as true. Therefore, interpreters from one language might accept that a sentence has a conceptually possible truth-value (since they can recognize its underlying presuppositions), but categorically deny that it has an actual truth-value (since the presuppositions, from the perspective of their own language, are false). The Leibnizians can grasp very well the truth-value conditions of Newtonian sentences in SN-SZ, namely, the existence of Newtonian absolute space, but categorically deny that it fits reality. Hence, the sentences in SN-SZ do not have actual truth-values to them. For the Newtonians, however, the sentences (9) and (10) have actual truth-values, no matter whether they are actually true or not, since their presupposition (namely, the existence of Newtonian absolute space) is held to be true. But for the Leibnizians, (9) and (10) have only conceptually possible truth-values, not actual truth-values since the same underlying presupposition is not held to be true for them. To demonstrate, we can surely grasp the set of presuppositions underlying the sentences in the fairy story Snow White without committing ourselves to the reality of the story. Therefore, we need to distinguish possible truth-values from actual truth-values. To say that a sentence has an actual truth-value in a fixed context C (usually within a context of a language) means that the sentence is actually used to make a statement (a statement is either true or false) in C because its presuppositions are actually held to be true in C. By definition,
A sentence has an actual truth-value for the interpreter who speaks a language L if the presuppositions underlying the sentence are held to be true from the perspective of L.
Between the two components of a P-language, a set of postulates of truth-value status <PT>, which are the M-Presuppositions of the language, is the hallmark of the P-language. Two P-languages differ in just this regard, having different M-presuppositions. The conceptual core of a P-language consists in its M-presuppositions, which are contingent factual presumptions about the world perceived by the language community. This is why Rescher calls them factual commitments. To claim M-presuppositions of a P-language to be factual presumptions, even to be factual commitments, assumes that a language can have assertorial content. However, one might challenge this basic premise: How can a language have assertorial content or a point of view? A language is supposed to be mere means of making statements. Statements, and only statements, are supposed to have assertorial contents and hence truth-values, while a language should be assertorially neutral. One language might lack some means (such as some words and/or a category system) to express something that can be expressed by another language. But it does not follow that the users of the language categorically deny anything that their language cannot describe. The absence of ‘tiger’ in one language does not mean that the user of the language does not admit the existence of tigers. Moreover, the fact that one language has some means (words and category systems) to describe a state of affairs does not mean that the users of the language commit themselves to the existence of the state of affairs described. The presence of ‘unicorn’ in one language does not imply a commitment to the existence of unicorns.
The above point may be illustrated by the following pair of imaginable languages.3 Imagine that LA is the language of a tribe living deep in the Amazon, while LF is the language of a group of Fiji islanders. LA but not LF has terms for the vegetation making up the jungle canopy, terms for monkeys, for snakes, for lizards, for ungulates and carnivores, terms for tree spirits (a superstitious belief), as well as the corresponding category systems for these jungle plants and animals. LF but not LA has a word for the ocean, terms for waves and tides, for ocean fishes, for sailboats, fishing nets, for sea dragons, etc. as well as the corresponding category systems for them. The Amazonians can describe a state of affairs of a monkey picking a fruit from a tree for which the Fiji islanders lack expressions to describe. In contrast, a Fiji islander can say ‘Utu’s fishing boat has a big sail’, which the Amazonians cannot say at all. It seems absurd to claim that the Amazonians categorically deny the existence of sea dragons while the Fiji islanders deny the existence of tree spirits. It is not fully justified also to claim that the Amazonians commit themselves to the existence of tree spirits while the Fiji islanders commit themselves to sea dragons. The moral behind this imaginable case has been theorized by I. Scheffler (1967, p. 36) in terms of the distinction between a vocabulary on the one hand and a body of assertions on the other, between categories or classes on the one hand and expectations or hypotheses as to category membership on the other. In Scheffler’s words, ‘categorization provides the pigeonholes; hypothesis makes assignments to them’. The existence of certain pigeonholes does not compel the user to direct letters to them.
