For those who are keen to know more about the sorites paradox – here is an essay I wrote about the fuzzy view of vagueness, which attempts to solve the problem. Be warned – it’s long and long winded.

A genuine solution to the sorites paradox must explain both why the argument it employs is flawed and why it was so compelling in the first place. In this essay I will provide an exposition of the standard fuzzy view as given by Machina, and the means by which he attempts to solve the paradox. I will then go on to argue that the notion of truth functionality is not determinative in assessing the success of fuzzy theories and that the debate should focus on their central notion: degrees of truth. I will then consider one brand of argument which attacks this notion and try to show why it is unsuccessful.

The best place to start is with Machina’s presentation of the fuzzy view. His presentation will then form the background against which the other accounts will later be considered.

Machina’s initial motivation grows out of his rejection of the principle of bivalence. This principle asserts the truth of any sentence to be just one of two values: true or false such that if it is not true it must be false and vice versa . But for many sentences, Machina notes, this is not the case. We are asked to consider Jones’ belief that Horatio planted petunias in the garden yesterday when in fact Horatio merely soaked the flower bed and threw the petunias on top. Of this belief Machina says: ‘I for one though, do not think it simply true under these circumstances that Horatio planted petunias in the garden yesterday. Nor is it simply false.’ This is perhaps not the best example. The lack of willingness to assent either way to the truth or falsity of a proposition can be seen in the application of any vague predicate to a borderline case.

Machina’s second motivation is to remain as conservative as possible with respect to the classical logical picture. This motivation comes out of the general recognition of the success of this picture in explaining many aspects of our language use. As such, the supervaluationist approach is rejected. Machina argues that while the supervaluationist manages to preserve all the logically true theorems of classical logic it fails to preserve the valid rules of inference . What’s more, it comes into conflict with Tarski’s convention and disallows the use of indirect proof. Machina urges that ‘we should search for another approach… which will preserve the limited version of indirect proof… as being reasonable.’

To this end he prescribes the normality constraint such that: ‘the logic of vagueness be normal, in the sense that the sentential connectives shall be defined in such a way that when operating on propositions with classical values they yield propositions with the usual classical values’ . He also wants to retain the truth functionality of the classical picture such that the truth values of the components of a sentence determine its over all truth value. This won’t be without cost. Machina is forced by this decision (combined with his rejection of bivalence) to abandon the law of excluded middle and the law of non-contradiction.

His suggestion is to say that sentences are capable of being more or less true to a variety of degrees beyond simply true or false. The classical values of true and false represent completely true and completely false i.e. the values 1 and 0 respectively. For the rest, a continuum of values given by the real numbers between 0 and 1 is used to represent the degrees of truth that are not clearly true or false simpliciter. The motivation to use a continuum is borne out of a desire to accurately represent borderline cases of vagueness which are neither true nor false, but nevertheless differ in degree of truth, which would not be possible with a finite number of truth values.

When outlining his treatment of the standard truth functional operators (‘and’, ‘or’, ‘not’), Machina stays true to his stated constraints. He notes that the classical operator ‘and’ has the value true where both conjuncts have the value true and false otherwise. Numerically speaking, given bivalence, this means the complex sentence involving ‘and’ always takes the minimum value of both conjuncts. So to with the fuzzy formulation: |p&q| = min(|p|, |q|). A similar principle is in play in the formulation of the other connectives ‘or’ and ‘not’ by means of: |pvq| = max (|p|, |q|), and |~p| = 1 – |p| respectively.

The definition of “if then??? is a little more complex. A complete adoption of a classical approach in this case is difficult as it leads to unintuitive results. Stating an equivalence between “if then??? and “or??? as in classical logic (|p q| = |~p v q|), gives us |p p| = ½ when |p| = ½ where we would expect it to take the value 1. Secondly, Machina considers the more important feature of classical logic worth preserving is the relationship between validity and the “if then??? operator. In the classical logic this amounted to the fact that of a valid argument a tautology could always be generated by making a conditional of the conjunction of the premises as antecedent and conclusion as consequent. In the fuzzy logic, the notion of validity to be employed is such that: ‘when an argument instantiates an argument form possessing the property that its conclusion must always be at least as true as the falsest premise, the argument will be fully valid.’ More on this notion of validity will be described immediately below – but the result of these constraints give a definition of “if then??? such that: |p q| = 1 where |q| > |p| or |p q| = (1 – |p| – |q| ) where |q| |p|.

