Is Evans’s Axiom (9) Useful?

1 Introduction

I will argue that Evans cannot use his axiom (9): (∀x) (The referent of ‘a’ = x iff [a] (x = a)) to show the sense of terms as he desires. [1, p. 38] I will start by outlining the historical problem of empty terms that Evans is seeking to address, and then discuss his argument to axiom (9) and the problems with that argument. I will conclude that axiom (9) is not useful, primarily because it does not show the senses of terms or otherwise elucidate them. In addition, I will argue that there are problems with the whole category of descriptive names that Evans introduces; that he does not improve on Frege because his approach is no longer Fregean; that there are logical problems which Evans notes but does not solve and finally that Evans’s approach is poorly motivated. For all five of these reasons, axiom (9) is not useful.

2 The Difficulty With Empty Terms

It is usual in probably all non-philosophical contexts and many philosophical ones to think that a word is associated with an object in the world. The meaning of a term will be closely bound up with the object to which the term refers. This leads to a problem with empty terms though. Since they do not refer to any object, they can have no meaning on this analysis. This runs counter to the very strong intuition we have that a meaning is nevertheless involved when someone fails to refer by accident, such as if they gesture at a tree and say ‘that lime tree is x’ when in fact the tree they aim to refer to is not a lime. Evans correctly observes that any theory must resolve the strong impression we have on such occasions that communication has occurred.

Frege [2] was considering related problems, including how it could be possible for true identity statements to be informative. Frege approached this by introducing two components of meaning: Sinn and Bedeutung, standardly translated as ‘sense’ and ‘reference’ respectively. ‘Reference’ is that property of a referring term by which it picks out or selects its referent; and its referent is the actual object in the world picked out. ‘Sense’ is the ‘mode of presentation’ of the object, or the way in which it is referred to. Since the introduction of sense allows


the same object to be referred to in multiple ways, we can now account for the informativeness of identity statements. This is because different expressions have different senses, and different senses may have different cognitive values, even if the expression refers to the same item.

Frege’s solution to his problem may create the space in which a solution could be found to the difficulty with empty terms, because now meaning has been separated into two components. Thus, while an empty term may have no referent, it could perhaps have a sense, which would mean that it could have at least that element of meaning. Perhaps that could account for the strong impression of communication. Evans believes that while there is much to be commended in Frege’s treatment, the latter goes wrong in his views on empty terms.

In Frege’s view, inaccuracy in a description destroys reference. If I aim to refer to a tree as a lime tree but it is a different type of tree, I have failed to refer and a sentence containing such an empty term is not truth-evaluable – it expresses no thought. Consider the situation where I aim to refer to the nearest tree to me, which is in fact an elm, but say ‘that lime tree’, when the second tree is in fact a lime. My hearers would misunderstand me, if they knew which tree was in fact the lime. We could not call this communication. If I wished to direct their attention to a squirrel in the lime tree, they would look in the wrong place, unless they shared my error. Relying on mutual error in that way should not be counted as communication; nor should we rely on the likely outcome that they would scan all trees in the approximate area indicated, because even if that is communication of some form, it is not communication proper of the thought as intended.

The problem with empty terms can also be glossed in terms of an obstacle in what would later become the two-step verificationist procedure for assessing whether someone understands a claim ‘a is F’. The first step is to find the object a and the second is to establish whether the predicate F applies to it. For empty terms, there is no object a and so the first step is impossible. Clearly it would be anachronistic to suggest that Frege was thinking in this way, but we might nevertheless wish to improve on his account if we wished to combine it with a more modern verificationist approach, which does have the merit of simplicity.

Frege thinks that when we fail to refer, we perhaps unknowingly enter the realm of fiction – and in this way we can have sense without a referent. For Evans, this response is unsatisfactory because it fails to account adequately for the phenomenology, being the strong impression we have on such occasions that we are successfully communicating with one another without entering a fictional realm.1 Also there seems to be something that constitutes understanding a sentence containing an empty term even though, strictly speaking, no procedure can be given for its verification. So Evans will aim to retain the sense and reference elements of Frege’s approach, but allow (contra Frege, in Evans’s view2 )

1 There may be some evidence that Evans is more sympathetic to this fictional line else- where, but I will consider only his views in [1] here.

