It has become clear that Collingwood’s and Strawson’s notion of semantic presupposition, along with its two twin notions—truth-value status and truth-valuelessness (truth-value gaps)—are the very foundation of my presuppositional interpretation of incommensurability. However, the notion of semantic presupposition has been under constant attack. The attacks come primarily from two directions.1 On one front, some critics attack the notion indirectly by undermining the central notion of any theory of semantic presuppositions: the notion of truth-valuelessness. It has been a contentious debate whether we should admit the existence of truth-valuelessness in our natural languages, and whether we should introduce it into semantics. For many philosophers, not only do our natural languages not admit truth-valuelessness, but also the notion itself is highly suspect. For some, the notion is ‘such a creeping infection that when we allow it to emerge in semantics, it will smite it’. On the other front, other critics attack the notion of semantic presupposition head-on, either arguing that the notion itself is not theoretically coherent or contending that the notion, although it is theoretically coherent, is in fact empty since it cannot be exemplified. It is thereby not a philosophically interesting notion. For these reasons, the notion of semantic presupposition is ‘a contemporary myth in semantics’ and ‘needs to be brought to light’ (Böer and Lycan, 1976).
In particular, Böer (1976) and Lycan (1976, 1984, 1987, 1994) present lengthy and sophisticated arguments against the concept of semantic presupposition from both directions and their arguments have not been sufficiently rebutted. Some of their arguments are now being repeated by others and having a great deal of influence.
To clear these barriers in the way of the presuppositional interpretation, I need to clarify and defend the notions of truth-valuelessness and semantic presupposition against a variety of objections, especially Böer and Lycan’s. To do so, I will first formally present a coherent and integrated notion of semantic presupposition in terms of a formal treatment of a three-valued language. I will then turn to two central arguments against the notion of semantic presupposition presented by Böer and Lycan. At last, the notion of truth-valuelessness is defended against two critical objections.
Consider the following three sets of sentences.
(i) Existential sentences and presuppositions
(16) The present king of France is bald.
(16n) The present king of France is not bald.
(16a) The present king of France exists.
(17) Some unicorns in the African jungle are hairless.
(17n) Some unicorns in the African jungle are not hairless.
(17a) At least one unicorn exists in the African jungle.
(ii) Sortal sentences and presuppositions
(18) My soul is red.
(18n) My soul is not red.
(18a) My soul is/can be colored.
(19) Some planets travel around the earth.
(19n) Some planets do not travel around the earth.
(19a) The earth is not a planet.
(iii) State-of-affair sentences and presuppositions
(20) Astronomical event e1 precedes event e2.
(20n) Astronomical event e1 does not precede event e2.
(20a) There is an absolute time ordering.
There are two common features to each of the above sets of sentences: (a) It is clear that the first two sentences in each group somehow strongly ‘suggest’ or ‘imply’ the third sentence. This kind of ‘felt implication’ relationship between these sentences needs explanation, (b) When asked whether the first two sentences are true or false when the third sentence is not true, responders usually hesitate to give an affirmative or negative answer. They cannot simply choose one of the two classical truth-values (true or false) on the spot. Either answer seems to set a trap for them.
It is well known that B. Russell (1905, 1957) and P. Strawson (1950, 1952) gave different interpretations of the two features observed above. The debate has continued over the truth-value status of vacuous existential sentences like (16) since then. Russell and Strawson agreed that appeal to ordinary intuitions is not sufficient to determine whether sentence (16) is false or neither-true-nor-false when sentence (16a) is not true. We need to appeal to some theoretical considerations:
(a) For Russell, the felt implication between (16) and (16a) is nothing but the classical logical entailment relation in a subtle way. For Strawson, in contrast, the felt implication is one of semantic presuppositions in the sense that whenever both (16) and (16n) are true or false, (16a) has to be true.
(b) For Russell, when (16a) fails to be true, (16) is simply false. There is no need to appeal to truth-value gaps. The classical bivalent semantics is thus preserved. For Strawson, in contrast, when (16a) fails to be true, both (16) and (16n) would be neither true nor false. Obviously, we have to introduce some kind of trivalent semantics to accommodate the occurrence of truth-value gaps in our languages.
Before we move further, let us make a few important distinctions. First, a sentence could be truth-valueless either due to the failure of one of its presuppositions or due to purely syntactic or semantic matters. Non-declarative sentences (such as questions, imperatives, and performatives), some declarative sentences containing unspecified hidden parameters (for instance, sentences with vague predicates such as ‘It is a heap’; sentences dealing with moral judgments such as ‘Burning a cat for fun is morally wrong’), and ill-formed meaningless sentences (‘sat Kanrog subbppp on’) are commonly accepted as truth-valueless. But the real controversy arises when we ask whether a well-formed, meaningful, declarative sentence free of unspecified hidden parameters could be truth-valueless. Since this kind of sentences look like fact-stating sentences, we can call them fact-stating sentences in short. In the following discussion, by ‘sentences’ I will usually mean fact-stating sentences unless clarified otherwise. With the above qualification in mind, the issue to be addressed here is: Are there (fact-stating) sentences that are truth-valueless due to the failure of semantic presuppositions underlying them?
Second, the term ‘presuppositions’ has been used to describe pragmatic as well as semantic presuppositions. Very roughly put, presuppositions are pragmatic ‘iff the implications in question arise only in virtue of contextual considerations, the roles of the relevant sentences in standard speech acts, Gricean conversational matters, simple matters of background knowledge on the part of particular speakers, etc’ (Lycan, 1984, pp. 79-80). For instance, the following notion of presupposition is pragmatic: A sentence A presupposes (or invites the inference of) B if and only if, given certain background beliefs that we have, we would have some warrant for assuming that if one utters A, then one will act as if one is willing to be regarded as having committed oneself to the truth of B. On the other hand, presuppositions are semantic ‘iff the implications in question are a function of semantic status, semantic properties, propositional content, or logical form’, but not a function of context (Lycan, 1984, p. 79). In this chapter, we restrict our discussion to semantic presuppositions only, especially existential presuppositions. Semantic presuppositions include logical presuppositions (defined by logical implication within an uninterpreted language such as existential presuppositions) and analytical presuppositions (defined by analytical implication within an interpreted language such as sortal and state-of-affair presuppositions) (Martin, 1979, p. 268 f).
Considering that it is widely accepted that semantics deals with the relationship between language and the world, I prefer the following definition of semantic presupposition:
A presupposition is semantic if and only if the implication in question is a contingent factual presumption about the way the world is around the speaker and hearer.
Since the contingent factual presumptions about the way the world is can manifest themselves in different forms, we can classify semantic presuppositions into different categories, such as existential presuppositions (about the entities existing in the world), sortal presuppositions (about categorical frameworks of the world), and state-of-affair presuppositions (about some specific state of affairs).
As the title of the chapter indicates, I will restrict our discussion to semantic presuppositions only. My purpose in this chapter is to argue that the basic notion of semantic presupposition is sound. For this limited purpose, I will only focus on one special kind of semantic presupposition, namely, the so-called existential presupposition. I do not intend to give a full analysis to other kinds of semantic presuppositions, such as analytical presupposition, which will be touched on in chapter 11. Nevertheless, the basic conclusion drawn from the analysis of existential presuppositions is applicable to other kinds of semantic presuppositions.
Let us first set up a formal treatment of a trivalent language.2 This treatment will serve as a basic framework for the formal presentation of a trivalent version of semantic presupposition in the next section.
Def. An uninterpreted language L is any pair <Syn, Val> such that Syn is a syntax and Val (a set of admissible valuations for L) is a set of functions mapping the sentences of Syn into truth-values.
Here, Syn is a structure containing (a) sets of expressions or descriptive terms that have no fixed meanings; (b) a connected set of logical terms that have fixed senses and are paired one-to-one in accordance with formation rules; (c) a series of formation rules that connect descriptive terms with logical terms to form well-formed formulae. ‘Val’ represents the set of all logically possible worlds/interpretations consistent with the intended reading of the logical terms of Syn. Val does not assign any specific meaning to descriptive terms in Syn.
We need to state a truth predicate explicitly in order to formulate the notion of semantic presupposition in language L. Unfortunately, the truth of sentences within a given language is undefined in that language according to the proof of Godel and Tarski. We can define truth for L in M, but not in L, truth for M in M′, but not in M, and so on. Therefore, we have to extend L to M in which a truth predicate can be explicitly stated. We can achieve this by adding a sentential operator to L, defined as follows:
TL(A) = df. It is true in L (from the viewpoint of L) that A.
