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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