The Sorites Paradox: a Behavioral Approach

The issues discussed in this chapter can be traced back to the Greek philosopher Eubulides of Miletus. He lived in the 4th century BCE, a contemporary of Aristotle whom he, according to Diogenes Laërtius, “was constantly attacking”. Eubulides belonged to what is known as the Megarian school of philosophy, founded by a pupil of Socrates by the name of Euclid(es). Besides his quarrels with Aristotle we know from Diogenes Laërtius that Eubulides was the target of an epigram referring to his “false arrogant speeches”, and that he “handed down a great many arguments in dialectics”. Most of the latter were trifle sophisms, of the kind ridiculed by Socrates in Plato’s Dialogues. For example, the Horned Man argument asks you to agree that ‘whatever you haven’t lost you have’, and points out that then you must have horns since you have not lost them. Two of the ‘arguments in dialectics’ ascribed to Eubulides, however, are among the most perplexing and solution-resistant puzzles in history. The first one is the Lying Man paradox, which demonstrates the impossibility of assigning a truth value to the statement ‘This statement is false’. The second of Eubulides’s ‘serious’ paradoxes is the Heap, and this is the one the present chapter is related to. This paradox can be stated as follows: (1) a single grain of sand does not form a heap, but many grains (say 1,000,000) do; (2) if one has a heap of sand, then it will remain a heap if one removes a single grain from it; (3) but by removing from a heap of sand one grain at a time sufficiently many times, one can eventually be left with too few grains to form a heap. This argument is traditionally referred to by the name ‘sorites’ (from the Greek σωρóς [soros] meaning ‘heap’), with the adjective ‘soritical’ used to indicate anything ‘sorites-related’. Thus, the Bald Man paradox which Diogenes Laërtius lists as yet another Eubulides’s argument in dialectics is a ‘soritical argument’, because it follows the logic of the sorites but applies it to the example of the number of hairs forming or not forming a full head of hair. This chapter is based on Dzhafarov and Dzhafarov (2010a-b) where we proposed to treat sorites as a behavioral issue, with ‘behavior’ broadly understood as the relationship between stimuli acting upon a system (the ‘system’ being a human observer, a digital scale, a set of rules, or anything whatever). We present here a sketch of this treatment, omitting some of the more delicate philosophical points.