Relationships in Truth Statements
Read these sections to learn more about relationships among truth statements and using and constructing logical proofs.
These sections review materially equivalent propositions and three other relationships among statements: tautological, contradictory, and contingent relationships. They also review the eight valid forms of inference: modus ponens, modus tollens, hypothetical syllogism, simplification, conjunction, disjunctive syllogism, addition, and constructive dilemma. They show how to construct proofs, including strategies for working forward or backward, depending on which is easier according to your premises. Finally, they summarize everything you have learned about sentential and propositional logic.
Complete the exercises as you study, then check your answers against the key.
Tautologies, contradictions and contingent statements
Can you think of a statement that could never be false? How about a statement
that could never be true? It is harder than you think, unless you know how to
utilize the truth functional operators to construct a tautology or a contradiction.
A tautology is a statement that is true in virtue of its form. Thus, we don't even
have to know what the statement means to know that it is true. In contrast, a
contradiction is a statement that is false in virtue of its form. Finally, a
contingent statement is a statement whose truth depends on the way the world
actually is. Thus, it is a statement that could be either true or false - it just
depends on what the facts actually are. In contrast, there is an important sense
in which the truth of a tautology or the falsity of a contradiction doesn't depend
on how the world is. As philosophers would say, tautologies are true in every possible world, whereas contradictions are false in every possible world.
Consider a statement like:
Matt is either 40 years old or not 40 years old.
That statement is a tautology, and it has a particular form, which can be
represented symbolically like this:
p v ~p
In contrast, consider a statement like:
Matt is both 40 years old and not 40 years old.
That statement is a contradiction, and it has a particular form, which can be
represented symbolically like this:
p ⋅ ~p
Finally, consider a statement like:
Matt is either 39 years old or 40 years old
That statement is a contingent statement. It doesn't have to be true (as
tautologies do) or false (as contradictions do). Instead, its truth depends on the
way the world is. Suppose that Matt is 39 years old. In that case, the statement
is true. But suppose he is 37 years old. In that case, the statement is false (since
he is neither 39 or 40). We can use truth tables to determine whether a
statement is a tautology, contradiction or contingent statement. In a tautology,
the truth table will be such that every row of the truth table under the main
operator will be true. In a contradiction, the truth table will be such that every
row of the truth table under the main operator will be false. And contingent
statements will be such that there is mixture of true and false under the main
operator of the statement.
The following two truth tables are examples of tautologies and contradictions, respectively.
| A | B | (A ⊃ B) v A |
|---|---|---|
| T | T | T T |
| T | F | F T |
| F | T | T T |
| F | F | T T |
| A | B | (A ⊃ B) v A |
|---|---|---|
| T | T | T F F F F |
| T | F | T F F F T |
| F | T | T F T F F |
| F | F | F F T F T |
Notice that in the second truth table, I had to do quite a lot of work before I could figure out what the truth values of the main operator were. I had to first determine the left conjunct (A v B) and then the right conjunct (~A ⋅ ~B), but in order to figure out the truth values of the right conjunct (which is itself a conjunct), I had to determine the negations of A and B. Constructing truth tables can sometimes be a chore, but once you understand what you are doing (and why), it certainly isn't very difficult.