Anatomists may deny that it is impossible for this to not be chance or
that complex arrangements are of common structures or forms of
organization. I guess we cannot deny that a river is like a blood
vessel either, but maybe when blood vessels evolved they just
incorperated the advantages of flow of liquid. That something is
similar doesn't make it necessary that it is identical, this can be
shown by counterexample. Your heading should read, "No anatomists can
deny these six Haversian canals at Phoenix’s foot share similar flow
patterns as human body parts"
PROCEDURE FOR CONSTRUCTING A COUNTEREXAMPLE:
A counterexample to an argument form is a substitution instance whose
premises are true and whose conclusion is false. Since the validity
of an argument depends on its form and since only an invalid argument
can have true premises and a false conclusion, a counterexample to an
argument form proves that every substitution instance of that argument
form is invalid.
1. Write out a well-crafted version of the argument--see section
2.2 of The Power of Logic, pp. 52-73 of the course reader.
2. Identify the form of the argument. Look for repeated references
to the same class (or to the same proposition) and then rewrite the
argument uniformly substituting a capital letter for each class (or
for each proposition). Indicate which capital letters stand for which
classes (or propositions).
3. Find English terms (or statements) that, if substituted for the
capital letters in the conclusion of the argument form, produce a well-
known falsehood. Substitute these English terms (or statements) for
the relevant capital letters uniformly throughout the argument form.
4. Find additional English terms (or statements) that, if
substituted uniformly for the remaining capital letters in the
argument form, produce premises that are well-known truths.
Important Note: If you have trouble coming up with a substitution
instance whose premises are well-known truths and whose conclusion is
a well-known falsehood, then come up with a substitution instance
whose premises are true and whose conclusion is false in some possible
world and then describe that possible world so as to make clear how,
in that world, the premises would be true and the conclusion would be
false.
Definitions:
*
* A well-crafted version of an argument in ordinary English is one
that is restated so as to make its logical features explicit by
employing uniform language, excluding excess verbiage, and making each
step of the argument explicit.
* A class refers to a collection or a set of things, such as the
set of things that are beagles or the set of things that are dogs.
* A term is word or phrase that stands for a class.
* A proposition is a truth or a falsehood that can often be
expressed by a number of different, but logically equivalent,
statements.
* A statement is a sentence that has a truth-value, that is, a
sentence that is either true or false.
* An argument form is a pattern of reasoning.
* An argument that results from uniformly replacing the capital
letters in an argument form with terms (or statements) is called a
substitution instance of that form.
Example 1:
If God exists, then life has meaning. But there is no God.
Therefore, life is meaningless.
Step 1:
1. If God exists, then life has meaning.
2. It is not the case that God exists.
So, 3. It is not the case that life has meaning.
Step 2:
G: God exists.
L: Life has meaning.
1. If G, then L.
2. It is not the case that G.
So, 3. It is not the case that L.
Step 3:
1. If G, then Al Gore is a U.S. citizen.
2. It is not the case that G.
So, 3. It is not the case that Al Gore is a U.S. citizen.
[False]
Step 4:
1. If Al Gore is the President of the U.S., then Al Gore is a
U.S. citizen. [True]
2. It is not the case that Al Gore is the President of the
U.S. [True]
So, 3. It is not the case that Al Gore is a U.S. citizen.
[False]
Thus the argument given in Example 1 is invalid.
Example 2:
Some catholic priests are bachelors. All bachelors are single. Fred
Feldman is married. So Fred Feldman isn't a catholic priest.
Step 1:
1. Some catholic priests are bachelors.
2. All bachelors are single.
3. Fred Feldman is not single.
So, 4. Fred Feldman is not a catholic priest.
Step 2:
C: catholic priests
B: bachelors
S: people who are single
F: people who are identical to Fred Feldman
1. Some C are B.
2. All B are S.
3. F is not S.
So, 4. F is not C.
Step 3:
1. Some animals are B.
2. All B are S.
3. Fido is not S.
So, 4. Fido is not an animal. [False] Imagine the possible world
where Fido is a dog.
Step 4:
1. Some animals are lizards. [True]
2. All lizards are reptiles. [True]
3. Fido is not a reptile. [True]
So, 4. Fido is not an animal. [False]
Step 5:
One possible world is one in which Fido is a dog and everything else
is as it is in the actual world. In that possible world, the premises
would be true and the conclusion would be false.
Thus, the argument given in Example 2 is invalid.
http://www.csun.edu/~dp56722/200spring04/04_200Handout_Procedure_for_Constructing_a_Counterexample...
http://216.239.57.104/search?q=cache:fQxj5bAYjPYJ:www.csun.edu/~dp56722/200sprin...
http://www.google.com/search?hl=en&ie=UTF-8&q=counterexample+possible+world&btnG...
