Categorical Syllogism
Warning: Web Page Notes are not intended
as a substitute for attending lectures.
A categorical syllogism is an argument which
has two categorical propositions
for premises and one categorical proposition
as the conclusion. (Each proposition has two
different terms.) A categorical syllogism has exactly three terms, each
occurring in two propositions. The term which occurs in both premises is
called the middle term. The term which occurs as the predicate term
in the conclusion is called the major term. The premise which has
the major term is called the major premise. To put the argument into standard
form, one must place the major premise first. The subject of the conclusion
is called the minor term and the premise with the minor term is
called the minor premise. In standard form the minor premise is placed
second. In other words, if the predicate term of the conclusion is in the
second premise, the argument is not in standard form and it must be rewritten
with the premises switched to put it in standard form.
Mood and Figure
Since categorical propositions come in 4
kinds, each premise and conclusion of a categorical syllogism may be
of 4 kinds. By putting syllogisms in standard form we can identify the
kinds of proposition each premise and conclusion is, merely by stating
the vowel names in order. This is called stating the mood of the
argument. For example if the mood of a categorical syllogism is given as
AEO,
we can know that the major premise is an A
statement, because the major premise comes first in standard form. Likewise
we can see that the minor premise would be an E
statement, and the conclusion an O
statement. By knowing the figure of the syllogism, in addition to
its mood, one can know the complete logical form of any categorical syllogism.
The logical form, not the content, completely determines whether the argument
is valid or invalid and so is very important. The
figure of the syllogism refers to the configuration of the middle terms
in the premises. If the middle term is the subject term of the major premise
and the predicate of the minor premise, then the figure is 1st. If the
middle term is predicate of both premises, then the figure is 2nd. If the
middle term is subject of both, then the figure is 3rd. If the middle term
is predicate of the major and subject of the minor, the figure is 4th.
In the following table M is always the middle
term; P is always the major term, and S is always the minor term. The words
which provide the copula, quality
and quantity have been deliberately
left out of this table as they are irrelevant to figure.

1st

2nd

3rd

4th

Major Premise

M P

P M

M P

P M

Minor Premise

S M

S M

M S

M S

Conclusion

S P

S P

S P

S P

There are 256 syllogistic forms given that each proposition may be one
of 4 forms (A, E, I,
or O) and the
figure may be one of 4 too (that is 4 X 4 X 4 X 4). At most only 24 of
these are valid.
Validity
The words 'valid' and 'invalid' are often used in a loose way. Someone
might say, "She makes a valid point." and mean perhaps that what she said
was true or even merely that what she said needs to be considered. It would
be less confusing if the word 'valid' were reserved for use in its more
technical sense. We shall use the word in this technical sense: an argument
is valid if and only if having a false conclusion is impossible when the
premises are true. In other words, in a valid argument the truth of the
premises necessitates the truth of the conclusion, indeed guarantees the
truth of the conclusion. In this technical sense validity is a property
of arguments, not of "points", sentences or statements. The relationship
between validity and truth and falsity is very specific and not entirely
what you might intuit or expect. The following table indicates which combinations
of truth and falsity are possible for the premises and conclusions of valid
and invalid arguments. When we speak of true premises we mean that all
the premises are true; when we speak of false premises we mean that at
least one premise is false.
Premises

Conclusion

Invalid Arguments

Valid Arguments

False

False

Yes

Yes

False

True

Yes

Yes

True

True

Yes

Yes

True

False

Yes

No

Only the combination of true premises with a false conclusion in a valid
argument is impossible. The validity of an argument is determined by its
logical form rather than by its content. If an argument having a certain
form is valid then all arguments having the same form are equally valid
no matter how different the content may be. Likewise if an argument having
a certain form is invalid then all other arguments with the same form will
be invalid. A sound argument is a valid argument with all true premises.
The Rules of Validity
An argument must meet all of the following conditions to be valid. Failing
to meet one or more conditions shows an argument to be invalid.

The middle term must be distributed
at least once.

If a term is distributed in the
conclusion, then it must be distributed
in its premise.

If one of the premises is negative,
then the conclusion must be negative, and if the conclusion is negative,
then one of the premises must be negative.

There must not be two negative premises.
For more information see Garth Kemerling's Web Pages on Categorical
Syllogism and
Their Validity