present phase of deductive theory is such that any attempt at
summary, even of the barest and most cursory kind, would
involve a lengthy and controversial disquisition. It is painful
to have to introduce a treatise that aims at simplicity and
clarity with a passage that will inevitably be somewhat obscure.

There can be no doubt that the general tendency of the
recent development of deductive logic has been to pare down
and whittle away the power of deduction to extend our know-
ledge of the facts of life. It has tended to re-enforce the view,
which may be expressed in old-fashioned terminology by saying,
that whatever is in the conclusion of a deduction is already
present in the premises.

The following sentences may overstate the detachment of
modern deductive logic from the real world. That logic sets
forth certain symbols. It frames rules for the transformation
of expressions by the substitution of symbols for one another.
Beginning with an appropriate grouping of symbols and apply-
ing the rules laid down, we find it possible to derive, or deduce,
the commonly accepted principles of deductive logic and the
whole of mathematics. This is an astonishing fact, and the
intellectual ingenuity of the pioneers of this system has been
rightly admired. The original impetus to the work appears
to have been given by the observation of a structural similarity
between certain logical principles and the basic theorems of
algebra.

The devising of symbols and the framing of rules is, in
this system, in a certain sense free. This contrasts with the
older theories of deduction in which the basic axioms were
subject to the requirement that they must be self-evident and
thus constitute intellectual intuitions. The modern tendency
has been to suspect, and even to reject, the notion of self-
evidence. This removes the old criterion for the selection,
out of possible groupings of symbols and transformation rules,
of a unique set which shall specify correct logical principles.
The selection may accordingly be made, either with the view
that in certain circumstances they would express a true move-
ment of thought, or without regard to this. I believe that there
is still difference of opinion as to how far in the initial stages
of the construction of the symbolic system, self-evidence con-
tinues to exert some implicit control.

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