Logic - Simplification Rule of Inference
I have been wondering if I could simplify the statement (p Λ q) → r to p → r using the simplification rule of inference. I can't really see why not since conjunction has precedence over the implication, so I thought I could adjust the conjunction statement before I get to the implication. Please correct me if I am wrong. Thank you!
$\endgroup$2 Answers
$\begingroup$Compare:
'If I am male and unmarried, then I am a bachelor'
'If I am male, then I am a bachelor'
The first statement is true, but the second statement is not
So this is a concrete counterexample to the purported validity of your inference.
But it also nicely shows what goes wrong conceptually when you 'weaken' the antecedent: you are taking away what may well be a necessary condition for something to be true.
$\endgroup$ $\begingroup$$p\to r$ is not a logical consequence of $(p\land q)\to r$. To see that, consider the truth assignment where $p$ is true while both $q$ and $r$ are false.
I suspect you've taken an inference rule of the form "from this, infer that" and tried to use "from a formula containing this as a subformula, infer the corresponding formula with that as a subformula." If so, you should confine yourself to taking rules of inference to mean literally what they say, nothing more.
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