2 Although Evans thinks that Frege does not allow sense to sentences containing empty terms, this is a minority view. [3, ‘Supplement: Evans on Frege’] Evans is right though, contra Bell [4], to explain Frege’s explanation of our frequent uses of empty terms in apparently communicative ways with the claim that we have lapsed into fiction. Ironically Bell accuses Evans of a mistranslation but is guilty of one himself. Bell claims that Frege’s term ‘Gedanken, dem Scheine nach’ is best translated as ‘thoughts, to all intents and purposes’ and uses this as evidence to attack Evans’s translation of ‘Scheingedanken’ as ‘mock thoughts’. This is wrong because ‘dem Scheine nach’ means ‘after the appearances’ i.e. in name only. So Evans is right to say that Frege’s term ‘Scheingedanken’ means mock or apparent thoughts – which are not thoughts. A very large number of examples of German terms prefixed with Schein that mean ‘false’ can be supplied; for example Scheinehe – false marriage; Scheinfirma – dummy firm; Scheinbild – simulacrum.


that there can be sense without a referent, and provide a better account of empty terms.

3 The Chain Of Evans’s Argument

Evans notes that for Frege, sense is a mode of presentation of the referent, which means that sense is a mode of presentation of the semantic value, since Frege has equated semantic value and referent. If ‘Aphla’ and ‘Ateb’3 have the same referent then we can see the equivalence of 1). and 2). below.

1. The semantic value (i.e. the referent) of ‘Aphla’ = Aphla

2. The semantic value (i.e. the referent) of ‘Aphla’ = Ateb

But we can only be shown the sense of the name by 1). because when competent users of the term ‘Aphla’ are asked for the way in which the term picks out its referent, they will indicate the object that is Aphla. They will do this while they may be completely ignorant of the term ‘Ateb’ and of the additional fact that they are also indicating the object Ateb. The term ‘Ateb’ has a different sense to the term ‘Aphla’ and we can not be shown the sense of the latter by the sense of the former even though the terms are co-referring.

Evans argues empty singular terms cannot be accommodated in the framework of a ‘typical truth-theoretic clause’ in the form of axiom (7) [1, p. 35] below:

(7): The referent of ‘a’ = α.

Evans will continue by proposing axioms (8) and (9) as improvements. I will discuss these below, but first I will outline Evans’s argument that (7) must be replaced.

Russell [5] does not countenance non-referring singular terms, because he wishes to avoid the problems described in §2 above. So Evans introduces the expression ‘Russellian singular terms’ for those singular terms that do refer i.e. that have a referent. Axiom (7) is clearly inadequate as an analysis for non- Russellian singular terms since they have no referent and so there is no object α, no truth conditions for sentences containing the term ‘a’ can be given, and

3 These are two names used by different groups of persons for the same mountain in a scenario outlined by Frege.


no thought is expressed or can be understood when such a sentence is stated.4

Therefore, if Evans can show that there are non-Russellian singular terms, given the inadequacy of (7) to cover those terms, then (7) is inadequate since it does not cover some terms that it should.5

Evans introduces the stipulation (3) below: (3): Let us call whoever invented the zip ‘Julius’.

Thereby he argues for the existence of non-Russellian singular terms via the claim that ‘Julius’ is one of them. Evans introduces the category ‘descriptive names’ for terms such as ‘Julius’; this simply means that they are names whose referent is determined by description rather than by baptism or otherwise.

It is unclear whether there is a single unique individual Julius, because the zip may have been invented by a number of people, or there could be no zips in a different possible world. We could indeed replace ‘zip’ by some invention which does not exist. In any case, in the scenario to be considered, ‘Julius’ may or may not have a referent. Evans claims convincingly that there is no problem understanding (4) below:

(4): Julius was an Englishman;

There is a clear sense to (4), and it is intelligible. These facts obtain irrespective of whether ‘Julius’ does or does not refer. If that is true, then (7) must be replaced, since we can have an empty term in a sentence but that sentence can nevertheless be understood.

Evans’s first proposal for a replacement is (8) below: (8): (∀x) (The referent of ‘a’ = x iff x is φ)6

The underlining notation is introduced by Evans to indicate uniqueness7 of x: it is the only entity which satisfies φ. Naturally, x must exist in order to so satisfy. Thus Evans is only allowing existing entities to be referents, thus ensuring that axiom (8) does not attempt to assign non-existing objects as referents of empty terms.

Evans now observes that there are two problems for classical logic which will mean that (8) must be modified. The first problem is a result of Existential Generalization, which licenses inferences in the following form:

( . . . a . . . ) → (∃x) (x = a & ( . . . x . . . ) )

This may be paraphrased as being the allowing of the inference from any sentence about an object a to the existence of some x such that x is a and also

4 Evans’s own gloss here on the problem is that for empty terms, no axiom such as (7) can be truly stated. Sainsbury [6, Ch. 1] alternatively describes the problem as being that axioms in this form entail the existence of the referent. This would follow from a ban on empty terms. Sainsbury also claims that the decision at this point about how to handle empty terms drives the later choice between classical and free logic, as I will outline later.

