Are “All-and-Some†Statements Falsifiable After All?: The Example of Utility Theory

Popper's well-known demarcation criterion has often been understood to distinguish statements of empirical science according to their logical form. Implicit in this interpretation of Popper's philosophy is the belief that when the universe of discourse of the empirical scientist is infinite, empirical universal sentences are falsifiable but not verifiable, whereas the converse holds for existential sentences. A remarkable elaboration of this belief is to be found in Watkins's early work (1957, 1958) on the statements he calls “all-and-some,†such as: “For every metal there is a melting point.†All-and-some statements (hereafter AS) are both universally and existentially quantified in that order. Watkins argued that AS should be regarded as both nonfalsifiable and nonverifiable, for they partake in the logical fate of both universal and existential statements. This claim is subject to the proviso that the bound variables are “uncircumscribed†(in Watkins's words); i.e., that the universe of discourse is infinite.