(Solved): 11 Hypothetical syllogism is a rule of implication, which means that its conclusion can be inferred ...

Hypothetical syllogism is a rule of implication, which means that its conclusion can be inferred from its premises but may not be logically equivalent to those premises. All rules of implication can be applied only to the main operator of the statements matching the form of the rule. Hypothetical syllogism states that if you have one conditional \( p \supset q \) on its own line, and if you have another conditional \( q \supset r \) the antecedent of which is the consequent of the first conditional on another line, then you can conclude the conditional \( p \leq r \) on a new line. You can think of hypothetical syllogism as linking two conditionals together in a chain. You can apply the hypothetical syllogism rule only if the two required conditional premises are on their own lines with the horseshoe as the main operator for each line. In other words, you cannot apply hypothetical syllogism to a part of a line. Remember that \( p \), \( q \), and \( r \) can stand for any statement, including compound statements with other operators (even other horseshoes) within. It does not matter which of the two conditional statement comes first in the lines of a proof. It only matters that you have two conditional lines that match the form of the two premises in the form for hypothetical syllogism, regardless of the order in which they appear. Consider the natural deduction proof given below. Using your knowiedge of the natural deduction proof method and the options provided in the dropdown menus, fin in the blanks to identify the missing information (premises, inferences, or justifications) that completes the given application of the pure hypotheticai syllogism (HS) rule. 1. \( \sim(M \vee \sim P)>\sim(H * B) \) 2. \( (\sim I \vee S) \sim \sim(M \vee \sim P) \) 3. \( (\sim l \sim \sim H)>(S \vee B) \quad /(\sim l \vee S)>\sim(H \cdot B) \) 4. \( (\sim I \vee S)>\sim(H \cdot B) \) 2,3 2 1 3 1,2 1,3