Wholistic reference

Wholistic reference is reference to the whole—with respect to the context. In its strongest, unqualified form, the principle of wholistic reference is the proposition that each and every proposition, regardless how limited the referents of its non-logical or content terms, refers to the whole of its universe of discourse. According to this principle every proposition of number theory, even an equational proposition such as 5 + 7 = 12, refers not only to the individual numbers that it happens to mention but to the whole universe of numbers. The relation verb ‘refers’ is being used in its broad sense (loosely “is about”) and not as a synonym for ‘names’ in the sense of “is a name of”.

George Boole (1815–1864) introduced this principle into modern logic: Even though he changed from a monistic fixed-universe framework in his 1840s writings to a pluralistic multiple-universe framework in 1854,^[1] he never wavered in his frank avowal of the principle of wholistic reference. Indeed, he took it as an essential accompaniment to his theory of concept formation and proposition formation. For Boole, the essential first step in the process of conceiving of a proposition preliminary to making a judgement of its truth or falsity – or even using it in a deduction, however hypothetically – was to conceive of the universe of discourse. See Boole 1854/2003, xxi, 27, 42, 43.^[2] One statement of his principle is in the sentence immediately following his definition of universe of discourse, which is his first use of the expression 'universe of discourse' and probably the first in the history of the English language. See the next section.

Similar views, perhaps not similarly motivated, are found in later logicians, including Gottlob Frege (1848–1925). Some recent formulations of standard one-sorted first-order logic seem to be in accord with a form of it, if they do not actually imply the principle itself.^[3]

Boole’s 1854 definition

In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined. The most unfettered discourse is that in which the words we use are understood in the widest possible application, and for them the limits of discourse are co-extensive with those of the universe itself. But more usually we confine ourselves to a less spacious field. Sometimes, in discoursing of men we imply (without expressing the limitation) that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse.^[4]

References