_Introduction._--Logic is the science of the processes of inference,
what, then, is inference? It is that mental operation which proceeds by
combining two premises so as to cause a consequent conclusion. Some
suppose that we may infer from one premise by a so-called "immediate
inference." But one premise can only reproduce itself in another form,
e.g. all men are some animals; therefore some animals are men. It
requires the combination of at least two premises to infer a conclusion
different from both. There are as many kinds of inference as there are
different ways of combining premises, and in the main three types:--
1. _Analogical Inference_, from particular to particular: e.g.
border-war between Thebes and Phocis is evil; border-war between Thebes
and Athens is similar to that between Thebes and Phocis; therefore,
border-war between Thebes and Athens is evil.
2. _Inductive Inference_, from particular to universal: e.g. border-war
between Thebes and Phocis is evil; all border-war is like that between
Thebes and Phocis; therefore, all border-war is evil.
3. _Deductive or Syllogistic Inference_, from universal to particular,
e.g. all border-war is evil; border-war between Thebes and Athens is
border-war; therefore border-war between Thebes and Athens is evil.
In each of these kinds of inference there are three mental judgments
capable of being expressed as above in three linguistic propositions;
and the two first are the premises which are combined, while the third
is the conclusion which is consequent on their combination. Each
proposition consists of two terms, the subject and its predicate, united
by the copula. Each inference contains three terms. In syllogistic
inference the subject of the conclusion is the minor term, and its
predicate the major term, while between these two extremes the term
common to the two premises is the middle term, and the premise
containing the middle and major terms is the major premise, the premise
containing the middle and minor terms the minor premise. Thus in the
example of syllogism given above, "border-war between Thebes and Athens"
is the minor term, "evil" the major term, and "border-war" the middle
term. Using S for minor, P for major and M for middle, and preserving
these signs for corresponding terms in analogical and inductive
inferences, we obtain the following formula of the three inferences:--
_Analogical._ | _Inductive._ | _Deductive or Syllogistic._
| |
S^1 is P | S is P | Every M is P
S^2 is similar | Every M is | S is M
to S^1 | similar to S |
:. S^2 is P. |:. Every M is P. | :. S is P.
The love of unity has often made logicians attempt to resolve these
three processes into one. But each process has a peculiarity of its own;
they are similar, not the same. Analogical and inductive inference alike
begin with a particular premise containing one or more instances; but
the former adds a particular premise to draw a particular conclusion,
the latter requires a universal premise to draw a universal conclusion.
A citizen of Athens, who had known the evils of the border-war between
Thebes and Phocis, would readily perceive the analogy of a similar war
between Thebes and Athens, and conclude analogously that it would be
evil; but he would have to generalize the similarity of all border-wars
in order to draw the inductive conclusion that all alike are evil.
Induction and deduction differ still more, and are in fact opposed, as
one makes a particular premise the evidence of a universal conclusion,
the other makes a universal premise evidence of a particular conclusion.
Yet they are alike in requiring the generalization of the universal and
the belief that there are classes which are whole numbers of similars.
On this point both differ from inference by analogy, which proceeds
entirely from particular premises to a particular conclusion. Hence we
may redivide inference into particular inference by analogy and
universal inference by induction and deduction. Universal inference is
what we call reasoning; and its two species are very closely connected,
because universal conclusions of induction become universal premises of
deduction. Indeed, we often induce in order to deduce, ascending from
particular to universal and descending from universal to particular in
one act as it were; so that we may proceed either directly from
particular to particular by analogical inference, or indirectly from
particular through universal to particular by an inductive-deductive
inference which might be called "perduction." On the whole, then,
analogical, inductive and deductive inferences are not the same but
three similar and closely connected processes.
