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If $p \rightarrow q$ and $p$ are given, then we can infer $q$. Example: If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet.

If $p \rightarrow q$ and $\neg q$ are given, then we can infer $\neg p$. Example: If it is raining, then the ground is wet. The ground is not wet. Therefore, it is not raining.

If $p \rightarrow q$ and $q \rightarrow r$ are given, then we can infer $p \rightarrow r$. Example: If it rains, the picnic will be canceled. If the picnic is canceled, we will watch a movie. Therefore, if it rains, we will watch a movie.

If $p \lor q$ and $\neg p$ are given, then we can infer $q$. Example: It is either raining or snowing. It is not raining. Therefore, it is snowing.

From proposition $p$, we can infer $p \lor q$, for any proposition $q$. Example: It is raining. Therefore, it is raining or I am hungry.

From $p \land q$, infer $p$. Example: The ball is red and round. Therefore, the ball is red.

From $p$ and $q$, infer $p \land q$. Example: The ball is red. The ball is round. Therefore, the ball is red and round.

From $(p \lor q)$ and $(\neg p \lor r)$, infer $q \lor r$. Example: It will rain or snow today. It will not rain or it will be cold. Therefore, it will snow or be cold.

From $\forall x, P(x)$, infer $P(c)$ for a particular $c$. Example: All birds can fly. Tweety is a bird. Therefore, Tweety can fly.