Theorists of argument suppose that arguments with definite conclusions that do not follow logically from their premiss or premisses have a “gap-filling” unexpressed premiss, whose identification and addition to the stated premiss or premisses would produce an argument whose conclusion does follow logically. A common explanation for the omission of a premiss, found from Aristotle to Quine and Copi, is that arguers leave unstated known information that the readers or hearers can supply for themselves.Traditional Aristotelian logic developed a method for supplying the supposedly omitted premiss in the case of incomplete categorical syllogisms. This traditional approach has two weaknesses. The first weakness is that not every argument that is supposed to have a gap-filling unstated premiss is an incomplete categorical syllogism. This weakness can be remedied by recognizing that filling out an incomplete categorical syllogism by adding the appropriate categorical statement is a special case of constructing a covering generalization of the argument. The second weakness is that there is indeterminacy about what covering generalization to supply, with respect to both which repeated components of the argument are to be subject to generalization and how broadly to generalize over them.This weakness can be remedied by adopting a policy of maximum generalization, subject to constraints of context and plausibility. A more fundamental objection to this approach is phenomenological: people reasoning and arguing in ways that are not logically compelling have no awareness of having omitted a premiss, even when they are reasoning something out for themselves. The whole approach of postulating an unexpressed gap-filler rests on a mistake, the mistake of supposing that there is a gap. Rather, logical consequence is a special case of a broader concept of consequence that includes material as well as formal consequence. The question to be asked in evaluating an argument with a definite conclusion is not how to expand it so as to make the conclusion follow logically but whether it has a non-trivially acceptable covering generalization that supports counterfactual instances. The broader concept of consequence has been recognized by Bolzano, Peirce, Ryle, Sellars,Toulmin, George, Brandom and others, but has not yet been recog-nized in introductory logic textbooks. It needs to be
