CONTENTS

1   INTRODUCTION

2   EXPOSITION

3   THE HIERARCHY OF REFERENCE

4   THE NEUTRALITY OF LANGUAGE

5   CONCLUSIONS

3.3 Truth-Values

Frege's claim that the referent of a sentence is one or other of the two truth-values, True or False is perhaps the most difficult part of his whole semantic theory. Kneale and Kneale write:

"Frege's doctrine of truth-values is strange and unacceptable because he never succeeds in explaining clearly what these objects are. In one place he says that the truth-value of a sentence (Satz) is the circumstance that it is true or false as the case may be. This seems intelligible, but it is quite inconsistent with the suggestion that all true sentences designate one truth-value and all false sentences another. For the circumstance that '5 is a prime number' is true (if we may allow ourselves to speak in this way) is not the same as the circumstance that '2+2=4' is true. In other places he talks simply of the True and the False. This phraseology is intended, no doubt, to suggest that he is dealing with two and only two objects; but what are they?"

In what follows, the arguments in favour of Frege's theory of sentence reference are split into two. First we look at the notion of truth-value and the arguments that possession of truth-value is the correct analogue of reference at the level of sentences. Then we proceed to examine a rival account of sentence reference, expressed using the notion of "circumstances" or "states of affairs", something Kneale and Kneale show that Frege was flirting with on occasion.

3.3.1 The Duality of Truth-Value

Frege's thought is full of radical departures from what went before, and from what unaided intuition might lead one to expect. Even within this context, the explication of sentence reference in terms of possession of truth-value is among the most radical and difficult, so it is little wonder if Frege himself had problems with it. At the outset, let us here agree with Dummett that Frege's later assimilation of truth-values to objects, and the consequent assimilation of sentences to names and predicates to functional expressions, was a serious mistake. Kneale and Kneale's question asking what sort of objects are the truth-values is pointing in quite the wrong direction. They are not objects at all, but quite a different form of referent.

We should not expect therefore to find an accurate representation of truth-value within natural language. The first reason for this is the extreme novelty of Frege's ideas here. Nothing extant in natural language was available for their expression. The second reason is, as discussed above, Frege's theory of reference is unsayable. We can have sayable models of reference, using the model predicate "x is a referent", but these fall short of the semantic reality. The proper noun expressions "the True" and "the False" are preferable to the predicates "x is true" and "x is false", because they point us in the direction of the reference of a complete expression, but to the extent they suggest this reference is to objects, they are misleading. The two truth-predicates are features of meta-language, within which they can be used as a part of a formal semantic model, but they have no final explanatory power, because to use them requires us to use the machinery of assertion, including truth-values, within the meta-language.

So instead of asking the unanswerable question "what are truth-values?", we should investigate the more sophisticated question, "what is the function of the notion of reference at the level of sentences, and why does reference have an intrinsic duality at that level?" This can be further split into the following three questions:

1. Why to we ascribe to sentences a duality of truth values?
2. Why should we describe the ascription of truth-values to sentences as being the reference of sentences?
3. Why should we not think of sentences as referring to something other than truth-values (the prime candidates for the "something other" being states of affairs)?

The first of these questions is addressed in the remainder of this subsection, and the other two in the following subsections.

The basic, prima facie argument for the duality of truth-value appeals to the role of sentences as items that can be asserted. There are other ways we can use sentences: as questions, commands, or as components of a work of fiction, but each of these usages is parasitic on assertion. When we use a sentence assertorically, we stop trying to put detail into the sentence and rely instead on simply displaying a token of the sentence in a context agreed socially to count as assertoric. This is what I have called elsewhere "the irreducible element of depiction right at the end of the process of saying".

Because we can at a particular opportunity only either display the token or not display it, the only distinction which can be made by the simple fact of assertion is a two-fold one. (Note that failure to assert a statement is not to deny it. If this were not so, the strict physical limitations on how much I can display at once would mean that I would be doing a huge amount of unintended denying all of the time. To deny something, I assert its negation - the negation operator is part of what can be expressed by saying.) Suppose we had some X-fold division of things we wanted to say about sentences, where X is greater than two. We would then have to say, "it is X that p". But saying "it is true that p" says nothing more than the bare assertion of p which we may have been trying to eliminate. So we can not replace this final assertoric step by adding new content to the sentence.

So we have here two different linguistic activities:

• modifying something in a sentence to say something more or different, and
• asserting that the sentence is true.

A key difference between them is that the former activity deals with the sentence-type, whereas the latter uses the sentence-token. It might be claimed that we could express more than truth-value with assertion if we were able to vary the way in which we presented the token - say, by altering the pitch of the voice. Here we would be saying that the pitch is an accidental feature of the sentence, that is, not a constituent of the identity criterion of the type. But it is central to the Fregean theory of assertion that this is not how we interpret the identity of sentences. Varying such features to express new senses constitutes making a new sentence-type. By contrast, features not intended as expressing sense can be regarded as accidental. In spoken English the absolute pitch of an utterance has no semantic significance, and so tokens of the same sentence-type uttered with a different pitch count as asserting the same thing.

