4.1 Language and Logic

	P is developed within L*, a first-order language obtained from L' by adding the formation rule (4.3). (4.3) An atomic formula results when a particular variable is substituted in an atomic formula for a complex variable. The definition of variable-binding and formulation of the quantifier rules in L* is as in L'. The deductive apparatus of P is that of C plus the inference rules I1 and I2. I1: From  to infer , where  results from  by substituting t for t.T at one or more places of its occurrence. I2: From  to infer , where  results from  by substituting  t.T for t at one or more places of its occurrence. I1 licenses taking an assertion about any C-complex into one about the corresponding particular, while I2 licenses taking an assertion about any particular into one about the corresponding C-complex.  The warrant for I1 and I2 is that the C-complex t.T and particular t are one and the same entity.  In HI, the properties of identity are determined by the reciprocal relations of individuals and haecceities. In C, these properties are determined by the reciprocal relations of individuals, haecceities and complexes. In P, particulars are C-complexes, and so the properties of identity (as this applies to particulars) are determined by the reciprocal relations of individuals, haecceities, and C-complexes.