form that makes it impossible for the premises to be true and the
conclusion nevertheless to be false. Otherwise, a deductive argument
is said to be invalid.
A deductive argument is sound if and only if it is both valid, and all
of its premises are actually true. Otherwise, a deductive argument is
unsound.
According to the definition of a deductive argument (see the Deduction
and Induction), the author of a deductive argument always intends that
the premises provide the sort of justification for the conclusion
whereby if the premises are true, the conclusion is guaranteed to be
true as well. Loosely speaking, if the author's process of reasoning
is a good one, if the premises actually do provide this sort of
justification for the conclusion, then the argument is valid.
In effect, an argument is valid if the truth of the premises logically
guarantees the truth of the conclusion. The following argument is
valid, because it is impossible for the premises to be true and the
conclusion to nevertheless be false:
Either Elizabeth owns a Honda or she owns a Saturn.
Elizabeth does not own a Honda.
Therefore, Elizabeth owns a Saturn.
It is important to stress that the premises of an argument do not have
actually to be true in order for the argument to be valid. An argument
is valid if the premises and conclusion are related to each other in
the right way so that if the premises were true, then the conclusion
would have to be true as well. We can recognize in the above case that
even if one of the premises is actually false, that if they had been
true the conclusion would have been true as well. Consider, then an
argument such as the following:
All toasters are items made of gold.
All items made of gold are time-travel devices.
Therefore, all toasters are time-travel devices.
Obviously, the premises in this argument are not true. It may be hard
to imagine these premises being true, but it is not hard to see that
if they were true, their truth would logically guarantee the
conclusion's truth.
It is easy to see that the previous example is not an example of a
completely good argument. A valid argument may still have a false
conclusion. When we construct our arguments, we must aim to construct
one that is not only valid, but sound. A sound argument is one that is
not only valid, but begins with premises that are actually true. The
example given about toasters is valid, but not sound. However, the
following argument is both valid and sound:
No felons are eligible voters.
Some professional athletes are felons.
Therefore, some professional athletes are not eligible voters.
Here, not only do the premises provide the right sort of support for
the conclusion, but the premises are actually true. Therefore, so is
the conclusion. Although it is not part of the definition of a sound
argument, because sound arguments both start out with true premises
and have a form that guarantees that the conclusion must be true if
the premises are, sound arguments always end with true conclusions.
It should be noted that both invalid, as well as valid but unsound,
arguments can nevertheless have true conclusions. One cannot reject
the conclusion of an argument simply by discovering a given argument
for that conclusion to be flawed.
Whether or not the premises of an argument are true depends on their
specific content. However, according to the dominant understanding
among logicians, the validity or invalidity of an argument is
determined entirely by its logical form. The logical form of an
argument is that which remains of it when one abstracts away from the
specific content of the premises and the conclusion, i.e., words
naming things, their properties and relations, leaving only those
elements that are common to discourse and reasoning about any subject
matter, i.e., words such as "all", "and", "not", "some", etc. One can
represent the logical form of an argument by replacing the specific
content words with letters used as place-holders or variables.
For example, consider these two arguments:
All tigers are mammals.
No mammals are creatures with scales.
Therefore, no tigers are creatures with scales.
All spider monkeys are elephants.
No elephants are animals.
Therefore, no spider monkeys are animals.
These arguments share the same form:
All A are B;
No B are C;
Therefore, No A are C.
All arguments with this form are valid. Because they have this form,
the examples above are valid. However, the first example is sound
while the second is unsound, because its premises are false. Now
consider:
All basketballs are round.
The Earth is round.
Therefore, the Earth is a basketball.
All popes reside at the Vatican.
John Paul II resides at the Vatican.
Therefore, John Paul II is a pope.
These arguments also have the same form:
All A's are F;
X is F;
Therefore, X is an A.
Arguments with this form are invalid. This is easy to see with the
first example. The second example may seem like a good argument
because the premises and the conclusion are all true, but note that
the conclusion's truth isn't guaranteed by the premises' truth. It
could have been possible for the premises to be true and the
conclusion false. This argument is invalid, and all invalid arguments
are unsound.
