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one premiss must be affirmative, at least one premiss must be universal, and if
either premiss is negative the conclusion must be negative. Aristotle’s rules, in
their totality, are sufficient to validate sound syllogisms and eliminate invalid ones.
They are sufficient, for instance, to accept inference (1) and reject inference (2).
Aristotle believed that his syllogistic was sufficient to deal with every possible
valid inference. This was an error; in fact, the system, though complete in itself,
was only a fragment of logic. It had two weaknesses. First, it did not deal with
inferences depending not on words like ‘all’ and ‘some’, which attach to nouns,
but on words like ‘if ’ and ‘then’, which link sentences. It was some centuries
later before anyone formalized patterns of inference such as ‘If it is not day, it is
night; but it is not day; therefore it is night’. Secondly, even within its own field,
Aristotle’s logic could not cope with inferences in which words like ‘all’ or ‘some’
(or ‘every’ and ‘any’) occurred not in the subject place but somewhere in the
grammatical predicate. The rules would not permit one to determine, for in-
stance, the validity of inferences containing premisses such as ‘every schoolchild
knows some dates’ or ‘some people hate all policemen’. It was not until twenty-
two centuries after Aristotle’s death that this gap was filled.
Logic is used in all the various sciences which Aristotle studied; perhaps it is
not so much a science itself as an instrument or tool of the sciences. That was the
view taken of Aristotle’s logical works by his successors, who called them the
‘Organon’ after the Greek word for tool.
The Posterior Analytics tells us how logic functions in the sciences. Those who
learnt Euclidean geometry at school will recall how many geometrical truths, or
theorems, were derived by deductive reasoning from a small initial set of other
truths called axioms. Though Euclid himself was not born until late in Aristotle’s
life, this axiomatic method was already familiar to geometricians, and Aristotle
believed it to be very widely applicable. Logic provided the rules for the derivation
of theorems from axioms, and each science would have its own special set of axioms.
The sciences could be ordered in hierarchies, with sciences lower down a hierarchy
treating as axioms propositions which might be theorems of a higher science.
If we take ‘science’ in a broad sense, Aristotle says, there are three kinds of
sciences: productive, practical, and theoretical. Productive sciences include engin-
eering and architecture, and also disciplines such as rhetoric and playwriting whose
products are less concrete. Practical sciences are ones which guide behaviour, most
notably ethics and politics. Theoretical sciences are those which have no product
and no practical goal, but pursue truth for its own sake.
Theoretical science, in its turn, is threefold. Aristotle names the three divisions
‘Physics, Mathematics, Theology’; but in this classification only mathematics is
what it seems to be. ‘Physics’ means natural philosophy or the study of nature
(physis); it includes, in addition to the disciplines which we would nowadays think
of as part of physics, chemistry and biology plus animal and human psychology.
‘Theology’ is, for Aristotle, the study of entities above and superior to human
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