Properties of Propositions: Tautologies, Contradictions, and Contingencies

We’ve seen how to use truth tables and the truth assignment test to determine whether an argument is valid or invalid. Those same tools also allow us to examine the logical properties of individual propositions and the logical relations between propositions. Logical properties of propositions are considered below; some important logical relations will be introduced on the next page.

Propositions can be classified into three categories: tautologies, contradictions, and contingencies. Whether a proposition is a tautology, contradiction, or contingency depends on its form—it’s logical structure.

• A tautology, or tautologous proposition, has a logical form that cannot possibly be false (no matter what truth values are assigned to the sentence letters).

The following propositions are tautologies:
• (A ⊃ A)
• (A ∨ ~A)
• ~(A • ~A)
• ((A • B) ⊃ (A ∨ B))
• ((A ∨ B) ≡ (B ∨ A))
• A contradiction, or self-contradictory proposition, has a logical form that cannot possibly be true (no matter what truth values are assigned to the sentence letters).

• (A • ~A)
• ~(A ∨ ~A)
• ~(A ⊃ A)
• ((A ∨ ~A) ⊃ (B • ~B))
• ~((A ∨ B) ≡ (B ∨ A))
• A contingency, or contingent proposition, has a logical form that can be either true or false (depending on what truth values are assigned to the sentence letters).

The following propositions are contingencies:
• A
• ~A
• (A ∨ B)
• ~(A • B)
• (A ⊃ B)
• ((A ∨ B) ⊃ (C • D))
• ((A • B) ≡ (C ∨ D))
Some propositions have logical structure that cannot be represented with truth-functional connectives, and for this reason some tautologies and contradictions look like contingencies when symbolized in propositional logic. We’ll deal with other kinds of logical structure in later chapters.

To determine whether a proposition is a tautology, contradiction, or contingency, we can construct a truth table for it. If the proposition is true in every row of the table, it’s a tautology. If it is false in every row, it’s a contradiction. And if the proposition is neither a tautology nor a contradiction—that is, if there is at least one row where it’s true and at least one row where it’s false—then the proposition is a contingency.

Consider the following proposition:

If roses are red and violets are blue, then roses aren’t red.

This may sound like a contradiction—a proposition that couldn’t possibly be true. However, our intuitions about logical properties are often mistaken. To find out which type of proposition it really is, let’s symbolize the sentence and construct a truth table for it:

RB((R • B) ⊃ ~R)
001
011
101
110

As we can see from the truth table above, the proposition is definitely not a contradiction. In fact, there are more ways for it to be true than there are ways for it to be false: it is true in every row except the last row. Since it is true in at least one row and false in at least one row, it is a contingency.

Let’s look at a few more examples. The following truth table shows the possible truth values for three compound propositions. One of the propositions is a tautology, one is a contradiction, and one is a contingency. Can you tell which is which?

ABC(~(A ∨ B) • B)((A • B) ∨ C)(B ⊃ (B ∨ C))
000001
001011
010001
011011
100001
101011
110011
111011

The proposition (~(A ∨ B) • B) is a contradiction, because it is false in every row. (There are no ones in its column of the table.)

The proposition ((A • B) ∨ C) is a contingency, because it is true in some rows and false in others.

The proposition (B ⊃ (B ∨ C)) is a tautology, because it is true in every row. (There are no zeros in its column.)

Alternatively, we can use the truth assignment method to determine whether a proposition is a tautology, contradiction, or contingency. Rather than constructing the entire truth table, we can simply check whether it is possible for the proposition to be false, and then check whether it is possible for the proposition to be true. Here’s a description of the procedure in a little more detail:

1. To check whether a proposition is a tautology, we need to determine whether it is possible for the proposition to be false. (Remember, a tautology has a form that can’t possibly be false.) So, we begin by assigning “0” to its main connective, then calculate the truth values of any other connectives and sentence letters that can be determined based on that assumption. If some letters cannot be calculated, try all possible combinations of values for those letters, to see whether there is any way for the proposition to be false. If there’s no way to make the proposition false, it’s a tautology. But if you succeed in finding a way for the proposition to be false, it isn’t a tautology, so you should proceed to step 2:
2. To check whether a proposition is a contradiction, begin by assigning “1” to its main connective, then calculate the truth values of any other connectives and sentence letters that can be determined based on that assumption. If some letters cannot be calculated, try all possible combinations of values for those letters. If there’s no way to make the proposition true, it’s a contradiction. But if you succeed in finding a way for the proposition to be true, it isn’t a contradiction.
3. If a proposition is neither a tautology nor a contradiction, as determined by steps 1 and 2, then it is a contingency.
Is the following sentence a tautology?

Either roses are red and violets are blue, or roses are red only if violets aren’t blue.

Remember, a tautology has a logical form that can’t possibly be false. So, to determine whether this proposition is a tautology, we need to check whether there is any possible way to make it false. To find out, let’s symbolize it and assign “0” to its main connective:

 (( R • B ) ∨ ( R ⊃ ~ B )) 0

There is only one way for the “∨” to be false, namely if both disjuncts are false:

 (( R • B ) ∨ ( R ⊃ ~ B )) 0 0 0

Moreover, there is only one way for the “⊃” to be false, namely if R is true and ~B is false (and thus B is true). This means that if the proposition as a whole is false (as we’ve assumed), then R and B must both be true. However, if we make R and B both true, then (R • B) isn’t false anymore, and hence the “∨” isn’t false anymore either:

 (( R • B ) ∨ ( R ⊃ ~ B )) 1 0 1 0 1 0 0 1

So, there is no possible way to make this proposition false. It is a tautology.