Machina employs an analogue of the notion of ‘preservation of designated values’ in order to define validity. In the classical logic the idea is that the value 1 is serves as the designation and a valid inference requires the preservation of the designated values from premises to conclusion such that ‘whenever the premises all have designated truth values, the conclusion must also have a designated truth value.’ Machina alters this slightly such that instead of a designation preserving argument form, we have a truth-preserving argument form which yields the definition of validity as stated above. This allows him to demonstrate the desired connection between “if, then??? and the notion of validity. He defines “p implies q to degree n??? to mean that |q| cannot dip below |p| by more than 1-n. Given this definition it can be shown that if q is a consequence of p to a degree of 1, then |p q| will also equal 1. Where q is a consequence of p to a degree less than 1 then |p q| can dip by just as far as n.

The model theory for the fuzzy view proceeds similarly to the classical model theory. An interpretation of the calculus is specified, the primary elements of which consist in a non empty set called the domain, an extension for each n-place predicate letter (a mapping to a subset of the domain) and a function which assigns to each individual constant an element of the domain. The only significant difference for the fuzzy theory is that it makes possible the membership, for any element of the domain, of an extension of a predicate, to be a matter of degree – the so called fuzzy set.

With an outline of the major features of Machina’s version of the fuzzy view in place we can now have a look at how this account of vagueness can be used to solve the sorities paradox. Machina presents the following version of the sorites paradox:

1) Horatio has no hair on his head.
2) Anyone who has no hair is bald.
3) Anyone who has just one more hair on his head than any bald man is also bald.

Therefore:

4) Anyone who has 107 hairs on his head is bald.

Machina explains that where the premises 1 and 2 are less than fully true, then the consequence that Horatio is bald can be less true than the least of the premises and hence the argument is not truth preserving and invalid. The argument is an instantiation of the argument form:

1) Nh
2) (x)(Nx Bx)

Therefore:

3) Bh

This argument would ordinarily (classically) proceed by modus ponens and the equivalence given for (2) such that |(x)(Fx)| = |Fa1 & Fa2 &… & Fan| for every n in the domain. By ‘& elimination’ we get (Nh Bh) – and the modus step is then available. But in Machina’s version of the fuzzy view – modus ponens is not a valid argument form . Hence, it is argued, the flaw of the argument is revealed: it involves an instantiation of an invalid argument form.

The deceptiveness of the argument is that premises (1) and (2) both actually take the value 1 and when they do, the value of the conclusion is also 1 – hence we are deceived into thinking the argument is correct.

It’s worth noting at this point that Machina is not yet successful in solving the sorites paradox. It’s not enough that he show that the argument involves the instantiation of an invalid argument form – for all valid arguments are the instantiation of the invalid argument form p therefore q. There is no effective procedure for the determination of invalidity in arguments for there may still be some valid argument form which it instantiates which we have not tested . To demonstrate the invalidity of the argument Machina needs to show how the actual values of the premises and conclusion are not truth preserving. He can’t do that with this section of the argument because (1) and (2) are fully true, and when the premises are fully true then modus ponens is truth preserving.

This is not so with the deduction that proceeds from premise (3). The truth value of the generalisation is given by the truth of least valued conjunct of the form: Mxy & By Bx – the conjuncts arising from a complete assignment of objects in the domain to x and y. The value if Bx is always going to be less than By by a tiny fraction ? and Mxy will be 1 so |Mxy & By Bx| will be 1 – (|By| – |Bx|) which gives us 1- ?. Hence the inductive premise is nearly fully true – but not completely. This is fine while By retains the value 1 – but we know that it doesn’t as we increase the number of hairs (in fact, as soon as we add a single hair to a bald head). As soon as this happens the value of the conclusion in each application of modus ponens is invalid. We can see this readily where |Bxn| = (1-(n*?), |By| = (1-((n-1)*?)). Since By will be the least true premise, |Bx| will always be smaller than the least true premise. Hence the argument has been demonstrated to be invalid (not just the argument form of which it is an instantiation). In this case – the argument is convincing because the inductive premise (3) is very nearly true.

Now that we have a version of the fuzzy view on the table we can step back a little and begin to take stock. We have seen that in creating his theory, Machina has sought to retain certain aspects of the classical picture, while discarding others – particularly, he has sought to retain the notion of truth functionality. What I will argue, however, is that the notion of truth functionality is not essential to the fuzzy view. Arguments for or against truth functionality in the fuzzy view often revolve around unclear appeals to intuition, or do so by claiming an advantage at having retained one aspect of logical theory over another (including truth functionality itself). No one seems to be able to gain a clear edge, however, leaving the concept of truth functionality to be somewhat irrelevant in the debate concerning the success of the fuzzy theory. I will show this to be the case by examining the arguments and showing them not to be determinative either way.