Though the above doctrine of assertorial neutrality of language might be applicable to some natural languages—languages that have nothing to do with scientific theorization—it is surely not applicable to the languages of our concern here, namely, scientific languages, or P-languages in general. Scientific languages behave like theories. When one adopts a scientific language, whether for communication, expressions, or description, one does make some assertorial commitments.
First, a scientific language has assertorial content by making assertorial commitments to the existence of certain theoretical entities. To adopt the language of phlogiston theory, one commits oneself to the existence of phlogiston. To adopt the Newtonian language of space, one commits oneself to the reality of Newtonian absolute space. To speak of the language of traditional Chinese medical theory, one makes a commitment to the reality and function of the yin and yang in the universe.
Second, some category systems adopted by some scientific languages do have assertorial content, for they make predictions and are therefore subject to the test of observation.4 By putting the earth, as a star, in the center of the universe, the Ptolemaic language makes different predictions and calculations of the movement of planets than the Copernican language. These predictions based on the Ptolemaic taxonomy are subject to empirical tests. In this sense, the Ptolemaic taxonomy has assertorial content by leading to testable observations. More importantly, as I will argue later, the taxonomy of a P-language actually functions as a set of sortal presuppositions of some substantial sentences of the language. The truth-values of these sortal presuppositions determine the truth-value status of these sentences. Because of this, we can even say, to some extent, that some category systems do qualify as being either true or false.
Third, a language can have assertorial content in the sense that it places restrictions on the possible events that can be expressed. By this, I do not mean the platitude that some possible states of affairs can be described by one language, but cannot be described by another because the latter lacks words to do so. The second language can be used to express the possible state of affairs in question if we simply enrich it by adding necessary words to it. Recall that the Amazonians cannot express an event of Utu’s fishing boat having a big sail. But that does not mean that the Amazonian’s language LA excludes the possible enrichment of the language such that the enriched language can be used to describe the state of affairs in question. What we should do is simply to add a set of terms for fishing boat, sail, ocean, etc. However, in many cases in which two languages involved are inconsistent5 with one another, some states of affairs that are describable by one language are not describable in principle by the other language or its enrichment. As Kuhn (1993a, pp. 330-31) has argued, with the Newtonian language of mechanics in place it does not make sense to speak of Aristotelian assertions in which terms like ‘force’ and ‘void’ play an essential role. This is because there is no way, even in an enriched Newtonian vocabulary, to convey the Aristotelian assertions regularly misconstrued as asserting the proportionality of force and motion or the impossibility of a void. Using the Newtonian lexicon, these Aristotelian assertions cannot be expressed. Therefore, some events or states of affairs described by the Aristotelian language are unable to be described by the Newtonian language. In this sense, the Aristotelian language places some restrictions on the possible events that can be described (recall Kuhn’s notion of possible worlds as lexicon-dependent).
In conclusion, it is no longer a novel idea that scientific languages and concepts are themselves laden with a variety of assertorial commitments about the world around us. They are no longer seen as a neutral vehicle for making substantive factual commitments but themselves are loaded with such commitments. A scientific language can and does have assertorial content.
The M-presuppositions of a P-language, functioning as semantic presuppositions, underlie many sentences of the language. But not all these sentences presuppose M-presuppositions directly, although some do. For example, sentence (30) (‘Element a contains more phlogiston than element b’) directly presupposes (30a) (‘Phlogiston exists’), which is a M-presupposition of the language of phlogiston theory. In contrast, sentence (32),
(32) Element a, when burning, releases more heat than element b,
when considered within the language of phlogiston theory, does not directly presuppose (30a). Instead, (32) when considered within the language of phlogiston theory directly presupposes (30). According to rule CP3 of Convention P, (32) indirectly presupposes (30a) within the language of phlogiston theory. Since the sentences that presuppose directly the M-presuppositions of a P-language are more close to the theoretical core of the language, I call them the core sentences of the P-language. By definition,
A sentence S of a P-language PL is a core sentence of PL if S directly presupposes some M-presuppositions of the language.