Here A represents any sentence.
In classical two-valued semantics, falsity is defined as the absence of truth by taking truth and falsity as contradictory concepts. Falsity is simply equal to non-truth. That is,
FL′(A) = df. It is not the case that it is true that A, or in symbols, not-TL(A).
In contrast, in our three-valued semantics, non-truth is further divided into falsity and neither-truth-nor-falsity. Falsity is defined as the truth of the negation of a sentence A. That is,
FL(A) = df. It is true that the negation of A, or in symbols, TL(not-A).
I adopt the definition of falsity in the three-valued semantics for obvious reasons. FL(A) is read as ‘It is false in L that A’ Table 8.1 is the truth-value table definition of the truth operator and the falsity operator in our three-valued semantics (‘n’ represents neither-true-nor-false):
Table 8.1 A truth-value table of truth and falsity operators
A |
T(A) |
F(A) |
F’(A) |
t |
t |
f |
f |
n |
f |
f |
t |
f |
f |
t |
t |
In our system, ‘A is true or false from L’s viewpoint’ can be expressed as TL(A) v FL(A). ‘A is neither true nor false from L’s point of view’ can be expressed as not-(TL(A) v FL(A)) or (not-TL(A) & not-FL(A)).
In our three-valued semantics, there are two notions of negation depending on what the designated truth-values are. Truth is always designated and falsity is never designated in any system of three-valued semantics. Whether neither-truth-nor-falsity is designated depends on whether one wishes to preserve truth or to preserve non-falsity in a valid inference. If one’s intention is to preserve truth, only truth is designated; if to preserve non-falsity, then both truth and neither-truth-nor-falsity are designated. If truth is the only designated truth value in L, we have a notion of unconditional negation:
Def. The unconditional negation of a sentence, briefly, ~A, is true if and only if the sentence denied is false.
Correspondingly, if non-falsity is the designated truth-value in L, we have a notion of conditional negation:
Def. The conditional negation of a sentence A, briefly, ¬A, is true if and only if the sentence denied is not true.
The corresponding truth-value table of these two concepts of negation is given in table 8.2.
Table 8.2 A truth-value table definition of conditional and unconditional negations
A |
~A |
¬A |
t |
f |
f |
n |
n |
t |
f |
t |
t |
In addition, let us extend the distinction between contradictories and contraries in traditional two-valued logic to our three-valued semantics.
Def. Two sentences are contradictories of one another if and only if they cannot both be true and they cannot both be false, although they may both be neither true nor false.
Def. Two sentences are contraries of one another if and only if they cannot both be true, but they can both be non-true.
Both unconditional and conditional negations are negations in the sense of contradictories.
Finally, conjunction, disjunction, and material implication can be defined in the following strong matrix (Kleene’s strong matrix):
Table 8.3 Conjunction, disjunction, and material implication
& |
v |
→ |
|
t f n |
t f n |
t f n |
|
t |
t f n |
t t t |
t f n |
f |
f f f |
t f n |
t t t |
n |
n f n |
t n n |
t n n |
Although logical entailment is essentially a notion of classical two-valued semantics, we can easily define it in our three-valued semantics:
Def. A entails B in a language L = <Syn, Val>, briefly, A ├ L B, iff for any V in Val, (a) if V(A) = T, then V(B) = T; and (b) if V(B) = F, then V(A) = F.
A crucial feature of entailment is that it preserves the principle of contraposition, that is:
A entails B if and only if ~B entails ~A, or in symbols, A ├ B iff ~B ├ ~A.
In addition, ├ L A means that A is unconditionally valid in L, or A is true in all valuations of L. For example, ├L TL(A) v FL(A) means that A is true or false in L unconditionally. Furthermore, we can use material implication and a truth operator to formulate the entailment relation as defined above as follows:
A ├ L B iff ├ L TL(A) → TL(B) and ├ L FL(B) → FL(A).
Corresponding to the entailment relation, which is essentially a notion of two-valued logic, we can introduce the notion of formal implication in our three-valued semantics to represent the logical inference relationship:
Def A formally implies B in a language L = <Syn, Val>, briefly, A ╞L B, iff for any V in Val, if V(A) = T, then V(B) = T.
Unlike entailment, formal implication does not preserve the principle of contraposition. ~B ╞ ~A does not necessarily follow from A ╞ B. Actually, the principle of contraposition is the principle of two-valued logic, which is dropped in any three-valued semantics. Furthermore, ╞LA means that A is unconditionally valid in L, or more precisely, A is never false in all valuations of L (A is always either true or neither-true-nor-false). ╞LTL(A) v FL(A) means that A is unconditionally true or false in L. Similarly, we can formulate formal implication in terms of material implication and a truth operator:
A ╞LB iff ╞LTL(A) → TL(B).
R1. |
╞ (F(A) → T(~A)) & T(~A) → F(A)) |
R2. |
╞ (F(~A) → T(A)) & T(A) → F(~A)) |
R3. |
If A ╞ C and B ╞ C, then A v B ╞ C |
R4. |
A ╞ B iff T(A) ╞ T(B) |
R5. |
T(A) v T(B) ╞ T(A v B) and T(A v B) ╞ T(A) v T(B) |
Especially, from R3, R4, and R5, we can infer that
if A ╞ B and not-A ╞ B, then ╞ (T(A) v T(not-A)) → T(B).
This inference will be very useful in our formulation of semantic presuppositions later. The same inference holds for entailment also:
if A ├ B and not-A ├ B, then ╞ (T(A) v T(not-A)) → T(B).
Some critics contend that the notion of semantic presupposition itself is theoretically trivial. According to an intuitive interpretation of the notion, a sentence B is a semantic presupposition of another sentence A if and only if whenever A is true or false B must be true. That means that both A and its negation entail B. However, this interpretation makes presupposition B tautologous: If A entails B and the denial of A entails B, then their disjunction, i.e., (A v ¬A), entails B. But (A v ¬A) is a tautology. Since a tautology cannot entail a non-tautologous sentence, B must be a tautology as well. Then a presupposition of any sentence can never be untrue (Böer and Lycan, 1976, p. 6). However, a presupposition should be contingent, not a tautology. Otherwise, any logical truth B is semantically presupposed by any sentence A; since a logical truth is entailed by any sentence. In other words, any sentence has an unrelated tautologous sentence as its presupposition. In this way, the notion of semantic presupposition is trivialized.
To rebut the above charge and other related criticisms, I need to construct a non-trivialized definition of semantic presupposition based on the trivalent semantics introduced in the last section. For obvious reasons, our following discussion will focus on logical presuppositions only.
Let us set up the following necessary conditions for any satisfactory notion of semantic presupposition.
The debate between Strawson and Russell on the notions of semantic presupposition and truth-valuelessness emerged from their different intuitive readings of sentences with non-denoting subject terms like (16). Both Russell and Strawson agree that (16) somehow implies (16a) in the sense that if (16) is true, then the truth of (16a) will necessarily follow. But they diverge when (16a) is false. Russell conceives the case in traditional two-valued semantics. Hence, the principle of contraposition holds between (16) and (16a) since (16) entails (16a). That means that (16) is necessarily false when (16a) is false. On the contrary, Strawson treats the case in a three-valued semantics in which the principle of contraposition does not hold. Sentence (16) should be neither true nor false when (16a) is not true. Furthermore, for Strawson, both (16) and the negation of (16), i.e., (16q), bear a special relation to (16a). If either (16) or its negation, (16q), is true, then (16a) is true as well.
For comparison, we can formulate Russell’s and Strawson’s intuitions, which I call Russell’s or Strawson’s rules, in table 8.4.
Table 8.4 Russell’s and Strawson’s rules of semantic presupposition
Strawson’s Rules |
Russell’s Rules |
Comparison |
Rule I: |
||
╞ T(16) → T(16a) |
├ T(16) → T(16a) |
agree |
Rule II: |
||
╞ T(16n) → T(16a) |
├ T(16n) → T(16a) or ├ T(16n) → F(16a) |
(partially) disagree |
Rule III: |
||
╞ F(16a) → ~(T(16) v F(16)) |
├ F(16a) → F(16) |
(completely) disagree |
Any satisfactory formal account of semantic presupposition has to validate Strawson’s rules.
Although the truth-value of (16), when (16a) is not true, is controversial, it is widely accepted that both sides should agree on the following claims. Given a sentence,5
(21) The current president of China is bald.
