Claude Gratton

: I explore the logic of counterexamples by possible conjunction in order to extend their use to estimate the degree of support of premises; address some problems with my proposal; discuss some ways of teaching this extended use; and argue that conditional probability fails to express the degree of support of premises. The scant literature on this topic sometimes presents the degree of support of premises P1…Pn for conclusion C in terms of conditional probability, Pr. I will argue that the degree of support is better expressed by the probability of the conditional statement expressing the inference, Pr; and prove that Pr is not equivalent to Pr.

I investigate various logical and contextual factors involved in the derivation and use of infinite regresses in infinite regress arguments. I discuss the concept of a regress; identify different kinds of infinite regresses; clarify the core structure of most infinite regress arguments; use the logic of binary relations to explain the derivation of the most common kind of infinite regress encountered in my research; explain how circular definitions and circular explanations entail infinite regresses; discuss the rhetorical features of infinite regress arguments that are presented or analyzed in terms of recurring questions and answers; examine different conditions under which an infinite regress is used to refute a statement; identify different structures of infinite regress arguments that are presented or analyzed in terms of recurring problems and solutions

Showing that the premises of an argument are not sufficient for (viz., do not entail) a conclusion a conclusion involves citing a counterexample that would show the premises of the argument to be true and the conclusion false. This paper distinguishes counterexamples by analogy (an argument with the same form as argument against which it is advanced but where it is clearly the case that the premises are true and the conclusion is false), counterexamples by possible conjunction (a conjunction of the premises and the negation of the conclusion), and counter-arguments (arguments directed against the truth of the conclusion). After detailing the logical differences between these concepts, the paper describes the pedagogical significance of this distinction and provides an assortment of test exercises for students.

Henry W. Johnstone (1996) attempts to use a notion of postponement to give a general account of viciousness of infinite regresses. Though some of his examples suggest that his notion applies to only beginningless regresses (...eRdRcRbRa), I will show that it also applies to endless ones (aRbRcRdRe...). Unfortunately, despite this expanded application, it does not apply to all vicious regresses, even to some of his own examples; it is cumbersome and unnecessary, and it fails to explain how some infinite regresses entail a contradiction.

Infinite regress arguments are part of a philosopher's tool kit of argumentation. But how sharp or strong is this tool? How effectively is it used? The typical presentation of infinite regress arguments throughout history is so succinct and has so many gaps that it is often unclear how an infinite regress is derived, and why an infinite regress is logically problematic, and as a result, it is often difficult to evaluate infinite regress arguments. These consequences of our customary way of using this tool indicate that there is a need for a theory to re-orient our practice. My general approach to contribute to such a theory, consists of collecting and evaluating as many infinite regress arguments as possible, comparing and contrasting many of the formal and non-formal properties, looking for recurring patterns, and identifying the properties that appeared essential to those patterns. Two very general questions guided this work: How are infinite regresses generated in infinite regress arguments? How do infinite regresses logically function as premises in an argument? In answering these questions I clarify the notion of an infinite regress; identify different logical forms of infinite regresses; describe different kinds of infinite regress arguments; distinguish the rhetoric from the logic in infinite regress arguments; and suggest ways of improving our discussion and our practice of constructing and evaluating these arguments.

I examine a number of infinite regress arguments whose infinite regresses are presented or described in terms of recurring questions and answers in order to determine whether such recurring questions have any role in generating these infinite regresses, or in disqualifying the recurring answers. I argue that despite the existence of such infinite regress arguments and the suggestions of some philosophers, these recurring questions have no such roles. Some ways of handling these infinite regress arguments are then proposed.