Against Determinable Universals

University of Durham, November 24, 2009

Dr Markku KeinänenUniversity of Turku

markku.keinanen@utu.fi

Introduction

• In this presentation, I argue against the existence of determinable universals.

• Properties and relations are assumed to be tropes.• Both determinate and determinable properties (if

considered as universals) are identified with sortal notions applying to tropes that are in certain formal relations to each other.

• I take up certain recent arguments for determinable universals (Armstrong 1997; Johansson 2000; Ellis 2001) and attempt to formulate a trope theorist answer to them.

General motivation

• Several recent advocates of determinate universals (e.g., Armstrong (1997), Johansson (2000), Ellis (2001) and Keller (2007)) defend the existence of determinable universals.

• Why to argue against determinable universals?

[1]: Assume that properties and relations are tropes (the claim which I will not defend). The postulation of determinable universals contradicts the trope nominalist claim that all properties and relations are particulars.

[2]: Second, if we need not introduce determinable universals and can account for the same facts by means of tropes, we gain qualitative economy.

General motivation

[3]: Determinable universals are prima facie redundant postulations: property tropes already suffice to determine the determinable characteristics of objects. We need not introduce separate determinable properties.

[4]: Property tropes seem to fall under a certain (highest) determinable because of their very nature and we need not introduce determinable universals to explain this.

Examples of determinates of determinables

Monadic natural properties

• Qualities: shapes, colours• Quantities: masses, lengths, charges

Natural relations

• Quantitative relations: distances, durations, space-time intervals

• Since quantities are best a posteriori candidates for natural properties, I will concentrate on quantities.

Conventions

• Let us stipulate that two objects share a determinate feature falling under determinable D (e.g., a specific mass) iff they are exactly similar in respect of D (mass).

• Moreover, we can say that object i has determinable characteristics D if the attribution of determinable D to i is true of i.

• In continuation, I will concentrate on the highest determinables (mass, charge) and on the lowest determinates (specific masses, charges).

General characteristics of the determinate determinable relation

• Downward entailment: If object i possesses a determinable characteristics D, i must possess some determinate feature F falling under D. For instance, an object having a mass must have some determinate mass.

• Upward entailment: If object i possesses some determinate feature F, i must possess the determinable under which F falls. For instance, a circular object must have a shape.

Relations between determinates of a determinable

• Exclusiveness: The determinate features falling under a common determinable D are exclusive of each other. For instance, a 1 kg object cannot have any other mass.

• Resemblance: The different objects possessing a determinate feature falling under determinable D resemble each other in respect of D. For instance, objects having specific masses resemble in respect of mass more or less closely. Assume that the determinate features are determined by the determinate properties of objects. The corresponding determinate properties falling under the determinable resemble more or less closely.

Different categorizations

• Determinate properties are commonly considered as first-order properties (universals or tropes).

• By contrast, determinable universals (if postulated) can be categorized in three alternative ways:

1) (Russellian) first-order universals, i.e., universal properties of objects.

2) (Russellian) second-order universals, i.e., properties of determinate properties (Armstrong 1997; Johansson 2000 (?)).

3) Neo-Aristotelian property kind universals, i.e., determinable kinds of tropes (Ellis 2001). According to this approach, determinate universals are determinate kinds of tropes (or, modes).

Different categorizations

• In what follows, I will take the third Neo-Aristotelian approach as granted for three reasons:

1) It is suitable for a theorist who assumes that properties of objects are tropes.

2) Gillett & Rives (2005) have already argued against the first approach fairly convincingly (redundancy argument).

3) We can construct all main arguments for determinable universals by assuming that determinable universals are kinds of property and relation tropes. Including the arguments Johansson (2000) presents.

The Neo-Aristotelian approach to determinable universals

• An important recent representative: Brian Ellis (2001, 2002).

• Properties of objects are particulars: tropes (or modes).• The distinct tropes (1 kg tropes) are exactly similar

because of being instances of the same determinate kind universal (of being a 1 kg trope).

• The distinct tropes (mass tropes) belong to the same determinable kind because of being instances of a determinable kind universal (of being a mass trope).

