Typed Properties

The discovery of his paradox (and then the awareness of related puzzles) led Russell to introduce a theory of types which institutes a total ban on self-exemplification by a strict segregation of properties into levels that he called ‘types’ (cf. Copi 1971). Actually his account involves a distinction of types and orders and is thus more complex and restrictive than this. More details can be found in the entry on Russell’s paradox (for a detailed reconstruction of how Russell reacted to the paradox, cf. Landini 1998).

7.3 Untyped Properties

Type theory has never gained unanimous consensus and its many problematic aspects are well-known (see, e.g., Fitch 1952, Appendix C; Bealer 1989). Just to mention a few, the type-theoretical hierarchy imposed on properties appears to be highly artificial and multiplies properties ad infinitum (e.g., since presumably properties are abstract, for any property P of typen, there is an abstractness of type n+1 that P exemplifies). Moreover, many cases of self-exemplification are innocuous and common (at least for realists who are not minimalists or conservative centrists). For example, the property of being a property is itself a property, so it exemplifies itself. There also seem to be transcendental relations. A transcendental relation likethinks about is one that can relate quite different types of things: Hans can think about Vienna and he can think about triangularity. But typed theories cannot accommodate transcendental properties without several epicycles.
Several recent accounts are thus type-free and treat properties as entities that can exemplify themselves. From this perspective, the picture of a hierarchy of levels is fundamentally misguided, if it is interpreted too rigidly; there are simply properties (which can be exemplified—in many cases by other properties, even by themselves) and individuals (which cannot be exemplified). One challenge here is to develop formal accounts that allow as much self-exemplification as possible without teetering over the brink into paradox (see §8).

7.4 Relations

What we now usually regard as genuine multi-place relations were not recognized as such by philosophers for quite a long time, or so it seems. Apparently, Aristotle and the Scholastics found no place for genuine irreducible relations in their ontology (see the entry on medieval theories of relations) and Leibniz is usually viewed as a philosopher who, in line with this tradition, tries to show effectively how relations can be reduced to monadic properties (Mugnai 1992). An accurate analysis of Leibniz’s technical use of expressions such as ‘insofar as’ (‘quatenus’) and ‘by the same token’ (‘et eo ipso’) in sentences such as ‘Paris loves and by the same token Helen is beloved’ suggests, however, that he did acknowledge somehow the existence of irreducibly relational facts (Orilia 2000a). Be this as it may, it was not before the second half of the 19th century (with the work of De Morgan, Schroeder, Peirce and, somewhat later, Russell) that irreducible relations began to be generally acknowledged. Some philosophers still hold that relations are reducible to properties in that they supervene on the monadic properties of their relata in a very strong sense that shows that relations are not actually real (some trope theorists hold this view; it is defended at length in Fisk 1972). But no one has been able to show that all relations do supervene on monadic properties, and there are strong reasons for thinking that at least some sorts of relations, e.g., spatio-temporal ones, do not. The view that there are relations but no monadic properties, or at least that the former have ontological priority over the latter, has also been considered. It is defended in different forms by Dipert 1997 and by various authors in the context of ontic structural realism (see, e.g., French & Ladyman 2003, Esfeld 2003 and §6 of the entry on structural realism). This view is far, however, from having gained some consensus (Ainsworth 2010). In sum, all in all most contemporary realists hold that there are both genuine monadic properties and genuine relations. (see, however, Marmodoro & Yates forthcoming, a collection of papers most of which deal in depth with all these issues).
In standard first-order logic predicates come with a fixed degree and in line with this relations are usually taken to have a fixed degree themselves (on abundant conceptions of properties, there are relations of every finite number of argument places, but on sparse conceptions it is an empirical question whether there are relations of any particular degree). In contrast with this, however, many natural language predicates appear to be multigrade or variably polyadic; they can be true of various numbers of things. For example, the predicate ‘robbed a bank together’ is true of Bonnie and Clyde, Ma Barker and her two boys, Patti Hearst and three members of the Symbionese Liberation Army, and so on. Multigrade predicates are very common (e.g., ‘work well together,’ ‘conspired to commit murder,’ ‘are lovers’). Moreover, there is a kind of inference, called ‘argument deletion,’ that also suggests that many predicates that prima facie could be assigned a certain fixed degree are in fact multigrade. For example, ‘John is eating a cake’ suggests that ‘is eating’ is dyadic, but since, by argument deletion, it entails ‘John is eating,’ one could at least tentatively conclude that ‘eating’ is also monadic and thus multigrade. Often one can resist the conclusion that there are multigrade predicates by resorting to one stratagem or another. For example, it could be said that ‘John is eating’ is simply short for ‘John is eating something.’ But it seems hard to find a systematic and convincing strategy that allows us to maintain that natural language predicates have a fixed degree. This has motivated the construction of logical languages that feature multigrade predicates in order to provide a more appropriate formal account of natural language (Gandy 1976; Graves 1993; Orilia 2000a; the latter two show that this can be done by appealing to thematic roles). Although any leap from language to ontology must be handled with care, all this suggests that relations, or at least some of them, are variably polyadic. Turning to naturalistic ontology, some support for this conclusion comes from the ingenious treatment of measurement in Mundy (1990), which is based on multigrade relations. In sum, it seems that a truly flexible account of properties should abandon not only the restrictive hierarchy of types but also the constraint that all properties come with a fixed number of argument places.
Relations pose a special problem, that of explaining from a very general, ontological, point of view the nature of the difference between states of affairs, such as Abelard loves Eloise andEloise loves Abelard, that at least prima facie involve exactly the same constituents, namely a non-symmetric relation and two other items (loving, Abelard, and Eloise, in our example). Such states of affairs are often said to differ in ‘relational order’ or in the ‘differential application’ of the non-symmetric relation in question, and the problem then is that of characterizing what this relational order or differential application amounts to. Russell (1903, §218) attributed an enormous importance to this issue and has attacked it repeatedly. In spite of this, until a few years ago, only a small number of other philosophers have confronted it systematically (e.g., Bergmann 1992; Hochberg 1987) and their efforts have been pretty much neglected. However, Fine (2000) has forcefully brought the issue on the agenda of ontologists and proposed a novel approach that has received some attention. Fine identifies a standard and a positionalist view (analogous to two views defended by Russell at different times (1903; 1984); cf. Orilia 2008). According to the former, relations are intrinsically endowed with a ‘direction,’ which allows us to distinguish, e.g., loving and being loved: Abelard loves Eloise and Eloise loves Abelard differ, because they involve two relations that differ in direction (e.g., the former involves loving and the latter being loved). According to the latter, relations have different ‘positions’ that can somehow host relata:Abelard loves Eloise and Eloise loves Abelard differ, because the two positions of the very same loving relation are differently occupied (by Abelard and Eloise in one case and by Eloise and Abelard in the other case). Fine goes on to propose and endorse an alternative, ‘anti-positionalist’ standpoint, according to which, relations have neither direction nor positions. The literature on this issue keeps growing and there are now various proposals on the market, including attempts to rescue positionalism from Fine’s criticism, and primitivism, a

Typed Properties

7.3 Untyped Properties

7.4 Relations

Typed Properties :: تعاليق

Typed Properties