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Logical paradox from vague predicates

The sorites paradox (/s//oʊ//'//r//aɪ//t//iː//z/; 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 to become a non heap, the paradox is to consider what happens when the process is repeated enough times that only one grain remains: is it still a heap? If not, when did it change from a heap to a non heap?

Quote|Then tension between small changes and big consequences gives rise to the sorites Paradox There are many variations [some of which allow] consideration of the difference between being (a question of fact) and seeming (a question of perception).

The continuum fallacy (also known as the fallacy of the beard, line drawing fallacy, or decision point fallacy) is an informal fallacy related to the sorites paradox. Both fallacies cause one to erroneously reject a vague claim simply because it is not as precise as one would like it to be. Vagueness alone does not necessarily imply invalidity. The fallacy is the argument that two states or conditions cannot be considered distinct (or do not exist at all) because between them there exists a continuum of states. Strictly, the sorites paradox refers to situations where there are many discrete states (classically between 1 and 1,000,000 grains of sand, hence 1,000,000 possible states), while the continuum fallacy refers to situations where there is (or appears to be) a continuum of states, such as temperature. Whether any continua exist in the physical world is the classic question of atomism, and while both Newtonian physics and quantum physics model the world as continuous, there are some proposals in quantum gravity, such as loop quantum gravity, that suggest that notions of continuous length do not apply at the Planck length, and thus what appear to be continua may simply be as yet undistinguishable discrete states. For the purpose of the continuum fallacy, one assumes that there is in fact a continuum, though this is generally a minor distinction: in general, any argument against the sorites paradox can also be used against the continuum fallacy. One argument against the fallacy is based on the simple counterexample: there do exist bald people and people who are not bald. Another argument is that for each degree of change in states, the degree of the condition changes slightly, and these slight changes build up to shift the state from one category to another. For example, perhaps the addition of a grain of rice causes the total group of rice to be "slightly more" of a heap, and enough slight changes will certify the group's heap status – see fuzzy logic.

One may object to the first premise by denying that 1000000 grains of sand makes a heap. But 1000000 is just an arbitrary large number, and the argument will apply with any such number. So the response must deny outright that there are such things as heaps. Peter Unger defends this solution. However, A. J. Ayer repudiated it when presented with it by Unger: "If we regard everything as being composed of atoms, and think of Unger as consisting not of cells but of the atoms which compose the cells, then, as David Wiggins has pointed out to me, a similar argument could be used to prove that Unger, so far from being non existent, is identical with everything that there is. We have only to substitute for the premise that the subtraction of one atom from Unger's body never makes any difference to his existence the premise that the addition of one atom to it never makes any difference either."

A common first response to the paradox is to term any set of grains that has more than a certain number of grains in it a heap. If one were to define the "fixed boundary" at 10000 grains then one would claim that for fewer than 10000, it is not a heap; for 10000 or more, then it is a heap. Collins argues that such solutions are unsatisfactory as there seems little significance to the difference between 9999 grains and 10000 grains. The boundary, wherever it may be set, remains arbitrary, and so its precision is misleading. It is objectionable on both philosophical and linguistic grounds: the former on account of its arbitrariness and the latter on the ground that it is simply not how natural language is used.

Timothy Williamson and Roy Sorensen claim that there are fixed boundaries but that they are necessarily unknowable.

Let be a classical valuation defined on every atomic sentence of the language , and let be the number of distinct atomic sentences in Then for every sentence , at most distinct classical valuations can exist. A supervaluation is a function from sentences to truth values such that, a sentence is super true (i. e. ) if and only if for every classical valuation ; likewise for super false. Otherwise, is undefined—i. e. exactly when there are two classical valuations and such that and
For example, let be the formal translation of "Pegasus likes licorice". Then there are exactly two classical valuations and on , viz. and So is neither super true nor super false. However, the tautology is evaluated to by every classical valuation; it is hence super true. Similarly, the formalization of the above heap proposition is neither super true nor super false, but is super true.

Another method is to use a multi valued logic. In this context, the problem is with the principle of bivalence: the sand is either a heap or is not a heap, without any shades of gray. Instead of two logical states, heap and not heap, a three value system can be used, for example heap, indeterminate and not heap. A response to this proposed solution is that three valued systems do not truly resolve the paradox as there is still a dividing line between heap and indeterminate and also between indeterminate and not heap. The third truth value can be understood either as a truth value gap or as a truth value glut. Alternatively, fuzzy logic offers a continuous spectrum of logical states represented in the unit interval of real numbers [0,1]—it is a many valued logic with infinitely many truth values, and thus the sand transitions gradually from "definitely heap" to "definitely not heap", with shades in the intermediate region. Fuzzy hedges are used to divide the continuum into regions corresponding to classes like definitely heap, mostly heap, partly heap, slightly heap, and not heap. Though the problem remains of where these borders occur; e. g. at what number of grains sand starts being 'definitely' a heap.

Another method, introduced by Raffman, is to use hysteresis, that is, knowledge of what the collection of sand started as. Equivalent amounts of sand may be termed heaps or not based on how they got there. If a large heap (indisputably described as a heap) is diminished slowly, it preserves its "heap status" to a point, even as the actual amount of sand is reduced to a smaller number of grains. For example, 500 grains is a pile and 1000 grains is a heap. There will be an overlap for these states. So if one is reducing it from a heap to a pile, it is a heap going down until 750. At that point, one would stop calling it a heap and start calling it a pile. But if one replaces one grain, it would not instantly turn back into a heap. When going up it would remain a pile until 900 grains. The numbers picked are arbitrary; the point is, that the same amount can be either a heap or a pile depending on what it was before the change. A common use of hysteresis would be the thermostat for air conditioning: the AC is set at 77 °F and it then cools the air to just below 77 °F, but does not activate again instantly when the air warms to 77.001 °F—it waits until almost 78 °F, to prevent instant change of state over and over again.