William Kneale

The logic of Saint Anselm

Saint Anselm, the 11th-century Archbishop of Canterbury, is well-known for his ontological argument for the existence of God. This book places his argument in the context of modal logic derived from Boethius, and also shows how linguistic analysis was developed extensively through Anselms work.

UNIVERSALITY AND NECESSITY

IN THIS paper I wish to consider some attempts which philosophers have made to define the notion of necessity by reference to universality. I think these attempts are mistaken, and I shall put forward some arguments in justification of my view; but my main purpose is not so much to argue a case against die suggested definitions as to bring them together and to point out some similarities and differences which have not been sufficiently noticed. I think that by so doing I may perhaps arouse doubts in the minds of some who have not felt doubts on the subject before. But apart from mat I shall be glad to hear a discussion of the subject, and that is my main reason for choosing it

Russell's Paradox and Some Others

I IN A RECENT paper [1972] I argue (i) that a language which makes possible the characteristically human form of social life must allow for talk not only about its own sounds but also about communication by means of those sounds, (2) that failure to recognise this had led many philosophers into a dangerous confusion between sentences and propositions, (3) that attempts to formulate logic as a theory of grammatically well formed sentences involve neglect of the token-reflexive device and misunderstanding of the role of definite descriptions, and (4) that the paradox of the Liar holds no terrors for those who realise how the notion of truth is

Aristotle and the Consequentia Mirabilis

In a passage of his Protrepticus mentioned by several ancient authors Aristotle wrote: eἰ μὲν φιλοσοφητέον φιλοσοφητέον, καὶ eἰ 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 Saccheris 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 Saccheris 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 Saccheris Euclides ), and J. -Łukasiewicz (in his ‘Philosophische Bemerkungen zu mehrwertigen Systemen des Aussagenkalkuls’, Comptes Rendus des seances de la societe 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 Aristotles Prior Analytics .

Numbers and Numerals

i In order to give an acceptable account of truth which avoids on the one hand the antinomy of the Liar and on the other the Tarski paradox that natural languages with pretensions to omnicompetence are all inconsistent, we must distinguish between propositions and the sentences or other propositional phrases by utterance of which propositions may be expressed (cf. Kneale [19721]). Similarly, in order to make clear what is involved in saying that predicates are true of (or satisfied by) things, and so to solve Russells paradox in the form in which he first presented it, we must distinguish between predicates in the logical sense presupposed by such talk and the predicative phrases of grammatical analysis by utterance of which predicates may be expressed (cf. Kneale [i971]). In this paper I wish to argue that numbers belong to the same realm as propositions and predicates, and that through recognition of this fact we can escape from some notorious difficulties in the philosophy of mathematics. Briefly my thesis is that numbers are to numerals as propositions are to sentences. But without further comment this is not very enlightening, and I must therefore begin by trying to explain it in some detail.

The Development of Logic.

The primary purpose of this book has not been to recount all that past scholars have said about the science, but rather to record the first appearances of those ideas which seem most important in the logic of our own day.