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An essay for an online course I took in Philosophy of Language via the Pathways to Philosophy site, Sheffield, U.K.
                   
Gordon Fisher

'A one ounce elephant is a small elephant. Adding one ounce to a small
elephant cannot make it not-small. Therefore, all elephants are small
elephants.' - Give a logical analysis of this
argument.

* * * * * * * *

First, I will give one of many possible reformulations of the ordinary language statement of the argument quoted above.

To begin with, I will assume that "not not-small" can be replaced with "not not small", and that "not not small" can be replaced with small. That is, I will assume that a standard use of double negation has been applied. I make this maneuver in order to avoid dealing with questions about whether or not the term "not-small" should or can be replaced with some term not synonymous with "large" (whatever "synonymous" may be taken to mean).

Reformulation 1: An elephant that weighs one ounce is a small elephant. If a small elephant gains one ounce of [body] weight, then the elephant is [still] a small elephant [not not-small]. Therefore all elephants are small.

I will ignore any questions involving time and tense suggested by my bracketed uses of "still" and "before they are born". Also, as indicated by my bracketed use of "body" and my use of "gains" instead of "added", I will assume that a change in weight of one ounce does not occur because something external is added to an elephant (an insect, a bird, whatever).

In order to use a version of standard first order propositional logic, I set

I(n) = "n is a positive integer"

E(x) = "x is an elephant"

W(x,k) = "x weighs k ounces"

S(x) = "x is small"

 

I want to try to show:

C. ([x)(E(x) S(x))

That is, I want to try to prove that it is true that every elephant is small. I will assert 3 premises.

P1. ([x)((E(x) and W(x,1)) S(x))

That is, I will assume that it is true that any elephant that weighs one ounce is small.

P2. ([n)([x)((I(n) and W(x,n) and S(x)) (W(x,n+1) S(x))

That is, I will assume that it is true that if an elephant that weighs n ounces is small, then if that elephant weighs n+1 ounces, then that elephant is small.

P3. (([x)(E(x) and W(x,1) S(x)) and (([n)([x)(I(n) and W(x,n) and S(x) W(x,n+1) S(x)) ([x)(E(x) S(x))

That is (I hope I have the parentheses right), I will postulate that it is true that mathematical induction can be applied here, using the indicated induction hypothesis. That is, I will assume that if all elephants that weigh 1 ounce are small, and also if all elephants that weigh n ounces are small then all elephants that weigh n+1 ounces are small, then one can conclude that all elephants are small.

So if P1-P3 are true, then I can conclude

C. ([x)(E(x) S(x))

That is, "all elephants are small" is true.

I will now try to show that the premise P1 cannot be true, so that the argument above is not valid.

I will work with my ordinary language formulation of P1, rather than the formal version. Suppose I declare that there is nothing which is small, or can properly be declared to be small, without being compared to something else, and that on account of this the statement cannot be true or not true. This is something like saying the term "small" is vague, in a sense of "vague" used by some philosophers when they discuss sorites paradoxes. As I understand it, in such a discussion, a term is said to be vague if when it is used in a certain kind of statement, the statement may be neither true nor not true, in some sense of the term "true". It is said that a vague term somehow possesses what are called borderline cases. However, I will say that that the using the term "small" makes certain statements incomplete, meaning that these statements are not eligible to be assigned a value of true or not true. This is different from saying that these statements are neither true nor not true, since this suggests that these statements are eligible to be to be assigned a value true or not true, but such an assignment can never be made.

Thus P1 can never be true, because it is not eligible to be true. Therefore, the argument I gave can never be valid.

In the light of this, in order to use "small" in premises which are eligible to be assigned a value true or not true, I will reformulate the proposed argument again.

Reformulation 2. An elephant that weighs one ounce is a small elephant when its weight is compared with my present weight. If a small elephant gains one ounce of [body] weight, then the elephant is [still] a small elephant [not not-small] when its weight is compared with my present weight. Therefore all elephants are small when their weights are compared with my present weight.

I now consider that argument I gave, except that I will replace S(x) with T(x), where T(x) stands for "x is small when its weight is compared to my present weight". I ask now if the premise ([x)((E(x) and W(x,1)) T(x)) can be true. That is, I will ask whether or not the statement "if x is an elephant that weighs 1 ounce, then x is small when its weight compared with my present weight" can be true. My present weight is k ounces for a certain k, and I will take "compared with" to be mean I will judge whether or not 1 < k, and I will say that indeed 1 < k, and that this entitles me to say that this premise can be true or not true, and in fact is true in a certain sense of "true".

Now I ask whether or not the premise

([n)([x)((I(n) and W(x,n) and T(x)) (W(x,n+1) T(x))

can be true or not true. That is, I will ask whether or not it can be true that for every positive integer n, if an elephant that weighs n ounces is small if its weight is compared with my present weight, then if that elephant weighs n+1 ounces, that elephant is small when its weight is compared with my present weight. It appears that this can be true if I say or assume it is true, in some sense of "true". However, because of my experiences with elephants and what I have heard and read about elephants, I will in fact say that the premise can be true or not true in some sense of "true", and in fact that the premise is not true. So this argument can never be valid.

