(Solved): 16 Simplification is a propositional logic rule of inference. It is a rule of implication, which mea ...

Simplification is a propositional logic rule of inference. It is a rule of implication, which means that its premise implies its conclusion but that the conclusion is not necessarily logically equivalent to the premise. Simplification, like all rules of implication, can be applied only to whole lines in a proof and not to parts of larger statements. Simplification is defined as follows: Simplification states that if you have a conjunction \( p \cdot q \) (which means \( p \) is true and \( q \) is true), then the first conjunct, \( p \), must be true on its own. You can use simplification whenever you have a conjunction on a preceding proof line. From the line containing the conjunction, you can add a new line containing only the left-hand conjunct, \( p \). Although \( q \) must be true also if \( \rho \cdot q \) is true, you cannot use simplification to conclude \( q \) directly from \( p \) - \( q \). If you need \( q \) on its own line, then you must first use a rule called "commutativity" (which you will leam in a later section from your text) to rewrite \( p \cdot q \) as \( q \cdot p \) before you use simplification to conclude \( q \) (which would then be the left-hand conjunct). For now, remember that you can only use simplification to obtain the left-hand conjunct, and not the right-hand conjunct, from a conjunction on its own proof line. Consider the natural deduction proof given below. Using your knowledge of the natural deduction proof method and the options provided in the dropdown menus, fill in the blanks to identify the missing information (premises, inferences, or justifications) that completes the given application of the simplification (Simp) rule.