Paradoxes and Their Resolutions

1.    What’s a heap?

The Sorites paradox is not a paradox, in the strict sense of the term, but a question. The question is sometimes put in a sophistical manner, so as to make it seem paradoxical. But it can be put in a more straightforward manner, in which case it is seen to be simple though not without importance. The term sorites is Latin, derived from the Greek sōros, meaning heap.

One way to express the Sorites paradox is: What is a ‘heap’ (of pebbles, say)? Or, how many pebbles (say) constitute a ‘heap’ of them? The obvious answer is there must surely be at least one pebble. If you have no pebbles, you do not have a heap, but a non-heap. But is one pebble enough? The obvious answer is: no, you need at least two pebbles to make a heap, since heap is a collective term, and one that additionally suggests that the pebbles are stacked one on top of the other (and you cannot stack a non-pebble on a pebble or a pebble on a non-pebble). A single pebble logically counts as a non-heap; heap is intrinsically plural.

Formulated like that, the question is not very problematic. But if it is formulated as follows, it becomes more complicated. If we have many pebbles (say, 100) piled up, we obviously have a heap. What happens if we remove one pebble, do we still have a heap? Yes, 99 make up a heap. What happens if we remove one more pebble, do we still have a heap? Yes, 98 make a heap. And so on, till we come to low numbers, at which stage the sophist wonders whether two pebbles constitute a heap, then one pebble, then no pebble. At this stage, the question appears paradoxical, rather artificially: should we conclude that no pebble makes a heap, or one pebble makes a heap? The sophist thinks we might; the real logician knows we cannot.

2.    The use of vague terms

Clearly, the problem here, insofar as there is one, has to do with the exact formulation of initially vague terms. If we do not at the outset step back and think about the exact intent of a vague term, of this sort (a term suggesting quantity, as it happens) or any other, we may find ourselves in difficulty further on in our discourse. So, we need to stop and think before use of such terms, and preempt any difficulties they might eventually create. In the present case, as we have seen, the term (heap) is inherently plural (and so inapplicable to less than two pebbles). A sophist prefers to complicate the matter, so as to put human reason and knowledge in doubt, as is his wont; but the matter is really simple enough.

In some cases, to be sure, there is a conventional element to the definition of a vague term. This can be illustrated with reference to another version of the so-called Sorites paradox: As of how many hairs is a man not bald? Obviously, a man with no hair at all is definitely bald. But would a man with only one hair, or a very small number of hairs, not be considered bald in ordinary discourse? Perhaps so – but only conventionally. If we define bald strictly, it implies zero hair; but if we define it loosely, it may include cases involving an arbitrary, though preferably small, number of hairs, determined by convention – for examples, five or twenty hairs[1]. Clearly, if we want to avoid confusion, nothing stops us from referring to this broadened sense of ‘bald’ as, more accurately, ‘bald or almost bald’.

This issue of vagueness is nothing special – it is not limited to terms giving rise to the Sorites paradox. For instance, a relative term like ‘small’ (or its relative, ‘large’) is inherently vague and can only be used with precision in specific situations by means of a conventional quantity. Again, when dealing with continua, we may need to set arbitrary dividing lines. For instance, there is no objective dividing line between one color and an adjacent color in the spectrum, and it may be necessary in some circumstances for us to imagine one (e.g. for legal or other practical purposes[2]). Conventional distinctions are part of human thought; but, it is important to stress, they are not all of human thought. There are always objective elements behind conventional ones. For example, the dividing line between blue and green would be somewhere in between what we see as clearly blue and what we see as clearly green – it would never be far on one side or the other, and much less between green and yellow or between blue and indigo or still further afield.

Returning to our alleged paradox, a few more comments are in order. As already stated, there is nothing paradoxical in the concept of a heap, if it is properly defined. Most simply, a heap can be defined as material items placed on top of each other in whatever way, implying that there must be two or more items. A more complex definition would assume that a heap must be pyramidal, i.e. requires at least four such items (three for the base and one on top); but this puts us in no difficulty, as it can be referred to more specifically as a pyramidal heap. Similarly, baldness stricto sensu refers to no hair at all; it is nevertheless applied to small quantities of hair, though only roughly-speaking.