The place and time of utterance are usually not regarded as being essential features of a sentence-type. This because more generally spatial and temporal position are taken as being intrinsically accidental. We do not include these features in the type definition, but want to be able to speak of the preservation of type identity through time and with spatial movement. But this does not work for tensed statements. The time of utterance has semantic significance. Usually this is expressed by saying that this form of sentence is token-reflexive - the meaning varies from token to token. It would however be more in keeping with the account of assertion given above to say that temporal position is now an essential feature of the sentence, by virtue of its semantic significance, and that different type-sentences are being uttered at different times.

Even when many-valued logics are advocated for one reason or another, it is necessary to divide the values into two types: the designated values and the non-designated values. As Dummett points out, "in most philosophical discussions of truth and falsity, what we really having in mind is the distinction between a designated and an undesignated value ...". Moreover "finer distinctions between different designated values of different undesignated ones, however naturally they come to us, are justified only if they are needed in order to give a truth-functional account of the formation of complex statements by means of operators." We shall return to this question briefly below, in the context of treating "not referring" as being a third truth value. The point to be made here is merely that even in many-valued logics, a duality re-appears, enforced by the needs of assertion.

It might be argued that the above argument from assertion to the duality of truth-values is circular. The premise of the argument is:

A statement X can be only asserted or not.

from which the conclusion is drawn:

The only thing about X which can be indicated by assertion is truth, as opposed to not-truth.

But the premise can be restated as:

"X is being asserted" is only either true or false.

This restatement demonstrates the circularity of the argument, because it shows the implicit appeal to the duality of truth-value (the conclusion) which was already present in the premise. Suppose there were three truth-values: True, False and Other. Then I could express the truth of X by acting in a way which made "X is being asserted" true. Similarly I could express the otherness of X by acting in a way which made "X is being asserted" other.

Rather than go into these deep matters further here, I shall short-circuit the arguments by saying I do not know how to act in this third way. In this state of ignorance, my acts of assertion are powerless to do other than work with a two-valued logic, which will then be the working hypothesis on which the rest of my philosophy is built. When someone more skilled in these matters teaches me the techniques of the third way, I shall consider a larger set of truth-values.

3.3.2 Truth-Values as Referents

To examine the claim that the truth-values are the two possible referents of all sentences, we test them with the following two theses about the reference of composite expressions.

1. If a component is replaced with another component with the same referent, then the referent of the composite is unchanged.
2. The composite has reference if and only if the components have reference.

Let us examine whether truth-value passes these two tests. If it does so it remains a candidate for being the reference of sentences. To complete the proof we would have to show that no other candidate passes these tests, and any others which might be implicit in the notion of reference.

The first test is passed with comparative ease. This is the original argument used by Frege for the identification of truth-value with the reference of sentences.

"If our supposition that the reference of a sentence is its truth value is correct, the latter must remain unchanged when a part of the sentence is replaced by an expression having the same reference. And this is in fact the case. Leibniz gives the definition: 'Eadem sunt, quae sibi mutuo substitui possunt, salva veritate'. What else but the truth value could be found, that belongs quite generally to every sentence if the reference of its components is relevant, and remains unchanged by substitutions of the kind in question?"

So truth value passes the test; we shall give a putative answer to Frege's rhetorical "what else?" question in the following subsection.

To apply the second test, let us split the biconditional in two.

Let us then start with 2a. Frege's theory of sentence reference says that, if any one of its components lacks a reference, then a sentences lacks a truth-value. An alternative approach is to say that if any component fails to refer then the sentence becomes false. This fails because the negation would then be true, and yet there would still be a non-referring component. To fix this we might try to say that any sentence headed by a negation defaults to true if a component does not refer. To test this attempted fix, let us consider

not (not A and not B)

and suppose that reference fails somewhere inside B. Then "not B" has to be true by our rule. Suppose "not A" is also true. Then the conjunction is true and its negation must be false. But by application of our rule to the whole sentence it must be false. Our rule for the negation of non-referential sentences leads to a contradiction and must therefore be abandoned.

An alternative is to introduce "non-referential" as a third truth-value. The standard truth tables are then extended with the rule that the negation of NR is NR and that for any binary operator the value is NR if either of the arguments is NR. This corresponds to the way failure of reference of a component infects any composite expression of which it is a part. This then leaves us with the question of what might be the relationship between this third truth-value and assertion.

To address this question, it is useful to start with the following argument, advanced by Dummett:

"A statement, so long as it is not ambiguous or vague, divides all possible states of affairs into just two classes. For a given state of affairs, either the statement is used in such a way that a man who asserted it but envisaged that state of affairs as a possibility would be held to have spoken misleadingly, or the assertion of the statement would not be taken as expressing the speaker's exclusion of that possibility. If a state of affairs of the first kind obtains, the statement is false; if all actual states of affairs are of the second kind, it is true. It is thus prima facie senseless to say of any statement that in such-and-such a state of affairs it would be neither true nor false."

