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G4G0-2/AI & Data Mining/Week 22/Chapter 22 Validity and Inference Rules.md
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1. A syllogism is an instance of a form of reasoning in which a conclusion is drawn from two given or assumed propositions; a common or middle term is present in the two premises but not in the conclusion, which may be invalid.
2. Aristotle
**Double Negation ¬ Elim ¬ ¬ p p**
| Propositions | Premises | Conclusion |
| ------------ | ------------ | ---------- |
| p | $\neg\neg p$ | p |
| T | T | T |
| F | F | F |
**Hypothetical syllogism; this says that if p implies q and q implies r, then it can be logically concluded that p implies r. p ⇒ q q ⇒ r p ⇒ r**
| Propositions | | | Premises | | Conclusion |
| ------------ | --- | --- | -------------- | -------------- | -------------- |
| p | q | r | $p \implies q$ | $q \implies r$ | $p \implies r$ |
| T | T | T | T | T | T |
| T | T | F | T | F | |
| T | F | T | F | T | |
| T | F | F | F | T | |
| F | T | T | T | T | T |
| F | T | F | T | F | |
| F | F | T | T | T | T |
| F | F | F | T | T | T |
1. Involves linking implications together in a sequential manner, much like the links in a chain.
**p q q Therefore, p**
| Propositions | | Premises | | Conclusion |
| ------------ | --- | ---------- | --- | ---------- |
| $p$ | $q$ | $p \lor q$ | $q$ | p |
| T | T | T | T | T |
| T | F | T | F | T |
| F | T | T | T | F |
| F | F | F | F | F |
**p ⇒ q q ⇒ p Therefore, p ∧ q**
| Propositions | | Premises | | Conclusion |
| ------------ | --- | -------------- | -------------- | ----------- |
| p | q | $p \implies q$ | $q \implies p$ | $p \land q$ |
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | T | F | F |
| F | F | T | T | F |
$p \implies q$
$r \implies s$
$p \lor r$ (p disjunction (or) r)
Conclusion: $q \lor s$
The "Constructive Dilemma": If the disjunction of the antecedent of two implications holds then the disjunction of the conclusions also must hold