Then,
(a) Although both (16) and (21) are non-true, (16) is non-true in a way which differs from the non-truth of (21). The non-truth of the former is due to failure of the denotation of the subject while the latter due to the falsity of the predicate of the subject.
Consequently,
(b) It is possible for the negation of (16) to be non-true if (16) is non-true (when (16) is neither true nor false). But the negation of (21) is true if (21) is non-true ((21) is actually false).
In general, there is an intuitively recognizable distinction to be drawn between two kinds of non-true sentences. Such a distinction needs an explanation. A satisfactory account of semantic presuppositions should regiment the data in (a) and (b).
The semantic presuppositions of a sentence should be contingent, not tautologous. In other words, a semantic presupposition of a sentence may fail to be true under some interpretations or models. This requirement is intended to exclude the possibility that any logical truth (sentences that are true under all possible interpretations) is semantically presupposed by any sentence.
The following are some typical definitions of semantic presupposition:
(P1) |
A semantically presupposes B if and only if both A and noWl imply B. |
(P2) |
A semantically presupposes B if and only if both A and its logical contrary imply B. |
(P3) |
A semantically presupposes B if and only if A entails B and the negation of A entails B. |
(P4) |
A semantically presupposes B if and only if both A and the negation of A materially imply B. |
(P5) |
A semantically presupposes B if and only if A necessitates B and ╞ F(B) → (~T(A) & ~F(A)). |
(P6) |
A semantically presupposes B if and only if A necessitates B and the negation of A necessitates B. |
(P7) |
A semantically presupposes B if and only if B is a necessary condition of the truth or falsity of A. Or, whenever A is true or false, B is true. |
Each of these definitions has some flaws and cannot meet our conditions of adequacy of the notion of semantic presupposition. For clarity, I will divide these definitions, according to their structures, into three groups and examine each group below.
P1, P2, P3, and P4 define semantic presupposition in a broadly similar way. They share a common structure represented by P1. Actually, P1 can be used as a schema to represent most of formal accounts of semantic presupposition. For this purpose, let us reformulate P1 as Schema P:
Schema P: A sentence A semantically presupposes B in a language L, briefly, A ⇒ B, if and only if both A and its negation, not-A, imply B in L.
Schema P not only represents some existing formal formulations of semantic presupposition, like P2, P3, and P4, but also covers some potential candidates for it. By discussing Schema P, we will touch on a rather broad range of possible interpretations of the notion of semantic presupposition.
Different theories of semantic presupposition differ radically as to how to unpack Schema P, especially in their interpretations of the negation ‘not’ and implication relation in it. This is because Schema P has two undetermined parameters. One is the sense of the negation ‘not-A’, the other the sense of ‘imply’. Let us examine each parameter in turn.
Suppose that A presupposes B. If ‘not-’ in Schema P is understood as a conditional negation, then from Schema P, we have:
(a) A implies B and (b) ¬A implies B.
If ‘implies’ means ‘formally implies’ (It will not affect our argument if ‘implies’ is read as ‘entails’ or ‘materially implies’), then (c) follows from (a) and (b):
(c) ╞(T(A) v T(¬A)) → T(B).
By contraposition, we have:
(c’) ╞ ¬T(B) → ¬(T(A) v T(¬A)).
Since T(¬A) ≠ F(A), when B is not true, A is not necessarily neither true nor false. This proves that the definition of semantic presupposition in terms of conditional negation does not conform to Strawson’s rule III. Therefore, this account is too weak.
However, perhaps the defender of conditional negation would protest that the notion of semantic presupposition in terms of conditional negation is intended to avoid truth-valuelessness. Then the fact that it does not conform to Strawson’s rule III should be regarded as its merit instead of a flaw. Let us accept this defense for the sake of argument. Nevertheless, this defense still cannot eliminate another serious problem faced by defining semantic presupposition in terms of conditional negation. If B is untrue, then the antecedent ¬T(B) is true. In order to make (c′) unconditionally valid, the consequent ¬(T(A) v T(¬A)) has to be true. But the consequent is logically false. That establishes that A’s presupposition, B, can never fail to be true. Thus, this notion of semantic presupposition based on conditional negation is trivialized.
As I have pointed out before, either unconditional or conditional negation of a sentence is the contradictory of the sentence (in the sense that a sentence and its contradictory cannot both be false or both be true). When faced with the threat of trivializing semantic presupposition, one way around it is to employ the notion of contrary, instead of contradictory, in defining semantic presupposition. This is the rationale of P2. The basic idea behind P2 is quite simple. Let us say that every sentence not only has a negation in the sense of logical contradictory (no matter whether as a conditional or as an unconditional negation), but also has a logical contrary. Then we can define semantic presupposition in terms of logical contrary instead of logical contradictory. If a statement and its logical contrary both imply a common statement, then they presuppose that statement. Let us formalize P2 in our formal system (taking ‘imply’ as formal implication for a reason that will become clear later). That is,
A presupposes B iff ╞ T(A) → T(B) and ╞ T(*A) → T(B).
Here ‘*A’ represents the logical contrary of A. From this definition, we have:
(d) ╞ (T(A) v T(*A)) → T(B) and (e) ╞ ~T(B) → ~(T(A) v T(*A)).
When B is untrue, the antecedent of (e), ~T(B), is true. Since A and *A can both be false, the consequent of (e), i.e., ~(T(A) v T(*A)), may be true when B is false. Therefore, it is possible for B to be untrue while (e) is unconditionally valid. No truth-value gap necessarily occurs. P2 appears to be a decent solution that avoids trivializing presupposition and preserves bivalent logic as well.
According to the notion of logical contrary in traditional two-valued logic, the definition of logical contrary is clear. It has to be defined in such a way that when two sentences are contraries of one another they can both be non-true although they cannot both be true. Now, the real problem is how to formulate the logical contrary, *A of a typical presupposing sentence A. G. Englebretsen (1973) suggests treating the logical contrary of a sentence A as the sentence with a negation occurring within A, which Russell calls the secondary occurrence of negation. In contrast, the logical contradictory of A, which Russell calls the primary occurrence of negation, is the sentence with a negation outside A. For example, the contrary of a universal subject-predicate sentence,
(22) A11S is P,
is
(*22) All S is not P, or No S is P.
But its logical contradictory would be
(~22) It is not the case that all S is P, or in symbols, not-(all S is P) = Some S is not P.
For a singular subject-predicate sentence,
(23) S is P,
its logical contrary is
(*23) Sis not P,
while its contradictory is
(~23) It is not the case that S is P, or in symbols, not-(S is P).
It will become clear that such a distinction between logical contradictory and contrary is nothing but the distinction between external negation and one kind of internal negation (the internal negation of a sentence as the contrary or as the subcontrary of that sentence).
Since I will address the problem of analyzing the notion of semantic presupposition with respect to the distinction between external and internal negations in detail later, I will leave my criticism of P2 until then. Nevertheless, the following two points should suffice to show the inadequacy of P2. On the one hand, not all presupposing sentences (such as particular subject-predicate sentences) have their corresponding contraries; on the other hand, the real trouble with P2 is that it does not conform to Strawson’s rule III. It makes no sense to call a ‘felt implication’ a semantic presupposition if it does not support the notion of truthvaluelessness; it would be more appropriate to regard such an implication as another version of entailment.
From the above analyses we know that the negation of a presupposing sentence A, i.e. not-A, cannot be either A’s conditional negation or its logical contrary. An appropriate candidate of the negation in question seems to be the unconditional negation of A: ~A within three-valued logic. Suppose that A presupposes B. Taking the negation of A as an unconditional negation, according to Schema P (taking ‘implies’ as formal implication), we have:
(f) ╞ (T(A) v T(~A) → T(B)
(g) ╞ ~T(B) → ~T(A) v T(~A)).
A has to be neither true nor false when B is not true.
Nevertheless, I have to point out that ~A is not the only candidate for the negation that could be used to define semantic presupposition. We know that some sentences not only have their contradictories but also have their subcontraries. We can define the notion of logical subcontrary as follows:
Def. Two sentences are subcontraries of one another if and only if they cannot both be false but can both be non-false.
For example, the internal negation of a particular subject-predicate sentence is the subcontrary of that sentence:
(24) Some S is P.
(#24) Some S is not P.
Suppose that A presupposes B. Let us take the negation of A in Schema P as the subcontrary of A and use the symbol ‘#A’ to represent it. Then, from Schema P (taking ‘implies’ as formal implication), we have:
(h) ╞ (T(A) v T(#A) → T(B)
(i) ╞ ~T(B) → ~T(A) v T(#A)).
