Introduction

Shaquille O'Neal is 7'1. He is tall. Tim Duncan is 7'0. He is also tall. Kevin Garnett is 6'11. He is tall. Amare Stoudemire is 6'10. He is tall. Rudy Gay is 6'9. He is tall. By the same token, LeBron James is 6'8, and he is also tall. Using this chain of reasoning, namely, that if I subtract an inch off of a given players height, it wouldn't make much of a difference because someone only one inch shorter than a tall person would, ostensibly, be tall himself. So as I use this chain of reasoning I can keep generating more premises, namely, that 6'7 Kobe Bryant is tall, as is the 6'6 Michael Jordan, the 6'5 Ray Allen, the 6'4 Jason Kidd, and so on, until I reach somebody like Nate Robinson, who is 5'9. Many people would hesitate to call Nate Robinson tall. I can reason even further to conclude that Earl Boykins, who is 5'5, and Muggsy Bogues, who is 5'3, are tall. Yet, these are players whom most, if not all, reasonable people would not classify as tall.

The reasoning above is an example of the sorites paradox, which is a paradox resulting from everyday vague predicates like "tall". This paradox can also be constructed using other vague predicates (e.g. the predicate "heap": 1,000,000 grains of sand is a heap of sand, so 999,999 grains is still a heap, as is 999,998, and so on, until I am forced to conclude that 1 grain of sand is still a heap). We can even reconstruct the "tall" version of the sorites paradox to work the opposite way--- If I reason that Muggsy Bogues, who is 5'3, is short, then a person who is only one inch taller would be considered short, and I would use this reasoning to work my way up to Shaquille O'Neal, who at 7'1, would be considered short. Here is the paradox in argument form:

Premise 1: Shaquille O'Neal, who is 7'1, is tall.

Premise 2: If Shaquille O'Neal is tall, then someone one inch shorter than Shaquille O'Neal is tall.

Premise 3: If someone one inch shorter than Shaquille O'Neal is tall, then somebody one inch shorter than this person is tall

.

.

.

.

Conclusion: Muggsy Bogues, who is 5'3, is tall.

This conclusion is paradoxical if we consider our everyday notions about terms like "tall" and "short" to be correct, or even meaningful. If we have a general consensus that Muggsy Bogues, who is 5'3, is short (which I believe we do agree on), then the conclusion which stems from the sorites argument, namely, that Muggsy Bogues must be tall, results in a contradiction, because Muggsy Bogues cannot have the properties "short" and "tall" at the same time.

Thus, we have three options if we are to avoid this paradox of vague predicates: We can deny the initial premise (that Shaquille O'Neal is tall); we can deny one of the other premises in the argument (that so-and-so, who is a particular height, is tall if a person an inch taller is tall) or reject the notion that the conclusion follows from the premises. I will discuss four different approaches to solving the paradox, including nihilist solution (rejecting the first premise), the epistemic solution (rejecting one of the other premises), the truth-value gap solution (of the same sort), and the "degrees of truth" solution (rejecting the validity of the argument). I will also discuss difficulties with accepting these solutions, and which solutions might actually be able to solve the sorites paradox.

The Nihilist Solution

One solution to the paradox is to reject the initial premise--- in this case, a nihilist solution would be to reject the claim that Shaquille O'Neal is tall. Furthermore, one might argue, a nihilist solution would result in the rendering of all vague predicates essentially meaningless. So, if we were to make to claim that Shaquille O'Neal was tall, a nihilist would claim that what we were saying was nonsense and meaningless. This seems like an extreme, yet plausible, solution. However, the nihilist solution has certain pragmatic difficulties.

The first difficulty with the solution is that, in our everyday language, it makes perfect sense to make the claim that Shaquille O'Neal is tall. When I make this claim, everybody knows what I am talking about, and the claim does have meaning. One might make the claim that O'Neal is nowhere near "tall", because the Statue of Liberty, at 151 feet, is tall. However, this claim is easily defeated when I point to something much taller than the Statue of Liberty-- for instance, the Sears Tower at 1,450 feet. And I can look at the Sears Tower, and point to the Tapei 101, the tallest building in the world. I can, furthermore, extend beyond this, and posit a hypothetically taller object. Eventually, my opponent will realize, this process could extend indefinitely, and thus for every "tall" thing one can bring up, there is always something that is, at least, conceivably taller.

What we generally mean when we say "Shaquille O'Neal is tall" is "Shaquille O'Neal is tall for a human being", so as to exclude other unbelievably tall objects. Almost no rational person would deny that Shaquille O'Neal is tall for a human--- thus, this statement does seem to have meaning. But, even with this more specific definition of "tall", we can still generate another sorites paradox:

Premise 1: Shaquille O'Neal is tall for a human

Premise 2: If Shaquille O'Neal is tall for a human, then somebody who is 6'11 is tall for a human

Premise 3: If somebody who is 6'11 is tall for a human, then somebody who is 6'10 is tall for a human

.