The first of these arguments comes from a classical purist, Williamson. He is convinced of the value of truth functionality for any theory of vagueness, and hence believes that if the adoption of the concept for a fuzzy theory can be shown to be problematic, then this is a significant reason to abandon the view. This is exactly what he attempts to do. He argues that the adoption of truth functionality for fuzzy theories of vagueness leads to unintuitive results. He asks us to consider the conjunction: ‘He is awake and he is asleep’. As the degree of truth of one conjunct falls the other should rise – at some point they should be even. By the truth functionality of Machina’s system the value of the conjunction will be ½. Williamson asks: ‘But how can that be? Waking and sleep by definition exclude each other.’ He finds similar examples for disjunction and implication.

A natural way for the fuzzy theorist to respond is to deny that Williamson’s intuitions are correct. This is the approach Smith takes. He makes the claim that the ordinary language user not trained in logic and philosophy would not simply reject the strange conjunction outright, but would at least hedge their response. But it’s difficult to negotiate between intuitions, and harder still to determine where the average language user would take their stand. What should be emphasised, however, is that it is the objector that loses out in such cases where disagreement over intuitions persists. If I have asserted a philosophical view of any kind, then the onus is on the objector to provide a clear reason why it is wrong. Until that time – I am under no obligation to accept the objection. I will continue to hold my view so long as I can reasonably stick to my own intuitions. So, returning to the debate between Williamson and Smith: even if we don’t share Smith’s intuitions on the matter, they certainly can’t be said to be unreasonable. Hence it is hard to see this kind of objection as successful.

Edgington, however, is able to provide a set of counter examples that are more compelling. But again, the appeal to intuition does not carry. Edgington proceeds as follows:

Let Ra, Rb and Rc be the statements that balls a, b and c are red, respectively, and Sa, Sb and Sc be the statements that they are small. Suppose:

|Ra| = 1, |Sa| = 0.5
|Rb| = 0.5, |Sb| = 0.5
|Rc| = 0.5, |Sc| = 0

And then she goes on to provide counter examples for all the truth functional connectives, excluding ‘not’. She begins with ‘and’ – |Ra & Sa| = |Rb & Sb| = 0.5, but isn’t a a better case for ‘red and small’ than b? Edgington provides a similar example for ‘or’, but also a case where the conjuncts are not independent. Let |Re| = 0.5 and |Rd| = 0.4. This gives |Rd & ~Re| as 0.4 which Edgington believes to be implausible and asks: ‘How could it be other than completely wrong, in any circumstance, to say “d is red and e is not???? |Rd & ~Re| should be zero.’ Edgington applies the same example to ‘if then’ and gets a similarly unintuitive response. As such, she concludes, it’s impossible to sustain a truth functional fuzzy view of the logical connectives.

Smith retorts that the counterexamples provided by Edgington are not strong enough to motivate us in the non-truth functional direction. He says:

We need to carefully distinguish two things: the distance of a sentence from the truth, in the sense of how much would have to change about its subject matter to render the sentence true; and the distance of a sentence from the truth, in the sense of how far its actual degree of truth is from the maximum truth value.

The idea here is that while Edgington is correct in saying that one ball is a better choice than another insofar as it is closer to the desired criterion, this doesn’t make it truer than the other. Edgington’s objection, we will see below, is one of a class of objections raised against fuzzy theories which rely on an equivalence between the measurement of a thing and its truth value. This class of objection will be considered below. It’s enough to note now, however, that these sorts of objection aren’t successful in establishing that a fuzzy theory of vagueness can’t be truth functional.

Given that in both these cases the appeal to intuition has been unsuccessful, the choice concerning whether or not a fuzzy theory of vagueness should, or can, be truth functional seems to come down to the different aspects a logical theory can contain. For example, in order to assert a continuum of truth values and the truth functionality of his logical system, Machina had to give up, among other things, the principle of bivalence. But if we were presented with a system that like Machina, promoted the continuum of truth values, preserved bivalence, but denied truth functionality, how would we decide?

This is exactly the choice we are faced with when comparing Edgington’s fuzzy theory with Machina’s. Edgington argues against the truth functionality of the fuzzy view, but presents a version which preserves bivalence. To do this, Edgington first introduces a non-truth functional system by means of an analogue to probability theory , i.e. the probability theorist’s notion of conditional probability as being the key to the probability of a conjunction. Using a similar approach she obtains a rule for ‘or’ as well:

|A & B| = |A| * |B given A|
|A v B| = |A| + |B| – |A & B|

A full treatment of the conditional is not considered (as everyone participating in debates on vagueness seems willing to allow) – but for now an equivalence to ~(A & ~B) is offered as it is the weakest notion of the conditional that generates the sorites paradox.

