Storrs McCall

The title of this paper may seem to involve a contradiction: my purpose is to show that it does not.Individuals fall into two categories; those which depend for their existence upon the existence of other individuals, and those which do not. In the second category are found such things as shoes, ships, cabbages, kings, and discrete bits of sealing wax. These may be calledindividual substances, and the way in which the existence of a cabbage depends upon water and earth, or in which Descartes says the existence of all things depends upon God, will not be in point here. The individuals of the first category are characterized by a much more obvious kind of dependence. They include the sound of an individual shoe falling on the floor, the sinking of theBismarck, the stupidity of George I, the centre of gravity of a bit of sealing wax. All these are individuals, though they are not individual substances. They are what I shall callabstract individuals, orabstract particulars.

The basis of a rigorous formal axiomatization of quantum mechanics is constructed, built upon Dirac's bra-ket notation. The system is three-sorted, with separate variables for scalars, vectors and operators. First-order quantification over all three types of variable is permitted. Economy in the axioms is effected by, e.g., assigning a single logical function * to transform (i) a scalar into its complex conjugate, (ii) a ket vector into a bra and a bra into a ket, (iii) an operator into its adjoint. The system is accompanied by a formal semantics. Further papers will deal with vector subspaces and projection operators, operators with continuous spectra, tensor products, observables, and quantum mechanical probabilities