After judgements, a
study of different modes of proof follow – this has been the
official order since someone decided to place Analytics
after On Interpretation
in Aristotelian corpus. Somewhat refreshingly, Hoffmann leaves the
classification of various types of proof to the practical part of
logic, where we shall find it later. In theoretical part, he instead
concentrates on some general features of all proofs and especially on
various principles followed in proving.
Just
like in case of judgements or propositions, proofs are not just
strings of propositions for Hoffmann, but instead, processes of human
thinking, in which we feel ourselves forced to move from some
judgements to a new one. Note that Hoffmann is not yet interested
whether these proofs have any connection to truth or not – this is
the topic of the next chapter. Indeed,
he admits outright that some proofs lead to false conclusions, even
if they are so seductive to follow. Yet, now it is time only to
describe the very act of following a proof, not give rules for
determining a validity of a proof.
What
is interesting in proofs is that we feel forced to follow them, no
matter what the matter, that is, the topic in question. If
propositions assumed just are of certain structure, then we feel
obliged to draw some conclusions of them. It is then the form of
proofs, embodied in some grounding principles, which are the topic
now – nowadays we might speak of rules of inference, and it is
important that Hoffmann clearly separates them from mere axioms,
which by itself or as mere
propositions do not force us to make any conclusions.
While
Hoffmann accepts a number of grounding principles, he thinks they can
all be based on three most basic principles. The first one should
amaze no one. What understanding finds the easiest to deal with,
Hoffmann begins his introduction of the principle, are those thoughts
that require least amount of analysis. Such thoughts are being and
non-being, which understanding at one notices to be impossible to
combine. This is the basis of the principle of non-contradiction.
Just like with Wolffians, the principle is for Hoffmann ontological,
while the corresponding logical principle ”nothing can be both true
and false”is based on the ontological principle.
For
Wolff, the principle of contradiction was not just ontological, but
apparently covered all cases of incompatible properties, since he
used ”wooden” and ”iron” as an example of contradictory
characteristics. Hoffmann, on the contrary, notes that, while
undoubtedly a rule that understanding follows, such denial of
incompatible properties must be based on another principle – what
is incompatible in understanding must be accepted as incompatible in
reality. Third is the corresponding positive principle – what is
necessarily conjoined in understanding must be accepted as conjoined
in reality.
This
is actually all that Hoffmann presents as a theory of proofs –
except he uses these basic principles to derive some further principles.
I shall leave most of them unmentioned and concentrate on the most
interesting of them, that is, the principle of sufficient reason or
ground.As we have seen earlier, Hoffmann doesn't buy the Leibnizian
principle as it is, but makes an important emendation – actions
started by an immediate use of a force that does not need any
external determination do not need to have any sufficient reason. The
principle of sufficient reason in its Hoffmanian guise is based on
the principle of necessary conjoinment – we must always connect an
event to some force starting it, thus, it must have begun by an
active force just described or by a force determined by something
external.
What
Hoffmann is suggesting is that a sort of Humean account of causality
as a string of events with no intrinsic connection is just so alien
to human thinking that we cannot accept it. Of
course, this does not yet prove whether this human way of thinking is
true – that type of question we have to leave to the next post.
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