Deductive Logic by St. George Stock

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DEDUCTIVE LOGIC

BY

ST. GEORGE STOCK, M.A.

PEMBROKE COLLEGE, OXFORD

PREFACE.

One critic, who was kind enough to look at this book in manuscript,
recommended me to abandon the design of Publishing it, on the ground
that my logic was too like all other logics; another suggested to me
to cut out a considerable amount of new matter. The latter advice I
have followed; the former has encouraged me to hope that I shall not
be considered guilty of wanton innovation. The few novelties which I
have ventured to retain will, I trust, be regarded as legitimate
extensions of received lines of teaching.

My object has been to produce a work which should be as thoroughly
representative of the present state of the logic of the Oxford Schools
as any of the text-books of the past. The qualities which I have aimed
at before all others have been clearness and consistency. For the task
which I have taken upon myself I may claim one qualification--that of
experience; since more than seventeen years have now elapsed since I
took my first pupil in logic for the Honour School of Moderations, and
during that time I have been pretty continuously engaged in studying
and teaching the subject.

In acknowledging my obligations to previous writers I must begin with
Archbishop Whately, whose writings first gave me an interest in the
subject. The works of Mill and Hamilton have of course been freely
drawn upon. I have not followed either of those two great writers
exclusively, but have endeavoured to assimilate what seemed best in
both. To Professor Fowler I am under a special debt. I had not the
privilege of personal teaching from him in logic,--as I had in some
other subjects; but his book fell into my hands at an early period in
my mental training, and was so thoroughly studied as to have become a
permanent part of the furniture of my mind. Much the same may be said
of my relation to the late Professor Jevons's Elementary Lessons in
Logic. Two other books, which I feel bound to mention with special
emphasis, are Hansel's edition of Aldrich and McCosh's Laws of
Discursive Thought. If there be added to the foregoing Watts's Logic,
Thomson's Outlines of the Laws of Thought, Bain's Deductive Logic,
Jevons's Studies in Deductive Logic and Principles of Science,
Bradley's Principles of Logic, Abbott's Elements of Logic, Walker's
edition of Murray, Ray's Text-book of Deductive Logic, and
Weatherley's Rudiments of Logic, I think the list will be exhausted of
modern works from which I am conscious of having borrowed. But, not to
forget the sun, while thanking the manufacturers of lamps and candles,
I should add that I have studied the works of Aristotle according to
the measure of my time and ability.

This work has had the great advantage of having been revised, while
still in manuscript, by Mr. Alfred Robinson, Fellow of New College, to
whom I cannot sufficiently express my obligation. I have availed
myself to the full of the series of criticisms which he was kind
enough to send me. As some additions have been made since then, he
cannot be held in anyway responsible for the faults which less kindly
critics may detect.

For the examples at the end I am mainly indebted to others, and to a
large extent to my ingenious friend, the Rev. W. J. Priest of Merton
College.

My thanks are due also to my friend and former pupil, Mr. Gilbert
Grindle, Scholar of Corpus, who has been at the pains to compose an
index, and to revise the proofs as they passed through the press.

And last, but not least, I must set on record my gratitude to
Commander R. A. Stock, R.N., one of Her Majesty's Knights of Windsor,
without whose brotherly aid this work might never have been written,
and would certainly not have assumed exactly its present shape.

OXFORD,

_October_ 22, 1888.

CONTENTS.

PREFACE.

INTRODUCTION, §§ 1-56.

PART I. Of Terms, §§ 57-171.

CHAP. I. Of the Term as distinguished from other words, §§ 57-76.

II. Of the Division of Things, §§ 77-85.

III. Of the Divisions of Terms, §§ 86-165.

IV. Of the Law of Inverse Variation of Extension and Intension, §§
166-171.

PART II. Of Propositions, §§ 172-185.

CHAP. I. Of the Proposition as distinguished from other Sentences,
§§ 172-185.

II. Of the Copula, §§ 186-201.

III. Of the Divisions of Propositions, §§ 202-273.

IV. Of the Distribution of Terms, §§ 274-294.

V. Of the Quantification of the Predicate, §§ 295-312.

VI. Of the Heads of Predicables, §§ 313-346.

VII. Of Definition, §§ 347-384.

VIII. Of Division, §§ 385-425.

PART III. Of Inferences, §§ 426-884.

