‘Puzzles’, ‘word games’, ‘logical anomalies’, whatever we call them, they perplex us and challenge our familiar patterns of reasoning. One of these puzzles, among many others, originated from the mind of an ancient Megarian logician, Eubulides of Miletus, and endures to the modern day.¹ Its name, ‘sorites’, can be traced to the Greek word soros, meaning ‘heap.’ The answer to whether one grain of sand ‘is a heap’ or ‘is not a heap’ seems quite simple: it is not a heap. However, as we add grains to the one, at what future point does the non-heap become a heap? Our decision is fraught with uncertainty. Are the objects or the language we are using to describe them vague?

In academic philosophy, the ancient Greek puzzle has gained the status of a paradox, as philosophers apply stoic and modern logic to these propositions considered to have vague predicates. The current debate has developed quite serious and wide-ranging implications, such as whether sorites issues provide adequate grounds for abandoning our standard ontology (or our understanding of what really exists), ² and (germinating into another discipline) whether vagueness in the language of legal rules can generate disagreement as to whether there are right answers to questions of law. ³

Several unique solutions to the paradox have been proposed, yet all suffer from specific inadequacies that might, upon further reflection, disappear in the event that we endeavour to produce a synthesis. When we say that we are attempting to derive a synthesis, we seek to combine elements of two solutions for the sake of creating a new (and hopefully, though not necessarily, a better) one. The paradox of the sorites, or ‘the heap’, appeals to us because it challenges our assumption that we may categorically describe clear cases and their negations, yet fail to distinguish the borderline ones. I will demonstrate that some combinations of solutions are easy fits while others are extremely weak and awkward as attempts at resolving the paradox. Nonetheless, the process of synthesis produces three noteworthy combinations.

The sorites paradox will be discussed in terms of categories, their negations and the extent to which our reasoning about them succeeds or fails to accord with the laws of logic. For example, to call something a ‘heap’, ‘bald’ or ‘short’ is to differentiate it from apparent cases of a thing that is a ‘non-heap’, ‘not bald’ or ‘not short.’ In keeping with formal logic and the law of non-contradiction, we might gainsay that a thing could evince both its positive instance and its negation, ‘P and not P’ or ‘P and ~P.’ According to the more debatable law of excluded middle, a thing must either display its positive property or the property’s negation (e.g. bald or not bald). Prima facie, denying the law of excluded middle has an appeal that rejection of the law of non-contradiction lacks. This claim originates with the intuition that while a thing cannot be both itself and its opposite [intuitively and by the law of non-contradiction], when we attempt to designate borderline cases, it might not clearly match the criteria for either category. Where do we specify that a thing has effectively crossed the categorical boundary between its positive and negative instance, e.g. from bald to not bald?

In more formal terms, we might state the Sorites as follows:

Since we cannot determine the point at which subtracting one grain of sand converts the heap to a non-heap, it seems inevitable that we must either accept the vagueness of the term ‘heap’ or slide back to the position that anything resembling sand [one grain or more] constitutes a heap. The paradox seems to revolve around the issue of whether there is an absolute boundary between categories and their negations, such that x elements makes a thing a member and x –1 makes it a non-member. In short, we would want to know whether there exists a grain of sand, a piece of hair or a one-millimetre [or other] difference between positive cases of something being a heap, bald or short and it being a non-heap, not bald or not short.

One attempt at a solution has already been hinted at in the process of defining the paradox. The notion that the linguistic term ‘heap’ is vague begs the question of whether or not further elaboration of our existing language, or the inclusion of a meta-language, could remove the vagueness. We might call this strategy the ‘semantic approach.’ While mathematics is precise in its form and operations, language invariably suffers in this respect because of its open texture; ⁴ the impreciseness of its meaning and of the rules that govern it invariably force the speaker’s hermeneutic license. Even though we may add infinitely many descriptive adjectives before a word, each with increasing levels of preciseness, it still proves impossible to escape a certain level of distortion inherent in practical language’s construction. Premised on this notion of the imperfection of practical language is Tarski’s view that we should embrace a meta-language encompassing greater logical rigor. [Tarski 1943: 603] However, it is uncertain whether the meta-language may prove useful for any purpose other than to avoid problems of self-reference.