5 An alternative approach would be to allow that different theories could cover empty and non-empty terms. Intuitions will differ here, but many will have sympathies with Evans’s general view against such ad hoc procedure.

6 Some might argue that the problems I outline in this essay are in fact already present in (8). While this may be so, I retain the reference to axiom (9) in the title of this essay since (9) is Evans’s finished product, as it were.

7 There is also a small vertical dash on the underline that points to x meaning that it is x that is unique.


such that the same sentence is true of x. This is clearly unacceptable when empty terms are admitted, since we could move from a sentence referring to Zeus, who does not exist, to a further sentence to the effect that Zeus does exist.

The second problem is similar; it results from Universal Elimination, which licenses sentences about an object a to be asserted on the basis of a universal statement such as:

(∀x)( . . . x . . . ) → [ a ] ( . . . a . . . ) )

Evans introduces the square bracket notation [ a ] to indicate that the object a exists – so therefore the term ‘a’ refers. This can be read as ‘there is a unique a such that . . . ’. So the inference above can be paraphrased as ‘if for all x a sentence is true of x, then there is some unique object referred to by the term a such that the sentence is true of that object. Again, this cannot be accepted because we do not wish to admit that sentences including an empty term ‘a’ can be true.

The solution to these two problems is to replace classical logic with a Free Logic which restricts the application of Existential Generalization and Universal Elimination. The rule of Universal Elimination may only be employed if there

is in addition a premise to the effect that (∃x) (x = a) i.e. the object a must exist so that the term ‘a’ can refer to it.

Evans notes in relation to this insistence that ‘[w]ith this in mind, we can see that there is no obstacle to using a name like ‘Julius’ to state its own semantic contribution’.8 So (8) should be replaced with (9):

(9): (∀x) (The referent of ‘Julius’ = x iff [Julius] (x = Julius)).

Again, the square brackets indicate that there must exist some unique satisfying object, so that (9) can be read as ‘for all x, the referent of the term ‘Julius’ is Julius iff there is some unique Julius such that x is Julius’.

In the Fregean theory, the significance of a sentence consists in its being true or false.9 The ‘semantic value’ of a term is its ability to affect the truth value of a sentence in which it occurs, because the semantic value of a sentence – a truth value – is composed of the semantic values of its components. The semantic value is for Frege determined by the association of the term with an extra-linguistic entity. So axiom (9) is telling us what that extra-linguistic entity is – namely the person Julius, should he exist – and that he must be the unique satisfier of the description we have in mind – namely being the inventor of the zip.

8 Various writers including McDowell [7], Evans himself and Sainsbury regard this as a ‘standard’ ‘homophonic’ approach, wherein the sense of Hesperus is shown by stating the referent as follows: The term ‘Hesperus’ refers to Hesperus. This is an ‘austere’ methodology, to adopt McDowell’s term, which is appropriate since semantic theory ‘should not aspire to provide detailed analyses of the meanings of individual words’ according to Sainsbury. [8, Ch. 2] This means that there is no better way of explaining the semantics of a term than to employ the same term and step back a meta-linguistic level. This Evans will now do for terms which may be empty.

9 This applies only to sentences which can be true or false, and so not to imperative sentences for example.


Axioms like these ‘give truth-theoretical expression to the alternative semantic values contemplated for names in 1.7’ – those alternative semantic values are the null set for empty terms and singletons10 otherwise. [1, p. 36] Now Evans allows that there can be sense without a referent – as is illustrated by his plau- sible claim that the statement ‘Julius was an Englishman’ has a sense and can be understood even if ‘Julius’ does not refer. He diagnoses the critical prob- lem with a neo-Fregean theory not as being the allowance of sense to sentences containing empty terms, but as the ‘equation between semantic value and referent’.11 [1, p. 32] For on that view, empty terms have no semantic value, which means they have no Meaning12 and so it is hard to see how they can have sense and be understood. Evans replaces this equation by holding that semantic val- ues are sets ; in the case of empty terms the semantic value would be the null set, meaning that an empty term has no referent because the null set has no members. The sense of ‘Julius’ is a mode of presentation of the semantic value of the term; and that semantic value is the null set if there is no Julius.13

If we view Evans’s axiom (9) solely as a truth-theoretic axiom, we will accept it because it produces the right truth conditions for all names, including Russellian ones. But there is a problem for Evans because he is also attempting to provide a theory of sense. In fact, Evans sees the truth theoretic axioms as showing the sense of the semantic value, which means also showing the sense of the referent – because the sense is the mode of presentation of both.14 But axiom (9) does not do this i) in the empty case because there is no referent and ii) in the non-empty case as the semantic value is whatever the semantic value of a description is, which is also not a referent. So Evans’s axiom for Julius cannot show the sense. We can see this by recalling the line taken by McDowell who reminds us [7, p. 169] that a theory of sense is meant to be a theory of understanding – we want to know how people know truths that are expressed

10 Evans actually says that these semantic values are ‘formally adequate’ which some claim is consistent with this not actually being his view. That seems strained, but in any case, this is the only line he gives.