• The determinate kinds of tropes are sub-kinds of the determinable kinds.

Arguments for determinable universals

[1]: The Argument from Resemblance (Ellis 2001: sec.2.3; cf. also Johansson 2000: sec.3)The distinct tropes belonging to the same determinate kind (e.g., two 1 kg tropes) are exactly similar to each other. Moreover, the tropes belonging to the same determinable kind (two mass tropes) resemble each other (are connected by quantitative distance), while the tropes falling under distinct determinables do not resemble (are not connected by quantitative distance).Property tropes resemble exactly because they are instances of the same determinate kind universal. Similarly, the best explanation of why the tropes belonging to a determinable kind resemble is that they are instances of a determinable kind universal.

Arguments for determinable universals

[2]: The Argument from the Hierarchy of Natural Kinds (Ellis 2001: 70ff.)The natural kinds of tropes form a hierarchy: the determinate kinds (e.g., such as the mass of 1 kg) are sub-kinds of the determinable kinds (such as mass). In turn, determinable kinds are sub-kinds of still higher natural kinds (such as the kind of causal powers).The best explanation of the objective and mind-independent division of tropes into determinate and determinable kinds (mass, electric charge, distance) is that tropes are instances of the corresponding property kind universals. This accounts for the hierarchy of the natural kinds of tropes existing in the world.

Arguments for determinable universals

[3]: The Argument from the Physical Magnitudes (cf. Johansson 2000: sec.4)

The physical quantities falling under the same determinable and related by quantitative distance form a physical dimension: we can add the determinate quantities in the same dimension and obtain new determinates falling under the same determinable. Secondly, many determinate quantities belonging to certain distinct dimensions can be multiplied by each other and we obtain determinates falling under a new determinable.

The best explanation of these facts is that (many) physical dimensions correspond to determinable universals, ontological determinables having the determinate quantities in question as their instances.

Arguments for determinable universals

[4]: The Argument from the Laws 0n Determinables (Johansson 2000: sec. 7)The determinable kinds of tropes are subjected to laws constraining the inter-determinable (e.g., determinable dependences) or intra-determinable relationships (e.g., determinate exclusions). These laws are made true by the determinable kind universals determining the essential nature of each determinable kind. Determinable kind universals must not be replaced with nominalist constructions.

Arguments for Determinable Universals

[5]: The argument from spectral kinds (Ellis 2001: 65-66, 79).Some of the determinable kinds of tropes (e.g., mass, distance) are spectral kinds. Spectral kinds allow for a continuous (or, a dense) variation among the intrinsic values of their instances. Spectral kinds cannot be constituted by their instantiated values: it is probable that every quantity which is a spectral kind has uninstantiated values (i.e., gappy instances). Thus, the quantity tropes belonging to a spectral kind do not suffice to constitute it.Hence, we need to introduce determinable quantity universals to act as spectral kinds having uninstantiated values.

Arguments for Determinable universals

• For instance, it can be argued that masses form a spectral kind.

1) It is consistent with physics that masses get arbitrarily small values (or, that there are infinitely many masses between any two distinct masses).

2) It is conceivable that there are arbitrarily small masses.3) Arbitrarily small instantiated masses are possible.4) Hence, the quantity mass gets arbitrarily small values,

many of which are uninstantiated.5) Consequently, the quantity mass cannot be construed

by means of property tropes in formal relations to each other.

Arguments for determinable universals

[6]: The Argument from Functional Laws (Armstrong 1997)Functional laws (such as Newton’s inverse square law of gravitation) establish necessary connections between determinate quantities falling under some distinct determinables (e.g., masses, distances and gravitational attractions by some force). In establishing the necessary connections, they are completely undiscriminative between determinates (they do not contain exceptions).Since (some of) these quantities have uninstantiated values, we cannot build the truthmakers of functional laws solely by of means of the instantiated determinate quantities. We must introduce determinable universals to act as the parts of the truthmakers of functional laws.

The trope nominalist account of determinables

• Tropes are particular properties. A trope is:

1) Categorially simple, i.e., it is either simple or all of its proper parts are further tropes (entities of the same category).

2) A concrete, i.e., spatio-temporal entity.