We have argued that truth-value status is language dependent. It is likely for one sentence S of one language L1, which is true or false within the context of L1? to lack a truth-value when considered within the context of the other competing language L2. But this way of speaking of a cross-language sentence and its truth-value status seems to cause some confusion: (a) Since S is not a sentence of L2, how can the interpreter from L2, who is not supposed to understand L1, consider its truth-value status? What is the semantic status of S anyway? Is S the original sentence in L1 the translation of the original sentence in L1 into a corresponding sentence S′ in L2, or the content or meaning of the original sentence?6 (b) Presumably, according to common wisdom, the semantic rules of a language, whatever they are, should be able to determine whether a sentence of the language has a truth-value. How, then, can the truth-value status of S in one language be determined by some facts or semantic rules about another language that is entirely external?7
To clarify this confusion, I have to clarify the notion of sentence and the meaning of ‘sentence’ that I have in mind. The term ‘sentence’ can be used either loosely and (one might argue) uncritically or strictly and critically in both philosophical discussion and everyday discourse. Strictly speaking, sentences are syntactic objects or well-formed linguistic symbols existing in a particular language. Sentences in this strict sense can be called uninterpreted sentences, such as {Snow is white}8 in which the term ‘snow’ does not have a fixed reference and the predicate ‘is white’ does not have a fixed extension. More precisely, ‘sentences’ here mean what we usually called sentence-types, syntactic forms that are exemplified by sentence-tokens (particular utterances, individual sounds and marks located in particular region of space and time). In contrast, loosely speaking, ‘a sentence’ can mean an interpreted sentence formed by assigning specific semantic values to the constituents of a corresponding uninterpreted sentence. For example, ‘Snow is white’9 is an interpreted sentence if the term ‘snow’ and the predicate ‘is white’ have fixed meanings as we use them in English. Furthermore, an interpreted sentence is either asserted or not asserted depending on the context in which it is considered. For example, the sentence ‘The present king of France is bald’ was asserted when it was uttered in the seventeenth century, but is not asserted when it is uttered today. The content asserted by an interpreted sentence is usually called an assertion, a statement, or a proposition.10 Different interpreted sentences, such as ‘John is loved by Jenny’ and ‘Jenny is falling in love with John’, can be used to make the same statement that Jenny loves John; and the same interpreted sentence, such as ‘I am in love’ uttered by Jenny and the same sentence uttered by John can be used to make different statements in different contexts.
Uninterpreted sentences have neither truth-values nor truth-value status. They are pure linguistic entities. But the notion of truth is a semantic notion that links language to the world. As far as the bearer of truth-value status is concerned, interpreted sentences have truth-value status, being either true-or-false or neither-true-nor-false. As far as the bearer of truth-values (in trivalent semantics) is concerned, interpreted sentences are true, false, or neither-true-nor-false. Statements, as asserted (interpreted) sentences, are always either true or false. In this sense, a statement has only one truth-value status, i.e., being true-or-false. So far, I have been using ‘sentence’ as interpreted sentence. I will continue to do so unless we specify otherwise.
Based on the above distinction, there is no confusion when we are only dealing with a single sentence within a single language. When we say that a sentence S of a language L is true or false, we actually mean that S is asserted (thereby used to make a statement) within L. When we say that S is neither true nor false, we actually mean that S is not asserted (thereby does not have a cognitive content) within L. But the troubles seem to emerge when we are dealing with cross-language sentences: If S refers to a sentence in L1 then how can interpreters who do not understand L! makes a judgment on the truth-value status of S (would S be trivially truth-valueless since it is just a noise for them)? If S refers to the translation of S into a corresponding sentence S’ in interpreters’ own language L2, then how can such translation get off the ground since they do not understand L1? Last, if S refers to the statement made by the sentence, then how can a statement lack a truth-value?
