Aristotle and the Consequentia Mirabilis

Abstract

In a passage of his Protrepticus mentioned by several ancient authors Aristotle wrote: εἰ μὲνφιλοσοφητέον φιλοσοφητέον, καὶ εἰ μὴ φιλοσοφητέον φιλοσοφητέον πάντως ἄρα φιλοσοφητέον (V. Rose, Aristotelis Fragmenta, 51. Cf. R. Walzer, Aristotelis Dialogorum Fragmenta, p. 22; W. D. Ross, Select Fragments of Aristotle, p. 27). That is to say, ‘If we ought to philosophise, then we ought to philosophise; and if we ought not to philosophise, then we ought to philosophise (i.e. in order to justify this view); in any case, therefore, we ought to philosophise’. So far as I know, this is the first appearance in philosophical literature of a pattern of argument that became popular among the Jesuits of the seventeenth century under the name of the consequentia mirabilis and inspired Saccheri's work Euclides ab Omni Naevo Vindicatus, in which theorems of non-Euclidean geometry were proved for the first time. The later history has been told by G. Vailati (in his article on Saccheri's Logica Demonstrativa, ‘Di un’ opera dimenticata del P. Gerolamo Saccheri’, reprinted in his Scritti, 1911, pp. 477–84), G. B. Halsted (in the preface to his 1920 edition of Saccheri's Euclides), and J. -Łukasiewicz (in his ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalküls’, Comptes Rendus des séances de la société des sciences et des lettres de Varsovie, Classe III, Vol. xxiii, 1930, p. 67). In this note I wish to consider only the early history of the argument and in particular a curious criticism of it which appears in Aristotle's Prior Analytics.