Secondly, a proof is given for the law of excluded middle based upon the definitions for ‘and’ and ‘or’ described above. Thirdly, Edgington provides examples that are supposed to convince us that no general objection to the definite truth of A v ~A where neither disjunct is definitely true. One of these examples concerns the case where someone, who is definitely a sibling, is in the process of a sex change. This person, Edgington claims, is definitely a brother or a sister, but not definitely either. And last, Edgington discriminates between a weaker and stronger sense of bivalence . The weaker sense is that the disjunction A v ~A can be definitely true when one or both of the disjuncts is not definitely true – this is the kind that her theory preserves. The stronger sense is that any proposition is definitely true or false. Edgington’s theory does not preserve this kind of bivalence.

We can now return to the question of deciding between Edgington’s theory and Machina’s – deciding between the truth functional or the bivalent fuzzy theory. What sorts of considerations are determinative? Not very much is offered by any author. But here are a few thoughts, none of them compelling either way.

Advocates of the truth functional approach often cite that this principle is essential in the systemisation of a theory. This is because the truth values of complex sentences can be determined systemically by the truth values of the components. Aware of the perception that the loss of truth functionality implies loosing the systematic treatment of the logical connectives, Edgington moves quickly to defend her position. As she says: ‘To abandon degree-functionality is not to give up on a systematic treatment of the logical constants.’ And she relies on the analogue she supplies to probability theory to back up this claim. Hence, neither theory seems to have an edge in this respect.

Edgington might argue that because she retains a form of systematisation along with bivalence, hers is the theory to prefer. But again, the advantage of her theory in this respect is not clear. The classicist might point out that this weak sense of bivalence is not what they meant by the term. Williamson, for example, quotes the principle of bivalence as: ‘If u says that P, then either u is true or u is false’ , which is clearly the strong sense of bivalence mentioned above. So, Edgington hasn’t retained the classicist principle at all.

Her reply might be to concede this point but argue that although it is not exactly the same concept, this weak version is enough to avoid the contradiction Williamson asserts any denial of bivalence implies. To see if this is so, we need to briefly examine Williamson’s argument.

Considering an utterance (u) that states a declarative sentence (P), Williamson puts down the denial of bivalence as a premise: not: either u is true or u is false. Using Tarski’s semantic definitions he can extract: not: either P or not P. By using De Morgan’s law he can get the conclusion: not P and not not P, and can remove the double negation to obtain the explicit contradiction: not P and P. Now Edgington can avoid this result because she can accept the definite truth of P or not P, and hence denies the premise which is needed to get the proof going.

This would be an advantage if it were clear that the truth functional fuzzy view is exposed to Williamson’s objection – but this is not the case. The application of De Morgan’s theorem is valid in the classical logic because of the principle of the substitutivity of logical equivalents. Two statements are known to be logically equivalent where their biconditional is a tautology. A statement is a tautology when its truth value is 1 under all interpretations. In classical logic the biconditional of ~(p v ~p) and (~p & p) does take the value 1 under all interpretations – hence the substitution is valid. But this is not so in Machina’s logic. Here |p q| is defined as |(p q) & (q p)|. Under an interpretation where P takes the value of 0.1, the biconditional of ~(p v ~p) and (~p & p) also takes the value 0.1 – considerably short of the value 1 which it would need in all cases to license De Morgan’s theorem . Hence there is no reason to suppose that Machina’s view is at a disadvantage for denying the principle of bivalence.

There is one last reply that Edgington might make – and that’s to claim that her theory is better than a truth functional variant because the truth functional, like Machina’s throws out the validity of modus ponens in order to solve the sorites paradox. Her theory, on the other hand, retains the validity of modus ponens and yet still solves the sorites paradox. Since modus ponens is such an intuitively strong form of reasoning, a system that retains it is to be preferred to one that doesn’t. It’s not necessary to show in detail how Edgington achieves this. Even if she has managed to retain the validity of modus ponens and solve the sorites paradox, this is not an advantage over the truth functional approach in general, if there is a truth functional version of the fuzzy view which can do the same – and as it turns out, there is.

Nicholas Smith provides a version of the fuzzy view which solves the sorites paradox and retains the validity of modus ponens. He does this by retaining the classical equivalence between |p q| and |~p v q| and by providing a notion of validity that does not rely on the notion of designated values. To quote:

I propose the following definition of validity. B is a fuzzy consequence of a set ???? of wfs just in case there is no interpretation M such that [A]M > 0.5, for every A in ???? , and [B]M < 0.5.