There are many vague terms of this sort in our common discourse[3], some of which may require a conventional definition for pragmatic reasons. For example, the term ‘crowd’ might be taken to refer to a gathering of three or more people, on the basis of the popular saying that “two’s company, but three’s a crowd;” or we might, say in software used by the police to monitor large groups of people, opt for a larger minimum (say, 50 or 500), set arbitrarily as cause for alarm. The word ‘mob’ might be preferred when the latter crowd goes on a rampage.

But in any case, there is no real logical problem in such unspecific quantitative expressions. They do not constitute a defect in ordinary language, requiring us to construct an “ideal language” where all terms have single precise meanings[4]. Much less do they call for treatment by means of abstruse symbolic logics (dearly loved by many modern logicians). On the contrary, they demonstrate the versatility and flexibility of ordinary thought and speech; and they witness the fact that much of our linguistic communication has non-verbal undercurrents, which we mostly comprehend very ably. Most of our daily use of vague terms involves no need for more clear-cut definition. They are used to suggest things approximately, and are not intended as precise and true affirmations. If the need for precision and truth does arise, it is then addressed in a way that preserves consistency (by explicit convention if necessary).

3.    Reasoning with vague terms

To be sure, vague terms can be perilous if we try to reason with them. But vague terms are often used without involving them in argumentative processes Moreover, reasoning with vague terms is not always invalid – there are contexts where the vagueness does not inhibit a reliable conclusion.

In categorical (or hypothetical) syllogism, the rule regarding vague terms (or theses) is the following: The middle item (term or thesis, as the case may be) cannot be vague, because it would provide no guarantee that its intent is the same in the two premises. If the middle item is vague, we cannot be sure of overlap and the conclusion is invalid. On the other hand, the major and minor items can be vague without affecting the argument, and that in all four figures. There is, however, an exception to this rule – when the middle term is vague, but not so vague that overlap is not guaranteed, a valid argument can still be made.

The latter is evident when we consider the following two arguments in the third figure, in which the middle term ensures overlap even though neither premise is universal. The expressions ‘most’ (more than half) and ‘few’ (less than half) are vague, insofar as they do not specify exact numbers. But notice the particularity (as against majority or minority) in the conclusion – i.e. the increased vagueness of the conclusion.

Most M are P

and Most M are S;

therefore, Some S are P

Few M are P (which implies that Most M are not P)

and Most M are S;

therefore, Some S are not P

In apodosis, the rules regarding vague terms or theses are the following:

Modus ponens:

If A is B, then C is D, and A is B (affirming the antecedent); then, C is D (consequent is affirmed).

The antecedent and the minor premise cannot be vague; else, the conclusion is invalid. However, the consequent could be vague, and the conclusion would still be valid (though also vague).

Modus tollens:

If A is B, then C is D, and C is not D (denying the consequent); then, A is not B (antecedent is denied).

The consequent and the minor premise cannot be vague; else, the conclusion is invalid. However, the antecedent could be vague, and the conclusion would still be valid (though also vague).

Here again, in both moods, exception is conceivable, if we know that the major and minor premises overlap, even if we don’t know precisely how much they overlap.

Similar rules may be formulated for other varieties of argument, such as dilemma or a fortiori.

The people who claim that vague terms are inherently paradoxical are dishonestly nitpicking, motivated by the desire to impress themselves or others by their ability to find and resolve (contrived) paradoxes, or (worse still) to demonstrate that human knowledge is inevitably paradoxical and therefore futile. Clearly, just as it is dishonest to call a single pebble or no pebble a heap, it is dishonest to call a person with one or more hairs bald[5]. If you indulge in such contradictions-in-terms to start with, you are bound to end up with paradoxes. People who behave thus are not real logicians but sophists. They spin and fabricate – they are not interested in finding ways to true knowledge.

4.    Making up fake paradoxes

The original formulations of both the conundrums described above, relating to a heap and to baldness, are attributed to Eubulides of Miletus (fl. 4^th Cent. BCE), the Megarian logician who also gave us the Liar paradox[6]. He was a student of Euclid, a teacher of Diodorus Cronus, and a contemporary and rival of Aristotle. These puzzles were perhaps not initially presented as paradoxes, but rather as illustrations of a question (viz. where should we draw the line?). This possibly reflected a dawning consciousness that there are vague terms that may require arbitrary definition in some circumstances. As above shown, this problem is solved easily enough. However, later thinkers tried to make a mountain out of a molehill, and presented the issue in the form of an argument-chain (or sorites, where the conclusion of the preceding argument serves as a premise for the next).

