------------------------------------------------------------------------
SUBJECT: POSITION PAPER #4 & VALIDITY
------------------------------------------------------------------------
As promised in lecture this morning, here are some further comments on
how to explain why you think an argument is valid or invalid.
I hope these comments will help you in writing your position paper.
1.
One preliminary comment:
------------------------------------------------------------------------
I noticed that all students who did *not* use the thinksheet also did
*not* evaluate the arguments for validity!
That suggests that the thinksheets have a value in spelling out exactly
what you need to do.
(However, there were some thinksheets that simply left those cells
blank, so I guess there's no guarantee!)
2.
Here are some of your responses to "Why?" the argument from premises
1,2,3 to conclusion 4 is valid or invalid, with my comments:
------------------------------------------------------------------------
a) The argument is valid This is, umm..., not very illuminating
because "Yes"
------------------------------------------------------------------------
b) The argument is valid This is true: An argument is, indeed,
because "Valid" valid iff it is valid. But this is not
very helpful.
------------------------------------------------------------------------
c) The argument is valid Also true: An argument is, indeed,
because "Follows logically" valid iff the conclusion follows
logically from the premises. And it's
a bit more useful than the previous
response, but it's not much more
illuminating--it immediately raises
the question: "Yes, but *why* does it
follow logically?"
------------------------------------------------------------------------
d) The argument is valid THIS IS INCORRECT
because (even if all premises are true):
"All premises are true"
The definition of "valid" (i.e., the definition of "follows logically")
is:
An argument is valid
iff
it is *impossible* for: all premises to be true
*and* for the conclusion to be false
Or: An argument is valid
iff
*whenever* all premises are true,
then the conclusion *must* also be true).
Or--here's a new version, which some of you may find clearer--
An argument is valid
iff
*any* situation that makes all premises true
*also* makes the conclusion true.
So, what's wrong with the answer that the student gave,
namely, that the argument is valid because "all premises are true"?
What's wrong is that you can have a VALID argument with FALSE premises.
I've given you examples before; here's another one:
All mice can fly.
Rudolph, the red-nosed reindeer, is a mouse.
.'. Rudolph can fly.
Both premises are false.
But it is impossible for both premises to be TRUE
*and* for the conclusion to be FALSE.
In other words, if all the premises *were* true,
then the conclusion *would have to be* true.
Or, if the world had been such that all mice could fly
and Rudolph were a mouse, then Rudolph would have been able to fly.
More precisely, this argument has the following valid form:
For anything, x, if x is a mouse, then x can fly.
Rudolph is a mouse (i.e., let x := Rudolph).
Then Rudolph can fly.
Or, set-theoretically, this argument's form would be:
The set of mice is a subset of the set of things that can fly.
Rudolph is a member of the set of mice.
So, Rudolph is a member of the set of things that can fly.
The form of this set-theory version is:
A is a subset of B
r is a member of A
.'. r is a member of B
------------------------------------------------------------------------
e) The argument is invalid THIS IS INCORRECT
because (even if the conclusion is false).
"The conclusion is false"
What's wrong with the answer that the student gave,
namely, that the argument is invalid because "the conclusion is false"?
What's wrong is that you can have a VALID argument with a FALSE conclusion.
Here's an example:
All students understand logic.
The person who said "the argument is invalid because
its conclusion is false" is a student.
.'. That person understands logic.
This argument has the same valid form as the one about mice and Rudolph.
But its conclusion is false (as I'm trying to explain to you).
By the way, because the argument IS valid, IF all premises WERE true,
then the conclusion would have to be true. And that means that,
because the conclusion ISN'T true, at least one of its premises also
isn't true. In this case, it's the first premise.
------------------------------------------------------------------------
f) The argument is valid THIS IS INCORRECT
because (even if the conclusion is true)
"The law says that the
conclusion is true"
This response has two problems with it:
First, as I mentioned in class and as Position Paper #4 was supposed to
make you aware of, the law is not always logical.
Maybe it should be, but it isn't.
Second, and more generally, you can have an INVALID argument with a TRUE
conclusion. Here's an example:
In NY State, if the Governor resigns,
then the Lt. Governor becomes Governer
David Paterson was elected Lt. Governor of NY State.
.'. David Paterson is Governor of NY State.
This is invalid, because it's possible for all premises to be true
(in fact, they are) *and* for the conclusion to be false.
Here, the conclusion is NOT false, but IT COULD HAVE BEEN FALSE EVEN IF
THOUGH THE PREMISES ARE TRUE.
The reason it could have been false is that it is possible that the
Governor had not resigned.
------------------------------------------------------------------------
g) The argument is valid THIS IS CORRECT
because "equivalence is
transitive, so the physical
machine is the hardwired
program and can be patented"
If a hardwired computer program is a physical machine (premise 2),
& if a physical machine can be patented (premise 3),
then a hardwired computer program can be patented. (conclusion 4)
Premise 2 can be interpreted as saying that two things are equivalent:
hardwired computer programs and physical machines.
Premise 3 can be interpreted as saying that two other things are
equivalent: physical machines and patentable things.
Conclusion 4 can be interpreted as saying that the first thing is
equivalent to the third thing. This follows by the transitivity of
equivalence: If A = B and if B = C, then A = C.
On another way of interpreting those statements, this is just the
transitivity of the implication relation, also known as hypothetical
syllogism.
Or, on a set-theoretic interpretation, this is just the transitivity
of the subset relation.
(Note that premise 1 really plays no role in this;
I probably could have omitted it. But extra premises do no harm.)