Smith is motivated in this way partly because he believes that Machina’s solution to the sorites is not a solution at all. He writes:

That’s a neat story, but it cannot be the correct account of why the Sorites is both compelling and mistaken. For on this approach, the fuzzy theorist is left without anything to say about the Sorites paradox as formulated in other ways.

Quoting Wright, Smith alerts us to this possible replacement of the traditional inductive premise with: (x)(red(x) & red(x’)). In this form, Machina can no longer necessarily explain how it is that the premise is almost true (which was why he said the argument was convincing), nor can he appeal to its invalidity as the reason for its failure.

Smith’s account of the fuzzy view can solve the sorites paradox. It demonstrates the flaw in the argument by pointing out that while it is valid, it is unsound. For given the definition of ‘if then’, at some point one of the premises in the series will be less than 0.5 true. As the definition of ‘sound’ is that none of the premises can be less than 0.5 true – the flaw is apparent. Smith’s explanation for the plausibility of the argument is complex and for reasons of space it is not possible to do it adequate justice here. What is important for my argument is that he does have a plausible account, and as such the claim that Edgington’s theory has an advantage of truth functional accounts because of her solution to the sorites is rendered mute.

One final argument might be made by Edgington to say that this version of the truth functional theory makes use of an implausible equivalence between |p q| and |~p v q|. But this brings us back to the realm of intuition which in this case (as per usual) is inconclusive. We have already seen the difficulty of adopting this view of ‘if then’ when we considered Machina’s account. |p p| takes the value of ½ when p is ½ where we would expect it to take the value 1. Smith, aware of the problem, considers further examples, he says:

…consider Bob, a borderline case of ‘bald’, and Bill, who has one less hair than Bob. Let us suppose ‘Bob is bald’ is 0.5 true and ‘Bill is bald’ is 0.51 true. Then ‘If Bob is bald, then Bill is bald’ is 0.51 true, according to my semantics, whereas on the standard fuzzy semantics, it would be 1 true—and isn’t the latter the more intuitive assignment?

His response to these kinds of objections is to point out that the conditional is an odd thing to assert where the antecedent is less than true. In order to galvanise our intuitions he gives us a case where the antecedent is clearly false: ‘If this is a
Spade. I will use it to dig a vegetable garden’ – spoken in a context where a person is unwrapping a present. If it is unknown what the present is then this is a perfectly natural thing to say, but if it is known that it is not a spade, then it becomes quiet strange. So to, Smith says, with the Bill and Bob example. It is not quite as odd because the antecedent is not completely false – but the more false the antecedent, the odder a thing it is to say.

Again, when it comes to conflicting intuitions, it is very difficult to arbitrate. I would say here what I said above when considering those counterexamples put forward by Williamson. So long as Smith can still be seen to be reasonable in holding to his intuitions (even if we don’t share them) then the counter example has not gone far enough. I think Smith succeeds in this respect here also.

As such it cannot be said that Edgington can demonstrate any advantage for have a non-truth functional theory. But neither have we seen an argument which clearly shows Edgington to be at a disadvantage for having chosen a non-truth functional system. The argument between fuzzy theorists on this point is mute. The only significant advantage displayed by any author hitherto considered is that of Smith over Machina, insofar as Machina fails to solve the sorites paradox conclusively. Any theorist in vagueness that fails in this respect has missed the boat entirely as this is an important criteria any theory of vagueness must be judged. But this does not bear on the issue of truth functionality since both theories are truth functional.

With respect to the claims made by advocates of the truth functional view (both inside and outside the fuzzy perspective) that truth functionality is an argument in itself – it’s difficult to reply, if only because the arguments for this view do not feature at all in this discourse, since it is widely held to just be a given. But it’s obviously not just a given since there are a number of theories out there which are not truth functional. That the discourse contains these alternatives means that the status of truth functionality and its value is not clearly known. I should like to see a positive argument – but I haven’t been able to find any. The best that Williamson can do is: ‘For some kind of truth functionality surely what is central to the semantics of the most basic logical operators…’ and is so confident that he neglects to say any more. For myself, I would accept that a positive argument to opposite effect should be forthcoming too – but I am short of space with respect to this essay and must leave it for another time.

Given these considerations, we have seen that there seems to be no argument on the table which decides conclusively that a fuzzy theory of vagueness either a) can’t be truth functional, or b) if truth functional, is more or less preferable to a non truth functional fuzzy system, or c) is at a disadvantage to other theories of vagueness that reject the notion of degrees of truth but preserve their truth functionality. In short, I conclude that the notion of truth functionality is not determinative for theories of vagueness, and in particular, is not a criterion on which the fuzzy view should be judged. I would tend to think that this conclusion might extend to a number of other logical criteria – but don’t have the space to consider this now. In general, however, I would say that the fuzzy view should be judged on its central notion: degrees of truth – all general and significant objections to the fuzzy view do, and it is to one of these that I will now turn.

