33.2: Tautology, Contradiction, and Contingencies

When we are looking to evaluate a single claim, it can often be helpful to know if it is a tautology, a contradiction or a contingency.

Tautologies are statements that are always true. The following are examples of tautologies:

• Either it will rain tomorrow, or it won’t
• It is what it is.
• There’s nothing you can do that can’t be done.

Contradictions are statements that are always false. The following are examples of contradictions:

• It is raining right now, and it isn’t raining right now.
• The glass is both full and empty.
• The triangle is a circle.

Contingencies, often called contingent statements, are true in some cases and not true in others. For example:

• If we go to the store, then we will buy some apples.
• If a high pressure zone meets a low pressure zone, there’s be a tornado.
• If you have a cat, you won’t have mice.

In all honesty, we don’t often need help determining if a sentence is a tautology, contradiction or contingency. We often say that tautologies are trivial, and contradictions are obvious. Certainly, this is true in the examples given here. That said, sometimes claims will be very complex, and it may be less obvious which category they fall in. This is where tables can help us. It is also the case that these are the easiest things we can test for using tables, so it is a good place to start, even if ultimately, we don’t use the test very often.

Since tautologies are always true, the way we test for them is to make a truth table for the statement and then to check every row of it to see if there are any Fs. If there are, then the statement is not a tautology. In other words, all Ts means that it is a tautology. ‘P v ~P’ is a tautology, as this truth table shows:

‘P v Q’ is not a tautology, as the following truth table shows:

Notice that on row four of the table, the claim is false. Even one F on the right side will mean that the claim is not a tautology (since there is at least one case in which it won’t be true).

Testing for contradiction works exactly opposite as testing for tautology. For a statement to be a contradiction, it has to always be false, so the table has to show all ‘F’s on the right side. So, if there are any ‘T’s in the table, then the statement is not a contradiction. ‘P & ~P’ is a contradiction, as the following table shows:

‘P v Q’ is not a contradiction, as the following table shows:

Notice on the first three rows of the table the claim is true, so it can’t be a contradiction.

A contingent statement will have a truth table with both true and false rows. As seen above, ‘P v Q’ is a contingent statement – there are instances where it is true (row 1, 2 and 3), and an instance where it is false (row 4).

Exercises

Construct truth tables to test the following sentences for tautology, contradiction and contingency.

1. P → Q
2. (P v ~P) & (Q & ~Q)
3. P ← → Q
4. ~ (P & ~P)
5. ~ (P v ~P)
6. ~ (P v P)
7. (P & ~P) v (Q & ~Q)
8. ~ (P v ~~Q) & (P & ~Q)
9. (P ← → Q) v (Q ← → P)
10. ~ [(P → Q) → R]
11. [P → (Q → R)] & (P → R)
12. (P & Q) → (P → Q)
13. ~P → P
14. ~P → (Q v P)
15. ~P & ~(~P v ~Q)
16. P v (Q → P)
17. (P ← → Q) & [(~P v ~Q) & P]
18. (P ← → Q) & (P → Q)
19. [P → (Q → R)] & (R → P)
20. (P ← → Q) v ~ [(~P & Q) v (P & ~Q)

Selected Answers

2. (P v ~P) & (Q & ~Q)

Contradiction. As you can see from the table below all rows are false.

7. (P & ~P) v (Q & ~Q)

Contradiction. As you can see from the table below all rows are false.

16. P v (Q → P)

Contingent. As you can see from the table below there is a mix of true and false rows.