2. Truth Tables
Consider conjunction:
| p | q | p ∧ q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
In truth-functional semantics, the behavior of conjunction is determined by the truth values of its inputs.
There are four possible combinations. The table contains all of them.
If both propositions are true, their conjunction is true.
Otherwise it is false.
With respect to the truth-functional cases represented here, the table makes the relevant possibilities explicit.
This gives it a useful kind of completeness.
We do not have to remember that perhaps another combination exists somewhere outside the table. Given two inputs that each have exactly two possible truth values, the four rows exhaust the combinations.
The usefulness of the table depends on something established beforehand:
- each proposition receives one of the two truth values;
- the listed cases exhaust those possibilities.
Once that background is accepted, the table provides a compact representation of the whole case structure.
Recursion tables begin from the same idea.
But instead of asking which values an expression can receive, we ask how its inputs can be constructed.