Of the many objections that attack the central notion of the fuzzy view, degrees of truth, I will consider only one. However, this single objection forms the base to a group of objections that contain a common seed – namely, that the notion of ‘degrees of truth’ somehow conflates to the notion of ‘degrees of qualities’, or that the relationship between the two is such that it makes it difficult to sustain the idea that truth comes in degrees. I will examine three such objections, one from Williamson, another from Keefe, and a second from Williamson on the notion of higher order vagueness. We’ll see that all three arguments share this common base, that Williamson’s first can be rejected on the basis of the closeness definition of vagueness and that all three can be put into doubt by undermining the assumption that theoretical concepts which applies to ‘degrees of qualities’ can apply to the notion of ‘degrees of truth’.

The first objection that I will consider comes from Williamson. He asserts that the notion of ‘degree of truth’ does not give us any better understanding of vagueness. He first asks us to consider how it is the fuzzy theorist expects us to understand ‘truer than’ – we do so, he says, by an equivalence with a comparative statement: ‘x is F is truer than ‘y is F’ if and only if x is Fer that y’. The problem with this is that it leads to cases where the degree of truth can be 1 even though the object is more or less of the quality. Williamson’s example asks us to consider the predicate – is acute. This is a precise predicate. Any angle with a degree less than 90 is acute. Hence 30 degrees and 60 degrees are both completely acute – and the first is more acute than the other. But by the equivalence relation one is truer than the other. He goes on to claim that: ‘If the notion of degree of truth is explained in comparative terms, then the occurrence of degrees of truth between perfect truth and perfect falsity in no way implies the occurrence of vagueness,’ and that this result undermines the fuzzy solution to the sorites paradox because it allows the construction of sorites series that are non-paradoxical and hence fails to locate the source of the problem.

It’s easy to see that this objection relies on providing an interpretation of how it is that the fuzzy view sees the relationship between the notion of ‘degrees of truth’ and ‘degrees of X’ness’. The idea is that if vague propositions or statements have degrees of truth, then these statements can be more or less true, and if they can be more or less true, there must be something that makes this so – that something is the Xness of the objects in question.

There are a number of reasons why we can ignore this argument. Firstly, Williamson hasn’t taken any effort to describe why he thinks this comparative relation is essential to the fuzzy view. Secondly, we can say that he has demonstrated that there is a problem with the notion of degrees of truth combined with the comparative relation – but he hasn’t demonstrated which of these two, or both, are the problem. When two ideas joined together equal a bad idea, it could be because of a flaw in just one of the thoughts, not necessarily both – why should the blame necessarily go to ‘degrees of truth’? He needs to supplement his discussion with an argument to this effect – or he hasn’t established that the problem is where he thinks it is.

Beyond stating that Williamson has failed to establish his argument, we can go further and provide an argument to say why he is wrong. In fact, the idea of degrees of truth does not imply the comparative relationship as Williamson describes it. The comparison equivalence that Williamson suggests is actually stronger than what is needed to capture vagueness – that is why it is so easy for him to shoot it down as being irrelevant. A better account of vagueness is given in the closeness relationship , such that: if a and b are very similar in F relevant respects, then ‘Fa’ and ‘Fb’ are very similar in respect of truth. Importantly, the closeness relationship is weaker than Williamson’s comparative relationship. While it asserts that those things that are similar in F relevant respects are similar in respect of truth, it does not assert the converse as does the comparative relationship – namely that those that are similar (or identical) in truth are similar in F relevant respects. As such, Williamson’s examples under this relationship are no longer problematic. Where x and y are both tall to degree 1, this does not necessitate that they must be the same height, though if they are both the same height, or are close in terms of height, then it should be true that they are tall to the same, or close degree. In the case of ‘acuteness’ – that it is true to a degree of 1 does not imply that their degree of acuteness is the same. Hence Williamson’s argument can be positively shown to be false.

There is another argument by Keefe that adopts a similar approach to Williamson. She attempts to draw an analogy between measurement theory and degree theories of truth; and by this analogy cast doubt on the notion of degrees of truth. Measurement theory, she states, makes use of a representation theorem which guarantees that a numerical property can faithfully represent the degree of a particular quality. The representation theorem will assert that an object will have the quality to a greater or equal degree with respect to another object, if and only if an assignment of numerical values can reflect this fact. Her argument is that there can be no numerical relationship of truth which satisfies a representation theorem and is also related to the phenomenon of vagueness. Hence, she claims, degree theories of truth should be rejected.