According to formula (i), A has to be neither true nor false when B is not true. The possibility of A to be true or to be false is ruled out when B is not true. Otherwise, B cannot be untrue, and is thereby trivialized.
The requirement of the negation of a presupposing sentence in Schema P as the subcontrary of the sentence is weaker than the requirement of the negation as the contradictory. But the problem with this is that for some presupposing sentences, there is no corresponding subcontrary (for example, a universal subject-predicate sentence, All S is P, does not have a subcontrary.). From now on, I will take the negation in Schema P, i.e., ‘not-A, as either the subcontrary of A, i.e., #A or as the contradictory of A, i.e., ~A, if the subcontrary is not available. Since the notion of subcontrary is more comprehensive than the notion of contradictory, we may read a contradictory as one case of subcontrary. In the following discussion, for clarity and simplicity of formal treatment,6 I will only use ~A in the related formulae unless indicated otherwise. But please remember that reading ‘not-A’ in Schema P as the subcontrary of A is more precise.
Table 8.5 A comparison of negations ‘not-A’
subcontrary/contrary |
contradictory |
|||
A |
#A |
*A |
~A |
¬A |
t |
t / f |
f |
f |
f |
n |
n |
n |
n |
t |
f |
t |
f / t |
t |
t |
P3 takes ‘implies’ in Schema P as entailment. The major problem with P3 is that if we define the notion of semantic presupposition in terms of entailment, then the notion would become trivialized since the presuppositions of a sentence based on such a definition can never be false. This can be shown easily by means of the formal system we have introduced earlier: If A ├ B and ~A ├ B, then by the definition of entailment, we have
(a) ├ T(A) → T(B)
(b) ├ F(B) → F(A)
(c) ├ T(~A) → T(B)
(d) ├ F(B) → F(~A)
By R2, the combination of (b) and (d) leads to
(e) ├ F(B) → (T(A) & F(A)).
That means that if B is a semantic presupposition of A, then B cannot be false. Otherwise, the formula (e) cannot be valid since its consequent, T(A) & F(A), can never be true (it is logically false). Furthermore, from (a) and (c) we have
(f) ├ ~T(B) → (~T(A) & ~F(A)).
(f) is what we expect. But it is not consistent with (e).
If a theory interprets the implication in Schema P as material implication, we may call such a presupposition material presupposition by the analogy of material implication. P4 defines such a material presupposition. The problem with P4 is easy to see. If ├ (A v ~A) → B, then, by contraposition, ├ ~B → (A & ~A). When B is false, the antecedent ~B is true, then the consequent A & ~A has to be true since the formula is unconditionally valid. But the consequent A & ~A cannot be true. That means that the presupposition B of A cannot fail to be true, and is thereby trivialized.
The real reason why a definition of semantic presupposition by virtue of entailment or material implication makes the notion trivialized is not very hard to see. This is because both relations, although they can be defined in three-valued logic, still preserve the principle of contraposition. The principle of contraposition is essentially the principle valid in classical two-valued logic. Any three-valued semantics rejects the principle of contraposition. It is plain now that the upshot from the failure of defining semantic presupposition in terms of classical entailment or material implication is that a notion of semantic presupposition requires a strict implication that does not preserve the principle of contraposition. A non-classical implication is called for. The formal implication defined in our three-valued semantics is what we need.
Böer and Lycan correctly diagnose that defining semantic presupposition in terms of entailment relation would trivialize presupposition because entailment supports contraposition. They claim that the proper way to analyze the notion of semantic presupposition is to employ a model-theoretic notion of strict implication that does not support contraposition. That is the notion of necessitation. It is clear that, in Böer and Lycan’s hands, the notion of necessitation functions as the notion of formal implication as defined earlier. ‘A sentence S1 necessitates a sentence S2, roughly, just in case there is no model relative to which S1 is true and S2 is untrue’ (Böer and Lycan, 1976). So, a sentence A necessitates another sentence B if and only if ╞ T(A) → T(B). Then P5 can be formulated as:
(a) ╞ T(A) → T(B)
(b) ╞ F(B) → (~T(A) & ~F(A)).
The real problem with P5 is that it cannot justify Strawson’s rule II, which can be restated here as ╞ T(not-A) → T(B). From (b), by contraposition, we have
(c) ╞ (T(A) v T(~A)) → ~F(B).
(c) holds if and only if both
(d) ╞ T(A) → ~F(B) and (e) ╞ T(~A) → ~F(B)
hold. In our three-valued logic, ~F(B) does not imply T(B). Hence, Strawson’s rule II cannot be derived from P5. In fact, when A is false but B is untrue, Strawson’s rule II is falsified while the same valuation validates (a) and (b). This establishes that, supposing that A presupposes B, according to P5, the truth of the negation of A does not necessarily imply the truth of B. This directly violates Strawson’s rule II.
If we read necessitation relation as that defined by Böer and Lycan, P7 can be easily derived from P6, and vice versa. From P6, we have
╞ T(A) → T(B) and ╞ T(~A) → T(B).
By R3, ╞ (T(A) v T(~A)) → T(B). By R2, P6 eventually becomes
╞ (T(A) v F(A)) → T(B).
Thus, it is nothing but P7.
P7 seems to have an appealing character free from the theory of negation. It appears that without explicitly mentioning negation, we would avoid much confusion caused by different readings of negation. However, this appealing feature is only superficial. It is more convenient to use the concept of falsity directly as far as Strawson’s rule III is concerned. But in formulating and testing any definition of semantic presupposition, we have to deal with Strawson’s rule II. In doing so, we need to explicitly use a specific reading of negation. For example, according to P7, sentence (16) presupposes (16a) if and only if both the truth of (16) and the falsity of it formally imply (16a). How can we find whether the falsity of (16) implies (16a), if the definition of formal implication only specifies the relation between the truth of an implying sentence and the truth of the implied sentence? We have to convert the falsity of (16) into the truth of its contradictory, namely, F(16) = T(~16). Therefore, the employment of negation in defining presupposition is inevitable. For this reason, we can regard P7 as one version of Schema P.
Besides, P7 does not specify the notion of falsity. As we have mentioned before, there are two senses of falsity: one in classical logic, the other within three-valued logic. If the falsity of A in P7 refers to the bivalent notion of falsity, i.e., F′(A), then P7 would be trivialized. For then P7 is equivalent to
╞ ~T(B) → ~(T(A) v F′(A)).
Since T(A) v F′(A) is a tautology (F′(A) = ~T(A)), the consequent ~(T(A) v F′(A)) is logically false. Thus, B cannot fail to be true and is trivialized. Therefore, we cannot leave the notion of falsity in the definition of semantic presupposition unspecified. The only way out is to define falsity in three-valued logic. This feature is reflected in Schema P by T(not-A) that represents the notion of falsity in three-valued logic.
The general conclusion drawn from the above analyses of PI, P2, P3, P4, P5, P6, and P7 is plain: The best candidates for the two parameters of Schema P are: (a) reading ‘not-A’ as the subcontrary (including contradictory) of A; (b) reading ‘imply’ as formal implication. Then Schema P can be refined as follows:
Schema P: A sentence A semantically presupposes a contingent sentence B7 in a three-valued language L, briefly, A ⇒ B, if and only if both A and its subcontrary, #A, (or its unconditional negation, ~A, if the subcontrary is not available) formally imply B in L.
That is, A ⇒ B iff ╞ (T(A) v T(#A / ~A)) → T(B).
From now on, I will use this modified Schema P as our formal definition of semantic presupposition.
Let us test Schema P against our three requirements of the notion of semantic presupposition. First, it is possible for B to fail to be true in our definition. Actually, when B is not true, no matter whether we take the negation of A as the subcontrary or as the contradictory of A, A has to be neither true nor false. That is,
╞ ~T(B) → ~(T(A) v T(~A / #A)).
Therefore, we have a non-trivialized notion of semantic presupposition. This not only takes care of the non-trivialization requirement, but also meets Strawson’s rule III. Second, sentence (16) (‘The present king of France is bald’) is neither true nor false; for its presupposition (16a) (‘The present king of France exists’) fails while sentence (21) (‘The current president of China is bald’) is false since a presupposition of (21) (‘The present president of China exists’) is true. In this way, we make a reasonable distinction between two kinds of non-true sentences by assigning them different truth-value status.
Third, in order to meet the rule II, we need to show that both a presupposing sentence and its negation (in the sense of unconditional negation or subcontrary) bear a special relation to a third sentence, i.e., their presupposition. If A is a particular subject-predicate sentence, say, (24), then it is obvious that both A and its subcontrary #A, say, (#24), formally imply their presupposition
(24a) There exists at least one S.
