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other hand； may assure us of the truth of the proposition； but it
cannot enable us to prehend the grounds of its possibility。 The
latter is； accordingly； rather an auxiliary to an argument； than a
strictly philosophical and rational mode of procedure。 In one respect；
however； they have an advantage over direct proofs； from the fact that
the mode of arguing by contradiction； which they employ； renders our
understanding of the question more clear； and approximates the proof
to the certainty of an intuitional demonstration。
The true reason why indirect proofs are employed in different
sciences is this。 When the grounds upon which we seek to base a
cognition are too various or too profound； we try whether or not we
may not discover the truth of our cognition from its consequences。 The
modus ponens of reasoning from the truth of its inferences to the
truth of a proposition would be admissible if all the inferences
that can be drawn from it are known to be true； for in this case there
can be only one possible ground for these inferences； and that is
the true one。 But this is a quite impracticable procedure； as it
surpasses all our powers to discover all the possible inferences
that can be drawn from a proposition。 But this mode of reasoning is
employed； under favour； when we wish to prove the truth of an
hypothesis； in which case we admit the truth of the conclusion…
which is supported by analogy… that； if all the inferences we have
drawn and examined agree with the proposition assumed； all other
possible inferences will also agree with it。 But； in this way； an
hypothesis can never be established as a demonstrated truth。 The modus
tollens of reasoning from known inferences to the unknown proposition；
is not only a rigorous； but a very easy mode of proof。 For； if it
can be shown that but one inference from a proposition is false；
then the proposition must itself be false。 Instead； then； of
examining； in an ostensive argument； the whole series of the grounds
on which the truth of a proposition rests； we need only take the
opposite of this proposition； and if one inference from it be false；
then must the opposite be itself false； and； consequently； the
proposition which we wished to prove must be true。
The apagogic method of proof is admissible only in those sciences
where it is impossible to mistake a subjective representation for an
objective cognition。 Where this is possible； it is plain that the
opposite of a given proposition may contradict merely the subjective
conditions of thought； and not the objective cognition； or it may
happen that both propositions contradict each other only under a
subjective condition； which is incorrectly considered to be objective；
and； as the condition is itself false； both propositions may be false；
and it will； consequently； be impossible to conclude the truth of
the one from the falseness of the other。
In mathematics such subreptions are impossible； and it is in this
science； accordingly； that the indirect mode of proof has its true
place。 In the science of nature； where all assertion is based upon
empirical intuition； such subreptions may be guarded against by the
repeated parison of observations； but this mode of proof is of
little value in this sphere of knowledge。 But the transcendental
efforts of pure reason are all made in the sphere of the subjective；
which is the real medium of all dialectical illusion； and thus
reason endeavours； in its premisses； to impose upon us subjective
representations for objective cognitions。 In the transcendental sphere
of pure reason； then； and in the case of synthetical propositions；
it is inadmissible to support a statement by disproving the
counter…statement。 For only two cases are possible； either； the
counter…statement is nothing but the enouncement of the
inconsistency of the opposite opinion with the subjective conditions
of reason； which does not affect the real case （for example； we cannot
prehend the unconditioned necessity of the existence of a being；
and hence every speculative proof of the existence of such a being
must be opposed on subjective grounds； while the possibility of this
being in itself cannot with justice be denied）； or； both propositions；
being dialectical in their nature； are based upon an impossible
conception。 In this latter case the rule applies： non entis nulla sunt
predicata； that is to say； what we affirm and what we deny； respecting
such an object； are equally untrue； and the apagogic mode of
arriving at the truth is in this case impossible。 If； for example；
we presuppose that the world of sense is given in itself in its
totality； it is false； either that it is infinite； or that it is
finite and limited in space。 Both are false； because the hypothesis is
false。 For the notion of phenomena （as mere representations） which are
given in themselves （as objects） is self…contradictory； and the
infinitude of this imaginary whole would； indeed； be unconditioned；
but would be inconsistent （as everything in the phenomenal world is
conditioned） with the unconditioned determination and finitude of
quantities which is presupposed in our conception。
The apagogic mode of proof is the true source of those illusions
which have always had so strong an attraction for the admirers of
dogmatical philosophy。 It may be pared to a champion who
maintains the honour and claims of the party he has adopted by
offering battle to all who doubt the validity of these claims and
the purity of that honour； while nothing can be proved in this way；
except the respective strength of the batants； and the advantage；
in this respect； is always on the side of the attacking party。
Spectators； observing that each party is alternately conqueror and
conquered； are led to regard the subject of dispute as beyond the
power of man to decide upon。 But such an opinion cannot be
justified； and it is sufficient to apply to these reasoners the
remark：
Non defensoribus istis
Tempus eget。
Each must try to establish his assertions by a transcendental
deduction of the grounds of proof employed in his argument； and thus
enable us to see in what way the cl