3) Countable and identifiable (has identity of its own indepedent of the identity any other entity).

4) Has a thin particular nature to determine a single feature of the object possessing the trope (e.g., the mass of 1 kg).

The trope nominalist account of determinables

• According to trope nominalism, two distinct tropes t1 and t2 (e.g., two distinct 1 kg tropes) are exactly similar simply because of being particular properties they are. Thus, exact similarity is an ungrounded internal relation (“strict internal relation”) between tropes t1 and t2.

• Consequently, we need not introduce any further entities to ground their exact similarity.

• We must not introduce such entities: since determinate 1kg tropes t1 and t2 are exactly similar because of their existence, any further entity (e.g., relational trope of similarity) that would ground their exact similarity would be a redundant postulation.

The trope nominalist account of determinables

• The exactly similar tropes (e.g., all 1 kg tropes) are all certain kinds of tropes (1 kg tropes). They belong to the same determinate kind because they are exactly similar to each other.

• The property tropes t1 and t2 are the truthmakers of the claim of their belonging to a determinate kind:

[1]: t1 and t2 are 1 kg tropes.

• The trope nominalist can identify the determinate kind of tropes (the kind of 1 kg tropes) with a kind term applying to a group of tropes exactly similar trope to each other. Consequently, she can reject the existence of determinate property kind universals

The trope nominalist account of determinables

• We obtain the trope nominalist account of determinables as a generalisation of the trope nominalist account of determinates (tropes belonging to a determinate kind).

• Prima facie, all quantity tropes belonging determinable kind D are connected by quantitative distance and (usually) ordered by greater than relation.

• However, we cannot express quantitative distances between tropes without recourse to the determinable kind itself (the specifications of quantitative distance such as trope t is 1kg greater than trope u contain reference to the determinable kind via the unit).

Trope nominalist account of determinables

• Proposal (inspired by Bigelow & Pargetter (1990)): all quantity tropes belonging to determinable kind bear the different formal relations of proportion to each other. Each relation of proportion can be expressed by a (real) number divided by another, for instance:

[3]: Two kilogram trope t3 is in the 2 : 1 relation to one kilogram trope t2.

• It is easy to observe that all mass tropes and distance tropes are connected by different formal relations of proportion. It seems that the idea can be genaralized to apply the other quantities.

The Trope Nominalist account of determinables

• Some quantities (such as electric charge) take positive and negative values and we must introduce both positive and negative relations of proportion.

• If the relations of proportion between determinate quantities falling under D are given, we can determine order between them and specify quantitative distances between them (given some conventionally chosen unit of the quantity).

• The formal relation relation of exact similarity is a special case of the proportion relations (1 : 1 proportion).

The trope nominalist account of determinables

• Thus, tropes t1 and t2 are truthmakers of claim [5]:

[5]: t1 and t2 are mass tropes

• The trope nominalist can identify each determinable kind of tropes (the kind of mass tropes) with a kind term applying to a group of tropes connected by the formal relations of proportion and (possibly) combinatorial exclusion, i.e., with a formal kind of tropes.

• Consequently, the trope nominalist can reject the existence of determinable kind universals.

The trope nominalist answer

• [A1]: According to the trope nominalist, certain property tropes are exactly similar to each other simply because of their existence. The introduction any further entity (e.g., a determinate kind universal) to account for the exact similarity of tropes leads to redundancy.

• Similarly, the quantity tropes falling under highest determinable D are connected by the relations of proportion because of their existence. Therefore, they resemble each other (i.e., are connected by quantitative distance). The tropes belonging to distinct determinable kinds are not connected by the relations of proportion. Again, the introduction of any further entities (e.g., determinable kind universals) to account for these facts leads to redundancy.

The trope nominalist answer

• [A2]: The division of tropes into determinate and determinable kinds is objective and mind-independent because tropes are in the required formal relations independent of us and our classifications. Hence, we need not introduce determinate or determinable kind universals to make true the attributions of natural kind to tropes.

• Determinate kinds are sub-kinds of determinable kinds because the formal relation determining the membership of trope in a determinate kind (i.e., exact similarity) is a special case of the formal relations that determine the membership of tropes in a determinable kind (i.e., the relations of proportion).

