Truth and Consequence in Mediaeval Logic (Studies in Logic and the Foundations of Mathematics, 11)

STUDIES IN LOGIC AND THE

FOUNDATIONS OF MATHEMATICS L.E.J.BROUWER I E. W.BETH I A.HEYTING EDITORS

Truth and Consequence • 1ft

Mediaev·al Logic ERNEST A. MOODY ORTH·HOLLA D PUBLISHING COMPA AMSTERDAM

Y

STUDIES IN L"OGIC AND

THE FOUNDATIONS OF MATHEMATICS

L. E. J. BROUWER E. W. BETH A. HEYTING Editors

~It

m ~

1953 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

~RUTH

AND CONSEQUENCE IN

MEDIAEVAL LOGIC

ERNEST A. MOODY Associate Professor of Philosophy Columbia University

1953 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM

'I I',

COPYRIGHT

1953

N. V. Noord-HollandBche Uitgeversmaatschappi} Amsterdam

(1

/

PRINTED IN THE NETHERLANDS DRUKKERIJ HOLLAND N.V.• AMSTERDAM:

PREFACE The study which follows, written with frequent interruptions over the course of nearly three years, represents the attempt of a mediaeval scholar, who is only an amateur in the field of contemporary formal logic, to interpret some of the rich content of fourteenth century logical literature in the logical language of the twentieth century. Both the vast extent of mediaeval writings in logic, and the difficulties of an exact understanding of its content, make the task of representing this logic in modern logical language one of great difficulty. Such attempts as can be made toward formalized representation of mediaeval logic must be viewed as tentative and partly conjectural translations of rules and arguments originally presented in a word language which, though perhaps quite precise for those who used it, is for us a foreign and only half understood language. The small space at my disposal has made it impossible to give adequate presentation of the textual foundations justifYing the interpretations which had to be chosen. References to the source materials, and occasional quotations of pertinent passages, are all that could be given. It is my hope that this introductory sketch, for all its shortcomings, may stimulate others to pursue the investigation of mediaeval logical literature with greater thoroughness and greater competence. Grateful acknowledgement is made to the many scholars who have recognized the interest of mediaeval logic, even for the twentieth century, and who have done arduous pioneer work in editing texts, analyzing their content, and grasping the significance of this material not merely for the history of logic, but for logical theory itself. E.A.M.

CONTENTS Page PREFACE

•••

Conspectus of Symbolic Notations. I.

3. 4. 5. 6. 7.

The Mediaeval Conception of Logic . . . . . . . The Formal and Material Constituents of Discourse Signification and Supposition . . . . . . . . . . Personal (Formal) and Material Supposition . . . Object Language, Metalanguage, and the Transcendental Terms ". . . . . . . . . . . . .

The Formal Classification of Propositions . The Logical Import of the Elementary Copula The Sentential Operators. Quantification . . . . . . Time Range and Modality

26

30 32 38 43 53

THE THEORY OF CONSEQUENCE

§ 13. The Meaning of ·Consequence' in Mediaeval Logic § 14. Formal and Material Consequences § 15. The Mediaeval Logic of Propositions V.

13 16 18 23

THE THEORY OF TRUTH CONDITIONS

§ 8. § 9. § 10. § 11. § 12.

IV.

1 10

LOGIC AND LANGUAGE

§ § § § §

III.

VII

INTRODUCTION

§ 1., Survey of the Development of Mediaeval Logic. . § 2. Aim and Scope of the Present Study . . . . . II.

V

64 70 80

TRUTH AND CONSEQUENCE

§ 16. The Aristotelian Definition of Truth . § 17. The Paradox of the Liar. . . . . .

101 103

Abbreviations for Works Cited, and Bibliography

III

CONSPECTUS OF SYMBOLIC NOTATIONS I.

VARIABLES:

Pronominal: x, y, z ,(Cf. § 5 and § 9) Nominal: F, G, H (Replaceable by a term; cf. § 4) Sentential: p, q, r (Replaceable by a sentence) When we wish to represent the name of whatever term, or whatever proposition, may replace the nominal or the sentential variables of a given formula, we enclose the variable in single quotation marks. When we wish to represent an ostensive use of whatever term may replace a given nominal variable of a formula, corresponding to the use of that term with a demonstrative pronoun prefixed to it, we add a subscript numeral to the nominal variable in question. II.

SENTENTIAL CONNECTIVES:

Negation: -p Definition: Conjunction: p.q (or pq) Possibility: Disjunction: p v q (Cf. "Simple" Implication: P-1q (Cf. "Simple" Equivalence: p.=.q (Cf. "As of now" Implication: p-:Jq (Cf. "As of now" Equivalence: p-:=q III.

§ 14 § 14 § 14 § 14

and and and and

§ 15) § 15) § 15)

§ 15)

SYMBOLS OF QUANTIFICATION- AND OF IDENTITY

Universal quantifier: Particular quantifier: Identity: IV.

p.=Df.q (Cf. § 14)

Op

(x), (y), (x,y) "For every " (Ex), (Ey), (Ex,y)"For some " .... = .. .. (E.g., "(x).x = x")

TEMPORAL OPERATORS:

These are required to indicate, in sentences whose copula is of past or future tense, the time (relative to the time the sentence is stated) for which a term of the sentence is indicated to stand for its values, or for those for which the other term stands. Where no temporal or modal operator is prefixed to a matrix expression in the formula of an analysed sentence, the supposition of the term represented by the variable in that matrix expression is

VIII

CONSPECTUS OF SYMBOLIC NOTATIONS

understood for the present time in which the sentence is stated. Of. § 12. ">Fx" means: "x was an F at some time prior to now" "' and ~ Fx.>Gx Every F'wasaG: : : :(Ex) :Fx v> Fx :.(x) :Fx v> Fx.""J. >Gx

12.2

Every term having supposition, as subject, with respect to a verb of future time, is ampliated to stand for that which exists or for that which will exist. Some F will be a G::-::(Ex):Fx v