Introduction to Logic
Truth, Validity, and Soundness
Abstract: The foundation-concepts of
deductive logic are explained--truth, validity, and soundness.
I. Truth, Validity, and Soundness: probably the three most
important concepts of the course.
A. First, let us briefly characterize these
concepts.
1. truth: a property
of statements, i.e., that they are the case.
2. validity: a property
of arguments, i.e., that they have a good structure.
For example, consider the structure of this argument:
All B-s are Cs. All As are Bs.
Therefore, All As are Cs.
(The premisses and conclusion
are so related that it is absolutely impossible for the premisses to be
true unless the conclusion is true also.)
Note that an argument can be valid even though its premisses are false.
Also note that an argument is not mistaken just on account of its
conclusion being false or ridiculous.
3. soundness: a property of both
arguments and the statements in them, i.e., the argument is valid and all the
statement are true.
Sound Argument: (1) valid, (2) true
premisses (obviously the conclusion is true as well by the definition of validity).
B. The fact that a deductive argument is valid
cannot, in itself, assure us that any of the statements in the argument are true; this
fact only tells us that the conclusion must be true if the premisses are true.
1. Let's look at the variety of valid
arguments that can be given as sorted by the truth of premisses and conclusion:
Cases —>
1
2
3
4
Premiss(es)
T
F
T
F
Conclusion
T
T
F
If it were possible to have true premisses and a
false conclusion, logic would be useless to prove anything.
2. Let's look at the variety of invalid
arguments that can be given as sorted by the truth of premisses and conclusion:
Cases —>
1
2
3
4
Premiss(es)
T
F
T
F
Conclusion
T
T
F
F
That is, all possibilities can be represented;
examples are given in the syllabus.
II. One way to summarize these concepts is to represent the
logical territory in a "tree-diagram."
Arguments
_________|___________
Deductive
Inductive
_____|_____
_____|_____
Valid
Invalid
correct > > > > incorrect
_______|________
Sound
Unsound
(all statements are true)
(at
least one premiss
is false)
III. Consider these examples from past quizzes
A. Tell whether the following statements are
true or false.
1. A sound argument is a deductive argument
which is valid and has true premisses.
2. It is impossible for a deductive argument to
be both valid and unsound.
3. If a deductive argument is valid, it cannot
be unsound.
4. If the premisses of a deductive argument are
true, then the argument must be sound.
5. If the conclusion of a deductive argument is
true, then the argument might be sound or it might be unsound.
6. If the premisses of a deductive argument are
true, then the conclusion must be true.
7. If a deductive argument is valid, then its
conclusion might be true or it might be false.
8. A sound argument is a deductive argument that
has valid premisses.
9. If a valid argument has a false conclusion,
then it must have at least one false premiss.
10. If a deductive argument is valid and at
least one of its premisses is false, then its conclusion is also false.
B. More rules, definitions, and examples are
presented in the syllabus on this topic. Also check your
understanding with the quiz and test on "Truth, Validity, and Soundness.