The three processes of inference, though different from one another,
rest on a common principle of similarity of which each is a different
application. Analogical inference requires that one particular is
similar to another, induction that a whole number or class is similar to
its particular instances, deduction that each particular is similar to
the whole number or class. Not that these inferences require us to
believe, or assume, or premise or formulate this principle either in
general, or in its applied forms: the premises are all that any
inference needs the mind to assume. The principle of similarity is used,
not assumed by the inferring mind, which in accordance with the
similarity of things and the parity of inference spontaneously concludes
in the form that similars are similarly determined ("similia similibus
convenire"). In applying this principle of similarity, each of the three
processes in its own way has to premise both that something is somehow
determined and that something is similar, and by combining these
premises to conclude that this is similarly determined to that. Thus the
very principle of inference by similarity requires it to be a
combination of premises in order to draw a conclusion.
The three processes, as different applications of the principle of
similarity, consisting of different combinations of premises, cause
different degrees of cogency in their several conclusions. Analogy
hardly requires as much evidence as induction. Men speculate about the
analogy between Mars and the earth, and infer that it is inhabited,
without troubling about all the planets. Induction has to consider more
instances, and the similarity of a whole number or class. Even so,
however, it starts from a particular premise which only contains many
instances, and leaves room to doubt the universality of its conclusions.
But deduction, starting from a premise about all the members of a class,
compels a conclusion about every and each of necessity. One border-war
may be similar to another, and the whole number may be similar, without
being similarly evil; but if all alike are evil, each is evil of
necessity. Deduction or syllogism is superior to analogy and induction
in combining premises so as to involve or contain the conclusion. For
this reason it has been elevated by some logicians above all other
inferences, and for this very same reason attacked by others as no
inference at all. The truth is that, though the premises contain the
conclusion, neither premise alone contains it, and a man who knows both
but does not combine them does not draw the conclusion; it is the
synthesis of the two premises which at once contains the conclusion and
advances our knowledge; and as syllogism consists, not indeed in the
discovery, but essentially in the synthesis of two premises, it is an
inference and an advance on each premise and on both taken separately.
As again the synthesis contains or involves the conclusion, syllogism
has the advantage of compelling assent to the consequences of the
premises. Inference in general is a combination of premises to cause a
conclusion; deduction is such a combination as to compel a conclusion
involved in the combination, and following from the premises of
necessity.
Nevertheless, deduction or syllogism is not independent of the other
processes of inference. It is not the primary inference of its own
premises, but constantly converts analogical and inductive conclusions
into its particular and universal premises. Of itself it causes a
necessity of consequence, but only a hypothetical necessity; if these
premises are true, then this conclusion necessarily follows. To
eliminate this "if" ultimately requires other inferences before
deduction. Especially, induction to universals is the warrant and
measure of deduction from universals. So far as it is inductively true
that all border-war is evil, it is deductively true that a given
border-war is therefore evil. Now, as an inductive combination of
premises does not necessarily involve the inductive conclusion,
induction normally leads, not to a necessary, but to a probable
conclusion; and whenever its probable conclusions become deductive
premises, the deduction only involves a probable conclusion. Can we then
infer any certainty at all? In order to answer this question we must
remember that there are many degrees of probability, and that induction,
and therefore deduction, draw conclusions more or less probable, and
rise to the point at which probability becomes moral certainty, or that
high degree of probability which is sufficient to guide our lives, and
even condemn murderers to death. But can we rise still higher and infer
real necessity? This is a difficult question, which has received many
answers. Some noölogists suppose a mental power of forming necessary
principles of deduction a priori; but fail to show how we can apply
principles of mind to things beyond mind. Some empiricists, on the other
hand, suppose that induction only infers probable conclusions which are
premises of probable deductions; but they give up all exact science.
