When Is a Heap Not a Heap? The Sorites Paradox and ‘Fuzzy Logic’

By Keith Tidman

 

Imagine you are looking at a ‘heap’ of wheat comprising some several million grains and just one grain is removed. Surely you would agree with everyone that afterward you are still staring at a heap. And that the onlookers were right to continue concluding ‘the heap’ remains reality if another grain were to be removed — and then another and another. But as the pile shrinks, the situation eventually gets trickier.

 

If grains continue to be removed one at a time, in incremental fashion, when does the heap no longer qualify, in the minds of the onlookers, as a heap? Which numbered grain makes the difference between a heap of wheat and not a heap of wheat? 

 

Arguably we face the same conundrum if we were to reverse the situation: starting with zero grains of wheat, then incrementally adding one grain at a time, one after the other (n + 1, n + 2 ...). In that case, which numbered grain causes the accumulating grains of wheat to transition into a heap? Put another way, what are the borderlines between true and not true as to pronouncing there’s a heap?

 

What we’re describing here is called the Sorites paradox, invented by the fourth-century BC Athenian Eubulides, a philosopher of the Megarian school, named after Euclides of Megara, one of the pupils of Socrates. The school, or group, is famous for paradoxes like this one. ‘Sorites’, by the way, derives not from a particular person, but from the Greek word soros, meaning ‘heap’ or ‘pile’. The focus here being on the boundary of ‘being a heap’ or ‘not being a heap’, which is indistinct when single grains are either added or removed. The paradox is deceptive in appearing simple, even simplistic, yet, any number of critically important real-world applications attest to its decided significance. 

 

A particularly provocative case in point, exemplifying the central incrementalism of the Sorites paradox, is concerns deciding when a fetus transitions into a person. Across the milestones of conception, birth, and infancy, the fetus-cum-person acquires increasing physical and cognitive complexity and sophistication, occurring in successively tiny changes. Involving not just the number of features, but of course also the particular type of features (that is, qualitative factors). Leading us to ask, what are the borderlines between true and not true as to pronouncing there’s a person. As we know, this example of gradualism has led to highly consequential medical, legal, constitutional, and ethical implications being heatedly and tirelessly debated in public forums. 

 

Likewise, with regard to this same Sorites-like incrementalism, we might assess which ‘grain-by-grain’ change rises to the level of a ‘human being’ close to the end of a life — when, let’s say, deep dementia increasingly ravages aspects of a person’s consciousness, identity, and rationalism, greatly impacting awareness. Or, say, when some other devasting health event results in gradually nearing brain death, and alternative decisions hover perilously over how much to intervene medically, given best-in-practice efforts at a prognosis and taking into account the patient’s and family’s humanity, dignity, and socially recognised rights.

 

Ot take the stepwise development of ‘megacomplex artificial intelligence’. Again, involving consideration of not just ‘how many features’ (n + 1 or n - 1), but also ‘which features’, the latter entailing qualifiable features. The discussion has stirred intense debate over the race for intellectual competitiveness, prompting hyperbolic public alarms about ‘existential risks’ to humanity and civilisation. The machine equivalence of human neurophysiology is speculated to transition, over years of gradual optimisation (and down the road, even self-optimisation), into human-like consciousness, awareness, and cognition. Leading us to ask, where are the borderlines between true and not true as to pronouncing it has consciousness and greater-than-human intelligence? 

 

In the three examples of Sorites ‘grain-by-grain’ incrementalism above — start of life, end of life, and artificial general intelligence — words like ‘human’, ‘consciousness’, ‘perception’, ‘sentience’, and ‘person’ provide grist for neuroscientists, philosophers of mind, ethicists, and AI technologists to work with, until the desired threshold is reached. The limitations of natural language, even in circumstances mainly governed by the prescribed rules of logic and mathematics, might not make it any easier to concretely describe these crystalising concepts.

 

Given the nebulousness of terms like personhood and consciousness, which tend to bob up and down in natural languages like English, bivalent logic — where a statement is either true or false, but not both or in-between — may be insufficient. The Achilles’ heel is that the meaning of these kinds of terms may obscure truth as we struggle to define them. Whereas classical logic says there either is or is not a heap, with no shades in the middle, there’s something called fuzzy logic that scraps bivalence.

 

Fuzzy logic recognises there are both large and subtle gradations between categorically true and categorically false. There’s a continuum, where statements can be partially true and partially false, while also shifting in their truth value. A state of becoming, one might say. A line may thus be drawn between concepts that lie on such continuums. Accordingly, as individual grains of wheat are removed, the heap becomes, in tiny increments, less and less a heap — arriving at a threshold where people may reasonably concur it’s no longer a heap.

 

That tipping point is key, for vagueness isn’t just a matter of logic, it’s also a matter of knowledge and understanding (a matter of epistemology). In particular, what do we know, with what degree of certainty and uncertainty do we know it, when do we know it, and when does what we know really matter? Also, how do we use natural language to capture all the functionality of that language? Despite the gradations of true and false that we just talked about in confirming or refuting a heap, realistically the addition or removal of just one grain does in fact tip whether it’s a heap, even if we’re not aware which grain it was. Just one grain, that is, ought to be enough in measuring ‘heapness’, even if it’s hard to recognise where that threshold is.

 

Another situation involves the moral incrementalism of decisions and actions: what are the borderlines between true and not true as to pronouncing that a decision or action is moral? An important case is when we regard or disregard the moral effects of our actions. Such as, environmentally, on the welfare of other species sharing this planet, or concerning the effects on the larger ecosystem in ways that exacerbate the extreme outcomes of climate change.

 

Judgments as to the merits of actions are not ethically bivalent, either — by which I mean they do not tidily split between being decidedly good or decidedly bad, leaving out any middle ground. Rather, according to fuzzy logic, judgments allow for ethical incrementalism between what’s unconditionally good at one extreme and what’s unconditionally bad at the other extreme. Life doesn’t work quite so cleanly, of course. As we discussed earlier, the process entails switching out from standard logic to allow for imprecise concepts, and to accommodate the ground between two distant outliers.

 

Oblique concepts such as ‘good versus bad’, ‘being human’, ‘consciousness’, ‘moral’, ‘standards’ — and, yes, ‘heap’ — have very little basis from which to derive exact meanings. A classic example of such imprecision is voiced by science’s uncertainty principle: that is, we cannot know both the speed and location of a particle with simultaneously equal accuracy. As our knowledge of one factor increases in precision, knowledge of the other decreases in precision.

 

The assertion that ‘there is a heap’ becomes less true the more we take grains away from a heap, and becomes increasingly true the more we add grains. Finding the borderlines between true and not true in the sorts of consequential pronouncements above is key. And so, regardless of the paradox’s ancient provenance, the gradualism of the Sorites metaphor underscores its value in making everyday determinations between truth and falsity.