.

.

.

Conclusion: Muggsy Bogues, who is 5'3, is tall for a human

Thus, if we hope to save the use of our everyday language and semantics as meaningful, then we must reject the nihilist approach as too extreme. Of course, it could very well be the case that our notions of vague predicates are meaningless, but I wish at this point to restructure the aim of this project: To solve the sorites paradox while retaining a reasonably similar usage of our everyday language.

The Epistemic Solution

The epistemic solution involves rejecting one of the other premises. For example, given the argument:

Premise 1: Shaquille O'Neal is tall for a human

Premise 2: If Shaquille O'Neal is tall for a human, then somebody who is 6'11 is tall for a human

Premise 3: If somebody who is 6'11 is tall for a human, then somebody who is 6'10 is tall for a human

.

.

.

.

Conclusion: Muggsy Bogues, who is 5'3, is tall for a human

We can reject one of the premises, which will provide us a way out of the paradox. For example, we can reject the premise (which will eventually result from the above argument), that if 6'3 Steve Nash is tall, then 6'2 Monta Ellis is tall. Or we can draw the line further down, and reject the premise that if 6'0 Chris Paul is tall, then 5'11 Damon Stoudamire is tall.

The epistemic solution involves creating a hard-line distinction, separating humans into two groups: "Tall" and "not-Tall". There would seem to be some cut-off point, whereby if someone who is (n'n) is tall, then it would not follow that somebody who is (n'n -1) is tall. Thus, if this distinction were to be in place, it would provide a barrier to the reasoning that led us to conclude in the first place that Muggsy Bogues was tall. If this distinction were in place, then the group of tall people would have stopped way before we got to Muggsy Bogues.

However, this solution also runs into much difficulty. First, the distinction would seem to be entirely arbitrary, and dependent on a particular person's conception of "tall". For example, I might want stipulate that all people who are 6'0 and above (which is one inch taller than me), are tall. My sister, who is 5'5, might want to stipulate that all people 5'6 and above are tall. Similarly, Michael Jordan might want to stipulate that ll people 6'7 and above are tall. All of these arbitrary lines are perfectly reasonable, relative to each person.

One could argue against arbitrariness by stipulating that we should consider those above the "average height for humans" as tall, and those below this height are short, and thus a fixed point would solve the problem. However, this stipulation also runs into (in addition to practical) hypothetical problems.

Firstly, the average height for humans is always changing. Humans today might be, on average, taller than humans were, say, 10,000 years ago. Thus, it would seem, somebody who would have been "tall" 10,000 years ago would be "short" now. But (if I can guess correctly) nobody wants to accept that someone who is tall can become short without even shrinking!

Secondly, if we are to reject one of the premises, then we must do so with good reason. We must give an account why, for example, it is better to reject the premise that if 6'3 Steve Nash is tall, then 6'2 Monta Ellis is tall, rather than the premise that if 7'1 Shaquille O'Neal is tall, then 7'0 Tim Duncan is tall. Is there a rational reason why we should reject the former premise rather than the latter?

Suppose, for science-fiction's sake, the case of a shrinking man. Suppose this shrinking man was Shaquille O'Neal, who shrank 1 inch per month, from a beginning height of 7'1. If we are to reject one of the premises, then we must also accept that there is an exact spatiotemporal location whereby Shaquille O'Neal transforms from somebody who is "tall" to somebody who is "not tall". Where is this point? 6'4? 6'3? 5'11? 5'5? Do we have good reason to privilege one of these heights over another? If we say we do, then we must give an account for which exact moment in space-time this transformation happens, and why it is rational to choose this moment rather than another. It seems that none of us can rationally privilege one of these moments over another, and, thus, it seems, the epistemic view runs into a roadblock.

The Truth-Value Gap Solution

One of the assumptions of the epistemic solution is that it posits only two possibilities for the height of humans: either one belongs to the group of people above the proposed line of distinction, and thus is tall; or, one belongs to the group below the line of distinction, and thus is not tall.

However, in our everyday language, it seems that the use of terms such as "tall" and "short" refer to extreme examples of height. For example, most everyone would agree that Shaquille O'Neal is tall, because it appears so obviously to be the case, as they would agree that Muggsy Bogues is not tall, because it appears so obviously to be the case.