CHAP. I. Of Inferences in general, §§ 426-441.

II. Of Deductive Inferences, §§ 442-448.

III. Of Opposition, §§ 449-478.

IV. Of Conversion, §§ 479-495.

V. Of Permutation, §§ 496-502.

VI. Of Compound Forms of Immediate Inference, §§ 503-532.

VII. Of Other Forms of Immediate Inference, §§ 533-539.

VIII. Of Mediate Inferences or Syllogisms, §§ 540-557.

IX. Of Mood and Figure, §§ 558-568.

X. Of the Canon of Reasoning, §§ 569-581.

XI. Of the General Rules of Syllogism, §§ 582-598.

XII. Of the Determination of the Legitimate Moods of Syllogism, §§
599-605.

XIII. Of the Special Rules of the Four Figures, §§ 606-620.

XIV. Of the Determination of the Moods that are valid in the Four
Figures, §§ 621-632.

XV. Of the Special Canons of the Four Figures, §§ 633-647.

XVI. Of the Special Uses of the Four Figures, §§ 648-655.

XVII. Of the Syllogism with Three Figures, §§ 656-666.

XVIII. Of Reduction, §§ 667-700.

XIX. Of Immediate Inference as applied to Complex Propositions, §§
701-730.

XX. Of Complex Syllogisms, §§ 731-743.

XXI. Of the Reduction of the Partly Conjunctive Syllogism, §§
744-752.

XXII. Of the Partly Conjunctive Syllogism regarded as all Immediate
Inference, §§ 753-759.

XXIII. Of the Disjunctive Syllogism, §§ 760-765.

XXIV. Of the Reduction of the Disjunctive Syllogism, §§ 766-769.

XXV. Of the Disjunctive Syllogism regarded as an Immediate
Inference, §§ 770-777.

XXVI. Of the Mixed Form of Complex Syllogism, §§ 778-795.

XXVII. Of the Reduction of the Dilemma, §§ 796-797.

XXVIII. Of the Dilemma regarded as an Immediate Inference, §§
798,799.

XXIX. Of Trains of Reasoning, §§ 800-826.

XXX. Of Fallacies, §§ 827-884.

EXERCISES.

INDEX.

INTRODUCTION.

§ 1. LOGIC is divided into two branches, namely--

(1) Inductive,

(2) Deductive.

§ 2. The problem of inductive logic is to determine the actual truth
or falsity of propositions: the problem of deductive logic is to
determine their relative truth or falsity, that is to say, given such
and such propositions as true, what others will follow from them.

§ 3. Hence in the natural order of treatment inductive logic precedes
deductive, since it is induction which supplies us with the general
truths, from which we reason down in our deductive inferences.

§ 4. It is not, however, with logic as a whole that we are here
concerned, but only with deductive logic, which may be defined as The
Science of the Formal Laws of Thought.

§ 5. In order fully to understand this definition we must know exactly
what is meant by 'thought,' by a 'law of thought,' by the term
'formal,' and by 'science.'

§ 6. Thought, as here used, is confined to the faculty of
comparison. All thought involves comparison, that is to say, a
recognition of likeness or unlikeness.

§ 7. The laws of thought are the conditions of correct thinking. The
term 'law,' however, is so ambiguous that it will be well to determine
more precisely in what sense it is here used.

§ 8. We talk of the 'laws of the land' and of the 'laws of nature,'
and it is evident that we mean very different things by these
expressions. By a law in the political sense is meant a command
imposed by a superior upon an inferior and sanctioned by a penalty for
disobedience. But by the 'laws of nature' are meant merely certain
uniformities among natural phenomena; for instance, the 'law of
gravitation' means that every particle of matter does invariably
attract every other particle of matter in the universe.

§ 9. The word 'law' is transferred by a metaphor from one of these
senses to the other. The effect of such a command as that described
above is to produce a certain amount of uniformity in the conduct of
men, and so, where we observe uniformity in nature, we assume that it
is the result of such a command, whereas the only thing really known
to us is the fact of uniformity itself.

§ 10. Now in which of these two senses are we using the term 'laws of
thought'? The laws of the land, it is plain, are often violated,
whereas the laws of nature never can be so [Footnote: There is a sense
in which people frequently speak of the laws of nature being violated,
as when one says that intemperance or celibacy is a violation of the
laws of nature, but here by 'nature' is meant an ideal perfection in
the conditions of existence.]. Can the laws of thought be violated in
like manner with the laws of the land? Or are they inviolable like the
laws of nature?

