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Sorites -- A New Analysis Based on Context
Aaron Sloman
Last updated: 6 Jan 2010
Installed: 6 Jan 2010 (Based on a message to Philos-L 4th Sep 2009)

From Wikipedia: Paradox of the heap
    The name "Sorites" derives from the Greek word for heap. The paradox
    is so-named because of its original characterization, attributed to
    Eubulides of Miletus. The paradox goes as follows: consider a heap
    of sand from which grains are individually removed. One might
    construct the argument, using premises, as follows:

        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 less
    grain), eventually forces one to accept the conclusion that a heap
    may be composed of just one grain of sand (and consequently, if one
    grain of sand is still a heap, then removing that one grain of sand
    to leave no grains at all still leaves a heap of sand).

I have been trying to develop an analysis of this and related puzzles, sometimes assumed to be due to vagueness, which I think are best analysed as application of higher order functions to specific goals and/or predicates/relationships which have to be induced from the context.

E.g. someone asked to get a heap of sand or stones to hold down a tarpaulin in a wind doesn't have to bother about how many grains or stones count as a heap because it is clear that the number required is determined by the need to cope with the wind.

I think there are many such examples of systematic context-dependence in words used in ordinary language (including good, better, efficient, dangerous, near, far, in front of, and many more).

For the analysis of 'better' see
A. Sloman, How to derive "Better" from "is", Jan, 1969, American Phil. Quarterly, vol 6, pp. 43-52,

Even 'conscious of' has this feature if the conditions for the truth of 'X is conscious of Y', and the implications depend systematically on what X and Y are (and other things).

For more on the polymorphism of ordinary uses of 'conscious' see this PDF presentation:
Why the "hard" problem of consciousness is easy and the "easy" problem hard. (And how to make progress)

All those words illustrate a general point which can perhaps be seen as closely related to Grice's maxims of communication, namely that in human communication (and thinking) compositional semantics is very often (perhaps always?) extended by context sensitivity.

I.e. the meaning of larger structure composed of smaller structures depends (recursively) not only on the smaller structures and how they are arranged, but also on some aspects of the context, which often can be inferred, but can sometimes lead to miscommunication.

This is related to the inadequacy of Frege's notion of predicates as functions, discussed in
Aaron Sloman, Functions and Rogators, 1965, in
Formal Systems and Recursive Functions: Proceedings of the Eighth Logic Colloquium Oxford, July 1963,
Ed. J. N. Crossley and M. A. E. Dummett, North-Holland Publishing Co, Amsterdam, pp. 156--175,
This paper extends Frege's concept of a function to "rogators", which are like functions in that they take arguments and produce results, but are unlike functions in that their results can depend on the state of the world, in addition to which arguments they are applied to.

If predicates were functions (or in Bertrand Russell's terminology, "Propositional functions" the truth or falsity of sentences produced by applying them would depend only on what the argument is, and not on the state of the world.

By allowing predicates (and relation expressions) to express rogators rather than functions, as part of a more general suggestion that context can make a difference at all levels of sentence structure, we produce a more plausible theory of how language works. I suspect that this is what Frege meant all along, since I am sure that he did not believe that the truth or falsity of most propositions expressed in ordinary language depended only on what was being referred to and what was being said about it, and not also on how things are in the world. Likewise, the value of a sentence like "The height of Fred" depends not only on who Fred is but what the state of the world is, which will change over time. So "the height of X" is not a function, but a rogator.

I argued in Explaining Logical Necessity that in some cases the construction of a complex expression can have the consequence that although the referent or truth-value can be discovered by examining the world, the examination is unnecessary because the structure permits only one possible value. Simple tautologies and contradictions are examples, though there are many more complex examples.

The fact that some predicates and functions implicitly or explicitly depend on what they are applied to to determine how they are to be evaluated is similar to the phenomenon of parametric polymorphism in some computer programming languages. Gilbert Ryle also used a similar, though less precise, concept concept of "polymorphism" to explain some of the vagaries of ordinary language.

The suggestion that complex linguistic expressions have to be understood in ways that allow context to play affect the semantics at all syntactic levels has implications for the information-processing architecture required for a language user. (I don't know if any Artificial Intelligence models of language comprehension include this.)

A paper showing how to eliminate the alleged vagueness of many concepts by interpreting them as requiring extra parameters normally provided by the context (originally provoked by problems concerning spatial predicates, then generalised) is here, and I'll be grateful for critical comments if anyone has time to read it:
Spatial prepositions as higher order functions: And implications of Grice's theory for evolution of language.
(HTML and PDF formats)

I argue against probabilistic interpretations of spatial prepositions, such as 'left of', 'right of', 'near', 'far', by showing that intelligent use of context can produce quit specific interpretations, as illustrated above by the strength of the wind and the use to which sand or stones will be put, as determining how much sand constitutes a heap, or how many stones constitute a pile.

A quite different example would be someone trying to look over a high wall and failing. It would then be perfectly clear what was intended if he asked someone to build a pile (or stack) of bricks or paving stones for him to stand on. The required height is whatever would suffice to allow the person (or people) involved see over the wall by standing on the pile or stack.

This suggestion also deals with the 'continuous' version of sorites, e.g. where someone asks for a 'long' piece of rope.

The paper also has reflections on the nature of conceptual analysis and how that has to overlap deeply with scientific research, if done well, as proposed also in
Two Notions Contrasted: 'Logical Geography' and 'Logical Topography'
Variations on a theme by Gilbert Ryle: The logical topography of 'Logical Geography'.

Maintained by Aaron Sloman
School of Computer Science
The University of Birmingham