Thus, the bald man puzzle became, in its positive formulation: Surely, a man with one hair is about as bald as one with no hair; and if a man with only one hair can still be called bald, then a man with two hairs qualifies as bald; and if a man with two hairs is bald, then a man with three hairs is bald; etc.; therefore, a man with a thousand hairs can still be considered bald (paradox). Alternatively, the argument could be stated in negative form: if a man with a thousand hairs is not to be regarded as bald, then one with 999 hairs is not to be so regarded either; and if 999 does not qualify, then 998 does not either; and so on… whence, a man with one hair only is not bald; therefore, a man with no hair is not bald (paradox).

Clearly, these arguments are forced – they involve some very doubtful and misleading premises. In the positive version, the false premise is that ‘a man with no hair can be called bald’ implies ‘a man with one hair can be called bald’. In the negative version, the false premise is that ‘a man with one hair cannot be called bald’ implies ‘a man with no hair cannot be called bald’. It is the same false claim of implication (in contraposite form). The paradox is created by the refusal to admit that ‘bald’, strictly-speaking, means ‘hairless’; which refusal is not based on honest logical insight, but on a willful act of illogic. Similarly, in the case of a heap, the trick consists in implying that a single pebble or even no pebble may be considered as a heap.

The inventors of these arguments do not pause and rationally reflect on the underlying issue (i.e. where is the dividing line?) before engaging in an apparent inference process, but instead attempt to bamboozle us into a paradoxical corner. The argument-chain proposed just serves as a smokescreen to conceal the crucial false claim being put over. They are the logical equivalents of pyramid sales, each sale supporting the next without solid foundation. People who are taken in by the tricky move are simply bad logicians, if not shamelessly dishonest. They then pretentiously weave massive and intricate theories around this phenomenon, untroubled by the initial error or lie in their discourse.

To those who argue that a single hair or pebble hardly makes any difference, I would suggest that they make the following simple physical experiment: take an accurate balance with the same weight on both sides, then add a single hair or pebble to one side and watch the scales tip! To those still unconvinced by this, because they dogmatically believe that logic is a matter of fancy and convention, I would suggest (tongue-in-cheek, more or less) that they place, under the heavier scale, a plunger connected to an explosive device strapped to their nose, and then watch Reality blow up in their face! That argument, I think, might finally convince them, if they survived.

As regards Eubulides, we might note in passing his other paradoxes. The most significant is of course the Liar paradox, which as I have shown in detail elsewhere[7] is exceptionally powerful due to the variety of difficulties it involves (but still quite resolvable). Another three paradoxes[8] deal with equivocations in the term ‘know’, specifically with failure to immediately recognize someone one normally recognizes immediately (such as a close relative or old friend), when the latter is masked or has been away too long or is not looked at attentively enough. Another, the Horns paradox claims that what you have not lost must be in your possession; whence, if you have not lost horns, you must have horns. This apodosis involves a false major premise, since something one has ‘never had’ may equally (as well as something one ‘still has’) be characterized as ‘not lost’ – so the consequent does not follow upon the antecedent; therefore, if one has not lost horns, one cannot be assumed to have horns (since one may well never have had any).

From this short list, it can be seen that Eubulides’ queries all give rise to some sort of reflection on logic – reflections on vague terms, on conventional definitions, on equivocations, on term-negations, on self-reference, and various other difficulties that may arise in human discourse. It would be wrong, I think, to assume the motive of such queries to have been teasing or obfuscation (although, to be sure, later skeptics did use such conundrums malignantly, as already mentioned). Rather, I’d say, they served as springboards for earnest reflections and discussions on logic – because it seems unlikely that they were formulated without any attempt to solve the problems they engendered. In some cases, valid explanations or resolutions were no doubt proposed (even though they may not have come down to us), while in other cases the difficulties may have seemed insurmountable. In other words, I doubt that Eubulides was merely sophistical – I’d class him rather as a serious logician.

From Topics (published only online so far).