I now consider another reformulation.

Reformulation 3. An elephant that weighs one ounce is a small elephant when its weight is compared with the present weight of our sun. If a small elephant gains one ounce of [body] weight, then the elephant is [still] a small elephant [not not-small] when its weight is compared with the present weight of our sun. Therefore all elephants are small when their weights are compared with the present weight of our sun.

Let U(x) stand for "x is small when its weight is compared with the present weight of our sun". Because of my experiences with and beliefs about elephants and about our sun, I will declare that no matter what an elephant weighs, its weight will be less than the weight of our sun. Thus I declare that, as before, the premise P1 with S(x) replaced with U(x) can be true or not true, and is in fact true. I will also declare that as a matter of fact, there is an integer k such that the weight of an elephant will never exceed k ounces, and that k ounces is less than the present weight of our sun in ounces.

In the light of this, I could amend the second premise by speaking of all positive integers up to a certain value k rather than speaking of all integers, and declare that the amended version can be either true or not true, and is in fact true. Thus, in this amended argument, the first two premises will be true. Now I ask about the third premise, which amounts to asserting that a postulate of mathematical induction can be used to lead to a conclusion. It appears that I will have to alter P3 with S(x) replaced with U(x) so that the induction hypothesis contains the altered version of P2. I will take it that this can be done, and that then the amended argument works to show that ([x)(E(x) U(x)) is true. That is, the amended argument will prove that for any positive integer n less than some integer k, an elephant which weighs n ounces is small, provided I take it to be true that mathematical induction can be used in dealing

Conclusion. Whether or not a person takes the argument as originally formulated to be valid or not depends on all sorts of auxiliary assumptions the person has to make in order to use a standard logic to determine validity. That is, the argument as originally formulated is vague, in some sense of the term "vague". This suggests that I try to relate this sense of "vague" with other uses, such as those used in discussing vague predicates.  However, I wont try that here.

I will close my analysis of the sizes of elephants now, except for two remarks that occurred to me while I was writing this essay.

Remark 1. Consider a circular disc D in a plane. In doing mathematics, one can say of a point of the plane that it is an interior point of D, according to some definition of the term "interior point", and one can say of a point of the plane that it is a boundary point of D according to some definition of the term "boundary point". These definitions are customarily chosen so that no point of the plane can be said to be both an interior and a boundary point of D, without introducing a logical contradiction. There will be points of the plane which are neither interior nor boundary points of D. Consider the statement "the points of a plane can be partitioned into two sets of points, the points which are inside a circular disk D in a plane and the points which are outside that circular disk D in that plane". A mathematician would object that this statement is vague, since there are boundary points of A, and there are common uses of the terms "inside" and "outside" which will make it such that one cant decide whether the boundary points are inside or outside of D. This goes some way toward explaining why mathematicians make certain definitions of "interior point" and "boundary point" the way they do, using such auxiliary terms as "neighborhoods in a plane" or "a topology for a (or the) plane". It also goes some way toward explain why in this connection they find it useful to introduce such terms as "limit point" and "open set" and "closed set" in the ways they do. I once fiddled around with a version of so-called fuzzy logic to see how to talk about fuzzy boundaries, of the sort which might be used by physicists or others in constructing mathematical models.

Remark 2. A field is defined by mathematicians to be a set consisting of elements which can be added, subtracted, multiplied and divided in the way people can learn to do with numbers (and/or numerals) they use. There are many kinds of fields. Many people learn to use a field of real numbers, or at least a field of rational numbers. It has been shown that in addition to the unique standard field of real numbers commonly used by many people, one can construct non-standard fields which contain the standard real numbers. All of these fields will be linearly ordered, that is, if r and s are any two elements of these fields and r doesnt equal s, then r < s or r >> s. A primary difference between this standard and these non-standard fields is that the standard field is archimedean and the non-standard fields are non-archimedean. A field is archimedean if and only if for any two elements r and s of the field such that 0 < r < s, there is a positive integer n such that nr > s. In the case of the standard real numbers, this requirement can be loosely (vaguely?) described by saying that the field is archimedean if and only if when r and s are positive numbers and r is less than s, one can get a number greater than s by adding r to itself n times, where n is some positive integer. In a non-standard field which contains the real number field, this will not be so. That is, there will be elements r and s of the field such that 0 < r < s, and one can never get a number greater than s by adding r to itself any number of times. Such an element r is called an infinitesimal. It has struck me that there may be some useful analogy between the notion of a field being archimedean or non-archimedean, and the contrast I discussed above between a comparison made using the term "small when its weight is compared with my present weight", and a comparison made using the term "small when its weight is compared with the present weight of the sun".

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