CS 100 - Week 14 Lecture 1 - 11-27-12
Chapter 9, continued - now using Venn diagrams to determine if
categorical syllogisms are valid
* NOTE: you DO need to READ and UNDERSTAND the Chapter 9 section on
TRANSLATING suitable logical statements into standard categorical
form.
* ...we're proceeding, in class, to using Venn diagrams to determine
if categorical syllogisms are valid;
* REMINDER: categorical syllogism:
a 3-line deductive argument with 2 premises and a conclusion
in which all of the statements are categorical statements
* Some categorical syllogisms:
* No doctors are pro wrestlers.
All cardiologists are doctors.
So, no cardiologists are pro wrestlers.
* All snakes are reptiles.
All reptiles are cold-blooded animals.
So, all snakes are cold-blooded animals.
* Some Californians are coffee-lovers.
All Californians are Americans.
So, some Americans are coffee-lovers.
* No Democrats are Republicans.
Some lifeguards are Republicans.
So, some lifeguards are not Democrats.
...and here's a non-valid categorical syllogism:
* All painters are artists.
Some magicians are artists.
So, some magicians are painters.
* USEFUL THING: Venn diagrams can be used quite nicely in testing the
validity of categorical syllogisms;
* when more than two categories, you have more than two overlapping
circles in the corresponding Venn diagram;
...these can get pretty intricate!
* BUT, for standard-form categorical syllogisms, -- such as
those above -- there are typically only
3 categories, so only 3 interlocking circles:
* BY CONVENTION, the two bottom circles are labelled with
the two categories in the conclusion;
so, for the syllogism (EXAMPLE 1):
* No doctors are pro wrestlers.
All cardiologists are doctors.
So, no cardiologists are pro wrestlers.
...since the conclusion involves cardiologists and
pro wrestlers, the two bottom circles in the Venn diagram
for this syllogism get those categories as their labels:
(and the other category gets the top circle -- thus, the
top circle is labeled as doctors above)
* now, you shade this diagram based on the two premises
in the categorical syllogism
* IF one of the premises is a "Some..." categorical statement
and the other is either a "No..." or "All..." categorical
statement,
THEN be sure to shade the "No..." or "All..." premise
FIRST, and THEN the "Some..." premise;
OTHERWISE, you can handle the premises in either order.
* SO, we'll handle "No doctors are pro wrestlers" first.
Thus, we shade the part of the Doctors circle overlapping
the Pro Wrestlers circle (it is empty, there are no
doctors who are also pro wrestlers):
* yes, fill in ALL of that overlap;
* Then, handle the second premise, "All cardiologists are
doctors", and shade that part of the cardiologists circle
that does NOT overlap with the doctors circle:
* yes, fill in ALL of that part of the cardiologists circle;
* Now, you don't shade any more -- you *consider* the conclusion.
* Here, it is that "No cardiologists are pro wrestlers."
We know that IF we were shading a Venn diagram for this,
we would shade in the overlap between the cardiologists and
the pro wrestlers circles;
To see whether this argument is valid, we see IF the premises
also LEAD to this shading --
IF SO, then the argument is valid;
IF NOT, then the argument is not valid/invalid;
* HERE, as a result of the premises, indeed
the entire overlap between the cardiologists circle
and the pro wrestlers circle IS shaded;
SO, this Venn diagram DOES demonstrate that this
categorical syllogism is valid.
* [remember! when we say that a deductive argument is valid, we are
saying that, IF its premises are true, its conclusion MUST be true
-- we are NOT necessarily saying that it must also be
*sound*. Sound means not only valid, but also with all-true
premises...]
* Another example, EXAMPLE 2
* All snakes are reptiles.
All reptiles are cold-blooded animals.
So, all snakes are cold-blooded animals.
* "snakes" and "cold-blooded animals" are the categories
involved in the conclusion, so those are the labels for
the bottom circles
(and "reptiles" is the label for the top circle)
* now, mark this Venn diagram based on the two premises --
both are "All..." statements, so can be marked in either order.
Let's mark "All snakes are reptiles":
...and then "All reptiles are cold-blooded animals"
* NOW -- do the premises result in the conclusion
DEFINITELY having to be the case?