Now the problem is whether both a singular subject-predicate sentence (23) and its contradictory (~23) (or its subcontrary (#23)) imply their presupposition (23a), respectively, that is,
(23a) S exists.
It is plain that when (23) is true, (23a) has to be true. So (23) ╞ (23a). However, there is a doubt whether the negation of (23) formally implies (23a) (Böer and Lycan, 1976). I will argue below that the negation of (23) does formally imply (23a).
Actually, our three-valued language L (which is a representation of our natural language) permits us to conclude that both (23) and (~23) formally imply (23a). What we need to do is to specify further its semantics Val.8 Val is often defined by virtue of a set of formed structures called models. A model is a subset of possible worlds or interpretations. Such a model consists of two parts: one is the domain D of the discourse, the other the function f that maps the predicates in Syn into the elements in D. Valuations are then defined to represent the models by assigning specific truth-values to each sentence under a specific model.
Following J. Martin (1975, p. 257), let us specify a model M for Syn of L as any pair <D, f>. Syn is the syntax of L with a singular subject-predicate sentence, ‘S is P’ Syn also contains a logical term ‘exists’, the existential predicate. Here, ‘D’ is a non-empty domain. ‘f’ is a function on all predicates and some subjects such that (a) for any predicate P, f(P) is a subset of D; (b) for any denoting subject S, f(S) is in D; (c) f(exists) = D. The set Val representing the model <D, f> maps sentences of Syn into truth-values in the following way: For any singular subject-predicate sentence, ‘S is P’, it is true if f(S) is in f(P) (‘S’ refers to something within the extension of P’); ‘S is P’ is false if f(S) is in D but not in f(P) (‘S’ refers to something that is not within the extension of CP’); ‘S is P’ is neither true nor false otherwise (‘S’ does not refer or f(S) is not in D). It follows from the above valuation that
S is P ╞ S exists or (23) ╞ (23a) and ~(S is P) ╞ S exists or (~23) ╞ (23a)
since they are theorems of L under the model M.
In conclusion, our definition of semantic presupposition meets all the three requirements of any satisfactory notion of semantic presupposition. This shows that the notion of semantic presupposition is theoretically coherent and integrated.
The remaining problem is whether a theoretically coherent notion of semantic presupposition as I have defined it can be exemplified as a practically feasible notion. Böer and Lycan believe that it cannot although they accept it as a theoretically coherent notion. In this and the next sections, I will consider Böer and Lycan’s two central arguments against the notion of semantic presupposition: the argument from the distinction between internal and external negation, and the argument from counterexamples. In terms of those two arguments, Böer and Lycan’s were intended to show that the notion of semantic presupposition, although it could be made to be theoretically coherent, is actually empty.
Böer and Lycan’s first critical argument against semantic presupposition has the following two basic components.
It is believed that negation in our natural language is ambiguous not only due to two different readings of negation with respect to their senses (i.e., unconditional negation and conditional negation), but also due to different scopes of negation. For example, Russell’s paraphrase of a grammatically simple sentence (16) is a logically complex sentence (16′),
(16′) There exists one and the only one person who is the present king of France, and this person is bald. Put in symbols, ∃x (Bald(x) & ∀y (King(y) ↔ x = y)).
According to Russell, the negation of (16) is ambiguous with respect to the scope of the negation. The negation can attach to the widest possible scope (the primary occurrence of negation). That is,
(ex-16′) It is not the case that there exists one and the only one person who is the present king of France, and this person is bald. In symbols, ~ ∃x (Bald(x) & ∀y (King(y) ↔ x = y)).
On the other hand, the negation can attach to the narrow scope (the secondary occurrence of negation). That is,
(in-16′) There exists one and the only one person who is the present king of
France, and this person is not bald. In symbols, ∃x (~Bald(x) & ∀y (King(y) ↔ x = y )).
Böer and Lycan adopt Russell’s two readings of negation with respect to scope, and call ex-(16) its external negation and in-(16) its internal negation.
The distinction between external and internal negation is a scope distinction, a negation being external when it has wide scope, internal when it occurs within the scope of the ‘presupposition’-generating locution. (Lycan, 1984, p. 91)
Presumably, based on our analysis of negation in Schema P, external negation corresponds to the logical contradictory. For this reason, I will use ~A to represent the external negation of A later on. The notion of internal negation corresponds to either the notion of contrary or the notion of subcontrary depending on the structure of the sentence in question.
Two essential requirements of presupposition Suppose A presupposes B. According to the adequacy requirement of the notion of semantic presupposition, there are two essential requirements for the negation of A. First, the negation of A has to formally imply B. That is, not-A ╞ B or ╞ T(not-A) → T(B). Second, the negation of A has to be the logical contradictory of A. If either one of these two conditions is not met, then A cannot be said to presuppose B.
The first requirement is obvious. The second appears to be convincing if we realize that in the following formula,
╞ ~T(B) → ~(T(A) v T(not-A))
‘not-A’ has to be the contradictory of A; otherwise, when B is not true, A would not necessarily be neither true nor false. Actually, if A and not-A can both be false at the same time, then A would be false when B is not true.
According to Böer and Lycan, the distinction between external and internal negation itself gives rise to an inescapable dilemma for the champion of semantic presupposition (Böer and Lycan, 1976, p. 77). Suppose that A presupposes B and A is a logically complex sentence. The alleged dilemma goes as follows:
(a) There are two essential requirements for the negation of A: it has to be the contradictory of A, and it must formally imply B.
(b) The negation of A can be read only in two ways, either as the external negation of A i.e., ~A, or as the internal negation of A, i.e., in-A.
(c) If ‘not-v4’ is read as the external negation of A, then it does not formally imply B.
(d) If ‘not-A’ is read as the internal negation of A, then it is not the logical contradictory of A.
Therefore,
(e) In either case, the two requirements of semantic presupposition cannot be fulfilled at the same time. Either way, semantic presupposition is ruled out.
The general conclusion drawn from the above dilemma is that the notion of semantic presupposition, although it is theoretically coherent, is in fact empty since it cannot be exemplified (Böer and Lycan, 1976, p. 10). We cannot even give any concrete sentence that presupposes another sentence in Strawson’s sense. Take sentence (16) again as an example. The external negation of (16), i.e., (~16), does not formally imply (16a) since the following sentence is consistent,
(~16&~16a) It is not the case that the present king of France is bald and there is not any present king of France (Böer and Lycan, 1976, p. 59).
On the other hand, the internal negation of (16) is not the contradictory of (16). Therefore, (16) does not presuppose (16a), but only entails it.
In my judgment, this argument presents one of the most serious challenges to the tenability and integrity of the notion of semantic presupposition. If it worked, then the notion of semantic presupposition would become useless. However, the argument does not work in the way Böer and Lycan expected. I argue below that the alleged dilemma is a fallacy. It does not rule out an interesting notion of semantic presupposition that can be properly exemplified in many interesting cases.
As I have argued earlier, treating the negation as contradictory is not the only appropriate reading for the negation in an appropriate definition of semantic presupposition (Schema P). Actually, taking the negation of a presupposing sentence A as the contradictory of A is too strong in many cases. A more appropriate reading of the negation of A is the subcontrary of A. If we take a negation of A as its subcontrary in general with its contradictory as one version of the subcontrary as I have suggested, then semantic presupposition can be properly exemplified. For instance, for a particular existential sentence (17), its subcontrary is,
(# 17) Some unicorns in the African jungle are not hairless.
Then, according to Schema P, both (17) and (#17) imply (17a), respectively. Furthermore, when (17a) is false, we have
╞ ~T(17a) → ~ (T(17) v T(#17)).
There is no case in which T(17) and T(#17) can both be false since one is the subcontrary of the other. Although (17) and (#17) can both be true, this possibility is ruled out. Otherwise, the formula cannot be unconditionally valid when (17a) is not true. Thus, the only possible truth-value for (17) is neither true nor false when (17a) is not true. It is clear that the relationship between (17) and (17a) meets our three requirements of semantic presupposition. Therefore, (17) semantically presupposes (17a). This shows that the premise (a) of the alleged dilemma is not justified.
