An Inquiry into Linguistic Vagueness
This essay looks at vagueness, primarily through the Sorites paradox. A non-vague term is one which is sharply defined in the sense that it neatly divides objects into those contained in the term's extension and those contained in the extension of its negation. A vague term is one whose correct definition permits the possibility of borderline cases. These are cases where it is not determined whether or not the term applies or fails to apply. Where there is vagueness there is genuine uncertainty concerning the application of expressions to certain objects. [1] Issues concerning vagueness to be looked at upon include the correct way to resolve the Sorites paradox, and the question of whether classical logic should be modified to countenance degrees of truth, corresponding to degrees of vagueness. I shall firstly examine various aspects of two different examples of the Sorites paradox in an attempt to elucidate why the paradox "raises a tension between classical logic and mathematical reasoning and the 'vague' predicates of natural language". [2] Discussion of the first example introduces fuzzy logic as an approach to resolving the paradox. After considering problems associated with this approach the discussion touches upon complications particular to that example. I subsequently consider a hypothetical situation which would do away with vagueness as exemplified in the second example and consequently raise the fundamental causal connection between limited sensory powers and vagueness. Briefly, I consider a 'merit' of vagueness before returning to fuzzy logic and demonstrating, in contrast to the introductory example, its more agreeable application to a Sorites-like situation. I intend to provide some reasons why this is so. Finally, in light of the essay's inquiries, I would like to offer a pragmatic perspective on vagueness adherent to our usage of language and dubiously insusceptible to the Sorites paradox.
In his brief discussion on the problem of vagueness in 'Logic: A very short introduction', Graham Priest introduces a system of fuzzy logic by demonstrating its application to an example of the Sorites paradox involving a five-year-old child named Jack and his transition from childhood to adulthood. The truth values of the relevant truth functions he uses are:
After demonstrating its application, he acknowledges that "with vagueness, nothing is straightforward" by asking "at what point in Jack's growing up does he cease to be 100% a child; that is, at what point does, "Jack is a child" change from having the value of exactly 1, to a value below 1?" and concludes by saying that "any place one chooses to draw this line would seem to be arbitrary" [3]
If one is going to determine 'how much of a child' or 'how much of an adult' a person is by employing a multi-valued logic, one apparent basis for truth values based on their age would a correlation between the sequence of truth values and a succession of elementary time intervals. I am not sure that such an approach would be possible though if time is continuous due to the indeterminacy of a truth value below 1 after the first elementary time interval has elapsed.
Furthermore, although theoretically appropriate, such a logical system is not a simple reflection of the way we use language; when referring to someone 4-years-old as a child and someone 5-years-old as a child, we tend not to think of the former as being any less of a child than the latter.
Irrespective of such problems, the classification of a person as being either a child or an adult is not merely based on time but also on a plethora of anatomical and psychological aspects. Any correlation between the development of such aspects and time can at best only be approximate. As a consequence of this complication, the development of a logical system to deal with such cases may be too abstract or complex for current consideration.
It is significant to note also that Priest has not specified a 'suitable level of acceptability', as he defines on pages 75 and 76, in his example. At what point of our sequence should the premises become unacceptable or modus ponens become invalid and how would selection of such a point be correlated with our perspective of actual situation being modeled? [4]
It is evident that the application of a binary opposite classification system to situations involving gradation is problematic, due to the indeterminacy 'surrounding the limits of application of predicates involved'. [5]
Another standard example of the Sorites paradox originally known as the falakros puzzle [6] concerns application of the predicate 'is bald'. As the definition of baldness is considerably simpler than that of biological childhood, an inquiry into fundamental vagueness-related problems should be afforded a clearer perspective. The aspect of baldness pertaining to men I consider is the general quantity of hairs atop the scalp of the man being described. A second aspect is the spatial location of the hair, though for the purposes of my discussion I need not consider this; at any rate I will assume that each additional hair is placed adjacent to the developing patch. The selection of a sole criterion (i.e. general quantity of hair) promotes the possibility of dealing with such an example by the introduction of a definitive classification standard; for the sake of discussion let us say that a man with 1000 hairs or less atop his scalp is bald and a man with an amount of hairs atop his scalp greater than 1000 is not bald. Some may retort that this seems arbitrary, though to do so would be a misunderstanding.