Per this example, it seems that there are people who are obviously tall, and people who are obviously not tall. But what would we say about the people who are in between these two groups? It would seem to follow, for example, that somebody like myself, who is 5'11, is neither "tall" nor "not tall". Rather, I am somewhere in between the two extremes. The truth-value of my height would thus be "undefined".

The truth-value gap solution rejects one or more of the premises by changing their truth-value from "true" to "undefined". For example, if we consider the premise:

If somebody who is 6'0 is tall for a human, then somebody who is 5'11 is tall for a human

it seems that it would not be true, because it is not obviously so, but by the same token it would not be false, either, because it is not obviously so. But, by designating a premise like this one as "undefined", the advocate of the truth-value solution would stop the sorites chain of reasoning, for it seems that "non-true" premises cannot lead to a true conclusion.

However, the truth-value gap solution seems to fall victim to the same type of objection that befell the epistemic solution. That is to say, rather than having to draw one line of distinction (as in the epistemic view), the advocate of the truth-value gap solution would have to draw two lines of distinction. He would have to draw the line between the "non-tall" group and the "undefined" group at the lower bound, and between the "undefined" group and the "tall" group at the upper bound. Since any line of distinction seems to be a merely arbitrary designation, the truth-value gap solution does not seem to provide a strong enough response to the sorites argument.

The Degree of Truth Solution

The final solution I will discuss is the "degree of truth" solution, which takes an unconventional approach towards the notions of truth and falsity, and seeks to invalidate the sorites argument.

One would not instantly see the problem with the argument. According to our inference rules for formal logic, it seems impossible to deny the reasonably simple syllogism: A, A--->B, B--->C, C--->D... Y--->Z : . Z.

However, advocates of the degree of truth solution argue that the sentences of our everyday language (e.g. Marques Camp, who is 5'11, is tall) are unlike propositions in formal logic. Whereas propositions of logic, by inference, are either true or false (i.e. the law of the excluded middle, P v ~P), it seems that sentences of our everyday language are true only to certain degrees.

For a clear example, let us consider the vague predicate "bald". We can agree, at least in a broad, general sense, that baldness is defined as "having no hairs on one's head". Yet, most people would consider my father (who has some, but not a lot, of hairs on his head) to be bald. We all know that it is not exactly the case that my father has no hairs on his head, but there is a significant consensus that he fits the rough parameters of "baldness", as he has less hairs on his head than most people do. Thus, when we say sentences of the sort "Marques' father is bald", we really seem to be saying, effectively, that "Marques' father is reasonably close to the strict definition of baldness, closer than most humans".

In the particular case of baldness, it seems quite reasonable to delineate the line for "baldness" as such: "baldness" is defined as having zero hairs on your head; "non-baldness" is defined as having one or more hairs on your head. Thus, it seems, we are back in the world of the strict binary--- The bald vs. the non-bald. It seems, therefore, that the vagueness has been cleared up.

Yet, at the same time, we still want to be able to say, with meaning, that my father is bald. After all, there is a significant consensus. With the degree of truth solution, we can say, for example, that the statement "Marques' father is bald" is true to some degree (say, .95 ), but not entirely true, because he still has a few hairs on his head. We can still make the claim that my mother is bald, although that statement would not hold the same degree of truth as the previous one. My mom has a full head of brown hair, yet there are many people out in the world with much more hair than she. Thus, the truth that "Marques' mother is bald" would in some sense be true, but only to a degree, of, say .39. This degree of truth would seem to indicate that she is closer to the paradigm of non-baldness than to baldness, yet, in some sense, she is more "bald" than people who have more hair than she does. It would seem, therefore, that the degree of truth for "baldness" depends on how many hairs you have on your head; as the number goes up, the degree of truth goes down, and as the number decreases, the degree of truth goes up.

With regard to the "tallness" argument, I can make the claim that Steve Nash, who is 6'3, is tall. I can say the degree of truth for this claim is about .70, as he seems to be closer to the paradigm for human tallness than the paradigm for shortness. If I repeat the line of reasoning for the argument, I can say that Monta Ellis, who is 6'2, is tall. This premise would be true, but slightly less so than the previous premise, to, say, a degree of .69. As I continue the line of reasoning, the shorter the person gets, the less true the premises about their tallness will be. So it would seem to be the case, that, in some sense, Muggsy Bogues is tall, but only to a degree of, for example, .58 (these figures are arbitrary, yet, whichever scale is used, the basic notion remains the same). The statement would not be true to the same degree as the statement "Shaquille O'Neal is tall", and thus, the conclusion of the argument would not follow from the premises.

The sorites paradox began by positing that if Shaquille O'Neal is tall, and we reason that Muggsy Bogues is also tall, we are saying that these two men possess the same property of "tallness", and the truth of these statements are true to the same degree. However, the degree of truth solution rejects this notion and instead redefines the notion of what it means to be "true", as well as make the claim that a less-true conclusion could not validly follow from more-true premises.