The power of the alleged dilemma depends upon a basic assumption that there is a distinction between external and internal negation for every presupposing sentence. However, the distinction is not universally applicable to many presupposing sentences. For a singular subject-predicate sentence (23), the alleged distinction between external and internal negation is blurred. As every student of logic knows, we cannot read (23) as a particular sentence (24), otherwise (23) would lose its universal aspect. The more proper way is to read (23) as the conjunction of a corresponding universal sentence (22) and a particular sentence (24). That is,
(23) S is P = def. (All S is P) and (Some S is P).
Then the external and the internal negation of (23) are respectively (~23) and in (23),
(~23) ~(S is P) = ~(All S is P) or ~(Some S is P) = (Some S is not P) or (No S is P)
(in-23) in-(S is P) = in-(All S is P) or in-(Some S is P) = (All S is not P) or (Some S is not P) – (No S is P) or (Some S is not P)
This shows that there is no real difference between the external and the internal negation for a singular subject-predicate sentence. If so, then the external negation of (23) (or (16)) formally implies (23a) (or 16a) just as any internal negation of a presupposing sentence formally implies the same presupposition as that sentence itself does (I have proved this earlier). In addition, (~23) (or (~16)) is the contradictory of (23) (or (16)). According to Schema P, (23) (or (16)) presupposes (23a) (or (16a)).
The opponents of the notion of semantic presupposition might point out that if we adopt Russellian treatment to paraphrase (23), which looks like a grammatically simple sentence, into a logically complex sentence, then the distinction between the internal and the external negation of (23) should be very clear. It is the distinction between internal and external negation with respect to scope as follows.
(23′) ∃x (P(x)& ∀y (S(y) ↔ x = y))
(in-23′) ∃x (~P(x)& ∀y (S(y) ↔ x = y))
(~23′) ∃x (P(x)& ∀y (S(y) ↔ x = y))
When its presupposition (23 a), or in symbols,
(23a′) ∃x ∀y (S(y) ↔ x = y))
is false, (23′) and (in-23′) are both false. Then (23′) entails, instead of presupposes, (23a′) Thus the dilemma stands.
The same strategy can be used against the two kinds of internal negations I have made, namely, internal negation as contrary or as subcontrary. Take sentence (17) (the same for sentence (24)) as an example. We should read (17), which appears to be a grammatically simple sentence, as a logically complex sentence. In other words, we should transfer the short surface form of (17) into the following logical form of conjunction:
(17′) There exist some unicorns in the African jungle and they are hairless, or in symbols, ∃x (Unicorn(x) & Hairless(x)).
Then the so-called subcontrary of (17′) would be,
(#17′) There exist some unicorns in the African jungle and they are not hairless, or in symbols, ∃x (Unicorn(x) & ~ Hairless(x)).
Besides, we have
(17a′) ∃x Unicorn(x).
We can see from the two formulas (17′) and (#17′) that if (17a′) is false, (17′) and (#17′) can both be false at the same time. That means that (#17′) is not the subcontrary of (17′)- There is hence no real case for subcontrary. The requirement of negation as subcontrary fails. If so, even when we read the negation in Schema P as the subcontrary, (17′) is still not necessarily neither true nor false when (17a′) is not true since (17′) can be false in the following formula,
╞ ~T(17a′) → ~(T(17′) v T(#17′)).
Hence (17′) fails to presuppose (17a′).
Generally put, the essence of the above treatment of a presupposing sentence is as follows. Suppose that a sentence A presupposes B. We can always treat A as exponible into a conjunction with its presupposition B as one conjunct and a sentence C about the property of B as the other conjunct as shown in sentence (25),
(25) B & C.
Furthermore, we can treat the internal negation of (25) as attaching a negation not to the presupposed conjunct B, but to the other conjunct C9 as shown in (in-25),
(in-25) B & ~ C.
It is obvious that both (25) and (in-25) imply B. Then if the presupposition B of (25) and (in-25) is false, (25) and (in-25) must both be false. In this way, (25) is false when its presupposition B does not hold (J. Martin, 1979, pp. 251-2, 268).
I have two responses to this objection. First, there is a grave defect in the syntactic structure of exponible sentences. The method of exponibilia treats a simple grammatical form as masking a complex logical form. For example, for a simple identity sentence (26),
(26) The king of France is the king of France, or in symbols, k = k
if we adopt the Russellian reading, we should paraphrase (26) as (27),
(27) The one and only one person has the property ascribed to the king of France and that person is self-identical, or in symbols, ∃x (x = x & ∀y (King(y) ↔ x = y )).
It is objected, by D. Kaplan (1975) and others, that we should not invoke hidden complexity unless there is a good reason to do so. That is, we should not read (26) as (27) until we have investigated the options of identifying (27) with (26) and found (26) to be unworkable (S. Lehmann, 1994, p. 309). In addition, all things being equal, if we have to paraphrase a simple grammatical sentence, its logical form should correspond as closely as possible to the surface form of the sentence. It is after all the surface form of a sentence in our natural language that is being used and explained. However, Russellian treatment frequently requires such extensive rewriting of the surface form of a simple sentence that its syntax becomes too complicated to be understood. For example, it is not convincing to construe the simple sentence (26) as the very complex sentence (27). A theory that treats (26) as a simple identity sentence would be better (J. Martin, 1979, p. 253).
A more crucial problem for the issue in hand is that treating the simple grammatical form of a sentence as the disguised complex logical form is a typical method employed by classical logic only. In fact, after a simple presupposing sentence is translated into a conjunction with its presupposition as one conjunct, it naturally follows that the sentence must be false when its presupposition is false. Therefore, accepting the Russellian reading of a presupposing sentence would amount to adopting Russell’s treatment of a non-denoting sentence. In this sense, whether we should accept the method of exponibilia is a crucial issue at stake. Adopting it without any convincing argument is to beg the question from Strawson’s point of view. For this reason, non-classical theories of presuppositions should not employ this method, not just because of its grave defect in syntax level, but because adopting it amounts to dropping Strawson’s notion of semantic presupposition from the outset.
Finally, the premise (c) of the argument is false. Suppose A formally implies B. Whether the external negation of A formally implies B depends on specific structures of sentences A and B. We cannot claim in general that any external negation of A does not formally imply B. Here is a counterexample. As I have argued earlier, the external negation of a singular subject-predicate sentence, namely, ~(S is P), formally implies the sentence ‘S exists’ as the original sentence ‘S is P’ does.
Another major critical argument raised by Böer and Lycan against the notion of semantic presupposition claims that it is easy to provide many perfect counterexamples to an enormous number of alleged semantic presuppositions. They contend that semantic presupposition as species of formal implication must hold universally without conceivable counterexamples. So if they can give some counterexamples in which the so-called semantic presuppositions are cancellable, then the notion of semantic presupposition itself cannot be held consistently. In fact, there would be no genuine instance of semantic presuppositions if such prefect counterexamples can be found. The strategy of Böer and Lycan is then to make up counterexamples in which the alleged semantic presuppositions can be canceled.
This argument runs as follows. Suppose that A presupposes B. That means, according to the general definition of semantic presupposition (the initial Schema P) that both A and its negation formally imply B. That is,
(a) ╞ T(A) → T(B) and (b) ╞ T(not-A) → T(B).
Let us focus on the formula (b) only. It is obvious that B cannot be false if not-A is true since not-A formally implies B. That means that the possibility that
(c) The negation of A is true but B is false, or in symbols, T(not-A) & F(B)
is ruled out by the very definition of semantic presupposition since (c) is self-contradictory or logically false if A really presupposes B. Therefore, if we can show some cases of alleged presuppositions in which (c) can be held without contradiction, then the alleged semantic presupposition B is canceled.
Böer and Lycan give a few counterexamples in which (c) can be held without contradiction. Consider the following set of sentences:
(28a) |
It is false that the present king of France is bald because10 there is not any present king of France. |
(28b) |
It is false that it was John who caught the thief because no one caught the thief. |
(28c) |
It is false that my soul is red because my soul is not colored. |
(28d) |
It is false that John managed to solve the problem because this problem is so easy to solve. |
According to Böer and Lycan, it is important to notice that these sentences are fully intelligible and are clearly not contradictory. In this way, the various ‘presuppositions’ carried by these negations of original presupposing sentences can be easily canceled. No semantic presuppositions are involved in these cases. Again, they reach the same conclusion as they drew from the argument from the distinction between internal and external negation: The notion of semantic presupposition is empty since it cannot be exemplified.
I grant that semantic presuppositions should be held universally. Hence, (c) would be a contradiction and a counterexample to semantic presuppositions if A did presuppose B. However, I am wondering whether the sentences given are really valid counterexamples, or whether they are genuine instances of (c). I will argue that these cases given in (28a, 28b, 28c, and 28d) are not genuine instances of (c). Therefore, they are not valid counterexamples to semantic presuppositions.
