Abstract

Consider the following. The first is a one-premise argument; the second has two premises. The question sign marks the conclusions as such. Matthew, Mark, Luke, and John wrote Greek. ? Every evangelist wrote Greek. Matthew, Mark, Luke, and John wrote Greek. Every evangelist is Matthew, Mark, Luke, or John. ? Every evangelist wrote Greek. The above pair of premise-conclusion arguments is of a sort familiar to logicians and philosophers of science. In each case the first premise is logically equivalent to the set of four atomic propositions: “Matthew wrote Greek”, “Mark wrote Greek”, “Luke wrote Greek”, and “John wrote Greek”. The universe of discourse is the set of evangelists. We presuppose standard first-order logic. As many logic texts teach, the first of these two premise-conclusion arguments—sometimes called a complete enumerative induction— is invalid in the sense that its conclusion does not follow from its premises. To get a counterargument, replace ‘Matthew’, ‘Mark’, ‘Luke’, and ‘John’ by ‘two’,’four’, ‘six’ and ‘eight’; replace ‘wrote Greek’ by ‘are even’; and replace ‘evangelist’ by ‘number’. This replacement converts the first argument into one having true premises and false conclusion. But the same replacement performed on the second argument does no such thing: it converts the second premise into the falsehood “Every number is two, four, six, or eight”. As many logic texts teach, there is no replacement that converts the second argument into one with all true premises and false conclusion. The second is valid; its conclusion is deducible from its two premises using an instructive natural deduction. This paper “does the math” behind the above examples. The theorem could be stated informally: the above examples are typical.