What logical connectives do you know?
| \top | \bot | \wedge | \vee | \rightarrow |
|---|---|---|---|---|
| a \wedge b | a \vee b | a \rightarrow b | ||
| truth | falsehood | conjunction | disjunction | implication |
| “trivial” | “impossible” | a and b | a or b | a gives b |
| shouldn’t get | got both | got at least one | given a, we get b |
How can we define them? Think in terms of derivation trees:
\frac{ \frac{\frac{\,}{\text{a premise}} \; \frac{\,}{\text{another premise}}}{\frac{\,}{\text{some fact}}} \; \frac{\frac{\,}{\text{a premise}} \; \frac{\,}{\text{another premise}}}{\frac{\,}{\text{another fact}}}} {\text{final conclusion}}
To define the connectives, we provide rules for using them: for example, a rule \frac{a \; b}{c} matches parts of the tree that have two premises, represented by variables a and b, and have any conclusion, represented by variable c.
Introduction rules say how to produce a connective. Elimination rules say how to use it. Text in parentheses is comments. Letters are variables: stand for anything.