The crucial issue here is how to interpret the negation ‘not-A’ in our definition of semantic presupposition. As we have mentioned earlier, there are a variety of readings of the negation ‘not-A’ in our natural language. For instance, ‘not-A’ may be read as the external negation or the contradictory of A that in turn includes an unconditional negation or a conditional negation; ‘not-A may be read as the internal negation of A that may again be read either as the contrary of A or as the subcontrary of A. It seems to be clear that, for Böer and Lycan, ‘not-A’ here is read as the external negation of A or the contradictory of A. If so, this argument shares the same assumption with the argument from the distinction between external and internal negation: not-v4 has to be the contradictory of A. However, if not-A is the contradictory or the external negation of A, then not-A would not imply B. Therefore ¥(B) & T(not-A) would not involve self-contradiction.
However, as I have argued, ‘not-A’ in the definition of semantic presupposition should be read as the subcontrary of A or the contradictory of A if A’s subcontrary is not available. Let us examine what the genuine instances of (c) are in our reading of ‘not-A’, starting with (28b). The cleft sentence, ‘It was John who caught the thief, can be read as ‘Someone who caught the thief was John’. The subcontrary of the sentence is, ‘Someone who caught the thief was not John’. Then the instance of (c) with respect to the sentence, ‘It was John who caught the thief, would be,
(28b′) Someone who caught the thief was not John because no one caught the thief.
(28b′) involves self-contradiction since the first conjunct formally implies the falsity of the second conjunct. In contrast, (28b) is not the genuine instance of (c) because the first conjunct of (28b) is the external negation of the original sentence, ‘It was John who caught the thief, instead of the internal negation of that sentence as it should be. The first conjunct, ‘It is false that it was John who caught the thief, does not imply the falsity of the second conjunct. This is the reason why (28b) does not involve self-contradiction.
Turn now to the singular sentence (28a). If we understand ‘not-A’ as the internal negation of A (more precisely, the subcontrary of A), then the genuine instance of (c) with respect to the sentence, ‘The present king of France is bald’, is,
(28a′) The present king of France is not bald because there is not any present king of France.
Since the first conjunct of (28a′) formally implies the falsity of the second conjunct, (28a’) is a self-contradictory sentence. On the other hand, as I have argued before, there is no real distinction between external and internal negation with respect to singular sentences like (28a). The external reading of (28a), that is, ‘It is false that the present king of France is bald’, still formally implies the falsity of the second conjunct. Hence, (28a) is a self-contradictory sentence. Either way, (28a) does not constitute a valid counterexample of semantic presuppositions. A similar analysis can be applied to other alleged counterexamples given by the critics.
The conclusion drawn from the above analyses is plain: The set of sentences (28) does not provide valid counterexamples to semantic presuppositions. No refutation of semantic presupposition is established.
The notions of truth-valuelessness and semantic presupposition are twin concepts. The former is engendered by the latter. Explaining the occurrence of truth-valueless (fact-stating) sentences is an essential utility of semantic presuppositions. On the other hand, truth-valuelessness is an inevitable notion in any semantic theory of semantic presupposition within trivalent semantics. However, for many philosophers, the notion of truth-valuelessness is highly suspect since it is not in line with common wisdom in semantics: it contradicts either the accepted notion of falsity or the received principle of bivalence.
Let us first consider a rather simple-minded argument against the notion of truthvaluelessness: By falsity we mean non-truth (lack of truth). To say that a sentence is neither true nor false is to say that it is true and untrue, which is a self-contradiction. Therefore, the notion of truth-valuelessness is not a coherent notion. Although the argument is overly simplistic as it is, many really do take it seriously. In defense of his minimal theory of truth, P. Horwich (1990, p. 80) gives us a sophisticated version along the same line of reasoning. According to Horwich, the simplest deflationary strategy is to define falsity as the absence of truth as stated in (MT1):
(MT1) A proposition that P is false if and only if P is not true; or in symbols, F(P) ↔ ¬T(P).
Given any logic that licenses the principle of contraposition, from (MT1) we have
(MT2) P is not false → P is not not true; namely, ¬F(P) ↔ ¬ ¬T(P).
Therefore, if P is neither true nor false, then
(MT3) P is not true and not false → P is not true & P is not not true; that is, (¬T(P) & ¬F(P)) → (¬T(P) & ¬¬T(P)).
Thus we cannot claim of a proposition that has no truth-value; for to do so would imply a contradiction.
It is obvious that the above argument begs the question since it already assumes the notion of falsity in classical bivalent semantics (the two-valued falsity). If we define the notion of falsity in three-valued semantics (the three-valued falsity),
(MT4) A sentence S is false if and only if the unconditional negation of S is true; or in symbols, F(S) ↔ T(~,S),
then, when S is neither true nor false, we have:
(MT5) (~T(S) & ~F(S)) → (~T(S) & ~T(~S)).
There is no contradiction involved at all.
Of course, Horwich realizes clearly that the contradiction in (MT3) does not derive solely from his minimal theory of truth, but depends also on the notion of falsity and negation in bivalent semantics. The reasons why he endorses such a notion of falsity are, first, that it conforms to ordinary usage; and second, that there can be no theoretical reasons, from his theory of minimal truth, to prefer a threeway distinction of truth-values. Although I disagree with his justification of the endorsement of the notion of falsity in bivalent semantics, I share with Horwich the same insight that we need an independent justification for the endorsement of a specific notion of falsity. We cannot in general suppose that we have given a proper account of a concept of falsity by simply describing the usage of the word ‘falsity’, or by describing those circumstances in which we do, and those in which we do not, make use of the word. We must give an account of the point of the concept ‘falsity’ as we use it. We need to explain what functions we use the word for and what philosophical consequences we can draw from this usage. Otherwise, the debate between Russell and Strawson, as Russell contends, is nothing but a verbal dispute. I will argue that it is far from clear why falsity should be defined as non-truth. In fact, there are a number of convincing or at least equally weighty conceptual reasons for dividing non-truth into two subcategories—falsity and neither-truth-norfalsity—as there are for bivalence.
As I have mentioned earlier, it is very natural to distinguish between two kinds of non-truths of a singular subject-predicate sentence: The sentence is untrue because either the subject has a denotation but the predicate does not apply to it or the subject lacks a denotation. Even the critics of semantic presupposition admit that there is an intuitive difference between two sorts of non-truths. For example, Böer and Lycan admit that respondents hesitate in issuing classical truth-values (truth or falsity) to sentence (16) (‘The present king of France is bald’). The common sense of falsity in bivalent semantics does not work under such a situation. To simply say that sentence (16) is false is inappropriate and misleading (since it violates Grice’s Maxim of Strength), but to say that sentence (21) (‘the current president of China is bald’) is false will be appropriate. Although we cannot draw, from the hesitation of respondents in issuing truth-value judgment, a conclusion that the respondents have a natural notion of truth-valuelessness, we can safely say that the respondents have an intuition of the difference between two kinds of non-truths. Such an intuitively recognizable difference in non-truths needs explanation.
I have shown that the opponents of the notion of semantic presupposition are unable to give a satisfactory explanation of two kinds of non-truths without falling into self-contradiction. Trivalent semantics, by classifying non-truths into falsity and neither-truth-nor-falsity, not only makes a sound distinction between two sorts of non-truths, but also is able to locate the real source of such a distinction based on the notion of semantic presupposition. This shows that the debate between Russell and Strawson over the truth-value status of non-denoting sentences is not merely a verbal dispute, but is theoretically productive. To distinguish different non-truths gives us one sufficient theoretical reason to introduce the notion of truth-valuelessness.
One objection to introducing the notion of three-valued falsity is that this notion does not conform to our ordinary usage of falsity in common situations. We have no ‘pure intuitions’ of truth-valuelessness. It is not an ordinary, commonsensical notion, but a theoretical artifact of linguistic and philosophical semantics. It contradicts our standard ordinary usage.