The introduction of such a border would be for semantic or definitional purposes and is not intended to be a reflection of the deep structure of reality; besides, in a reality of inherent gradation it is doubtful whether there does objectively subsist a definite border, particularly between two extremes of which at least one is indefinite (what is the maximum amount of hairs of a hirsute man?). Hence I believe that it would be entirely acceptable to incorporate such a numerical bipartition into our definition of baldness and implement it as a standard.
Given this definition of baldness, to determine whether a man is bald or not, all that needs to be done is a count of the number of hairs atop his scalp. If he has 1000 hairs or less he is bald. If he has more than 1000 hairs, he is not bald. Despite the precision of such a definition, it is unviable. Although in accordance with this definition we can assert that a man with 1000 hairs atop his scalp is by the given definition bald and a man with 1001 hairs atop his scalp is not bald, upon an indiscriminate empirical observation it is not possible to differentiate between quantities of 1000 and 1001 hairs. As a result, when looking at a man with 1001 hairs atop his scalp (although we are not aware of this fact), we are just as likely to describe the man as being bald than as not being bald, clearly a consequence of our inability to attach numerical quantities to our indiscriminate empirical observations. As initially mentioned, we could perform a count of the number of hairs, although this is absurdly impractical and clearly unacceptable for an adjective that needs to be expeditiously employed in ordinary discourse. If we possessed extraordinary sensory abilities whereby we could extract quantitative information from general sensory perceptions, vagueness as being discussed would cease to be a problem; though as such abilities are not in our possession, vagueness as discussed in this section is an inevitable phenomenon.
To accentuate this point, consider the following example, that although is not strictly a member of the Sorties family, is not entirely unrelated. A group of three people does make a 'trio', irrespective of whether they are children or adults, bald or hirsute. Let stand for, 'x people make a trio'. The assertion 'three people make a trio' is an analytic truism; therefore if one were to construct the following Sorites-style chain of reasoning, in which they must necessarily incorporate the proposition, the argument would obviously be invalid.
As a brief aside, consider a word such as 'group'. It is not at all clear what minimum quantity of people make a group of people, yet this lack of precision does not inhibit purposeful employment of the word in normal discourse. Indeed, it is precisely the malleability of this term that is a factor in its significance. Although its use is regulated to a degree (i.e. one person or two people do not make a group), for the most part we are allowed to employ the term when referring to an unspecified amount of people, without concern as to this lack of specificity and borderline cases. So we have an example where the vagueness associated with a word is not unfavourable.
Another lurking problem, as touched upon earlier, accompanying many, if not all, examples of the Sorites paradox is one concerning boundary. In regards to an example such as the 'paradox of the heap', there exists a specific lower bound of zero (or one) grains of wheat but there is no identifiable upper bound of x grains of wheat and the existence of an upper bound is not beyond doubt. This actuality prevents a plausible and intuitive attempt to define some type of border as the median of the lower and upper bounds. This point could straightforwardly be ascertained by successively adding one constituent to what is initially the lower bound and subtracting one constituent from what is initially the upper bound until they converge to a point. Such a point, if it were possible to ascertain, in some situations may reasonably be considered as "the point in a sorites series where the relevant predicate ceases to apply and its negation does" [7]
I would now like to consider an ideal (or least problematic) Sorites-like case with which to apply a many-valued logic as introduced at the beginning of this essay. The situation involves the emptying (or filling) of a cylindrical container with water. I shall provide some reasons why this case is ideal and contrast it with problematic cases considered earlier.
Firstly, with regards to the sequence of emptying a container, there exist definite upper and lower boundaries respectively corresponding to maximum and minimum volumes. Secondly, the two binary opposite adjectives 'full' and 'empty' in their absolute sense respectively refer to these bounds whereas the binary opposite nouns 'child' and 'adult' do not have such absolute senses. Thirdly, we are able to base our determination of intermediate cases on a well-defined gauge. Fourthly 'full' and 'empty' are not relative terms whereas terms such as young and old are; when does a young person become old? These factors are probably why it seems acceptable to say of a glass that it is 35% percent full yet unusual to say of a human being that they are 35% an adult or 35% tall.
Our container has a volume of 1 litre and we apportion it into 10 sections, each of volume 0.1 litres (theoretically these portions will be infinitesimal).