The trope nominalist answer

• [A3]: The present approach does not directly explain the (alleged) truth of certain existence claims concerning determinates of determinables (Johansson (2000).

• Since certain monadic determinate quantities (such as masses and charges) are additive, the determinate tropes falling under D possessed by two objects a and b determine the quantitative feature falling under D possessed by the object composed by a and b. (Those who like complex properties, can introduce a complex trope corresponding this feature.)

• Second, we can either postulate tropes belonging to a further determinable kind that form a physical dimension of some derived physical quantity or try to explain that dimension away as a determinable kind of tropes.

The trope nominalist answer

• [A4]: Johansson’s examples of laws concerning determinables fail to show that we need to postulate determinable universals.

• First, determinate tropes falling under a determinable are in the relations of proportion to each other, we can give them certain quantized values (given some chosen unit). These values can be added and (in some cases) subtracted.

• Secondly, the determinate tropes falling under a determinable are exclusive of each other because they are connected by formal relation of combinatorial exclusion.

• Thirdly, the tropes acting as properties of a certain kind of object can be generically dependent on the tropes belonging to another determinable kind.

The trope nominalist answer

• [A5]: Spectral kinds cannot be constructed by means of the present approach. However, some proposed examples can be explained away:

1) Masses: the argument that masses form a spectral kind was rather weak based on the considerations of conceivability (understood as consistency with physics). Counter-proposal: The fundamental values of quantity (rest) mass are given by the rest masses of the fundamental particles. Although the quantity mass does not come in natural units, it does not get arbitrarily small values. All masses of objects are multiples of the fundamental masses.

2) Distances (durations and space-time intervals): if space is infinitely divisible, distances have a stronger claim for a spectral kind.

The Trope Nominalist answer

• [A6]: Pace Armstrong, we need not identify instances of functional laws (i.e., determinate laws) with relations between universals. According to dispositionalist approach, the statements of determinate law are made true by the property tropes possessed by objects.

• However, many determinate laws seem to describe how the objects possessing these property tropes would behave in counterfactual situations.

Conclusion

• The Trope Nominalism identifies the determinable kinds of tropes with kind terms applying to groups of tropes connected by the formal relations of 1) proportion, and (usually) 2) combinatorial exclusion.

• The existence of property kind universals is rejected. • Tropes are in the formal relations of proportion because

of being the tropes they are (as formal relations proportions are not relational entities).

• Burden: spectral kinds must be explained away.• Benefit: similarities between the determinates of a

determinable are given by means of the relations of proportion. This helps us to give a more unified list of formal relations.

References• Armstrong, D. M. (1978): A Theory of Universals, Vol. 2 of Universals and Scientific Realism

(Cambridge: Cambridge University Press).• Armstrong, D. M. (1997): A World of States of Affairs, (Cambridge: Cambridge University Press).• Bigelow, J. & Pargetter, R. (1990): Science and Necessity, (Cambridge: Cambridge University

Press). • Campbell, K. K. (1990): Abstract Particulars. (Oxford: Basil Blackwell).• Fales, E. (1990): Causation and Universals. (London & New York: Routledge).• Gillett, C. & Rives, B. (2005): “The Non-existence of Determinables: Or, a world of Absolute

Determinates as a Default Hypothesis”, Nous 39:3, 483-504.• Ellis, B. (2001): Scientific Essentialism, (Cambridge: Cambridge University Press).• Ellis, B. (2002): Philosophy of Nature, (Chesham: Acumen).• Johansson, I. (2000): ”Determinables as Universals”, The Monist 1, 101-121.• Maurin, A-S. (2002): If Tropes, (Dordrecht: Reidel).• Russell, B. (1912): The Problems of Philosophy,(Oxford: Oxford University Press).• Sanford, D. (2006): “Determinates and Determinables”, Standford Encyclopedia of Philosophy• Shoemaker, S. (1980): “Causality and Properties”, in van Inwagen (ed.): Time and Cause,

(Dordrecht: Reidel).• Simons, P. (2003): ”Tropes, Relational”, Conceptus.

Thank You!