It is true that the notion of three-valued falsity is not a commonsensical notion. But from this, it does not follow that the notion contradicts our standard ordinary usage or our intuitions of the notion. The reason why we have not gotten used to this notion is that we usually do not have occasions to develop relevant intuitions and to practice the usage of three-valued falsity, not because it contradicts our common sense. We seldom, for example, even encounter sentences like (16), (17), or (18) in our everyday conversation. However, we do have a strong intuition of two kinds of non-truths whenever we encounter a pair of sentences like (17) and (21). We feel something is wrong to simply answer, ‘it is false’, in response to (16) although we feel secure to give the same answer to (21). This intuition can be used as a starting point to build up a new standard usage of the notion of three-valued falsity. Imagine a linguistic society in which people encounter sentences whose presuppositions do not hold all the time. In order to ensure successful communication, they have to make a distinction between two kinds of non-truths (perhaps the distinction was made initially by some linguistics). After long-time development of the usage, the notion of truth-valuelessness would become a perfectly commonsensical notion in this society eventually.
Furthermore, the assumption that there is a standard ordinary usage of falsity in general (falsity as lack of truth) is very doubtful. All cases we have for the alleged standard usage of falsity are easier cases in which the corresponding presuppositions are held, in which there is no need to employ the other subcategory of non-truth, i.e., the notion of neither-truth-nor-falsity. But for other cases in which the corresponding presuppositions fail, we do not have a chance to form a standard usage yet. In other words, there is no standard way of dealing with sentences with false presuppositions. Therefore, it is unfair to regard the standard way of dealing with bivalent sentences as the standard usage of falsity in general.
Since there is no standard usage of falsity in general, we have to depend on the theoretical usage of it. Nevertheless, there is no established standard theoretical usage of falsity in general either, although there have been long traditions both for and against treating falsity as lack of truth since Aristotle. I conclude that the notion of three-valued falsity is not in violation of both ordinary and theoretical usage of ‘falsity’.
Perhaps the most serious challenge to the notion of truth-valuelessness comes from the tradition of bivalence in semantics. According to the principle of bivalence, each well-formed declarative sentence is either true or false. If so, it is impossible for a sentence to be neither true nor false. In fact, an attempt to save the principle of bivalence from the threat of truth-valuelessness is what really motivates many critics of the notion of truth-valuelessness and semantic presupposition.
I am not intending to give a comprehensive examination and critique of the principle of bivalence here; it is not necessary to do so for our current limited purpose. In the following, I will focus on only one logical argument for the principle of bivalence that is intended to reject the notion of truth-valuelessness.
Many critics of the notion of truth-valuelessness argue that the acceptance of truth-valuelessness amounts to rejecting the principle of bivalence. The rejection of the principle would force us to give up one basic classical logical law—the law of the excluded middle. This is because the principle of bivalence is equivalent or necessarily follows from the law of the excluded middle. But we have no convincing reasons to give up the law of the excluded middle. So we had better stick to the principle of bivalence. For example, one of the arguments for the principle of bivalence presented by P. Horwich (1990, pp. 80-82) runs as follows. The logical law of the excluded middle claims that, if P is a declarative sentence, the proposition that (P ∨ not-P) is necessarily true. That is,
(M) ├ P ∨ ¬P11
According to Tarski’s Convention T, for each sentence S of an object language Lo, a metalinguistic sentence (T-sentence) in the corresponding metalanguage LM can be given in the following form:
(T) S is true in L iff P.
Here ‘S’ stands for a specific sentence of Lo and ‘P’ stands for a corresponding sentence of LM. The T-sentence gives us the truth conditions of sentence S in Lo. If LM includes Lo and ‘true’ is treated as a sentential operator, then the T-sentence becomes:
(T) It is true that P iff P, or T(P) iff P.
If ‘iff’ in (T) is understood as material equivalence, then we have schema T’
(T′) ├ T(P) ↔ P and ├ T(¬P) ↔ ¬P.
By the rule of constructive dilemma, from (M) and (T′), we have
├ T(P) v T(¬P).
Since T(¬P) = T(¬T(P)) = F(T(P)), and by (T′), F(T(P)) = F(P), T(¬P) = F(P). Then,
(B) ├ T(P) ∨ F(P)
can be derived from (M) and (T). (B) reads ‘It is true that P or it is false that P′ which is presumably the principle of bivalence. This establishes that it is inconsistent to deny the principle of bivalence but retain the law of the excluded middle since the latter entails the former.
The above argument is misleading due to a misinterpretation of T-sentence. As a matter of fact, whether we can derive (B) from (M) in terms of Convention T all depends on how we read T-sentence. Normally, there are at least two different interpretations of T-sentence due to two different readings of ‘if and only if’. In one interpretation, ‘if and only if’ means material equivalence as shown in schema T’. In the other, ‘if and only if’ means deductive equivalence and can be read as implication relation (either classical entailment or formal implication). Then the corresponding schema T” would be
(T″) T(P) ╞ P(or T(P) ├ P) and P ╞ T(P) (or P ├ T(P)).
One problem with schema T′ is that it is only valid in classical two-valued logic. When we move from two-valued logic to three-valued logic, (T′) fails. In fact, when P is neither true nor false, (T′) is not valid since the left hand of the equivalence, sentence T(P), is false but the right hand of the equivalence, sentence P, is neither true nor false. Furthermore, schema T′ says that P and T(P) have exactly the same truth-value in any situation. This in turn, by the truth-conditional theory of meaning, will be the case if and only if P and T(P) have the same meaning or express the same proposition. But many philosophers have denied that there is an identity of meanings between P and T(P). For instance, ‘Tom is tall’ and ‘It is true that Tom is tall’ have different meanings on the grounds that the latter is in some sense metalinguistic while the former is not. This shows that Schema T′ is not an appropriate formulation of T-sentence. Besides, there is another more appropriate reading of T-sentence, namely, Schema T″. Schema T″ only says that when the premise P or T(P) is true so is the conclusion T(P) or P; and it says nothing about the truth-value of the conclusion if the premise is not true (false or neither-true-nor-false). For this reason, Tarski’s convention T is satisfied even in three-valued logic.
According to schema T″, P ├ T(P) and ¬P ├ T(¬P). From schema T″ and ├ (P v ¬P), we have ├ T(P) v T(¬P). But within three-valued logic, T(¬P) = F(T(P)) ≠ F(P). Therefore, ╞ T(P) v F(P) cannot be derived from (M). This establishes that the semantic principle of bivalence must be distinguished from the logical law of the excluded middle. It is a tenable position to deny the former while retaining the latter within three-valued semantics.
1 1 For some criticisms of the notion, please see M. Bergmann, 1981, S. Böer and W. Lycan, 1976, W. Lycan, 1984, 1987, G. Englebretsen, 1973, R. Kempson, 1975, W. Sellars, 1954, D. Wilson, 1975, J. Orenduff, 1979, G. Gazdar, 1979, J. Atlas, 1989, and many others.
2 The present chapter was directly inspired by J. Martin’s 1979, which has helped me a great deal in straightening out my own thought on the topic. Especially in my formal treatment of the notion of semantic presupposition, I borrow many analytic tools from it.
3 From now on, for simplicity, whenever I use T(A), it is implicitly assumed that A is a sentence in a language L. So I will omit the explicit mention of L in TL(A). The similar treatment applies to ‘╞L’ and ‘├L’.
4 See M. Bergmann, 1981.
5 Let us put the issue of vague predicates aside, and suppose that ‘. . . is bald’ is not a vague predicate. Actually, we can easily avoid a vague predicate by using other predicates, such as, ‘. . . is female’.
6 The notions of subcontrary and contrary, unlike that of contradictory, are not well-defined truth-functional operators, though we could make them so by splitting ‘#’ (or ‘*’ into ‘#t’ and ‘#f’ (or ‘*t’ and ‘*f’).
7 The requirement of B as a contingent sentence is intended to exclude an extremely trivialized notion of semantic presupposition according to which any logically true sentence B is presupposed by any sentence A.
8 S. Lehmann points this out to me.
9 Treating the internal negation of (25) as attaching a negation to the presupposed conjunct B, namely, (~ B & C), would make the internal negation of (25) not imply B.
10 ‘Because’ is not a well-defined truth-functional operator (in fact, it is not even a truth-functional operator). Presumably Böer and Lycan use ‘because’ here as the conjunction ‘and’. I assume the reasoning behind this usage is something like this: ‘C because D’ implies ‘C and D’. Hence, if’C and D’ are contradictory (a logically false sentence), so must ‘C because D’ be. For this reason, I will treat ‘because’ as ‘and’ in analyzing their argument.
11 The reason for using conditional negation ‘¬i’ instead of unconditional negation ‘~’ in the formulation of the law of the excluded middle is obvious. In fact, if’not-’ is read as unconditional negation within three-valued logic, the proposition (P ∨ not-P) would not be necessarily true; since when P is neither true nor false the proposition is neither true nor false.