Section 3: Using truth tables
Tautologies, contradictions, and contingent sentences
Recall that an English sentence is a tautology if it must be true as a matter of logic. With a complete truth table, we consider all of the ways that the world might be. If the sentence is true on every line of a complete truth table, then it is true as a matter of logic, regardless of what the world is like.
So a sentence is a tautology in sl if the column under its main connective is 1 on every row of a complete truth table.
Conversely, a sentence is a contradiction in sl if the column under its main connective is 0 on every row of a complete truth table.
A sentence is contingent in sl if it is neither a tautology nor a contradiction; i.e. if it is 1 on at least one row and 0 on at least one row.
From the truth tables in the previous section, we know that (\(H\)&\(I\)) → \(H\) is a tautology, that [(\(C\) ↔ \(C\)) → \(C\)]&¬(\(C\) → \(C\)) is a contradiction, and that \(M\) &(\(N\)∨\(P\)) is contingent.
Logical equivalence
Two sentences are logically equivalent in English if they have the same truth value as a matter logic. Once again, truth tables allow us to define an analogous concept for SL: Two sentences are logically equivalent in sl if they have the same truth-value on every row of a complete truth table.
Consider the sentences ¬(\(A\)∨\(B\)) and ¬\(A\)&¬\(B\). Are they logically equivalent? To find out, we construct a truth table.
| \(A\) | \(B\) | ¬(\(A\)∨\(B\)) | ¬\(A\) & ¬\(B\) |
|
1 1 0 0 |
1 0 1 0 |
0 1 1 1 0 1 1 0 0 0 1 1 1 0 0 0 |
0 1 0 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 1 0 |
Look at the columns for the main connectives; negation for the first sentence, conjunction for the second. On the first three rows, both are 0. On the final row, both are 1. Since they match on every row, the two sentences are logically equivalent.
Consistency
A set of sentences in English is consistent if it is logically possible for them all to be true at once. A set of sentences is logically consistent in sl if there is at least one line of a complete truth table on which all of the sentences are true. It is inconsistent otherwise.
Validity
An argument in English is valid if it is logically impossible for the premises to be true and for the conclusion to be false at the same time. An argument is valid in sl if there is no row of a complete truth table on which the premises are all 1 and the conclusion is 0; an argument is invalid in sl if there is such a row.
Consider this argument:
¬\(L\) → (\(J\) ∨\(L\))
¬\(L\)
.˙. \(J\)
Is it valid? To find out, we construct a truth table.
| \(J\) | \(L\) | ¬\(L\)→(\(J\)∨\(L\)) | ¬\(L\) | \(J\) |
|
1 1 0 0 |
1 0 1 0 |
0 1 1 1 1 1 1 0 1 1 1 0 0 1 1 0 1 1 1 0 0 0 0 0 |
0 1 1 0 0 1 1 0 |
1 1 0 0 |
Yes, the argument is valid. The only row on which both the premises are 1 is the second row, and on that row the conclusion is also 1.