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Published online by Cambridge University Press: 01 January 2020
It is a compelling idea that the calculus of individuals proposed by Leonard and Goodman articulates the concept of parthood.1 However, the persistence of material objects through some changes of parts, and their failure to persist through other changes in the organization of their parts present a serious challenge to this idea.
This paper is based on my PhD dissertation, ‘Identity and Quantification’ (M.I.T., 1985). I am indebted to Judith Thomson and George Boolos, and to John Heintz and Scott Soames for comments on earlier drafts.
1 Henry S. Leonard and Nelson Goodman, ‘The Calculus of Individuals and Its Uses,’ Journal of Symbolic Logic 5, 2 (June, 1940) 45-55
2 I have changed their notation to conform to the notation in the present essay.
3 Another approach to this puzzle appeals to temporal parts. Advocates of temporal parts will presumably reject both (5) and (4) on the ground that since the temporal parts of an object are its parts and some temporal parts of the members of a are not parts of the silver chain on my table at t1 some parts of the members of a are not part of the silver chain on my table at t1. Rejection of (5) gives rise to a variant of the puzzle I discuss in section V. But there are other problems with this proposal as well. If some parts of the members of a, the thirty links and a clasp from which the silver chain on my table at t1 is made, are not part of the silver chain on my table at t1, then these members of a are not part of the silver chain on my table at t1. This is implausible. For a discussion of temporal part and arguments against the existence of temporal parts see J.J. Thomson's ‘Parthood and Identity Across Time,’ The Journal of Philosophy 80, 4 (April, 1983) 201-20.
4 This conclusion may also be reached from the inconsistency of the set of (5.2), (9.2), and (10.2). Barring appeal to temporal parts, if the axioms of the calculus are true under their itended interpretation, then at t1 (5.2) expresses a true proposition; and if at t1 (9.2) also expresses a true proposition, then at t1 (10.2) must express a false proposition. But at t2 (10.2) expresses a true proposition; and since the same object satisfies the open sentence ‘w is a silver chain on my table at t2’ at both t1 and t2, the object which satisfies the open sentence ‘w is a fusion of a’ at t1 is not identical with the object which satisfies ‘w is a fusion of a’ at t2. Hence, the object which fuses a at t1 is not identical with the object which fuses a at t2.
5 Saul Kripke, ‘Semantical Considerations on Modal Logic,’ Acta Philosophica Fennica 16 (1963), 83-94. A model structure is defined as a triple < G,K,R > together with a function, 𝛹, where K is a set (the set of instants or moments), R is a reflexive relation on K, G (the present moment) is a member of K, and 𝛹(H) is a set for each HЄK. Intuitively, 𝛹(H) is the set of things which exist at H. Let U = UHЄK 𝛹(H), and let Un be the nth cartesian product of U with itself. A model on a model structure < G,K,R > is a binary function ɸ(Pn,H), where ‘Pn’ ranges over n-adic predicate letters, ‘H’ ranges over members of K, and ɸ(Pn,H) ⊆ Un, where n>l, otherwise ɸ(Pn,H) = T or F. The clauses of the inductive definition are as follows:
(i) For an atomic formula
ɸ(Pn(x1 … ,xn), H) = T with respect to an assignment a1 … ,an of elements of U to x1, … ,xn, if and only if <a1, … ,an>Єɸ(Pn,H).
(ii) ɸ(-A(x1 … ,xn),H) = T with respect to an assignment a1,...,an of elements of U to x1, … ,xn, if and only if ɸ(A(x1, … ,xn),H)≠T with respect to that assignment.
(iii) ɸ((A(x1, … ,xn) & B(y1, … ,yn)), H)=T with respect to an assignment of a1 … ,an of elements of U to x1, … Xn, and b1,...,bn of elements of U to y1, … ,yn, if and only if both ɸ(A(x1, … ,xn), H)= T and ɸ(B(Y1,...,yn(,H)=T with respect to that assignment.
(iv) ɸ(LA(x1, … ,xn),H)=T with respect to a given assignment if and only if ɸ(A(x1, … ,xn),H’)=T with respect to that assignment, for every H’ such that HRH'.
(v) ɸ(∀xA(x,y1, … ,yn),H)=T with respect to an assignment a1 … ,an of elements of U to y1, … ,yn if and only if for every aЄ𝛹(H), ɸ(A(x,y1, … ,yn),H)=T with respect to an assignment of a,a1 … ,an to x, y1, … ,yn.
For the intended interpretation of ‘L,’ we take R to be a transitive and a symmetric relation; thus intuitively, all moments of time accessible from a time are mutually accessible.
6 Suppose that x and y are distinct objects, s is the set of x and y, and t1, t2, and t3 are points of time. Suppose further that x exists at t1 and t2 but not at t3, and y exists at t2 and t3 but not at t1. Assuming that s exists at t1, t2, and t3, given the axioms of the revised calculus, at t1 there exists an object x1 such that x1 fuses s at t1 and x1 is identical with x, and, at t3 there exists an object y1 such that y1 fuses sat t2 and y1 is identical withy, and, at t2 there exists an object z such that z fuses s at t2 and z is distinct from both x and y.
However, if K, the set of times specified in the semantics of ‘L,’ is such that the fusion of every non-empty subset of K is a member of K, then it is provable in the revised calculus that if an object x fuses a set s at a time t and y fuses sat another time t', and the same members of sexist at t and t', then x is identical with y. Suppose that t1 and t2 are distinct points of time, and t fuses the set of t1 and t2. Suppose also that an object x fuses a set s at t1 and y fuses s at t2. Assuming that s exists at t, given the fusion axiom, at t there exists an object z which fuses s at t. But since t1 and t2 are parts of t, if the same members of s exist both at t1 and t2, z fuses s at t1 and also at t2. But then, given the identity axiom, at t1 z is identical with x and at t2 z is identical with y. Hence x is identical with y. Reasoning like this can be used to show that if the semantics of ‘L’ specifies that the fusion of any non-empty subset of K is a member of K, then given that Alpha fuses b at t1 and the silver chain on my table at t2 fuses b at t2 and all the members of b exist both at t1 and t2, if the axioms of the revised calculus are true under their intended interpretation, then Alpha is identical with the silver chain on my table at t2.
7 For a similar argument, see Richard Cartwright's ‘Scattered Objects’ in Keith Lehrer, ed., Analysis and Metaphysics (Dordrecht, Holland; Boston, MA: Reidel 1975) 153-71.
8 Thomson (‘Parthood and Identity Across Time,’ 220) proposes the following analogue of (32):
∀x ∀y (⁓ ∀t ((x exists at t v y exists at t) → (x is part of y at t & y is part of x at t)) → x=y).
