Question

Provide a solution to the following proof: 1. C→(T→L) 2. ∼L 3. ∼E→C 4. LV∼E

Answer 1

Based on the given premises:

`\[1. C \rightarrow (T \rightarrow L)\]`

`\[2. \sim L\]`

`\[3. \sim E \rightarrow C\]`

`\[4. L \lor \sim E\]`

we can conclude:

`\[5. C \rightarrow T \quad \text{(From 1 by Simplification)}\]`

`\[6. \sim C \lor T \quad \text{(From 5 by Material Implication)}\]`

`\[7. \sim \sim C \rightarrow T \quad \text{(From 6 by Double Negation)}\]`

`\[8. \sim \sim C \lor \sim E \rightarrow T \quad \text{(From 7 by Constructive Dilemma with 4)}\]`

`\[9. C \rightarrow \sim E \quad \text{(From 3 by Contrapositive)}\]`

`\[10. C \lor \sim E \rightarrow T \quad \text{(From 8 by Disjunctive Syllogism with 9)}\]`

Therefore, the conclusion is
`\(C \lor \sim E \rightarrow T\)`.

Step-by-step explanation:

The solution involves applying logical inference rules to derive a conclusion from the given premises. The steps include simplification, material implication, double negation, constructive dilemma, and disjunctive syllogism. The process ensures a valid transformation of the statements according to the rules of propositional logic.

In detail, we use simplification to extract
`\(C \rightarrow T\)` from the first premise. Then, through material implication, we derive
`\(\sim C \lor T\)`. Applying double negation and constructive dilemma allows us to infer
`\(\sim \sim C \lor \sim E \rightarrow T\)` with the fourth premise. Finally, by contrapositive and disjunctive syllogism, we obtain the conclusion
`\(C \lor \sim E \rightarrow T\)`.

The logical steps ensure a coherent and valid solution to the given proof.