These questions arise from confusion between scientific languages or P-languages and natural languages. To see this, let us consider two different cases of cross-language understanding based on the distinction between scientific language and natural language. In one case, suppose that two scientific languages L(T1) and L(T2) (say, the Newtonian language and the Einsteinian language) are coded in the same natural language L (say, English). In this case, to say that a sentence S of L(T1) (which is also a sentence of L), which has a truth-value within the context of L(T1), has no truth-value when considered within the context of L(T2) is to say that the same sentence S, which is asserted or used to make a statement within the context of L(T1), cannot be asserted or used to make a statement within the context of L(T2). In the other case, suppose that two scientific languages PL1 and PL2 (say, the language of contemporary Western medical theory and that of traditional Chinese medical theory) are coded in two different natural languages L1 and L2, respectively (say, the former in English and the latter in Chinese). Then when we claim that a sentence S in PL1, which has a truth-value in PL1, is neither true nor false, we actually mean that the translation of S of L1 into a corresponding sentence S′ in L2 (not the translation into PL2) cannot be asserted or used to make a statement within the context of PL2.
These formulations clear up the confusions as long as we realize the following two points: (a) The interpreter who does not understand the language of a scientific theory T, namely, L(T), can know very well the natural language L in which T is coded, (b) The translation we need in the second case is between two natural languages (L1 and L2) used to code two scientific languages (L(T1) and L(T2)), not the translation between the two scientific languages. For interpreters from one natural language L1 to translate another natural language L2 into L1 might require them to understand L2. Nevertheless, such a translation does not require understanding of the scientific theory L(T2) coded in L2.
At last, let us turn to the problem of truth-value status of cross-language sentences. It is true that whether or not a sentence S of a scientific language L(T) has a truth-value, when considered within the context of the same language L(T), is determined by the semantic rules of L(T). But when the same sentence S (or the translation of S into the natural language in which another scientific language L(T′) is coded) is considered within the context of L(T′), due to the different semantic rules involved in L(T′), S will have a different truth-value status. We are not saying that the truth-value status of a sentence S of L(T) is determined by another language L(T′). What we are claiming is that when S is considered within the context of L(T′), S cannot be used to make a statement in L(T′). To claim that the truth-value status of sentences is language dependent is to claim that whether sentences can be used to make statements or to assert is language dependent.
1 M. Devitt (1984) seems to commit himself to this misconception when he emphasizes that the semantic dimension of truth should be distinguished from the epistemic dimension and that the former is more fundamental than the latter.
2 Davidson, 1984, pp. 152, 168-9, 200-201.
3 Austen Clark points out this imagined case to me.
4 This point is made in detail by H. Hung, 1981a and 1981b.
5 Inconsistency here means conflict in assertorial contents.
6 Anne Hiskes raises this concern.
7 An issue raised by Austen Clark.
8 I use { ... } to mark uninterpreted sentences.
9 I use single quotations ‘ ... ‘to mark interpreted sentences.
10 For my limited purpose, I will not make a further distinction between statement, assertion, and proposition here. There are two possible ways to make a distinction between proposition and statement/assertion, either by means of the theory of speech acts or in terms of the theory of intentionality. According to J. Searle’s theory of speech acts, a proposition is the common content expressed by a few closely related (interpreted) sentences. For example, (interpreted) sentences (a), (b), and (c),
(a) John will love Jenny.
(b) Jenny will be loved by John.
(c) John will fall in love with Jenny.
express the same proposition that John will love Jenny. A proposition does not have any special force attached to it, while a statement is a speech act that is a proposition with an assertorial force attached to it. For instance, the following three speech acts,
(d) I state that John will love Jenny.
(e) I question whether John will love Jenny.
(f) I promise that John will love Jenny.
express the same propositional content that John will love Jenny. Only (d) is a statement. In other words, a statement/assertion is a proposition the truth of which the speaker has committed him or herself to based on reasonable evidence.