Transient Things and Permanent Stuff

Abstract

A view of individuals as constituted of quantities of matter, both understood as continuants enduring over time, is elaborated in some detail. Constitution is a three-place relation which can't be collapsed to identity because of the place-holder for a time and because individuals and quantities of matter have such a radically different character. Individuals are transient entities with limited lifetimes, whereas quantities are permanent existents undergoing change in physical and chemical properties from time to time. Coincidence, considered as a matter of occupying the same place, is developed, alongside sameness of constitutive matter, as a criterion of identity for individuals. Quantities satisfy the mereological criterion of identity, applicable to entities subject to mereological relations and operations such as regions of space and intervals of time. A time-dependent analogue of mereological parthood is defined for individuals, in terms of which analogues of the other mereological relations can be defined. But it is argued that there is no analogue of the mereological operation of summation for individuals.

Notes

¹Following Cartwright [1970] and Roeper [1983] in the use of the term ‘quantity’. See Needham [1996: 206–7] for a discussion of the choice of term.

²Specifically, with the variables π, ρ, σ, … ranging over quantities, separation (|) taken as primitive, and part (⊆), proper part (⊂) and overlap (o) defined as usual, the principles governing these relations are captured by the axioms (i) π ⊆ ρ ∧ ρ ⊆ π .⊃ π = ρ and (ii) π o ρ ≡ ∼(π | ρ). Further existential axioms are needed governing the mereological operations, which may be dyadic, like the dyadic sum, π ∪ ρ, of π and ρ, or they may include the general sum, Σπϕ(π), of quantities satisfying the condition ϕ(π). The existence of general sums of quantities, which are allowed in my account of quantities, is given by the axiom schema (iii) ∃πϕ (π) ⊃ ∃ρ∀σ(σ|ρ ≡ ∀π(ϕ(π) ⊃ σ|π). Uniqueness, which follows from the axioms, allows the general sum to be defined by ∃πϕ(π) ⊃ . σ = Σπϕ(π) ≡ ∀ρ)ρ|σ ≡ ∀π(ϕ(π) ⊃ ρ|π)), from which the general product and the dyadic operations, including difference, can in turn be defined. As will transpire from the discussion in §5, mereology as I am using the term includes the algebraic operations properly so called.

³This is described in much greater detail in Needham [2007].

⁴See Needham [2002]. Exactly how distinctions of substance are drawn can be controversial. Some of the issues are discussed in Needham [2008]. Microessentialism inspired by Kripke and Putnam offers little insight into these matters [Needham 2009].

⁵In the system of Tarski [1983], for example, defining two separate regions where there is a sphere overlapping both but none other separate from both as abutting is adequate for regions occupied by bodies. To include regions touching at a point, let regions with no sphere in common abut if every sphere in the set of concentric spheres defining some point (in Tarski's system) overlaps both regions.

⁶Intervals are taken to be temporally connected, i.e. without gaps, sustaining the mereological relations and binary operations.

⁷Formally, where ‘x is constituted of π at t’ is symbolized Const(x, π, t), ‘⊆’ is the mereological part relation and, for an open sentence ϕ(π), Σπϕ(π) is the sum of ϕ-ers, the principle of accumulation is

⁸Arguably, this example fails because timbers have a variable water content which will have been exchanged during what ordinarily counts as preservation of timbers. But in order to conform with the way variations on the ship of Theseus are usually described, this circumstance is ignored.

⁹Formally, ∃ t (∃πConst(x, π, t) ∧ ∀ t′ (t|t′⊃∼∃πConst(x, π, t′))), where as before, Const(x, π, t) is read ‘x is constituted of π at t’, and ‘|’ is the mereological relation of separation.

¹⁰Formally, where ‘x occupies p at t’ is symbolized Occ(x, p, t),

¹¹Formally, Occ(x, p, t) ⊃p = Σq ∃ t′ ⊆ t Occ(x, q, t′).

¹²In order to avoid any misunderstanding of the scope of the quantifiers, this definition is formalized by:

¹³This wouldn't be accepted by those who, like Johanna Seibt [2000: 256–7] object to the transitivity of being a component. Spatial parthood provides an interpretation of what Simons [1987: 158] understands as constitution when he says ‘constitution is transitive’. Holding times fixed, constitution as understood here is not transitive because it is always a quantity that constitutes an individual.

¹⁴Formally, symbolizing ‘x is a proper spatial part of y at t’ by x < _t y, the definition is

where ‘⊃’ is the proper part relation of mereology.

¹⁵Formally, x≤ _t y ≡ . x < _t y∨Coin(x, y, t), where x≤ _t y is read ‘x is a spatial part y at t’.

¹⁶Formally, x# _t y ≡ ∃ z (z≤ _t x ∧ z≤ _t y), where x# _t y is read ‘x spatially overlaps y at t’.

¹⁷Using notation introduced in earlier footnotes.

¹⁸This would remain true even if individuals were defined, for the purposes of reducing the four-sorted language to a single-sorted language, in terms of constitution as entities x such that for some quantity π and some time t, x is constituted of π at t, because of the existential quantification.

¹⁹I would like to acknowledge the helpful comments of two anonymous referees. Research on which the paper is based was supported by the Swedish Research Council.