2. Such a particular is as portrayed by John Baker, in "Particulars: Bare, Naked, and Nude," Noûs 1 (1967), 2, pp. 211-212. 

"Particulars are nude in that they have no nature, that is, they are not necessarily connected to any specific property or set of properties. A nude particular has no nature, and is to be distinguished from the naked particular which has no properties. Those who claim that there are bare particulars, Russell, Bergmann, Allaire, et al., claim that they are nude of natures [...]" (p. 211)

3. By a minimal identity-relation, I understand one whose properties (symmetry, transitivity, weak reflexivity, indiscernibility for identicals and identity for self-identical indiscernibles) are criterial for identity.

4. The classical and non-classical relations differ in respect of the size of their domains, the non-classical relation including non-existents and the classical relation excluding them. 

5. Pick any particular. Corresponding thereto is a property that your particular has if anything does (see § 4.2). This property is a 'haecceity'. My notion of a haecceity and Adams' ([1979],[1981]) notion of a 'thisness' are compared in note 22.

6. A haecceity needs a support, or substratum, if it is to belong to an actual thing.  I call such a support an 'individual'.

7. In L, particular variables occur only in quantifier expressions, binding individual and haeceity variables as indicated in § 2.1.

8. My treatment of variable-binding in L is an adaptation of the treatment of variable-binding for first-order logic in Rogers [1971].

9.  is possibly null.

10. Let (t_k,T_l/T_k,t_l) be the result of replacing t_k and T_l everywhere in  by T_k and t_l, and then replacing every resultant occurrence of T_m ex t_n by t_m ex T_n. If  is a theorem, * indicates that (t_k,T_l/T_k,t_l) is also a theorem.  T1* thus indicates that (i) and (ii) are theorems.

(i) x(y(x = y)  (x = x))

(Identity is weakly reflexive for individuals.)

(ii) x(y(X = Y)  (X = X))

(Identity is weakly reflexive for haecceities.)

12. T9* indicates that (i) and (ii) are theorems.

(i) xy(y ex X) xy(x ex Y)

(Every haecceity is exemplified only if every individual exemplifies.)

(ii) xy(x ex Y) xy(y ex X)

(Every individual exemplifies only if every haecceity is exemplified.)

14. Embodiment relates a particular to a haecceity just in case there is some complex to which the haecceity and constituents of the particular belong (see T32).

15. Containment relates a particular to an individual just in case there is some complex to which the individual and constituents of the particular belong (see T33).

16. Let (t_i.T_k//T_k.t_i) be obtained from  by simultaneously replacing t_i and T_k everywhere in  by T_i and t_k; "cont" and "emb" by "emb" and "cont"; and then T_k.t_i by t_i.T_k.  If  is a theorem, # indicates that (t_i.T_k//T_k.t_i) is also a theorem.  T15# thus indicates that (i) and (ii) are theorems.

(i) vuwxy(v.U = w.X  (v.U emb Yw.X emb Y))

(Complexes are indiscernible in respect of the haecceities they embody.)

(ii) vuwxy(v.U = w.X  (v.U cont yw.X cont y)) 

(Complexes are indiscernible in respect of the individuals they contain.)

18. The following is also a theorem of C:

xy[¬(x.Y = x.Y)  ¬wz(x.Y = w.Z)]

19. The wffs of the classical and non-classical theories coincide.  Their theorems however differ, the Reflexive Law of Equality and logically equivalent principle of the Identity of Indiscernibles being provable in the classical theory, while their contradictory negations are provable in the non-classical.  The non-classical theory is thus a deviation, as Susan Haack defines "deviation" in her (1974), of classical identity theory.

20. The individual variables of classical identity theory I refer to as particular variables, a particular being construed here as an entity whose constituents are individuals and haecceities.