Let be the sentence 'is full with a volume of litre/s. has the truth value 0.9, indicating that the glass is being emptied. Also, as it is the case that the glass is being emptied, we can make a definite contextual selection as to whether we are to employ a predicate or its negation so that when the truth value of we can decide to express the state of the container as 'half empty', as opposed to indecision as to whether it is 'half full' or 'half empty'.Finally, with regards to vagueness in natural language; I believe that it is best to accept vagueness as an ineliminable feature of a natural language replete with empirically derived concepts and adopt a pragmatic approach. As a response to the Sorites paradox, fuzzy logic, although theoretically attractive, is unsuitable as a system that reflects the basic manner in which we expeditiously employ language. We do not intuitively attach a truth value to our assertion that a certain person is tall and furthermore we only possess a limited range of adverbs to express degrees. A basic approach retains elements of classical logic and semantics yet also incorporates some formally unconventional notions. Consider the vague complementary adjectival predicates 'is short' and 'is tall'. "Tall is a vague term because a man who is 1.8 metres in height is neither clearly tall nor clearly non-tall". [8] This prompts the "rejection of a simple dichotomy between tall and short" [9] and our usage of a third predicate such as 'is of average height' to describe intermediate heights. This third predicate is not to be used for the formation of a 3-valued logic; rather it serves to extend the binary opposite system to one of ternary opposites. By ternary opposites I mean that each person is to be classified as short, average or tall with the disjunction being exclusive. It is practically admissible to disregard resultant higher-order vagueness issues by establishing overlapping short-average and average-tall regions. The boundaries of both overlapping regions will contain a reasonable portion of transitional heights, a portion chosen so as to assure adaptability to discrepant height estimate-to-description judgements. So 1.8 metres is a height within this region whereas 2 meters is not. Actually, it would be more appropriate to consider human perceptions of heights rather than numerical measurements of heights; so the human perception of a height corresponding to 1.8 metres is a height within this region. "If I regard you as a borderline case of 'tall man', I can assert the hedged claim 'you are tall or of average height'". [10] So if someone falls within the average-tall region then a sentence asserting that they are of either height would be correct. This is not to be confused with some type of truth glut though and does not permit the concurrent assertion of both heights. It only accommodates the fact that either predicate may be applied to describe that person's height; so we have some type of contextual basis for truth values ). Of course, consistency on a context-by-context basis is necessary. If the heights of John and Mary are contained within the average-tall region and upon concurrent observation it is evident that John is taller than Mary, if Mary is described as tall then so must John. Also, for example, the short-average region upper boundary, denoted as point b (see diagram), is an established border point that would break a Sorites chain commencing from a minimum height.
Discussion of the first two examples in this essay raised several significant points, least of all being that each example of the Sorites paradox warrants its own scientific investigation into cognitive limitations. It was argued that "vagueness is an inevitable feature of any language used by creatureswith our sensory powers and limitations and the Sorites paradox is an inevitable consequence of this vagueness". [11] A term that is perfectly precise would generate no borderline cases, and although this is often presented as a theoretical ideal it is extremely unclear any learnable, speaking language could begin to meet it. Gradation is an inherent aspect of reality; many of our intuitive and empirically derived linguistic concepts involve binary opposites. Consequently, we are able to distinguish between two extremes of a continuum, but unable to establish a definite point of division within intermediate regions as our empirical and numerical evaluations diverge. Fuzzy logic, as looked at in this essay, is an attractive though inconclusive response. It was subsequently applied to a situation less problematic than the introductory example and some significant points were offered as to why it is less problematic. I have implied that vagueness is not necessarily an imperfection of natural language and acceptance of it as an ineliminable aspect of natural language prompted the suggestion of a pragmatic attitude towards vagueness.
References
[1] Burns, Linda, Vagueness, (Dordrecht: Kluwer Academic Publishers, 1991), pg 3
[2] Blackburn, Simon, Oxford Dictionary of Philosophy (Kent: Oxford University Press, 1994), [Entry on Sorties Paradox].
[3] Priest, Graham, Logic. A Very Short Introduction, (New York: Oxford University Press, 2000), pg 77
[4] Priest, Graham, Logic. A Very Short Introduction, (New York: Oxford University Press, 2000),Chapter 10
[5] Stanford Encyclopedia of Philosophy entry on 'Sorites Paradox' accessed at http://plato.stanford.edu/entries/sorites-paradox/
[6] Ibid
[7] Ibid
[8] Burns, pg 2
[9] Priest, Graham, An Introduction to Non-Classical Logic, (Cambridge: Cambridge University Press, 2001), pg 213
[10] Stanford Encyclopedia of Philosophy entry on 'Vagueness' accessed at http://plato.stanford.edu/entries/vaguness/
[11] Burns, pg 84