To analyse this attack on the notion that sentences with well-formed senses may lack a truth-value let us simplify it somewhat. (This involves stripping away the reference to possibility, which seems at best confusing and at worst wrong. I may assert "Mars is red" perfectly validly while envisaging the possibility that it could have been green, as long as I am envisaging it as counterfactual. I might even envisage that one day Mars might turn green - for example as the result of human cultivation - in which case I would change my assertion to reflect the change in the state of affairs.) The simplified form of the argument is the following.

A statement p divides states of affairs into two types:

• the falsifying ones such that, if S is a falsifying state for affairs relative to p, then asserting p and believing that S is actual counts as using p in a misleading way;
  
  
• the verifying ones, such that asserting p does not exclude them.

The states of affairs in which some part of p lacked reference would be included among the falsifying ones. Certainly if one asserted a sentence, knowing that parts of it lacked reference, it is reasonable to characterise this behaviour as misleading.

The first thing to question about this argument is which universe of possible states of affairs is envisaged here. Does it include ones where elements of the description of the state lack reference, although they could counterfactually refer. Do we include the circumstances that certain expressions for which we have a sense fail to refer. If we restrict the states of affairs in Dummett's argument to the C-possible ones, then we have no problem. The only way states of affairs can be falsifying is for the corresponding sentences to be false. But it seems clear that Dummett is employing a wider realm of states than this.

The idea that the false includes the non-referential immediately runs into the negation argument given above. A falsity including non-reference is not the falsity of standard sentential logic. Rather we need a third truth-value - bearing out Dummett's claim that while we are talking about assertion, we need only a two-fold division, whereas inquiring as to the behaviour of sentential operators sends us in the direction of having more than two truth-values. Then both classical-false and non-reference comprise the non-designated values, both of which preclude assertion in the rules of the language game. This evades the argument from assertion to two-valuedness, because duality re-appears as the distinction between designated and non-designated values.

We can contrast two ways to react to failure of reference within a sentence: the Frege way and the Strawson way as characterised by Dummett (the third way, due to Russell, is of great interest but is of less relevance here). For Frege the failure of reference means that a coherent semantics of the language can not be given, while for Strawson failure of reference could happen within a semantically well-formed language. Dummett goes on to demand of Strawson a three-valued explanation of the sentential connectives to show how they handle failure of reference.

My contention is that while the three-valued logic elaborations of the Strawson position are technically possible, they are not true to the theory of reference. C'est magnifique, mais ce n'est pas la référence. It is quixotic to regard not having a referent as a special case of having a referent; the absence of truth-value is not itself a truth-value. In this, I side with Frege. The primary semantic function of language is to refer, and if this does not happen then we have not really started doing semantics. We should not start to play the assertion game until we are guaranteed that all our terms have reference, even though technically we could devise a game formally like assertion using the devices of three-valued logic, to cover states of affairs where parts of our sentences fail to refer.

The above contentions need a qualification to the effect that there are valid linguistic activities which work purely at the level of sense. When we are unsure of how to articulate certain facts of which we have become aware, we need to experiment with senses, not knowing along the way whether or not they will manage to latch on to anything in the world (philosophical analysis). We can take well-defined senses and systematically explore their logical consequences, either as an adjunct to the attempt to express facts (applied mathematics) or as an end in itself (pure mathematics). Even in pure mathematics the status of reference as the ultimate end of semantics is implicit, because the logical transformations are defined in terms of the preservation of truth in those cases where everything manages to refer.

At the level of sense there certainly is a three-fold system of thought predicates:

it is true that (...)
it is false that (...)
(...) lacks a reference

such that one and only one is true of any given thought. (Another way of putting this is to say that the negation of any of these predicates is synonymous with the conjunction of the other two.) The third predicate can be true of thoughts because the sense forming operator is opaque - the name of the sense of an expression can refer even if the expression itself does not.

Now we proceed to the second half of the biconditional, namely: if all components of a sentence refer, then the sentence must have a truth-value. This runs into problems with non-universal ranges of applicability. Consider the sentence:

A     The Empire State Building is greater than the number eight.

At first sight we would want to reject sentences like this as nonsense. But the components possess not only sense, but also reference. (Here let us make the assumption that arithmetic does indeed refer to objects called numbers. Elsewhere I question this assumption, but for the time being let us persist with it.) We could make these "category mistakes" simply false. We then have to test this hypothesis by looking at its effects on negation. This would mean that:

B    The Empire State Building is not greater than the number eight

is true. Taking the normal meaning of "x is greater than h", we would then say that

C    The Empire State Building is less than or equal to the number eight

is true, whereas this is another example of the name and relational expression not being appropriate for each other, which by our original hypothesis means the result is false. To save the theory from this contradiction we have to deny that "less than or equal to" is the same as the negation of "greater than". Instead we have to say:

Ø(x is greater than h)

= (x is less than or equal to h) Ú (x does not stand in a size relation to h)

In practice what we tend to do is divide the world up into separate realms of reference, within which the conditional holds, with "mixed sentences" being forbidden. Ultimately a better understanding of things might lead to a unified ontology in which Frege's account of reference is true everywhere. (We might, for example, reject the naive interpretation of arithmetic as referring to a range of objects.)

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