6.3 Deduction rules

• $X, Y \vdash X \wedge Y$: this means that X is true, Y is true, we can deduce that the conjunction is true as well.
• $X \rightarrow Y, X \vdash Y$: this means that X implies Y, X is true, so we can deduce that Y is true.
• $X \rightarrow Y, \neg Y \vdash \neg X$: this means that X implies Y, Y is false, so we can deduce that X is false.
• $X \wedge Y \vdash X$: this means that the conjunction of X and Y is true, we can deduce that X is true.
• etc.

How do we get these deduction (argument) rules? They can be derived from truth tables. For example, let us consider the rule $X \rightarrow Y, X \vdash Y$.

Now, we look up the truth table of implication. Because deduction assumes all the propositions (separated by commas) are true, we look up all the row that has $X \rightarrow Y$ being true, and $X$ being true. The only row that corresponds to this combination indicates that $Y$ is true. Because $Y$ is always true when $X \rightarrow Y$ and $X$ are both true, the deduction rule holds.

Now, can we have a deduction rule that states that $X \rightarrow Y, Y \vdash X$? Let's look up the table again. This time, we need to look for rows in which $X \rightarrow Y$ and $Y$ are both true. We can locate two such rows. However, $X$ is true in one row, and false in the other row. This means that we cannot guarantee that $X$ is true, given that $X \rightarrow Y$ and $Y$ are true. As a result, we cannot have the deduction rule of $X \rightarrow Y, Y \vdash X$.

Now, as an exercise, verify that $X \rightarrow Y, \neg Y \vdash \neg X$.