11 Note that for Evans such an allowance is not a problem for Frege’s actual theory because Evans believes, as previously noted, that Frege does not allow sense to empty terms. Morris [9, p. 37] though goes so far as to claim that Frege used sense specifically to handle the problem of empty names. This is a significant problem for Evans as his motivation for denying the equation is to make it possible to allow sense to empty terms. So he is solving a problem for Frege that Frege does not have.

12 ‘Meaning’ is Evans’s preferred translation for Bedeutung, elsewhere translated as ‘reference’. Evans can correctly claim normal German usage as a justification here, but opponents could urge that it is a term of art for Frege. I judge the opponents correct here because ‘meaning’ is really what Frege is trying to elucidate with his theory; it is not an aspect of that theory.

13 Note that Evans is not vulnerable to Sainsbury’s criticism [8, p. 53] that it is not appropriate to model failure of reference on successful reference to a special entity because Evans continues to hold that empty terms have no referent even though they do have a semantic value.

14 Alternatively, Evans wishes only to show the sense in his terms as being the mode of presentation of the semantic value and not the mode of presentation of the referent. These are the same for Frege since Frege equates semantic value and referent. Since Evans denies that equation, he is enabled to restrict his claim to sense in his terms. But then he is not explaining Fregean senses.


by sentences, not how they know that the sentences are true. The former task can only be completed by examining senses, by considering the ways in which people refer to Julius. If they cannot so refer, because ‘Julius’ does not refer, we will not be able to consider how they refer.

So we have shown that Evans’s approach fails to provide a theory of sense. There is in addition a further problem with the whole category of descriptive names as well, as I will now outline.15

Geach [10] illustrates the problem with his example of a name – ‘Fifi’ – given to a particular diamond specified by description, such as ‘the diamond in a certain pendant’. ‘Fifi’ would still name that diamond even if it had fallen out of the pendant and no longer fitted the description.16 So Geach denies that there are any descriptive names at all because in his counter-example, the name continues to refer to an item which no longer meets the original description. And since the sense is also given by the original description, the sense of the name is also now detached from the object.

This difficulty is only avoided by Evans because the description Evans employs relates to a past achievement of Julius which, if it took place, cannot be undone and thus always continues to be available to refer to Julius, if he exists. So the question becomes whether Evans is entitled to restrict descriptions to those which will always function to select a particular referent where it exists.

We can imagine that an object is named at time tname and that a descriptive name is given, with a particular description supplied. That description will relate to circumstances at a time tdes . In the case of Julius, tname is later than tdes and so the name cannot subsequently become detached from its original object. Geach notes there is no reason at all for tname and tdes to occur in that order. ‘Fifi’ applies to the diamond to begin with because it meets the description but then the name stays with the diamond when it no longer does. There are other problems of this type. I could on Thursday adopt a name for the egg that I will eat for breakfast on Friday morning (i.e. tname is earlier than tdes ). One might argue that the no effective name giving has taken place in those circumstances, because it is possible that I will in fact eat no eggs on Friday. But if I do, the name applies, surely, to that egg. So Evans needs a large number of temporal restrictions within his category of descriptive names. These restrictions appear ad hoc and there is no obvious reason to permit them to Evans.

Evans claims to be operating in a neo-Fregean framework, modifying that approach within compass. But his treatment of empty terms is not consistent with that of Frege. For Frege, the sense of an expression is the condition that must be met by the referent.17 The natural gloss on Frege’s position is that the sense of an empty term is then the condition that would be met by the

15 It might be urged that a problem with descriptive names is just that, and not necessarily also a problem for axiom (9). Be that as it may, descriptive names are key to Evans’s argument for axiom (9) and so we are entitled to question the latter if we can question the former.

16 Note that ‘Fifi’ is not a term equivalent to ‘whichever diamond is in the pendant’; it is a ‘descriptive baptism’, so to speak.