A crucial step in the argument is that in order to prove a representation theorem, a connectedness axiom must be true: (CT ) either p T q or q T p. But Keefe argues that this axiom is not true for degree theories of truth as there are multi-dimensional predicates like ‘nice’ which have a number of different (and often unclear) factors which determine their degree. Hence nothing will settle the question as to who is nicest where people are nice in different ways. More clear is her example where p = “a is tall??? and q = “b is red??? and she says:

Here we have no single comparative on which “true to a greater degree??? can piggy-back. The comparison may be read as “a is more clearly tall??? than “b is red??? and if a is clearly tall and b is clearly not red, then this will be true. But in a wide variety of cases (e.g. with a 5’10??? man and a reddish-orange path) neither disjunct of (CT ) will be true… we cannot assume that there is always a fact of the matter about which of two borderline sentences is more true.

Given that the connectedness axiom is not true, Keefe argues, the representation theorem cannot follow and, she believes, her case has been made.

We can again readily see that, as in the case of Williamson above, the idea of ‘degree of truth’ is being conflated with ‘degree of Xness’ – particularly in the idea that a theorem that applies to the measurement of the latter goes just as well for the former. As it was with Williamson we see that Keefe has given no positive argument for this claim. Why should ‘degrees of truth’ be considered to be a kind of measure? Why is this analogy appropriate? It’s incumbent upon Keefe to make this positive claim, rather than just assume it – and given that she hasn’t, and we hold the opposite belief, that ‘degree of truth’ is not a kind of measure – then we are not compelled to accept the rest of her argument.

We should do better than this, however, and see if we can find a positive reason for disbelieving that ‘degree of truth’ is a kind of measure. First, however, let’s ask why someone might think that what goes for qualities and their measurement should go for degrees of truth. Well first off, as in Williamson, a natural thing to want to do, once we have declared that there are degrees of truth, is to make comparisons between the degrees of truth of different propositions. For example, of two objects x and y we ask which is truer: “x is red??? or “y is red??? and feel that if there are degrees of truth, there should be some sort of meaningful answer to this question. And since we can ask similarly structured questions about propositions concerning qualities it’s natural to think the two are analogous. And if the two are analogous, then what applies theoretically to one, applies to the other.

But let’s think the analogy through a little further. When we make a comparison concerning qualities, what kind of question do we ask? Something like: which is F’er a or b? Notice that it’s between objects that the comparison is made - when we ask after degrees of truth, the comparison is made between propositions. Do we still think they are analogous? Should that which can be applied to objects be applied to propositions? Let’s ask another question: how do we establish the answer to the question: ‘which is F’er a or b?’ Well, we compare them. We can do this in a number of ways depending on the objects. Sometimes we can put them side by side and see the difference, other times we need to pull out a measure – something individuated precisely enough to make the difference apparent. Do we do anything like this when we compare the truth of propositions? Someone might reply – yes we do, when someone claims that “Bill is tall??? is truer than “Bob is tall??? we look at Bill and compare him to Bob and see if he is taller. To this we should reply: ok, by doing this you would be able to answer the question: ‘Is Bill taller than Bob???? – but this would not provide an answer to the question under consideration simply because if Bill is 7 feet tall and Bob is 8 feet tall, then the answer to the latter question is yes, but the answer to the former is no – because they are both clearly tall. In fact there seems to be no process by which we can make the comparison. The analogy here completely breaks down. We can’t hold up two propositions next to each other, in any sense, and compare them for truth. It would be nonsense to say that we can.

This situation goes for a two valued truth system as much as it does for a continuum valued logic. Take two propositions that can be only one of two values 1 and 0. 1 is a greater value than 0 – so any proposition with this value can be said to be truer than one that takes the value 0 (if the idea of comparison here does make any sense). If we ask how we determine whether ‘Bill is tall’ is truer than ‘Bob is tall’ we are at a similar loss. There is no process whereby we can compare the propositions.

Someone will probably reply – sure there is, if we know whether Bill is tall and we know whether Bob is tall then we can make the comparison between the truth of the two propositions. The process, they say, by which we make the comparison is epistemological. But if this is the case, then the analogy breaks down as well. When we are making a comparison between two objects we perform a process whereby we measure the qualities as they appear in the object – if we can find no measure fine enough to discriminate a difference between the two objects in respect of that quality then we judge that they have that quality to the same degree. But this is not the case in an epistemological judgement. I do not study two propositions and apply some measure to the propositions – a truth measure. I look to the world to see if it is the case that such and such is the case. Most importantly, I have discovered no similarity in the propositions when I discover they have the same truth value. Or conversely, if I compare two distinct propositions, there is no amount of similarity between them that would make me think they have the same truth value.

