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Games, graphs and circular arguments

The traditional logical fallacy called petitio principii (begging the question, arguing in a circle, circulus probandi, etc.) as traditionally conceived has been an elusive target for logical analysis. It has never been made clear why ostensibly circular arguments are in some precisely specifiable sense "fallacious". From a point of view of formal logic, we would scarcely want to deny that a proposition is a deductive consequence of itself. However, we hardly want to rule that every equivalence proof in mathematics that argues from A to B, and then back from B to A, has to thereby be fallacious. It is shown in [11] that one traditional account of the petitio, called the equivalence conception, rules that an argument is circular, just in case the conclusion is the same as a premiss. But it has never been made clear what is meant by`sameness', for strict identity would be too narrow a workable criterion, while logical equivalence (in classical first-order logic) would be too wide. A second traditional account, called in [11] the dependency conception, postulates that an argument is circular if some premiss rests on or depends on the conclusion. The notion of dependency seems to reflect an asymmetrical concept of directionality of proof, in the sense of`starting at A and going to B', which is unfamiliar as an item in the logician's vocabulary. How essential is this notion of`precedence' of propositions for the study of circular argument? Some historical remarks on the traditional conceptions of circular reasoning will set the stage for our argument that`precedence' is essential in examining circles. The complete and careful history of the petitio remains to be done, as a topic of importance in its own right. For the present, we want to garner some guidance from traditional accounts that might be helpful in working towards a useful model of circular argument. Hamblin [3, p. 32] states that the petitio principii might be explained