17 Bell [4, p. 275] takes this line.


referent, were it to exist. But for Evans, the sense of an empty term is a mode of presentation of the null set [1, p. 32] which is too dramatic a departure from Frege. I have already discussed the general disagreement with Evans’s claim that Frege does not allow sense to empty terms and how this leaves a question mark over the motivation of the whole enterprise. Moreover, Sainsbury [8, p. 66] observes that Evans is working with negative Free Logic (“NFL”) – which Sainsbury also recommends – but that Frege himself used his own different version. In NFL, all sentences containing empty terms are false. This is true even for statements of self-identity. For example, if ‘Vulcan’ does not refer, then even ‘Vulcan = Vulcan’ is false; these types of sentences might be thought the strongest candidates for true sentences containing empty terms. For Frege, sentences containing empty terms lack truth values. Evans cannot claim that axiom (9) enables him to produce a basically Fregean framework with an improved handling of empty terms if he has distorted Frege too greatly in the attempt.

Sainsbury [8, p. 69] notes problems with NFL which Evans will naturally also need to deal with. Evans agrees that NFL will say that all sentences containing empty terms are false. We can see that Evans is on board with this because he agrees that ‘Julius = Julius’ is false just because it might not refer. This means that both of the following sentences are false.

1. ‘Vulcan is distinct from Vulcan’

2. ‘Vulcan is identical to Vulcan’

If sentence 2 is false, then ¬2 is true. But ¬2 has the same meaning as

‘Vulcan is not identical to Vulcan’, so that is also true. But that is synonymous with item 1, so we have different truth values for the same sentences. This argument relies on a Russellian scope ambiguity between the negation of (a is

F) being ¬(a is F) or (a is ¬F) because Sainsbury needs the synonymity of

¬‘Vulcan is identical to Vulcan’ and ‘Vulcan is not identical to Vulcan’, which

may be questionable. To be fair, we know Evans is aware of these problems and that he is also aware of problems arising from negation in NFL since he also uses the ‘global negation operator’ N [1, p. 52] that creates the difficulties. He observes that insisting on applying only one of the types of negation available – attaching the ¬ to the predicate – is equivalent to insisting that all singular terms are Russellian, which begs the question. But this awareness does not constitute a solution.

There are in addition problems with the motivation for Evans’s approach. The problem of empty terms occurs if we combine non-referring terms with standard rules of logic. Frege and Russell solve this by excluding the terms in different ways. For the former, sentences containing empty terms are fictional; for the latter they are ‘nonsense’. Evans solves it by retaining the terms but adjusting the rules of logic. How do we know this is superior? Evans does indeed avoid the problem of empty terms but at the cost of adjusting logic. It is hard to show that that price is lower than the costs he avoids. Indeed, many writers would hold that logic is more fundamental than language and that adjusting


logic therefore carries a higher price. Quine [11] has suggested that there may be a core and a periphery of our web of beliefs, with the logical laws forming part of the core for revision of which we would require extraordinary persuasion. Evans has not provided such persuasion.


[1] G. Evans and J. McDowell, The varieties of reference. Clarendon Paper- backs Series, Clarendon Press, 1982.

[2] G. Frege, “Sense and reference,” The Philosophical Review, vol. 57, no. 3, pp. pp. 209–230, 1948.

[3] G. Fitch and M. Nelson, “Singular propositions,” in The Stanford Encyclo- pedia of Philosophy (E. N. Zalta, ed.), Metaphysics Research Lab, CSLI, Stanford University, spring 2009 ed., 2009.

[4] D. Bell, “How ‘russellian’ was frege?,” Mind, vol. 99, no. 394, pp. pp. 267–277, 1990.

[5] B. Russell, “On denoting,” Mind, vol. 14, no. 56, pp. pp. 479–493, 1905. [6] C. Macdonald and G. Macdonald, McDowell and his critics. Philosophers and their critics, Blackwell Pub., 2006.

[7] J. McDowell, “On the sense and reference of a proper name,” Mind, vol. 86, no. 342, pp. 159–185, 1977.

[8] R. Sainsbury, Reference Without Referents. Oxford University Press, USA, 2007.

[9] M. Morris, An introduction to the philosophy of language. Cambridge in- troductions to philosophy, Cambridge University Press, 2007.

[10] P. T. Geach, “The varieties of reference by gareth evans edited by john mcdowell oxford: Clarendon press, 1982, xiii + 418 pp., £15.00, £5.95 paper,” Philosophy, vol. 61, no. 238, pp. 534–, 1986.

[11] W. V. Quine, “Main trends in recent philosophy: Two dogmas of empiri- cism,” The Philosophical Review, vol. 60, no. 1, pp. pp. 20–43, 1951.