Given these considerations we can see that the analogy these authors are using to raise this kind of objection does not hold, and as such there is no reason to think that the theoretical particulars that apply to the theory of measurement should apply to the theory of truth – whether it comes in degrees or not.

The incredulous may continue to ask: does this mean that those questions that concern truth and are comparative in nature are meaningless? But how could they be meaningless when they have distinct truth values which are greater, lesser or equal to one another? Surely when a sentence has the truth value 0.8 and another 0.2, then one is truer than the other and this claim has a sense. Call it a comparison if you like – but it can’t be of the same kind as the kind of comparison that occurs between qualities. In fact, as the inquiry stands now, I see no reason not to simply say that ‘truer than’ means nothing more than ‘has a greater truth value’. Has any reason been adduced to the effect that it shouldn’t?

This answer to Keefe’s objection applies equally to Williamson’s objection considered first – but in that case closeness was enough to solve the problem because we were only asking after comparisons of like terms, whereas in Keefe’s objection we asked after the meaning of ‘truer than’ when the comparison was being made between unlike things. Although involved in closeness is the substantive claim that there is a distinction between degrees of qualities and degrees of truth, it’s not a substantive reason why theoretical approaches to one do not apply to the other. That’s why Keefe retorts in her reply to Smith that whether or not degree theorists conflate the two senses of degrees, they still suffer from the problem that she raised – and the problem, as she restates in a nutshell:

…if a degree theorist is to assign numbers to vague sentences in such a way as to capture vagueness via degrees of truth, it must be the case that any two sentences are always comparable as regards their degree of truth.

We have now shown that if this claim is true, then it certainly can’t have the kinds of implications she thinks it has – and further that it is actually the objector Keefe who is performing the conflation insofar as she insists that what is theoretically applicable to degrees of qualities is applicable to the notion of degrees of truth. There is good reason to suppose that the notion of degrees of truth does not need a connectedness axiom in order to be appropriate in the case of vague predicates. And when we do make comparisons of truth it is simply to say that one has a greater, equal or less truth value than another and no more.

There is one more objection that relies on this conflation or analogy between the two kinds of degrees – the problem of higher order vagueness for degree theories, as raised by Williamson. He argues that sentences of a kind: “’It is wet’ is truer than ‘It is cold’??? are also vague. This vagueness in the metalanguage, he says, works its way up to the meta meta language and so on up the hierarchy of meta-languages and we have the usual regress. But we now have the resources to deny outright that these sentences are vague. If ‘truer than’ means simply that one proposition has a greater truth value than another, then there can be nothing vague at all concerning the meta-propositions – ‘truer than’ is perfectly precise, since all we mean by ‘truer than’ is that the truth value of one statement is higher than another. The difficulty we have in determining the truth of these statements is epistemological insofar as it is rare that we know the exact truth value of the object language propositions.

To conclude – I have made two distinct arguments. The first was that the notion of truth functionality should not form part of the criteria upon which the fuzzy theory of vagueness should be assessed. I made this argument be examining different fuzzy theories and by showing that with respect to their truth functionality, or lack of it, no theory gained or lost in plausibility thereby. I also argued that purveyors of theories which denied the notion of degrees of truth but retained their truth functionality had not successfully made the case that their theories were any better for it. Secondly I argued against a class of objections that are underpinned by a conflation of the notion of degrees of truth with degrees of qualities, or by the idea that theoretical approaches to one notion goes equally well for the other. This does not exhaust the objections to the fuzzy view – not even close, but I hope these considerations go at least some way to supporting the case for its overall plausibility.

Bibliography:

Machina, Kenton F. “Truth Belief and Vagueness??? in Vagueness ed. Keefe and Smith, MIT Press, 1996

Williamson, Timothy, ‘Vagueness’ Routledge, New York, 1994

Edgington, Dorothy, ‘Vagueness by Degrees’ in Vagueness, ed Keefe and Smith, MIT Press, 1996

Keefe, Rosanna, ‘Vagueness by Numbers’ in Mind: Jul 1998; 107, 427; Academic Research Library

Keefe, Rosanna ‘Unsolved Problems with Numbers: Reply to Smith’ in Mind: March 2003; 112, 446; Academic Research Library

Smith, Nicholas ‘Worldly Vagueness’ (Forthcoming)