While some attempts at amalgamating solutions to the paradox produce natural unions, others result in less rewarding and often strange combinations. For example, joining the supervaluationist account with the epistemic solution would only create obfuscation since aspects of both are clearly antithetical to each other: the former holds that there is no definite boundary between a category and its negation, unless it is arbitrarily stipulated, while the latter states that such a boundary exists regardless of any stipulation. The same would hold true of an attempted pairing of the degrees of truth theory with the epistemic view. To suggest that the law of excluded middle should be rejected in favour of assigning degrees of truth admits that the boundary cannot be sharp, though it can be known to what extent it is vague; the epistemic position, however, defends a conception of the boundary that it is neither vague nor known. On the epistemic account, there are definite boundaries, e.g. one hair, one grain of sand or one-millimetre differences, between categories and their negations, yet it is impossible for us to know where these boundaries lie. The ‘supervaluationist’, ‘degrees of truth’ and ‘rejecting the categorical’ accounts, on the other hand, synthesise easily with the ‘semantic approach’ because each of them reveals to some extent that the sorites owes its paradoxical quality to the limitations of natural language.

In the supervaluationist model, boundaries between positive categories and their negations blur into an area of uncertainty. Supervaluationists do not intend that vagueness should correspond to an object as a property does, but instead believe that vagueness resides in language. Vague terms correspond to things in such a way that they either do or could partake in the penumbra, an area where we could say something is or is not the case without designating a truth-value. [Sainsbury 1995: 33] More specifically, in the case of the heap, for any specific number of grains of sand x which is in the penumbra, ‘x grains of sand make a heap’ and ‘x grains do not make a heap’ are both propositions which are neither true nor false. But ‘there is some number of grains that make a heap where that number minus one grain does not make a heap’ is a proposition that comes out as true. According to the supervaluationist’s reasoning, vagueness is avoided only when human beings arbitrarily stipulate meaning that serves to sharpen the category from its negation in the penumbral area. For example, a person may be clearly short in the positive extension, clearly not short in the negative extension, or either short or not short in the penumbra depending on where one freely decides to make a sharp distinction.

To its merit, the supervaluationist position seems to accord best with our intuition that absolute boundaries—like one millimetre, one hair or one grain differences between categories and their negations—do not exist except by stipulation. However, when put in concert with the semantic approach, the strong criticism that the approach levels at language, and specifically at any attempt to precisely delineate meaning, seems to backfire on the supervaluationist. According to the semantic approach, natural language is inherently defective to such a degree that even descriptively elaborate or scientifically exact terminology may fail to formulate an effective distinction or absolute boundary between a category and its negation. Does the supervaluationist’s attempt at a sharpening, or precisification, in the penumbra likewise prove impossible? ⁵

We can only say that a statement is true, e.g. ‘x is a heap’, if it is true for all sharpenings, false if it is false for all sharpenings. Statements in the penumbral area, on the other hand, are neither true nor false until we delineate a clearer, or less ambiguous, meaning. Still, since we can make sharpenings wherever we like, the truth or falsity of a statement falling within the penumbra will only be so relative to a sharpening, which any further sharpening may controvert. If we decide that a panel of the sunset having predominant tones of purple is yellow by way of a sharpening then we are completely justified in stating that any panel which is yellower than it, though still predominantly purple, is yellow. In addition, higher order vagueness has a way of infecting the tripartite nature of the model of the supervaluationist. We may ask whether the division between the penumbra and the positive and negative extensions is itself vague, so that a higher order uncertainty spreads outwards and entreats us to stipulate whether the sky is ‘certainly yellow’ or ‘certainly not yellow’ in order to avoid further vagueness. [Sainsbury 1995: 39]

Laurence Goldstein addresses this particularly unfavourable result of applying the sorites to events of differentiating colour gradation. He claims that the problem is based on a mistake assumption about the revisability of our judgements: