Indeterminacy and Fuzziness

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Indeterminacy and Fuzziness

We conclude with some remarks on a question that was briefly mentioned above in connection with the Special Composition Question but that pertains more generally to the underlying notion of parthood that mereology seeks to systematize. All the theories examined so far, from M toGEM and its variants, appear to assume that parthood is a perfectly determinate relation: given any two entities x and y, there is always an objective, determinate fact of the matter as to whether or not x is part of y. However, in some cases this seems problematic. Perhaps there is no room for indeterminacy in the idealized mereology of space and time as such; but when it comes to the mereology of ordinary spatio-temporal particulars (for instance) the picture looks different. Think of objects such as clouds, forests, heaps of sand. What exactly are their constitutive parts? What are the mereological boundaries of a desert, a river, a mountain? Some stuff is positively part of Mount Everest and some stuff is positively not part of it, but there is borderline stuff whose mereological relationship to Everest seems indeterminate. Even living organisms may, on closer look, give rise to indeterminacy issues. Surely Tibbles's body comprises his tail and surely it does not comprise Pluto's. But what about the whisker that is coming loose? It used to be a firm part of Tibbles and soon it will drop off for good, yet meanwhile its mereological relation to the cat is dubious. And what goes for material bodies goes for everything. What are the mereological boundaries of a neighborhood, a college, a social organization? What about the boundaries of events such as promenades, concerts, wars? What about the extensions of such ordinary concepts as baldness, wisdom, personhood?
These worries are of no little import, and it might be thought that some of the principles discussed above would have to be revisited accordingly—not because of their ontological import but because of their classical, bivalent presuppositions. For example, the extensionality theorem of EM, (27), says that composite things with the same proper parts are identical, but in the presence of indeterminacy this may call for qualifications. The model in Figure 9, left, depicts xand y as non-identical by virtue of their having distinct determinate parts; yet one might prefer to describe a situation of this sort as one in which the identity between x and y is itself indeterminate, owing to the partly indeterminate status of the two outer atoms. Conversely, in the model on the right x and y have the same determinate proper parts, yet again one might prefer to suspend judgment concerning their identity, owing to the indeterminate status of the middle atom.

اقتباس :: Figure 9. Objects with indeterminate parts (dashed lines).

Now, it is clear that a lot here depends on how exactly one understands the relevant notion of indeterminacy. There are, in fact, two ways of understanding a claim of the form

(64)	It is indeterminate whether a is part of b,

depending on whether the phrase ‘it is indeterminate whether’ is assigned wide scope, as in (64a), or narrow scope, as in (64b):

(64a)	It is indeterminate whether b is such that a is part of it.
(64b)	b is such that it is indeterminate whether a is part of it.

On the first understanding, the indeterminacy is merely de dicto: perhaps ‘a’ or ‘b’ are vague terms, or perhaps ‘part’ is a vague predicate, but there is no reason to suppose that such vagueness is due to objective deficiencies in the underlying reality. If so, then there is no reason to think that it should affect the apparatus of mereology either, at least insofar as the theory is meant to capture some structural features of the world regardless of how we talk about it. For example, the statement

(65)	The loose whisker is part of Tibbles

may owe its indeterminacy to the semantic indeterminacy of ‘Tibbles’: our linguistic practices do not, on closer look, specify exactly which portion of reality is currently picked out by that name. In particular, they do not specify whether the name picks out something whose current parts include the whisker that is coming loose and, as a consequence, the truth conditions of (65) are not fully determined. But this is not to say that the stuff out there is mereologically indeterminate. Each one of a large variety of slightly distinct chunks of reality has an equal claim to being the referent of the vaguely introduced name ‘Tibbles’, and each such thing has a perfectly precise mereological structure: some of them currently include the lose whisker among their parts, others do not. (Proponents of this view, which also affords a way of dealing with the so-called “problem of the many” of Unger 1980 and Geach 1980, include Hughes 1986, Heller 1990, Lewis 1993a, McGee 1997, and Varzi 2001.) Alternatively, one could hold that the indeterminacy of (65) is due, not to the semantic indeterminacy of ‘Tibbles’, but to that of ‘part’ (as in Donnelly 2014): there is no one parthood relation; rather, several slightly different relations are equally eligible as extension of the parthood predicate, and while some such relations connect the loose wisker to Tibbles, others do not. In this sense, the dashed lines in Figure 9 would be “defects” in the models, not in the reality that they are meant to represent. Either way, it is apparent that, on a de dicto understanding, mereological indeterminacy need not be due to the way the world is (or isn't): it may just be an instance of a more general and widespread phenomenon of indeterminacy that affects our language and our conceptual apparatus at large. As such, it can be accounted for in terms of whatever theory—semantic, pragmatic, or even epistemic—one finds best suited for dealing with the phenomenon in its generality. (See the entry on vagueness.) The principles of mereology, understood as a theory of the parthoodrelation, or of all the relations that qualify as admissible interpretations of the parthood predicate, would hold regardless.[26]
By contrast, on the second way of understanding claims of the form (64), corresponding to (64b), the relevant indeterminacy is genuinely de re: there is no objective fact of the matter as to whether a is part of b, regardless of the words we use to describe the situation. For example, on this view (65) would be indeterminate, not because of the vagueness of ‘Tibbles’, but because of the vagueness of Tibbles itself: there simply would be no fact of the matter as to whether the whisker that is coming loose is part of the cat. Similarly, the dashed lines in Figure 9 would not reflect a “defect” in the models but a genuine, objective deficiency in the mereological organization of the underlying reality. As it turns out, this is not a popular view: already Russell (1923) argued that the very idea of worldly indeterminacy betrays a “fallacy of verbalism”, and some have gone as far as saying that de re indeterminacy is simply not “intelligible” (Dummett 1975: 314; Lewis 1986b: 212) or ruled out a priori (Jackson 2001: 657). Nonetheless, several philosophers feel otherwise and the idea that the world may include vague entities relative to which the parthood relation is not fully determined has received considerable attention in recent literature, from Johnsen (1989), Tye (1990), and van Inwagen (1990: ch. 17) to Morreau (2002), McKinnon (2003), Akiba (2004), N. Smith (2005), Hyde (2008: §5.3), Carmichael (2011), and Sattig (2013, 2014), inter alia. Even those who do not find that thought attractive might wonder whether an a priori ban on it might be unwarranted—a deep-seated metaphysical prejudice, as Burgess (1990: 263) puts it. (Dummett himself withdrew his earlier remark and spoke of a “prejudice” in his 1981: 440.) It is therefore worth asking: How would such a thought impact on the mereological theses considered in the preceding sections?
There is, unfortunately, no straightforward way of answering this question. Broadly speaking, two main sorts of answer may be considered, depending on whether (i) one simply takes the indeterminacy of the parthood relation to be the reason why certain statements involving the parthood predicate lack a definite truth-value, or (ii) one understands the indeterminacy so that parthood becomes a genuine matter of degree. Both options, however, may be articulated in a variety of ways.
On option (i) (initially favored by such authors as Johnsen and Tye), it could once again be argued that no modification of the basic mereological machinery is strictly necessary, as long as each postulate is taken to characterize the parthood relation insofar as it behaves in a determinate fashion. Thus, on this approach, (P.1) should be understood as asserting that everything is definitely part of itself, (P.2) that any definite part of any definite part of a thing is itself a definite part of that thing, (P.3) that things that are definitely part of each other are identical, and so on, and the truth of such principles is not affected by the consideration that parthood need notbe fully determinate. There is, however, some leeway as to how such basic postulates could be integrated with further principles concerning explicitly the indeterminate cases. For example, do objects with indeterminate parts have indeterminate identity? Following Evans (1978), many philosophers have taken the answer to be obviously in the affirmative. Others, such as Cook (1986), Sainsbury (1989), or Tye (2000), hold the opposite view: vague objects are mereologically elusive, but they have the same precise identity conditions as any other object. Still others maintain that the answer depends on the strength of the underlying mereology. For instance, T. Parsons (2000: §5.6.1) argues that on a theory such as EM cum unrestricted binary sums,[27] the de re indeterminacy of (65) would be inherited by

(66)	Tibbles is identical with the sum of Tibbles and the loose whisker.

A related question is: Does countenancing objects with indeterminate parts entail that composition be vague, i.e., that there is sometimes no matter of fact whether some things make up a whole? A popular view, much influenced by Lewis (1986b: 212), says that it does. Others, such as Morreau (2002: 338), argue instead that the link between vague parthood and vague composition is unwarranted: perhaps the de re indeterminacy of (65) is inherited by some instances of

(67)	Tibbles is composed of x and the loose whisker.

(for example, x could be something that is just like Tibbles except that the whisker is determinately not part of it); yet this would not amount to saying that composition is vague, for the following might nonetheless be true:

(68)	There is something composed of x and the loose whisker.

Finally, there is of course the general question of how one should handle logically complex statements concerning, at least in part, mereologically indeterminate objects. A natural choice is to rely on a three-valued semantics of some sort, the third value being, strictly speaking, not a truth value but rather a truth-value gap. In this spirit, both Johnsen and Tye endorse the truth-tables of Kleene (1938) while Hyde those of Łukasiewicz (1920). However, it is worth stressing that other choices are available, including non-truth-functional accounts. For example, Akiba (2000) and Morreau (2002) recommend a form of “supervaluationism”. This was originally put forward by Fine (1975) as a theory for dealing with de dicto indeterminacy, the idea being that a statement involving vague expressions should count as true (false) if and only it is true (false) on every “precisification” of those expressions. Still, a friend of de re indeterminacy may exploit the same idea by speaking instead of precisifications of the underlying reality—what Sainsbury (1989) calls “approximants”, Cohn and Gotts (1996) “crispings”, and T. Parsons (2000) “resolutions” of vague objects. As a result, one would be able to explain why, for example, (69) appears to be true and (70) false (assuming that Tibbles's head is definitely part of Tibbles), whereas both conditionals would be equally indeterminate on Kleene's semantics and equally true on Łukasiewicz's:

(69)	If the loose whisker is part of the head and the head is part of Tibbles, then the whisker itself is part of Tibbles.
(70)	If the loose whisker is part of the head and the head is part of Tibbles, then the whisker itself is not part of Tibbles.

As for option (ii)—to the effect that de re mereological indeterminacy is a matter of degree—the picture is different. Here the main motivation is that whether or not something is part of something else is really not an all-or-nothing affair. If Tibbles has two whiskers that are coming loose, then we may want to say that neither is a definite part of Tibbles. But if one whisker is looser than the other, then it would seem plausible to say that the first is part of Tibbles to alesser degree than the second, and one may want the postulates of mereology to be sensitive to such distinctions. This is, for example, van Inwagen's (1990) view of the matter, which results in a fuzzification of parthood that parallels in many ways to the fuzzification of membership in Zadeh's (1965) set theory, and it is this sort of intuition that also led to the development of such formal theories as Polkowsky and Skowron's (1994) “rough mereology” or N. Smith's (2005) theory of “concrete parts”. Again, there is room for some leeway concerning matters of detail, but in this case the main features of the approach are fairly clear and uniform across the literature. For let π be the characteristic function associated with the parthood relation denoted by the basic mereological primitive, ‘P’. Then, if classically this function is bivalent, which can be expressed by saying that π(x, y) always takes, say, the value 1 or the value 0 according to whether or not x is part of y, to say that parthood may be indeterminate is to say that π need not be fully bivalent. And whereas option (i) simply takes this to mean that π may sometimes be undefined, option (ii) can be characterized by saying that the range of π may include values intermediate between 0 and 1, i.e., effectively, values from the closed real interval [0, 1]. In other words, on this latter approach π is still a perfectly standard, total function, and the only serious question that needs to be addressed is the genuinely mereological question of what conditions should be assumed to characterize its behavior—a question not different from the one that we have considered for the bivalent case throughout the preceding sections.
This is not to say that the question is an easy one. As it turns out, the “fuzzification” of the core theory M is rather straightforward, but its extensions give rise to various issues. Thus, consider the partial ordering axioms (P.1)–(P.3). Classically, these correspond to the following conditions on π :

(P.1π)	π(x, x) = 1
(P.2π)	π(x, z) ≥ min(π(x, y), π(y, z))
(P.3π)	If π(x, y) = 1 and π(y, x) = 1, then x = y,

and one could argue that the very same conditions may be taken to fix the basic properties of parthood regardless of whether π is bivalent. Perhaps one may consider weakening (P.2π) as follows (Polkowsky and Skowron 1994):

(P.2π′)

If π(y, z) = 1, then π(x, z) ≥ π(x, y).

Or one may consider strengthening (P.3π) as follows (N. Smith 2005):[28]

(P.3π′)

If π(x, y) > 0 and π(y, x) > 0, then x = y.

But that is about it: there is little room for further adjustments. Things immediately get complicated, though, as soon as we move beyond M. Take, for instance, the Supplementation principle (P.4) of MM. One natural way of expressing it in terms of π is as follows:

(P.4π)

If π(x, y) = 1 and x ≠ y, then π(z, y) = 1 for some z such that, for all w, either π(w,z) = 0 or π(w, x) = 0.

There are, however, fifteen other ways of expressing (P.4) in terms of π, obtained by re-writing one or both occurrences of ‘= 1’ as ‘> 0’ and one or both occurrences of ‘= 0’ as ‘< 1’. In the presence of bivalence, these would all be equivalent ways of saying the same thing. However, such alternative formulations would not coincide if π is allowed to take non-integral values, and the question of which version(s) best reflect the supplementation intuition would have to be carefully examined. (See e.g. the discussion in N. Smith 2005: 397.) And this is just the beginning: it is clear that similar issues arise with most other principles discussed in the previous sections, such as Complementation, Density, or the various composition principles. (See e.g. Polkowsky and Skowron 1994: 86 for a formulation of the Unrestricted Sum axiom (P.152).)
On the other hand, it is worth noting that precisely because the difficulty is mainly technical—the framework itself being fairly firm—now some of the questions raised in connection with option (i) tend to be less open to controversy. For example, the question of whether mereological indeterminacy implies vague identity is generally answered in the negative, especially if one adheres to the spirit of extensionality. For then it is natural to say that non-atomic objects are identical if and only if they have exactly the same parts to the same degree—and that is not a vague matter (a point already made in Williamson 1994: 255). In other words, given that classically the extensionality principle (27) corresponds to the following condition:

(27π)

If there is a z such that either π(z, x) = 1 or π(z, y) = 1, then x = y if and only if, for every z, π(z, x) = π(z, y),

it seems perfectly natural to stick to this condition even if the range of π is extended from {0, 1} to [0, 1]. Likewise, the question of whether mereological indeterminacy implies vague composition or vague existence is generally answered in the affirmative (though not always; see e.g. Donnelly 2009 and Barnes and Williams 2009). Van Inwagen (1990: 228) takes this to be a rather obvious consequence of the approach, but N. Smith (2005: 399ff) goes further and provides a detailed analysis of how one can calculate the degree to which a given non-empty set of things has a sum, i.e., the degree of existence of the sum. (Roughly, the idea is to begin with the sum as it would exist if every element of the set were a definite part of it, and then calculate the actual degree of existence of the sum as a function of the degree to which each element of the set is actually part of it).

The one question that remains widely open is how all of this should be reflected in the semantics of our language, specifically the semantics of logically complex statements. As a matter of fact, there is a tendency to regard this question as part and parcel of the more general problem of choosing the appropriate semantics for fuzzy logic, which typically amounts to an infinitary generalization of some truth-functional three-valued semantics. The range of possibilities, however, is broader, and even here there is room for non-truth-functional approaches—including degree-theoretic variants of supervaluationism (as recommended e.g. in Sanford 1993: 225).]

Indeterminacy and Fuzziness

Indeterminacy and Fuzziness

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Indeterminacy and Fuzziness