Deductive
Arguments: Validity and Truth
(Baronett, Section 1 FG)
When we evaluate arguments, we begin by
evaluating the inferential claim, then move to the factual claim.
Validity and Soundness apply to deductive
arguments.
A valid deductive argument is an
argument such that it is impossible for the premises to be true and the
conclusion false. The conclusion follows with strict necessity from the
premises.
On the other hand, an invalid deductive
argument is a deductive argument such that it is possible for the
premises to be true and the conclusion false. The conclusion does not follow
with strict necessity from the premises.
Note that there is no middle groundan
argument cannot be "almost" valid.
Examples:

All television networks are media companies. NBC is a television network. Therefore, NBC is a media company. 
Imagine that the premises are true (and they
are), and then ask: is it possible for the conclusion to be false? Not in this
case.

All automakers are computer manufacturers. United Airlines is an automaker. Therefore, United Airlines is a computer manufacturer. 
Again, imagine that the premises are true
(they are not), and then ask: is it possible for the conclusion to be false?
Again, not in this case, even though the premises themselves are false.
Note: The truth or falsity of the individual
premises and conclusion has nothing to do with the validity of the argument! It
is valid or invalid regardless of the truth value of its premises. In a valid
argument, we say that if the premises are true, then the conclusion must
be true. Logical analysis comes first, then truth value analysis.
Another example:

All banks are financial institutions. SmithBarney is a financial institution. Therefore, SmithBarney is a bank. 
True premises, false
conclusion. This form is invalid.
The relationship between the premises and conclusion determines
validity. Note that any (EVERY!) deductive argument with actually true premises
and an actually false conclusion is, by definition, invalid.
A sound argument is a valid argument
with true premises. Notice that, by definition, a sound argument will have a
true conclusion as well. An unsound
argument is a deductive argument that is invalid, has one or more false
premises, or both. Review table page 28.
The validity
of a deductive argument is determined by its form. Consider two examples:
A. All dogs
are cats. All cats are snakes. Therefore, all dogs are snakes.
B. No
mammals are beagles. No mammals are dogs. Therefore, no beagles are dogs.
Okay, all of
the premises and conclusions are false, but that doesn’t matter for our
analysis of logical form. To isolate the form, we’ll substitute letters for
terms. Thus, the first argument has this form:
All D are C
All C are S
All D are S
The second argument has this form:
No M are B
No M are D
No B are D
We can
provide substitution instances of the argument form, so long as the logical
form remains intact. The question is, can the form
yield a substitution instance which yields true premises and a false conclusion?
Consider the table on page 29. The first form is valid; the second is invalid.
Why?
Counterexamples:
Logical
analysis shows a deductive argument to be valid or invalid. When we add the
results of truth analysis, deductive arguments are either sound or unsound.
A
substitution instance having true premises and a false conclusion is called a
counterexample. We can use this method in a systematic fashion to demonstrate
and argument’s invalidity. To use the method, we first isolate the form of the
argument, and then construct a substitution instance having true premises and a
false conclusion, which, as we recall, is the very definition of invalidity.
For
categorical syllogisms. Consider:

All adlers are bobkins. All crockers are bobkins. Therefore,
all adlers are crockers. 
Which
has this form:

All A are
B All C are B All A are
C 
Why is this
form invalid?
Note the
substitution instance:

All women
are featherless bipeds All men
are featherless bipeds All women
are men 
True True False 
Thus, this form
is invalid, and any uniform substitution instance of
that form results in an invalid argument. Here is another counterexample:

All cats
(A) are animals (B). All dogs
(C) are animals (B). All cats
(A) are dogs (C). 
This
argument has true premises and a false conclusion. Thus, this is proven
invalid.
Note: For
categorical syllogisms, use dogs, cats, mammals, fish, and animals to
substitute for the terms. Also remember that in logic, “some” means “at least
one.” So, “some dogs are mammals” is a true statement. We cannot infer from “some dogs are mammals” that some of them are not
mammals.
To
facilitate the process, begin by making the conclusion false, then fill in the premises.
But, the
categorical syllogism is but one form of deductive arguments. Consider:

If the
government imposes import restrictions, the price of automobiles will rise.
Therefore, since the government will not impose import restrictions, it
follows that the price of automobiles will not rise. 
To represent
this form, we need a different approach. We use letters to represent entire
statements rather than terms. In this case, we get:

If G, then
P. Not G. Therefore,
not P. 
If we give
the following substitution instance:
G = Abraham
Lincoln committed suicide.
P = Abraham
Lincoln is dead.
We get:

If Abraham
Lincoln committed suicide, then Abraham Lincoln is dead. Abraham
Lincoln did not commit suicide. Therefore,
Abraham Lincoln is not dead. 
We have true
premises and a false conclusion. Yes, that first premise is true.
Identifying
basic argument forms requires that we be familiar with basic deductive forms.
First, write the argument with premises first and conclusion last. Then identity the form by noting the "form words."
Leave the form words as they are and substitute the nonform words with
letters. For categorical syllogisms, form words are "all,"
"no", "are", and "not." For hypothetical
syllogisms, "if," "then," and "not" are form
words. For other arguments, we have "either", "or",
"both", and "and."
Another
example:
All
movie stars are actors who are famous, because all movie stars who are famous
are actors.
If we
replace “movies stars,” “actors,” and “famous” with the letters M, A, and F, we
get this argument form:
All M who are F
are A.
All M who are A
are F.
Counterexample:
All humans who are fathers are men.
Therefore, all humans who are men are
fathers.
Remember:
The counterexample method cannot prove validity, only invalidity.
Inductive
Arguments: Strength and Truth
Strength and Cogency apply to inductive
arguments.
A strong inductive argument is an
inductive argument such that it is improbable that the premises be true
and the conclusion false. A weak inductive argument, on the other hand,
is one in which the conclusion probably does not follow from the premises.
Again, assume that the premises are true, and
then we determine whether, based on that assumption, the conclusion is probably
true.
Examples:

All crows we have seen to date are black. Therefore, probably the next crow we see will be black. 

All meteorites found to this day have contained sugar. Therefore, probably the next meteorite found will contain sugar. 
Both are strong. The first argument, though,
is cogent while the second is uncogent.
Cogent: A strong inductive argument with all true premises.

During the past fifty years, inflation has consistently reduced the value of the American Dollar. Therefore, industrial productivity will probably increase in the years ahead. 
Notice: Both of these may well be true, but
the premise provides no evidence for the conclusion. It is weak. Again, the
strength of an argument is simply the degree to which the premises, when
assumed true, support the conclusion.
Review the table on page 39.
Note: For both inductive and deductive
arguments, the only arrangement of truth and falsity relevant to validity or
soundness is when we have true premises and a false conclusion. Difference: the
strength of an inductive argument is a matter of degree. Validity is either/or.