§ 11. In appearance they can be, and manifestly often are violated-for
how else could error be possible? But in reality they can not. No man
ever accepts a contradiction when it presents itself to the mind as
such: but when reasoning is at all complicated what does really
involve a contradiction is not seen to do so; and this sort of error
is further assisted by the infinite perplexities of language.

§ 12. The laws of thought then in their ultimate expression are
certain uniformities which invariably hold among mental phenomena, and
so far they resemble the laws of nature: but in their complex
applications they may be violated owing to error, as the laws of the
land may be violated by crime.

§ 13. We have now to determine the meaning of the expression 'formal
laws of thought.'

§ 14. The distinction between form and matter is one which pervades
all nature. We are familiar with it in the case of concrete things. A
cup, for instance, with precisely the same form, may be composed of
very different matter-gold, silver, pewter, horn or what not?

§ 15. Similarly in every act of thought we may distinguish two
things--

(1) the object thought about,

(2) the way in which the mind thinks of it.

The first is called the Matter; the second the Form of Thought.

§ 16. Now Formal, which is another name for Deductive Logic, is
concerned only with the way in which the mind thinks, and has nothing
to do with the particular objects thought about.

§ 17. Since the form may be the same, whilst the matter is different,
we may say that formal logic is concerned with the essential and
necessary elements of thought as opposed to such as are accidental and
contingent. By 'contingent' is meant what holds true in some cases,
but not in others. For instance, in the particular case of equilateral
triangles it is true to say, not only that 'all equilateral triangles
are equiangular,' but also that 'all equiangular triangles are
equilateral.' But the evidence for these two propositions is
independent. The one is not a formal consequence of the other. If it
were, we should be able to apply the same inference to all matter, and
assert generally that if all A is B, all B is A, which it is notorious
that we cannot do.

§ 18. It remains now for the full elucidation of our definition to
determine what is meant by 'science.'

§ 19. The question has often been discussed whether logic is a science
or an art. The answer to it must depend upon the meaning we assign to
these terms.

§ 20. Broadly speaking, there is the same difference between Science
and Art as there is between knowing and doing.

Science is systematized knowledge;
Art is systematized action.
Science is acquired by study;
Art is acquired by practice.

§ 21. Now logic is manifestly a branch of knowledge, and does not
necessarily confer any practical skill. It is only the right use of
its rules in thinking which can make men think better. It is
therefore, in the broad sense of the terms, wholly a science and not
at all an art.

§ 22. But this word 'art,' like most others, is ambiguous, and is
often used, not for skill displayed in practice, but for the knowledge
necessary thereto. This meaning is better conveyed by the term
'practical science.'

§ 23. Science is either speculative or practical. In the first case we
study merely that we may know; in the latter that we may do.

Anatomy is a speculative science;
Surgery is a practical science.

In the first case we study the human frame in order that we may
understand its structure; in the second that we may assist its
needs. Whether logic is a speculative or a practical science depends
entirely upon the way in which it is treated. If we study the laws of
thought merely that we may know what they are, we are making it a
speculative science; if we study the same laws with a view to deducing
rules for the guidance of thought, we are making it a practical
science.

§ 24. Hence logic may be declared to be both the science and the art
of thinking. It is the art of thinking in the same sense in which
grammar is the art of speaking. Grammar is not in itself the right
use of words, but a knowledge of it enables men to use words
correctly. In the same way a knowledge of logic enables men to think
correctly, or at least to avoid incorrect thoughts. As an art logic
may be called the navigation of the sea of thought.

§ 25. The laws of thought are all reducible to the three following
axioms, which are known as The Three Fundamental Laws of Thought.

(1) The Law of Identity--

Whatever is, is;

or, in a more precise form,

Every A is A.

(2) The Law of Contradiction--

Nothing can both be and not be;
Nothing can be A and not A.

(3) The Law of Excluded Middle--

Everything must either be or not be;
Everything is either A or not A.

§ 26. Each of these principles is independent and self-evident.