Between these extremes there is room for a third theory, empirical yet
providing a knowledge of the really necessary. In some cases of
induction concerned with objects capable of abstraction and
simplification, we have a power of identification, by which, not a
priori but in the act of inducing a conclusion, we apprehend that the
things signified by its subject and predicate are one and the same
thing which cannot exist apart from itself. Thus by combined induction
and identification we apprehend that one and one are the same as two,
that there is no difference between a triangle and a three-sided
rectilineal figure, that a whole must be greater than its part by being
the whole, that inter-resisting bodies necessarily force one another
apart, otherwise they would not be inter-resisting but occupy the same
place at the same moment. Necessary principles, discovered by this
process of induction and identification, become premises of deductive
demonstration to conclusions which are not only necessary consequents on
the premises, but also equally necessary in reality. Induction thus is
the source of deduction, of its truth, of its probability, of its moral
certainty; and induction, combined with identification, is the origin of
the necessary principles of demonstration or deduction to necessary
conclusions.
Analogical inference in its turn is as closely allied with induction.
Like induction, it starts from a particular premise, containing one or
more examples or instances; but, as it is easier to infer a particular
than a universal conclusion, it supplies particular conclusions which in
their turn become further particular premises of induction. Its second
premise is indeed merely a particular apprehension that one particular
is similar to another, whereas the second premise of induction is a
universal apprehension that a whole number of particulars is similar to
those from which the inference starts; but at bottom these two
apprehensions of similarity are so alike as to suggest that the
universal premise of induction has arisen as a generalized analogy. It
seems likely that man has arrived at the apprehension of a whole
individual, e.g. a whole animal including all its parts, and thence has
inferred by analogy a whole number, or class, e.g. of animals including
all individual animals; and accordingly that the particular analogy of
one individual to another has given rise to the general analogy of every
to each individual in a class, or whole number of individuals, contained
in the second premise of induction. In this case, analogical inference
has led to induction, as induction to deduction. Further, analogical
inference from particular to particular suggests inductive-deductive
inference from particular through universal to particular.
Newton, according to Dr Pemberton, thought in 1666 that the moon moves
so like a falling body that it has a similar centripetal force to the
earth, 20 years before he demonstrated this conclusion from the laws of
motion in the _Principia_. In fact, analogical, inductive and deductive
inferences, though different processes of combining premises to cause
different conclusions, are so similar and related, so united in
principle and interdependent, so consolidated into a system of
inference, that they cannot be completely investigated apart, but
together constitute a single subject of science. This science of
inference in general is logic.
Logic, however, did not begin as a science of all inference. Rather it
began as a science of reasoning ([Greek: logos]), of syllogism ([Greek:
syllogismos]), of deductive inference. Aristotle was its founder. He was
anticipated of course by many generations of spontaneous thinking
(_logica naturalis_). Many of the higher animals infer by analogy:
otherwise we cannot explain their thinking. Man so infers at first:
otherwise we cannot explain the actions of young children, who before
they begin to speak give no evidence of universal thinking. It is likely
that man began with particular inference and with particular language;
and that, gradually generalizing thought and language, he learnt at last
to think and say "all," to infer universally, to induce and deduce, to
reason, in short, and raise himself above other animals. In ancient
times, and especially in Egypt, Babylon and Greece, he went on to
develop reason into science or the systematic investigation of definite
subjects, e.g. arithmetic of number, geometry of magnitude, astronomy of
stars, politics of government, ethics of goods. In Greece he became more
and more reflective and conscious of himself, of his body and soul, his
manners and morals, his mental operations and especially his reason. One
of the characteristics of Greek philosophers is their growing tendency,
in investigating any subject, to turn round and ask themselves what
should be the method of investigation. In this way the Presocratics and
Sophists, and still more Socrates and Plato, threw out hints on sense
and reason, on inferential processes and scientific methods which may be
called anticipations of logic. But Aristotle was the first to conceive
of reasoning itself as a definite subject of a special science, which he
called analytics or analytic science, specially designed to analyse
syllogism and especially demonstrative syllogism, or science, and to be
in fact a science of sciences. He was therefore the founder of the
science of logic.
Among the Aristotelian treatises we have the following, which together
constitute this new science of reasoning:--
1. The _Categories_, or names signifying things which can become
predicates;
2. The _De Interpretatione_, or the enumeration of conceptions and
their combinations by (1) nouns and verbs (names), (2) enunciations
(propositions);