HOW TO SOLVE SYLLOGISM

Rule 1: The middle term must be
distributed at least once.
Fallacy: Undistributed middle
Example:
All sharks are fish
All salmon are fish
All salmon are sharks
Justification : The middle term
is what connects the major and
the minor term. If the middle
term is never distributed, then
the major and minor terms
might be related to different
parts of the M class, thus giving
no common ground to relate S
and P.
Rule 2: If a term is distributed in
the conclusion, then it must be
distributed in a premise.
Fallacy: Illicit major; illicit minor
Examples :
And:
All horses are animals
Some dogs are not horses
Some dogs are not animals
All tigers are mammals
All mammals are animals
All animals are tigers
Justification : When a term is
distributed in the conclusion,
let’s say that P is distributed,
then that term is saying
something about every member
of the P class. If that same term
is NOT distributed in the major
premise, then the major premise
is saying something about only
some members of the P class.
Remember that the minor
premise says nothing about the
P class. Therefore, the
conclusion contains information
that is not contained in the
premises, making the argument
invalid.
Rule 3: Two negative premises
are not allowed.
Fallacy: Exclusive premises
Example:
No fish are mammals
Some dogs are not fish
Some dogs are not mammals
Justification : If the premises
are both negative, then the
relationship between S and P is
denied. The conclusion cannot,
therefore, say anything in a
positive fashion. That
information goes beyond what
is contained in the premises.
Rule 4: A negative premise
requires a negative conclusion,
and a negative conclusion
requires a negative premise.
(Alternate rendering: Any
syllogism having exactly one
negative statement is invalid.)
Fallacy: Drawing an affirmative
conclusion from a negative
premise, or drawing a negative
conclusion from an affirmative
premise.
Example:
All crows are birds
Some wolves are not crows
Some wolves are birds
Justification : Two directions,
here. Take a positive conclusion
from one negative premise. The
conclusion states that the S class
is either wholly or partially
contained in the P class. The
only way that this can happen is
if the S class is either partially or
fully contained in the M class
(remember, the middle term
relates the two) and the M class
fully contained in the P class.
Negative statements cannot
establish this relationship, so a
valid conclusion cannot follow.
Take a negative conclusion. It
asserts that the S class is
separated in whole or in part
from the P class. If both
premises are affirmative, no
separation can be established,
only connections. Thus, a
negative conclusion cannot
follow from positive premises.
Note: These first four rules
working together indicate that
any syllogism with two
particular premises is invalid.
Rule 5: If both premises are
universal, the conclusion cannot
be particular.
Fallacy: Existential fallacy
Example:
All mammals are animals
All tigers are mammals
Some tigers are animals
Justification : On the Boolean
model, Universal statements
make no claims about existence
while particular ones do. Thus,
if the syllogism has universal
premises, they necessarily say
nothing about existence. Yet if
the conclusion is particular,
then it does say something about
existence. In which case, the
conclusion contains more
information than the premises
do, thereby making it invalid.
The Aristotelian Standpoint
Any syllogism that violates any
of the first four rules is invalid
from either standpoint. If a
syllogism, though, violates only
rule 5, it is then valid from the
Aristotelian standpoint,
provided that the conditional
existence is fulfilled. Thus, in the
example above, since tigers
exist, this syllogism is valid from
the Aristotelian point of view.
On the other hand, consider this
substitution instance:
All mammals are animals
All unicorns are mammals
Some unicorns are animals
Since "unicorns" do not exist,
the condition is not fulfilled,
and this syllogism is invalid
from either perspective.
In order to determine the
needed condition, you can
simply consult the chart (but not
on the exam!). But there are two
other ways. First, as we learned
in section 5.2, you can draw a
Venn diagram and find the
circle with only one open area.
The term that that circle
represents is the required
existent thing. Second, you can
check the distributions and, in
these cases, there will always be
one term that is superfluously
distributed. That is, there will be
one term that is distributed
more than is necessary to insure
the validity of the syllogism.
Examples:
All Md are P
All Sd are M
Some S are P
No Md are Pd
All Md are S
Some S are not Pd
All P d are M
All Md are S
Some S are P