Prove that Hypothetical Syllogism is redundant using the language of propositional logic. Build a proof for hypothetical syllogism without using hypothetical syllogism in the proof.
The Correct Answer and Explanation is :
To prove that Hypothetical Syllogism is redundant using the language of propositional logic, we need to show that we can derive the conclusion of a Hypothetical Syllogism (i.e., ( p \rightarrow q ) and ( q \rightarrow r ) implies ( p \rightarrow r )) without directly using the rule of Hypothetical Syllogism.
Hypothetical Syllogism:
The rule of Hypothetical Syllogism states:
[
\frac{p \rightarrow q \quad q \rightarrow r}{\therefore p \rightarrow r}
]
That is, if ( p \rightarrow q ) and ( q \rightarrow r ), we can conclude ( p \rightarrow r ).
Proof Without Using Hypothetical Syllogism:
We will prove ( p \rightarrow r ) from the premises ( p \rightarrow q ) and ( q \rightarrow r ) by breaking down the logical steps.
- Assume ( p ) (Assumption for Conditional Proof).
- From ( p \rightarrow q ), derive ( q ) (Modus Ponens):
- Since we have ( p \rightarrow q ) and ( p ), we can apply Modus Ponens to derive ( q ).
- From ( q \rightarrow r ), derive ( r ) (Modus Ponens):
- Now, using the derived ( q ) and the premise ( q \rightarrow r ), we can again apply Modus Ponens to derive ( r ).
- Conclude ( p \rightarrow r ) (Conditional Proof):
- Since assuming ( p ) led us to derive ( r ), we can conclude ( p \rightarrow r ) by the Conditional Proof rule.
Thus, we have derived ( p \rightarrow r ) from ( p \rightarrow q ) and ( q \rightarrow r ), but we did not directly use the rule of Hypothetical Syllogism. Instead, we used Modus Ponens twice and the Conditional Proof method.
Explanation:
The key idea behind this proof is that Hypothetical Syllogism is simply a shorthand for applying Modus Ponens in a chain of reasoning. By assuming ( p ), applying Modus Ponens to derive ( q ), and then applying Modus Ponens again to derive ( r ), we can reach the same conclusion that Hypothetical Syllogism would give, but we avoid directly invoking the rule. Therefore, Hypothetical Syllogism is redundant because the same reasoning can be accomplished through basic inference rules like Modus Ponens and Conditional Proof.