Proving arguments are valid using rules of inference Use the rules of inference and the laws of propositional logic to prove that each argument is valid. Number each line of your argument and label each line of your proof "Hypothesis" or with the name of the rule of inference used at that line. If a rule of inference is used, then include the numbers of the previous lines to which the rule is applied.
Example 1:
p → q
q → r
¬r
∴ ¬p
Solution to example 1
1. q → r Hypothesis
2. p → q Hypothesis
3. p → r Hypothetical syllogism, 1, 2
4. ¬r Hypothesis
5. ¬p Modus tollens, 3, 4.
Example 2:
(p ∧ q) → r
¬r
q
∴ ¬p
Solution to example 2
1. (p ∧ q) → r Hypothesis
2. ¬r Hypothesis
3. ¬(p ∧ q) Modus tollens, 1, 2.
4. ¬p ∨ ¬q De Morgan's law, 3
5. q Hypothesis
6. ¬¬q Double negation law, 5.
7. ¬q ∨ ¬p Commutative law, 4
8. ¬p Disjunctive syllogism, 6, 7.
1. p → (q ∧ r)
¬q
∴ ¬p
2. p ∨ q
¬p ∨ r
¬q
∴ r