While it is accepted by most contemporary logicians that logical
validity and invalidity is determined entirely by form, there is some
dissent. Consider, for example, the following arguments:
My table is circular. Therefore, it is not square shaped.
Juan is bachelor. Therefore, he is not married.
These arguments, at least on the surface, have the form:
x is F;
Therefore, x is not G.
Arguments of this form are not valid as a rule. However, it seems
clear in these particular cases that it is, in some strong sense,
impossible for the premises to be true while the conclusion is false.
However, many logicians would respond to these complications in
various ways. Some might insist–although this is controverisal–that
these arguments actually contain implicit premises such as "Nothing is
both circular and square shaped" or "All bachelors are unmarried,"
which, while themselves necessary truths, nevertheless play a role in
the form of these arguments. It might also be suggested, especially
with the first argument, that while (even without the additional
premise) there is a necessary connection between the premise and the
conclusion, the sort of necessity involved is something other than
"logical" necessity, and hence that this argument (in the simple form)
should not be regarded as logically valid. Lastly, especially with
regard to the second example, it might be suggested that because
"bachelor" is defined as "adult unmarried male", that the true logical
form of the argument is the following universally valid form:
x is F and not G and H;
Therefore, x is not G.
The logical form of a statement is not always as easy to discern as
one might expect. For example, statements that seem to have the same
surface grammar can nevertheless differ in logical form. Take for
example the two statements:
(1) Tony is a ferocious tiger.
(2) Clinton is a lame duck.
Despite their apparent similarity, only (1) has the form "x is a A
that is F". From it one can validly infer that Tony is a tiger. One
cannot validly infer from (2) that Clinton is a duck. Indeed, one and
the same sentence can be used in different ways in different contexts.
Consider the statement:
(3) The King and Queen are visiting dignitaries.
It is not clear what the logical form of this statement is. Either
there are dignitaries that the King and Queen are visiting, in which
case the sentence (3) has the same logical form as "The King and Queen
are playing violins," or the King and Queen are themselves the
dignitaries who are visiting from somewhere else, in which case the
sentence has the same logical form as "The King and Queen are
sniveling cowards." Depending on which logical form the statement has,
inferences may be valid or invalid. Consider:
The King and Queen are visiting dignitaries. Visiting dignitaries
is always boring. Therefore, the King and Queen are doing something
boring.
Only if the statement is given the first reading can this argument be
considered to be valid.
Because of the difficulty in identifying the logical form of an
argument, and the potential deviation of logical form from grammatical
form in ordinary language, contemporary logicians typically make use
of artificial logical languages in which logical form and grammatical
form coincide. In these artificial languages, certain symbols, similar
to those used in mathematics, are used to represent those elements of
form analogous to ordinary English words such as "all", "not", "or",
"and", etc. The use of an artifically constructed language makes it
easier to specify a set of rules that determine whether or not a given
argument is valid or invalid. Hence, the study of which deductive
argument forms are valid and which are invalid is often called "formal
logic" or "symbolic logic".
In short, a deductive argument must be evaluated in two ways. First,
one must ask if the premises provide support for the conclusion by
examing the form of the argument. If they do, then the argument is
valid. Then, one must ask whether the premises are true or false in
actuality. Only if an argument passes both these tests is it sound.
However, if an argument does not pass these tests, its conclusion may
still be true, despite that no support for its truth is given by the
argument.
Note: there are other, related, uses of these words that are found
within more advanced mathematical logic. In that context, a formula
(on its own) written in a logical language is said to be valid if it
comes out as true (or "satisfied") under all admissible or standard
assignments of meaning to that formula within the intended semantics
for the logical language. Moreover, an axiomatic logical calculus (in
its entirety) is said to be sound if and only if all theorems
derivable from the axioms of the logical calculus are semantically
valid in the sense just described.
For a more sophisticated look at the nature of logical validity, see
the articles on "Logical Consequence" in this encyclopedia. The
articles on "Argument" and "Deductive and Inductive Arguments" in this
encyclopedia may also be helpful.
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