Deductive Validity

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Deductive Validity
 
In this tutorial you will learn how to determine
whether deductive arguments are valid or invalid.
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Perhaps the most important concept in logic is
the concept of deductive validity.
Deductive arguments are either valid or invalid.
A valid argument is a deductive argument in which
the conclusion follows logically (i.e., with
strict logical necessity) from the premises.
In other words, a valid argument is a deductive
argument in which it would be contradictory to
assert all the premises as true and yet deny the
conclusion.
An invalid argument is a deductive argument in
which the conclusion does not follow logically
from the premises.
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Chapter 3 introduces you to a helpful (but not
foolproof) method for testing arguments for
validity called the "The C's Test." .
 The Three C's Test involves three steps  
  1. Check to see whether the premises are
actually true and the conclusion is actually
false. If they are, then the argument is invalid.
(By definition, no valid argument can have all
true premises and a false conclusion.) If they
are not, or if you don't know whether the
premises are true and the conclusion is false,
then go on to step 2.    
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2. See if you can conceive a possible scenario in
which the premises would be true and the
conclusion false. If you can, then the argument
is invalid. If you can't, and it is not obvious
that the conclusion follows necessarily from the
premises, then go on to step 3.
3. Try to construct a counterexample--a special
kind of parallel argument--that proves that the
argument is invalid. Constructing a
counterexample involves two steps (1) Determine
the logical form of the argument you are testing
for invalidity, using letters (A, B, C, etc.) to
represent the various terms in the argument. (2)
Construct a parallel argument that has exactly
the same logical pattern as the argument you are
testing but that has premises that are clearly
true and a conclusion that is clearly false. If
you can successfully construct such a
counterexample, then the argument is invalid. If,
after repeated attempts, you cannot construct
such a counterexample, then the argument is
probably valid.
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If Michelangelo painted the Mona Lisa, then he's
a great painter. Michelangelo is a great
painter. So, Michelangelo painted the Mona Lisa.
Is this argument valid or invalid? How can we use
the Three C's Test to determine if it is valid or
invalid?
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If Michelangelo painted the Mona Lisa, then he's
a great painter. Michelangelo is a great
painter. So, Michelangelo painted the Mona Lisa.
This argument is invalid.
  We can most readily see that the argument is
invalid by applying the first step of the Three
C's Test.
The premises of the argument are, in fact,
true, and the conclusion of the argument is, in
fact, false. Since no valid argument can have
true premises and a false conclusion, we know
straight away that the argument is invalid.
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If Bill Clinton is president, then he lives in
the White House. Bill Clinton is not
president. So, Bill Clinton doesn't live in the
White House.
Is this argument valid or invalid? How can we use
the Three C's Test to determine whether it is
valid or invalid?
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If Bill Clinton is president, then he lives in
the White House. Bill Clinton is not
president. So, Bill Clinton doesn't live in the
White House.
This argument is invalid.
The first step of the Three C's Test is not
applicable here, because both the premises and
the conclusion are actually true.
However, the second step of the Three C's Test
shows that the argument is invalid.  
It's easy to conceive of circumstances in which
the premises and the conclusion is false. This
would be the case, for example, if Clinton became
an advisor who lived in the White House.
Because we can imagine circumstances in which the
premises could be true and the conclusion false,
the conclusion does not follow from the premises
with strict logical necessity. This shows that
the argument is invalid.
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All Alphans are Betans. Some Betans are
Deltans. So, some Deltans are Alphans.
  Is this argument valid or invalid? How can we
use the Three C's Test to determine whether it is
valid or invalid?  
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All Alphans are Betans. Some Betans are
Deltans. So, some Deltans are Alphans.
With this argument, the first step of the Three
C's Test is useless, because the terms are just
made up, and thus the statements are neither true
nor false.
It's also difficult to apply the second test,
since the logic of the argument is complex.  
Thus, let's apply the third test the
counterexample method of proving invalidity.  
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All Alphans are Betans. Some Betans are
Deltans. So, some Deltans are Alphans.
  To apply the counterexample method, we first
must determine the logical pattern, or form, of
the argument, using letters (A, B, C, etc.) to
represent the various terms.  
  What is the logical form of this argument?
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1. All A's are B's. 2. Some B's are D's. 3. So,
some D's are A's.
The second step in the counterexample method is
to try to construct a second argument--one that
has exactly the same logical form as the argument
we are testing for validity but that has premises
that are obviously true and a conclusion that is
obviously false.
  If we can successfully construct such an
argument, that will show that our first argument,
the argument being tested for validity, is
invalid. For if any argument with a certain
logical form is invalid, then all arguments with
that form are invalid.  
  Can you construct a counterexample to the
argument form given at the top of this page?
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All dogs are carnivores. Some carnivores are
cats. So, some cats are dogs.
  Bingo! This argument has the same logical form
as the first argument, but in this argument the
premises are both clearly true and the conclusion
is clearly false. This shows that arguments with
this pattern of reasoning are not guaranteed to
have true conclusions if the premises are true.
And this shows that all arguments that have that
pattern of reasoning are invalid.  
This is the end of this tutorial