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