1
Handout
PHIL 100
§§207209
TA: Wm Kallfelz
September 23, 2005
I.
LOGIC (Cont.)
Recall the “challenge exercise” I suggested in the previous handout: using Rules AE,
how can we determine the total number of
valid
standard form categorical syllogisms
(SFCS)?
Recall the total number of possible combinations is:
4
4
= 256
cases
Let’s ask a “simpler” question: What are the total number of
invalid
cases
,
independent of syllogistic figure
?
We’ve gained enough familiarity with Rules AE
to list explicitly their invalid cases:
Rule A:
No SFCS is valid that has two negative premises.
According to this rule, cases like
EOX, OEX
are therefore forbidden (where
X
means
any
syllogistic form.)
Why?
Recall that
E
and
I
are negative categorical forms
(universal and particular, respectively.)
Listing these cases explicitly:
EOX
= {
EOA, EOE, EOI, EOO
}
OEX
= {
OEA,OEE, OEI, OEO
}
Therefore, in set notation (i.e., in notation using operations involving sets) the total
number of
forbidden
cases according to Rule A = 8, listed explicitly as
1
:
EOX
∪
OEX
= {
EOA, EOE, EOI, EOO, OEA,OEE, OEI, OEO
}
Rule B:
No SFCS is valid that has either
a negative premise but does not have a
negative conclusions or
vice versa.
2
1
Note: The symbol: “
∪
”, if you’re not familiar with it, means: “union.” That is to say, the “union” of two
sets is defined (informally) as follows:
A
∪
B
= {all elements
x
such that
x
is in
A
or
x
is in
B
}.
It’s
extremely important
to understand the sense of “or” used here: it’s
inclusive
, and therefore
not exclusive
!
What does this mean?
Here’s an example: “Either I am alive or I am dead” is an example of “exclusive
or
,” (i.e., I cannot obviously be both alive and dead.)
On the other hand, “either she will name her newborn
daughter ‘Sue’ or she will name her newborn daughter ‘Ann’” is an example of
inclusive ‘or.’
Why?
Because there’s no logical reason why
the mother to her new daughter cannot name her child: “Sue Ann.”
In other words, inclusive ‘or’ involves the maximum number of cases with respect to ‘or.’
That is to say,
inclusive ‘or’ occupies both the regions of
A
alone and
B
alone and
where they may happen to overlap.
Unless otherwise (explicitly) noted, in logic we always
use “or” in this inclusive
sense.
2
Note that the sense of “or” used in Rule B is clearly exclusive.
Why?
Explain this to yourself.
(Hint:
Consider the introduction of “
either
…
or
…”
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Clearly this forbids the following cases:
EX
{
A
or
I
}
or
OX
{
A
or
I
}
or
XE
{
A
or
I
}
or
XO
{
A
or
I
}
Or:
(
EXA
∪
EXI
)
∪
(
OXA
∪
OXI
)
∪
(
XEA
∪
XEI
)
∪
(
XOA
∪
XOI
)=
{
EAA, EEA, EIA, EOA
,
EAI, EEI, EII, EOI
}
∪
{
OAA
,
OEA
,
OIA
,
OOA
,
OAI,OEI,OII,OOI
}
∪
{
AEA
,
EEA
,
IEA
,
OEA
,
AEI,EEI,IEI,OEI
}
∪
{
AOA
,
EOA
,
IOA
,
OOA
,
AOI,EOI,IOI,OOI
}
Whew!
But are all these cases mutually exclusive? No!
Performing the (pre)union, I
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 Fall '07
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 Philosophy, Logic, Syllogism, EEI, IAE IAE IAO IAO IEA IEE IEI IEO IIA IIA IIE IIE IIE, negative premise, IAO IIO IIA IIE III IIO AIA EIA IIA OIA AOA

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