21. C-complexes in C, C+ and C- are intended both as realizations of minimal, classical and non-classical identity theory, and as schematizations of contrasting systems of real relations of particulars and their constituents.  C-complexes thus constitute a model in both senses Jean Ladrière distinguishes for this term in his (1979).  For Ladrière, a model is:

[...] une construction idéale, intermédiaire entre une théorie au sens strict (considerée comme ensemble de propositions munie d'une structure déductive) et un domaine concret dont il s'agit d'analyser le fonctionnement [...] (p. 183)

22. Between my haecceity and Robert Adams' (1979, 1981) thisness there are two key differences.  First, a thisness depends for its features upon Adams' conception of what a thisness is.  In contrast, every feature of a haecceity--including its being an identity property*--is engendered by the mutual relations of complexes and their constituents.  While the features of a thisness are thus the artefacts of arbitrary legislative postulation, the features of a haecceity are conferred upon it by the role it plays in the system of complexes.

 Second, Adams makes it clear that a thisness is not "a special sort of metaphysical component of [a particular]":

I am not proposing to revive this aspect of [Scotus'] conception of a haecceity, because I am not committed to regarding properties as components of [particulars]. ([1979], p.7)

23. A comparison of the strongly non-classical system P-- with the weakly classical system P+- may be of interest.  In P--, no haecceity is exemplified (P10'*).  From P10'* and company it follows that in P-- nothing is identical with anything else.  So, every complex in P-- is non-self-identical; so much was to be expected. However, non-self-identical complexes also occur in P+-, in which particulars, individuals, and haecceities are self-identical*.

Does kripke exemplify THE AUTHOR OF WAVERLEY?  Unlike P++, P+- doesn't require this. So let's say kripke doesn't exemplify THE AUTHOR OF WAVERLEY.  Presto! From P7 it follows that kripke.THE AUTHOR OF WAVERLEY is not self-identical*.

24. The conception of a natured particular clashes with a fundamental trait, according to André Lichnerowicz (1972), of contemporary mathematical thought: "l'absence de toute métaphysique de l'identité et de la chose en soi" (p. 1502).  A kindred anti-metaphysical strain in neo-positivist philosophy of logic has perhaps been responsible for the reluctance of  linguistically oriented analytical philosophers to posit a logical nature for particulars, these tending to be regarded, as Manuel Sacristán points out in his 1984, as "individuos puntuales sin intrincación ontológica" (p. 249).

25. x(x = xy(x = y))

26. This and the ensuing remarks are based on section 48.0 of Gerald Massey's [1970].

27. The mathematical existence of such entities I take to turn upon the following stipulations of Hilbert (as stated by O. Becker [1927] and cited by Fernando Gil [1971]):
 

Déf. 1:  on appelle mathématiquement existants les objectités dont on fait le thème ("Thema") d'une théorie mathématique et qui peuvent fonctionner sans contradiction dans cette théorie.  Déf. 2:  on appelle mathématiquement existants les objets qu'avec des moyens déterminés avec précision peuvent être construits à partir de points de départ fondés.

28. In 'The Paradox of Identity', I refer the difference between 'a = a' and 'a = b' to features of a and b.  In 'Let X = X But Not Necessarily', I show that the failure of (e.g.) nine and the number of planets to share their modal properties and relations supervenes on features of identity and identicals.  Finally, in 'Where Does PM 14.05 Come From?', I show that a key clause in Russell's Theory of Descriptions likewise follows from the principles governing complexes and their constituents.

29. The only argument of which I am aware for these restrictions is Georg Kreisel's argument in his (1969, 1971) that "true in all models" and "true in all set-theoretic models" are extensionally equivalent.  But Kreisel's argument rests on the "intuitive validity" of the axioms of predicate logic with identity; and as I suggest here, "x(x = x)" is not valid.  For Tory restatements of Kreisel's argument, see John Etchemendy (1990: p. 144 f) and Daniel Quesada (1985: p. 154 f).  For a contrasting view, see 'The Paradox of Identity,' Appendix One.