Abstract
- A form of Mathematical Argument that is guaranteed to be Valid if all its Premise are true
- Tools that enable Deductive Reasoning (演繹推理)
Basics
Modus Ponens
Modus Tollens
- Denying the consequent
Generalization
Case 1
- Premise:
p - Conclusion:
p v q
Case 2
- Premise:
q - Conclusion:
p v q
Specilisation
- Allow us to discard some information to focus on things that we are interested
Case 1
- Premise:
p AND q - Conclusion:
p
Case 2
- Premise:
p AND q - Conclusion:
q
Elimination
Case 1
- Premise:
p v q,~q - Conclusion:
p
Case 2
- Premise:
p v q,~p - Conclusion:
q
Transitivity
- Premise:
p -> q,q -> r - Conclusion:
p -> r
Division into Cases
- Premise:
p v q,p -> r,q -> r - Conclusion:
r - The above only 2 cases, we can have more than 2 cases
Contradiction Rule
- Premise:
~p -> false - Conclusion:
p - The core of Proof by Contradiction (反证法)
- If a assumption leads to a contradiction, then that assumption must be false
With Predicate Quantifier
Universal Modus Ponens
- Premise: For all x, if x makes P(x) true, then x makes Q(x) true
- Premise: a is an element of x, a makes P(x) true
- Conclusion: a makes Q(x) true