But consider the following statement: "It is raining". According to the degree of truth solution , this statement would have a degree of truth of .50, while its negation, "It is not raining" would also have a degree of truth of .50. If we want the truth of a conjunction of partially true statements, it seems reasonable to conclude that the truth of the conjunction is determined by each of its component parts. Thus, the following two conjunctions,

It is not raining and it is not raining

It is raining and it is not raining

would seem to have the same degree of truth. But it seems impossible that a contradiction can have the same degree of truth as a perfectly reasonable statement. Thus, there seems to be a problem for the degree of truth solution.

However, one can answer this objection by pointing to the false dichotomy of "raining" and "not raining". We seem to falsely have the idea that it must be the case that either it is "raining" or it is "not raining", much in the same way we did when we believed that it must be the case that somebody is either "tall" or "not tall".

It would help to compare this kind of statement to the example of "baldness". Suppose we drew a distinction: If there are zero raindrops falling from the sky, then it is not raining; if there are one or more raindrops falling from the sky, then it is raining. The former would have a degree of truth (in the context of the statement, "It is not raining") of 1, whereas the degrees of truth for "raining" would vary, according to how many raindrops were falling. Thus, when we make the claim that "It is not raining" when there are technically a very small number of insignificant raindrops falling, much like with the bald person, we really mean "It is reasonably closer to the paradigm of "non-raininess" than it is to "raininess", to an x degree". Thus, as more raindrops start to fall, the statement, "It is not raining" would gradually become less true, and 'It is raining" would become more true.

It seems, with a degree of truth language, that the conjunction "It is not raining and it is not raining" (given the assumption that we are predicating the same location at the same time) could mean different things in different contexts. It would not be the case that it is, in all cases, a conjunction of two statements whose truth value is each .50. For example, if there were a very insignificant amount of raindrops falling at a given time, and the degree of truth for the statement "It is not raining" is, say, .95, then the degree of truth for the conjunction, "It is not raining and it is not raining" would not have the same degree of truth as the conjunction "It is not raining and it is raining". The relationship between the degrees of truth for the first conjunction would be (.95 and .95) (to which the degree of truth for the conjunction, it seems, is .95), while the relationship between the degrees of truth for the second (ostensibly contradictory) conjunction would be (.95 and .05). Furthermore, in this latter case, the degrees of truth for each of the components of the second conjunction are inversely proportional to each other, and their sum will always add up to a degree of truth of 1.

Thus, with the degree of truth theory, there seems to be no logical contradiction involved in stating, "It is not raining and it is raining", because, in a sense, we are asserting this with regards to all claims about rain. Every time we make a claim, for example, that "It is not raining, and this is true to a degree of .95", what we are implicitly asserting is "It is not raining, and this is true to a degree of .95 and it is raining, and this is true to a degree of .05". This particular theory seems to have the strength of eliminating all contradictions involving vague predicates (unless, of course, one attempts to assert that it is not raining with a degree of truth of .95 and it is raining with a .95 degree of truth--- this still appears to be a manifest contradiction). Thus, another strength of the degree of truth theory.

A Solution to the Paradox

The degree of truth solution appears to be the closest solution that keeps aim with the intentions of this project. It allows us to keep our everyday use of language meaningful (and clarifying what we really mean when we say that Shaquille O'Neal is tall), while at the same time challenging our implicit assumptions of truth, falsity, and contradictions which led to the paradox in the first place. The contradiction resulting from the conjunction of a sorites argument and our everyday use of language results when our language isn't strict enough. We are so outraged at the conclusion that Muggsy Bogues possesses the property of tallness in the same way that Shaquille O'Neal possesses the property of tallness. But the degree of truth solution provides us a way to clarify our everyday language. Of course, using the sorites argument and our inference rules for formal logic, it is easy enough to validly conclude that Muggsy Bogues is tall; however, the language of the argument does not provide us a manner in which we can describe the "tallness" of Muggsy Bogues in the context of the "tallness" of Shaquille O'Neal. By just appealing to the language of logic, it would appear that these two statements mean the exact same thing. Adopting the degree of truth solution provides just this context; it does not deny that Muggsy Bogues is, in some way, tall (for that matter, any object with any measurable height could technically be considered tall), but rather, shows how Muggsy Bogues is not "tall" in the exact same way, with the exact same degree of truth, as Shaquille O'Neal. This is perhaps the closest analogy to what we really mean when we use vague predicates, so, to avoid the sorites paradox, we should know exactly what we mean when we make certain claims using our everyday language.