Rules of inference - Is my application of simplification in this proof, correct?
By Emma Terry •
Could someone verify that my proof is valid as the question did not have a solution?
- $(\lnot R \lor \lnot F) \to (S \land L)$
- $(S \to T)$
- $(\lnot T)$
- $(\lnot S)$ 2,3 Modus tollens
- $\lnot (\lnot R \lor \lnot F) \lor (S \land L)$ 1, implication equivalence
- $(R \land F) \lor (S \land L)$ 5, double negation
- $(R \lor S)$ 6, simplification
- $(R)$ 7,4 Disjunctive syllogism
My main concern is with line 7 with the use of simplification, have I applied the rule correctly?
I understand that with simplification if you have $(P \land Q)$ and apply it, it returns $(P)$.
For my proof, you had to show that lines 1-3 (the hypotheses) entail $R$.
$\endgroup$ 51 Answer
$\begingroup$As correctly said by Mauro Allegranza, your usage of simplification is wrong.
As an alternative to the proofs suggested by Mauro Allegranza (which are perfect), consider the following proof:
- $(\lnot R \lor \lnot F) \to (S \land L)$ assumption
- $(S \to T)$ assumption
- $(\lnot T)$ assumption
- $(\lnot S)$ 2,3 Modus tollens
- $(\lnot S \lor \lnot L)$ 4, addition
- $\lnot(S \land L)$ 5, De Morgan
- $\lnot(\lnot R \lor \lnot F)$ 1, 6 Modus tollens
- $(R \land F)$ 7, double negation (De Morgan)
- $(R)$ 8, simplification
