Formal Language and Postulates for "The Paradox of Identity" and "A Theory of Complexes"

Theory P

Predicates P is formulated in a first-order, multi-sorted language L*. The vocabulary of L* includes four predicates;
(1.1) "ex" for exemplification
(1.2) "=" for identity
(1.3) "emb" for embodiment
(1.4) "cont" for containment

Operations an operation symbol;
(1.5) the operation symbol: .

Variables four styles of variables;
(2.1) variables for individuals: v, w, x, y, z, v₁, w₂,...
(2.2) variables for haecceities: V, W, X, Y, Z, V₁, W₂,...
(2.3) variables for particulars: v, w, x, y, z, v₁, w₂,...
(2.4) variables for complexes, as specified by the rule: if t_i is an individual variable and T_k a haecceity variable, t_i.T_k is a complex variable.

Connectives the usual sentential connectives;
(3.1) ¬, , & , ,

Quantifiers and the universal and existential quantifiers.
(3.2) ,

Metalanguage In the metalanguage of L*, t₁, t₂,... range over individual variables; T₁, T₂,... over haecceity variables; t₁, t₂,... over particular variables; and , ,... over expressions.

Atomic
Formulas An atomic formula of L* is any expression provided for by the following conditions:
(4.1) t_i ex T_k, t_i = t_k and T_i = T_kare atomic formulas.
(4.2) t_i.T_k emb T_m, t_i.T_k cont t_m, and t_i.T_k = t_m.T_n are atomic formulas.
(4.3) An atomic formula results when a particular variable is substituted in an atomic formula for a complex variable.

Formulas (5.1) Every atomic formula is a formula.
(5.2) If is a formula, so is ¬.
(5.3) If and are formulas, so are (), ( & ) and ().
(5.4) If is a formula, so are t and t.

Theory P

Predicates	P is formulated in a first-order, multi-sorted language L. The vocabulary of L includes four predicates; (1.1) "ex" for exemplification (1.2) "=" for identity (1.3) "emb" for embodiment (1.4) "cont" for containment
Operations	an operation symbol; (1.5) the operation symbol: .
Variables	four styles of variables; (2.1) variables for individuals: v, w, x, y, z, v₁, w₂,... (2.2) variables for haecceities: V, W, X, Y, Z, V₁, W₂,... (2.3) variables for particulars: v, w, x, y, z, v₁, w₂,... (2.4) variables for complexes, as specified by the rule: if t_i is an individual variable and T_k a haecceity variable, t_i.T_k is a complex variable.
Connectives	the usual sentential connectives; (3.1) ¬, , & , ,
Quantifiers	and the universal and existential quantifiers. (3.2) ,
Metalanguage	In the metalanguage of L*, t₁, t₂,... range over individual variables; T₁, T₂,... over haecceity variables; t₁, t₂,... over particular variables; and , ,... over expressions.
Atomic Formulas	An atomic formula of L* is any expression provided for by the following conditions: (4.1) t_i ex T_k, t_i = t_k and T_i = T_kare atomic formulas. (4.2) t_i.T_k emb T_m, t_i.T_k cont t_m, and t_i.T_k = t_m.T_n are atomic formulas. (4.3) An atomic formula results when a particular variable is substituted in an atomic formula for a complex variable.
Formulas	(5.1) Every atomic formula is a formula. (5.2) If is a formula, so is ¬. (5.3) If and are formulas, so are (), ( & ) and (). (5.4) If is a formula, so are t and t.

[cont.]