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Is The Denial Of Bivalence Absurd?
“To be, or not to be: that is the question”
This essay will examine Williamson’s argument against the denial of bivalence (Williamson, 1994), the reply offered by Pelletier & Stainton (Pelletier & Stainton, 2003) and come to the conclusion that such a response is lacking.
Williamson (Williamson 1994 p. 187) formulates bivalence as:
(B) If u says that P, then either u is true or u is false.
Some authors do not distinguish between (B) and the Law of the Excluded Middle (the following version is that formulated by Haack (Haack 1974, p. 66):
(LEM) ‘P v ~P’ is a theorem of the system
Such a distinction is important to note in this paper as a non-standard logic could deny either one and accept the other. Another reason to keep the distinction in mind is that one of the main driving forces of Williamson’s argument is how we are to understand truth in the object language, not simply whether we may derive formal contradictions from the denial of (LEM) & Tarski’s (T) schema. Or in other words, Williamson is trying to show that the standard meta-language we use for bivalence is the best way of understanding how we talk about truth and falsity every day.
Formally, Williamson’s argument is as follows:
(1) ~ (P ∈ [T] v ~P ∈ [T])
(‘P’ is a member of the set of true sentences or ‘~P’ is a member of the set of true sentences)
(2) ~ (P v ~P)
(Result of switching the left hand side of the Tarski biconditional  with the right hand side)
(3) ~P & ~ ~P
(Via De Morgan’s Law)
The argument is straightforward enough. Assuming Tarski’s (T) schema holds, denying (B) leads to a formal contradiction.
It is at this point that we can turn to what Williamson’s opponents make of the argument offered. Pelletier & Stainton argue that the multivalentist need not agree with the Tarski biconditional, taking issue with one of the directions of the conditional. Assuming the following ‘choice negation’ account of negation:
The conditional that is as follows should, argues Pelletier & Stainton [Pelletier & Stainton 2003, pp. 374-375), be rejected:
P ⊃ T[‘P’]
~P ⊃ T[‘~P’]
This is because if P is neither true nor false, but indeterminate, then it certainly does not follow that ‘P’ is true (or false). If this is the case then Williamson’s argument fails, as it is the biconditional that gets the multivalentist into contradiction. If the biconditional is thus denied, the steps to contradiction simply do not follow.
Without further elaboration, it would seem that Williamson’s argument is therefore not a particularly convincing one, and if it were the case that Williamson did not further elaborate then Pelletier & Stainton would have seemingly left Williamson dead in the water. As it so happens, Williamson does elaborate on his reasons for formulating the argument as he does which, curiously (or perhaps conveniently), Pelletier & Stainton do not particularly acknowledge.
Williamson himself acknowledges that the weight of the argument is thrown on accepting Tarski’s (T) schema.
“The rationale…is simple. Given that an utterance says that TW is thin, what it takes for it to be true is just for TW to be thin, and what it takes for it to be false is for TW not to be thin…To put the condition for truth or falsity any higher or lower would be to misconceive the nature of truth or falsity” (Williamson 1994, p. 190)
It is this point that I want to discuss for the rest of the essay. The point that I take Williamson to be making in his essay is just that the multivalentist is distorting how we understand truth. When formalised, multivalentist seems to be prima facie plausible, at least in that we can see the motivation for bringing in the third truth value. If we then translate this formal structure into English, the third value becomes puzzling and difficult to comprehend. What precisely does it mean or stand for?
A supervaluationist could say that ‘TW is thin’ is either definitely true, definitely false or in the penumbra (neither true nor false) (Sainsbury 1995, p. 35). The question then arises about how we are to understand ‘definitely’, specifically what does it add to ‘true’ to make it different enough from Tarski’s conception? As Williamson notes,
“…what more could it take for an utterance to be definitely true than just for it to be true?”
(Williamson 1994, p. 195)
‘Definitely’ is presumably intended for more than just straightforward emphasis; logically, an emphatic true and a quietly murmured true are identical.
For me, this is where I think Williamson’s argument builds considerable force; by asking the opponent how exactly we are to understand truth if we reject Williamson’s conception of bivalence.
Let us look closer at what an utterance is, and what it is not. If I were to say to you ‘That elephant is about to charge’ and no elephant is within the vicinity to be singled out, I have failed to make an utterance. There could be circumstances when an utterance of those exact words could come out true (or false); if there were an elephant in the same field as us, for example. It is thus confusing when Pelletier & Stainton accuse Williamson of stating that sentences of this type are either true or false tout court (to borrow their phrase) (Pelletier & Stainton 2003, p. 380), when Williamson gives a similar example to mine specifically to make the point that sentence types are not true tout court (Williamson 1994, pp. 196-197)!
Williamson’s first premise for his argument is that bivalence is, in his own words;
“…explicitly restricted to occasions when someone uses an utterance to say that something is the case” (Williamson 1994, pp.187)
Is someone who says ‘TW is thin’ saying something, or are vague words just as empty as ‘That elephant is about to charge’? At least, the opponent might object, that if it is vague whether or not TW is thin, then surely it is vague whether what has been said is true. Our (T) schema is not very precise; it has an element of vagueness about it. It is that very feature, Williamson exclaims (Williamson 1994, p.192), which is precisely (no pun intended) why we need the (T) schema! Hoping for a precise meta-language is a ‘dream [that is] unrealisable’ (Williamson 1994, p. 192).
By arguing that we need a vague metalanguage that uses Tarski’s (T) schema for truth, Williamson then puts the burden of proof on the multivalentist who, it seems, provides little. The force of Williamson’s argument lies not in his formal proof, but his sustained attack after it which is directed at the multivalentist’s lack of an account of the semantics of the object-language. By assuming a precise meta-language, it cannot deal with the vague utterances of the object-language. Pelletier & Stainton’s paper, therefore, misses the point of Williamson’s. By only addressing his formal argument and ignoring the reasons why he chose to frame the argument with such assumptions, of course it will appear to be begging the question against the multivalentist!
Taken as a whole, I think Williamson puts forward a convincing case that, as it stands, the standard bivalence is the best conception of truth we have to hand. If the multivalentist is to convince us otherwise they will need to come up with a way of explaining the semantics of our object-language through the non-standard, precise meta-language they insist upon.
 ‘T[‘P’] ≡ P’ and ‘T[‘~P’] ≡ ~P’, respectively.
 That is, not ignoring important parts of the argument as it is presented.
Haack, S. (1974). Deviant Logic. London. Cambridge University Press
Pelletier, F. J. & Stainton, R. J. (2003) On ‘The Denial Of Bivalence Is Absurd’, Australasian Journal Of Philosophy, Vol. 1, No.3, pp. 369-382
Sainsbury, R. M. (1995) Paradoxes, Second Ed. Cambridge University Press
Williamson, T. (1994). Vagueness. Routledge
This is an essay I wrote for one of my logic courses. It isn’t perfect, but I’m still quite pleased with it.