The Sorites paradox (), sometimes known as the paradox of the heap, is a paradox that results from vague predicates. A typical formulation involves a heap of sand, from which grains are removed individually. With the assumption that removing a single grain does not cause a heap not to be considered a heap anymore, the paradox is to consider what happens when the process is repeated enough times that only one grain remains and if it is still a heap. If not, then the question asks when it changed from a heap to a non-heap.
Original formulation The word sorites (Ancient Greek: σωρείτης) derives from the Greek word for heap (Ancient Greek: σωρός). The paradox is so named because of its original characterization, attributed to Eubulides of Miletus. The paradox is as follows: consider a heap of sand from which grains are removed individually. One might construct the argument from the following premises:
1,000,000 grains of sand is a heap of sand (Premise 1) A heap of sand minus one grain is still a heap. (Premise 2) Repeated applications of premise 2 (each time starting with one fewer grain) eventually forces one to accept the conclusion that a heap may be composed of just one grain of sand. Read (1995) observes that "the argument is itself a heap, or sorites, of steps of modus ponens":
1,000,000 grains is a heap. If 1,000,000 grains is a heap then 999,999 grains is a heap. So 999,999 grains is a heap. If 999,999 grains is a heap then 999,998 grains is a heap. So 999,998 grains is a heap. If ... ... So 1 grain is a heap. One grain of sand is not considered to be a heap of sand. So the argument, although seeming valid and with plausible premises, has a false conclusion, which makes it a paradox, according to a popular (though not universally accepted) academic definition of "paradox".
Variations There are many variations of the sorites paradox, some of which allow consideration of the difference between "being" and "seeming", that is, between a question of fact and a question of perception; this may be seen to be relevant when the argument hinges on each change being "imperceptible". Another formulation is to start with a grain of sand, which is clearly not a heap, and then assume that adding a single grain of sand to something that is not a heap does not cause it to become a heap. Inductively, this process can be repeated as much as one wants without ever constructing a heap. A more natural formulation of this variant is to assume a set of colored chips exists such that two adjacent chips vary in color too little for human eyesight to be able to distinguish between them. Then by induction on this premise, humans would not be able to distinguish between any colors. The removal of one drop from the ocean, will not make it "not an ocean" (it is still an ocean), but since the volume of water in the ocean is finite, eventually, after enough removals, even a litre of water left is still an ocean. This paradox can be reconstructed for a variety of predicates, for example, with "tall", "rich", "old", "blue", "bald", and so on. The version about baldness, where it is argued that adding a single hair does not make a bald man no longer bald, is known as the "falakros", from the Greek for "bald" (φαλακρός). Bertrand Russell argued that all of natural language, even logical connectives, is vague; moreover, representations of propositions are vague.
General conditional sorites A formal generalization of the paradoxical sorites argument is as follows:
F a 1 {\displaystyle Fa_{1}} . If F a 1 {\displaystyle Fa_{1}} , then F a 2 {\displaystyle Fa_{2}} . If F a 2 {\displaystyle Fa_{2}} , then F a 3 {\displaystyle Fa_{3}} .
⋮ {\displaystyle \vdots }
If F a n − 1 {\displaystyle Fa_{n-1}} , then F a n {\displaystyle Fa_{n}} .
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