Help me with this propositional logic demonstration

This is a simple propositional logic demonstration. I’d appreciate your help. I don’t know if my answer is correct, but the textbook used another demonstration.

The question

1. $$T \vee R$$
2. $$(T \vee R) \vee (S.P) \to (Q.S)$$, therefore $$Q.S$$

1. $$T$$ 1, Simplification
2. $$T \vee (S.P) \to (Q.S)$$ 2, Simplification
3. $$T \to Q.S$$ 2,4, Simplification
4. $$T$$ 5, Modus Ponens

I know this is a very basic question and my answer is not elegant, but since I’m studying logic all by myself, and have no teacher to ask, I’d appreciate an answer.

1. $$(T \vee R) \vee (S.P)$$ 1, Addition
2. $$Q.S$$ 2,3 Modus Ponens

You cannot apply Simplification to a Disjunction.

$$X\vee Y$$ does not logically entail $$X$$.

$$X\vee Y$$ means at least one from $$\{X, Y\}$$ is true, which does not guarantee that one will be $$X$$.

You also cannot apply the rule of Simplification to the antecedent of a Conditional statement.

The rule of Simplification is applied to a Conjunction, when that connective is the root operation in the statement.$$X\wedge Y~\vdash~X\\X\wedge Y~\vdash~Y$$

$$X\land Y$$ means both from $$\{X,Y\}$$ are true, and that entails that $$X$$ is true.

• Thank you! It was a simple and elementary mistake indeed! 🤦🏻‍♀️ – brigittethecat Apr 19 at 12:55