Contemporary British Philosophy

Personal Statements Third Series

H D Lewis

• Publisher: Routledge
• Year: 2014
• ISBN:
  9781317853275
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

This is Volume IX of twenty-two in a collection on 20th Century Philosophy. Originally published in 1964, this text is volume three of three on contemporary British Philosophy and includes personal statements from varied contributors of the time.

THE PROVINCE OF LOGIC

By WILLIAM KNEALE

Fellow and Tutor of Exeter College, Oxford

THE PROVINCE OF LOGIC

I

UNTIL recently logic has always been presented as the science of those common principles of reasoning which are valid for all possible studies, whatever their subject matter, and it is still conceived as such by mathematicians and other specialists who have most occasion to talk of logic in connection with their own work. It is therefore rather surprising to find in modern times two tendencies which are at variance with this conception.

On the one hand, it has become customary within the past forty years to present logic as a deductive science in which we have only one or two rules of inference but a host of asserted propositions, some introduced as axioms and the others derived from these in accordance with the rule or rules of inference. In antiquity already the principles of non-contradiction and excluded middle were allowed a special place as fundamental laws, and at a later date both Leibniz and Kant wrote sometimes as though the principle of non-contradiction were the sum of all truth in logical studies; but in general these laws have not been regarded as axioms from which the logician should derive theorems by deduction. The fashion of presenting logic as a deductive system in which all the asserted propositions have the same status as the so-called laws of thought is derived ultimately from Frege’s Begriffsschrift of 1879, and it has probably been accepted in this century without much question because it is associated historically with a great enrichment of the science. Frege, indeed, said that he favoured the use of the modus ponendo ponens as sole rule of inference only because he thought this restriction made for greater rigour; and in his Grundgesetze he allowed other patterns of inference for reasons of practical convenience. But he seems never to have questioned the need for some axioms. Presumably he thought he was committed to them by his programme of exhibiting arithmetic as an extension of logic.

On the other hand, there has arisen in British philosophy during the past twenty years a fashion for using the word ‘logic’ in a much wider sense, as though it meant the study of all interconnections of meaning between words. We find it said, for example, that a statement about the incompatibility of two colours belongs to the ‘logic of colour words’, that a philosophical enquiry about the status of minds is really a discussion of the ‘logic of psychological verbs’, and in general that philosophical questions are all problems of ‘logical grammar’ or, worse still, of ‘logical geography’. The philosophers who talk and write in this way are usually not very much interested in the development of logic started by Frege (though their fondness for the word ‘logic’ may be due to the prestige which logic has acquired from that development), and their departure from traditional usage is quite different from that of mathematical logicians. With them ‘logic’ is no longer the name of a science concerned with the principles of inference common to all studies, but rather a name for any collection of rules in accordance with which we may argue in some context. One philosopher of this persuasion has even said that every kind of statement has its own logic. In such a welter of metaphor and epigram it is difficult to know what we are expected to take seriously; but it seems clear that this way of talking involves abandonment of the notion that logic is concerned with form as opposed to subject matter.

In this essay I shall try first to show how the restricted calculus of propositional functions can be presented satisfactorily without axioms. This result is not very important in itself, since it amounts only to a proof that this part of the systems of Frege and Russell can be derived from something simpler and nearer to the traditional conception of logic. But it is interesting because it suggests a reason for delimiting logic in such a way as to exclude not only Frege’s talk about the reduction of arithmetic to logic, which led to the use of axioms in logic, but also modern talk about the logic of colour words. The attempt to work out this suggestion will occupy the last section of the essay.

II

We can dispense with axioms in logic if we can show that the propositions taken as axiomatic by Frege or his successors can all be derived from any propositions whatsoever by application of rules of inference valid for all subject matters. For to say that a proposition is necessary absolutely is surely the same as to say that it is necessary in relation to anything whatsoever, i.e. that no special premisses are required for the proof of it. But how in detail can we carry out this programme?

The author of the Quaestiones Exaclissimae in Universam Aristotelis Logicam, formerly attributed to Duns Scotus, gives in Quaestio 10 on Prior Analytics i the following five remarks about consequentiae:

(1a) ‘From any statement which contains a formal contradiction any statement whatsoever follows in a formal consequentia, e.g. from “Socrates exists and Socrates does not exist”, which contains a formal contradiction, there follows “A man is an ass” or “The stick is standing in the corner”.’

(2a) ‘From any impossible state...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Original Title Page
• Original Copyright Page
• Editor's Preface
• Table of Contents
• The Rational and the Empirical
• Political Justification
• Philosophical Scepticism
• On Seeing and Hearing
• Self-activity and its Modes
• Philosophical Knowledge
• The Necessity of Metaphysics
• An Examination of Tenses
• Identification and Existence
• What is to become of Philosophical Theology?
• The Province of Logic
• Worship and Idolatry
• Freewill and Punishment
• Ethical Intuition
• Fifty Years of Philosophy
• Three Views Concerning Human Knowledge
• The Argument from Illusion
• Belief and Action
• Sensation
• How I see Philosophy
• Biographical Notes
• Index