§ 27. If it were possible for the law of identity to be violated, no
violation of the law of contradiction would necessarily ensue: for a
thing might then be something else, without being itself at the same
time, which latter is what the law of contradiction militates
against. Neither would the law of excluded middle be infringed. For,
on the supposition, a thing would be something else, whereas all that
the law of excluded middle demands is that it should either be itself
or not. A would in this case adopt the alternative of being not A.

§ 28. Again, the violation of the law of contradiction does not
involve any violation of the law of identity: for a thing might in
that case be still itself, so that the law of identity would be
observed, even though, owing to the law of contradiction not holding,
it were not itself at the same time. Neither would the law of excluded
middle be infringed. For a thing would, on the supposition, be both
itself and not itself, which is the very reverse of being neither.

§ 29. Lastly, the law of excluded middle might be violated without a
violation of the law of contradiction: for we should then have a thing
which was neither A nor not A, but not a thing which was both at the
same time. Neither would the law of identity be infringed. For we
should in this case have a thing which neither was nor was not, so
that the conditions of the law of identity could not exist to be
broken. That law postulates that whatever is, is: here we have a thing
which never was to begin with.

§ 30. These principles are of so simple a character that the
discussion of them is apt to be regarded as puerile. Especially is
this the case with regard to the law of identity. This principle in
fact is one of those things which are more honoured in the breach than
in the observance. Suppose for a moment that this law did not
hold--then what would become of all our reasoning? Where would be the
use of establishing conclusions about things, if they were liable to
evade us by a Protean change of identity?

§ 31. The remaining two laws supplement each other in the following
way. The law of contradiction enables us to affirm of two exhaustive
and mutually exclusive alternatives, that it is impossible for both to
be true; the law of excluded middle entitles us to add, that it is
equally impossible for both to be false. Or, to put the same thing in
a different form, the law of contradiction lays down that one of two
such alternatives must be false; the law of excluded middle adds that
one must be true.

§32. There are three processes of thought

(1) Conception.

(2) Judgement.

(3) Inference or Reasoning.

§ 33. Conception, which is otherwise known as Simple Apprehension, is
the act of forming in the mind the idea of anything, e.g. when we form
in the mind the idea of a cup, we are performing the process of
conception.

§ 34. Judgement, in the sense in which it is here used [Footnote:
Sometimes the term 'judgement' is extended to the comparison of
nameless sense-impressions, which underlies the formation of
concepts. But this amounts to identifying judgement with thought in
general.] may be resolved into putting two ideas together in the
mind, and pronouncing as to their agreement or disagreement, e.g. we
have in our minds the idea of a cup and the idea of a thing made of
porcelain, and we combine them in the judgement--'This cup is made of
porcelain.'

§ 35. Inference, or Reasoning, is the passage of the mind from one or
more judgements to another, e.g. from the two judgements 'Whatever is
made of porcelain is brittle,' and 'This cup is made of porcelain,' we
elicit a third judgement, 'This cup is brittle.'

§ 36. Corresponding to these three processes there are three products
of thought, viz.

(1) The Concept.

(2) The Judgement.

(3) The Inference.

§ 37. Since our language has a tendency to confuse the distinction
between processes and products, [Footnote: E.g. We have to speak quite
indiscriminately of Sensation, Imagination, Reflexion, Sight, Thought,
Division, Definition, and so on, whether we mean in any case a process
or a product.] it is the more necessary to keep them distinct in
thought. Strictly we ought to speak of conceiving, judging and
inferring on the one hand, and, on the other, of the concept, the
judgement and the inference.

The direct object of logic is the study of the products rather than of
the processes of thought. But, at the same time, in studying the
products we are studying the processes in the only way in which it is
possible to do so. For the human mind cannot be both actor and
spectator at once; we must wait until a thought is formed in our minds
before we can examine it. Thought must be already dead in order to be
dissected: there is no vivisection of consciousness. Thus we can never
know more of the processes of thought than what is revealed to us in
their products.

§ 38. When the three products of thought are expressed in language,
they are called respectively

(1) The Term.

(2) The Proposition.

(3) The Inference.

§ 39. Such is the ambiguity of language that we have already used the
term 'inference' in three different senses--first, for the act or
process of inferring; secondly, for the result of that act as it
exists in the mind; and, thirdly, for the same thing as expressed in
language. Later on we shall have to notice a further ambiguity in its
use.