Conclusion is, "All snakes are cold-blooded animals" --
does the resulting diagram show that any snakes may not be
cold-blooded animals?
...NO, because the only region of snakes still shown
unmarked (non-empty) is also part of the cold-blooded
animals circle;
SO, this Venn diagram DOES demonstrate that this categorical
syllogism IS valid.
* Another example, EXAMPLE 3
* Some Californians are coffee-lovers.
All Californians are Americans.
So, some Americans are coffee-lovers.
* "Americans" and "coffee-lovers" are the categories
involved in the conclusion, so those are the labels for
the bottom circles
(and "Californians" is the label for the top circle)
* now, mark this Venn diagram based on the two premises --
here, the first premise is a "Some..." statement, but the
second premise is not,
so: HANDLE the SECOND premise FIRST!
so, mark "All Californians are Americans."
...and THEN handle the "Some..." premise,
"Some Californians are coffee-lovers.".
* when you are placing this X, since one premise
guarantees that a sub-area is empty, you SHOULD put the X
IN the remaining sub-area where the premise states
that at least one thing exists;
* NOW -- do the premises result in the conclusion
DEFINITELY having to be the case?
Conclusion is, "Some Americans are coffee-lovers" --
does the resulting diagram show that there must exist
at least one American that is a coffee-lover?
...YES, because there IS an X, indicating the existence
of at least ONE thing in that area, in an area that is
part of the overlap between the Americans circle and the
coffee-lovers circle;
SO, this Venn diagram DOES demonstrate that this categorical
syllogism IS valid.
* Another example, EXAMPLE 4
* No Democrats are Republicans.
Some lifeguards are Republicans.
So, some lifeguards are not Democrats.
* "lifeguards" and "Democrats" are the categories
involved in the conclusion, so those are the labels for
the bottom circles
(and "Republicans" is the label for the top circle)
* now, mark this Venn diagram based on the two premises --
here, the second premise is a "Some..." statement, but the
first premise is not,
so: HANDLE the FIRST premise FIRST!
so, mark "No Democrats are Republicans."
...and THEN handle the "Some..." premise,
"Some lifeguards are Republicans.".
* when you are placing this X, since one premise
guarantees that a sub-area is empty, you SHOULD put the X
IN the remaining sub-area where the premise states
that at least one thing exists;
* NOW -- do the premises result in the conclusion
DEFINITELY having to be the case?
Conclusion is, "Some lifeguards are not Democrats" --
does the resulting diagram show that there must exist
at least one lifeguard that is NOT a Democrat?
...YES, because there IS an X, indicating the existence
of at least ONE thing in that area, in an area that is
part of the Lifeguards circle but NOT part of the Democrats
circle;
SO, this Venn diagram DOES demonstrate that this categorical
syllogism IS valid.
* Another example, EXAMPLE 5 - because should demo one that
demonstrates that a particular categorical syllogism is NOT
valid, also!
* All painters are artists.
Some magicians are artists.
So, some magicians are painters.
* "magicians" and "painters" are the categories
involved in the conclusion, so those are the labels for
the bottom circles
(and "artists" is the label for the top circle)
* now, mark this Venn diagram based on the two premises --
here, the second premise is a "Some..." statement, but the
first premise is not,
so: HANDLE the FIRST premise FIRST!
so, mark "All painters are artists."
...and THEN handle the "Some..." premise,
"Some magicians are artists.".
* IMPORTANT: see how, in this case, there are essentially
two sub-areas in the "Some magicians are artists" overlap?
* ...we don't know which sub-area MUST have at least one
magician who is also an artist, BUT this statement
asserts that there does exist such a magician;
* For a case like this, you draw the X ON the boundary
between two such sub-areas, to show that there is at
least one instance in one OR the other, but we can't
guarantee which;
* NOW -- do the premises result in the conclusion
DEFINITELY having to be the case?
Conclusion is, "Some magicians are painters" --
does the resulting diagram show that there MUST exist
at least one magician that is also a painter?
...NO! The Venn diagram shows that there is at least one
magician that is also an artist, BUT it does NOT show
that there DEFINITELY exists a magician painter who is also an
artist -- that boundary-X is NOT "strong" enough to
guarantee the given conclusion, in this case.
SO, this Venn diagram demonstrates that this categorical
syllogism is NOT valid.