§ 40. It has been declared that thought in general is the faculty of
comparison, and we have now seen that there are three products of
thought. It follows that each of these products of thought must be the
result of a comparison of some kind or other.

The concept is the result of comparing attributes.
The judgement is the result of comparing concepts.
The inference is the result of comparing judgements.

§ 41. In what follows we shall, for convenience, adopt the phraseology
which regards the products of thought as clothed in language in
preference to that which regards the same products as they exist in
the mind of the individual. For although the object of logic is to
examine thought pure and simple, it is obviously impossible to discuss
it except as clothed in language. Accordingly the three statements
above made may be expressed as follows--

The term is the result of comparing attributes.
The proposition is the result of comparing terms.
The inference is the result of comparing propositions.

§ 42. There is an advantage attending the change of language in the
fact that the word 'concept' is not an adequate expression for the
first of the three products of thought, whereas the word 'term' is. By
a concept is meant a general notion, or the idea of a class, which
corresponds only to a common term. Now not only are common terms the
results of comparison, but singular terms, or the names of
individuals, are so too.

§ 43. The earliest result of thought is the recognition of an
individual object as such, that is to say as distinguished and marked
off from the mass of its surroundings. No doubt the first impression
produced Upon the nascent intelligence of an infant is that of a
confused whole. It requires much exercise of thought to distinguish
this whole into its parts. The completeness of the recognition of an
individual object is announced by attaching a name to it. Hence even
an individual name, or singular term, implies thought or
comparison. Before the _child_ can attach a meaning to the word
'_mother_,' which to it is a singular term, it must have
distinguished between the set of impressions produced in it by one
object from those which are produced in it by others. Thus, when
Vergil says

Incipe, parve puer, risu cognoscere matrem,

he is exhorting the beatific infant to the exercise of the faculty of
comparison.

§ 44. That a common term implies comparison does not need to be
insisted upon. It is because things resemble each other in certain of
their attributes that we call them by a common name, and this
resemblance could not be ascertained except by comparison, at some
time and by some one. Thus a common term, or concept, is the
compressed result of an indefinite number of comparisons, which lie
wrapped up in it like so many fossils, witnessing to prior ages of
thought.

§ 45. In the next product of thought, namely, the proposition, we have
the result of a single act of comparison between two terms; and this
is why the proposition is called the unit of thought, as being the
simplest and most direct result of comparison.

§ 46. In the third product of thought, namely, the inference, we have
a comparison of propositions either directly or by means of a
third. This will be explained later on. For the present we return to
the first product of thought.

§ 47. The nature of singular terms has not given rise to much dispute;
but the nature of common terms has been the great battle-ground of
logicians. What corresponds to a singular term is easy to determine,
for the thing of which it is a name is there to point to: but the
meaning of a common term, like 'man' or 'horse,' is not so obvious as
people are apt to think on first hearing of the question.

§ 48. A common term or class-name was known to mediæval logicians
under the title of a Universal; and it was on the question 'What is a
Universal 7' that they split into the three schools of Realists,
Nominalists, and Conceptualists. Here are the answers of the three
schools to this question in their most exaggerated form--

§ 49. Universals, said the Realists, are substances having an
independent existence in nature.

§ 50. Universals, said the Nominalists, are a mere matter of words,
the members of what we call a class having nothing in common but the
name.

§ 51. Universals, said the Conceptualists, exist in the mind alone,
They are the conceptions under which the mind regards external
objects.

§ 52. The origin of pure Realism is due to Plato and his doctrine of
'ideas'; for Idealism, in this sense, is not opposed to Realism, but
identical with it. Plato seems to have imagined that, as there was a
really existing thing corresponding to a singular term, such as
Socrates, so there must be a really existing thing corresponding to
the common term 'man.' But when once the existence of these general
objects is admitted, they swamp all other existences. For individual
men are fleeting and transitory--subject to growth, decay and
death--whereas the idea of man is imperishable and eternal. It is only
by partaking in the nature of these ideas that individual objects
exist at all.

§ 53. Pure Nominalism was the swing of the pendulum of thought to the
very opposite extreme; while Conceptualism was an attempt to hit the
happy mean between the two.

§ 54. Roughly it may be said that the Realists sought for the answer
to the question 'What is a Universal?' in the matter of thought, the
Conceptualists in the form, and the Nominalists in the expression.

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