### **Slide Notes:**

---

#### **Slide 0: Learning Objectives**
- Identify genuine variables and variables standing for unknowns.
- Describe the role of inference rules for eliminating and introducing quantifiers.
- Justify constraints on handling variables when eliminating and introducing quantifiers.
- Conduct proofs in predicate logic.
- Prove the validity of an argument presented in English.

---

#### **Slide 1: Contents**
- Recap on inference rules
- Deductive proof in predicate logic
- Genuine variables versus unknown variables
- Extra inference rules for predicate logic
- Constraints on variables
- Summary, reading and references

---

#### **Slide 2: Recap on Inference Rules**
| **Connective**     | **Introduction Rule**                                                 | **Elimination Rule**                                         |
| ------------------ | --------------------------------------------------------------------- | ------------------------------------------------------------ |
| ∧ (and)            | ∧ Intro:$p, q \vdash p \land q$                                       | ∧ Elim:$p \land q, p \vdash q$                               |
| ∨ (or)             | ∨ Intro:$p \vdash p \lor q$,$q \vdash p \lor q$                       | ∨ Elim:$p \vdash \varnothing$,$p \lor q \vdash \varnothing$x |
| ¬ (not)            | N/A                                                                   | ¬ Elim:$\neg p, q \vdash p$                                  |
| ⇒ (implies)        | ⇒ Intro:$p, p \Rightarrow q \vdash q$                                 | ⇒ Elim (Modus Ponens):$p \Rightarrow q, p \vdash q$          |
| ⇔ (if and only if) | ⇔ Intro:$p \Rightarrow q, q \Rightarrow p \vdash p \leftrightarrow q$ | ⇔ Elim:$p \leftrightarrow q, p, q \vdash \varnothing$        |

---

#### **Slide 3: Deductive Proof in Predicate Logic**
1. Reduce formulae to propositional form by removing quantifiers.
2. Manipulate formulae using inference rules and logical laws of propositional logic.
3. Reintroduce quantifiers at the end, depending on the goal.

---

#### **Slide 4: Genuine Variables vs. Unknown Variables**
- **Genuine variable**:
  - A free variable whose universal quantification yields a true formula.
- **Unknown variable**:
  - A free variable whose existential quantification yields a true formula.

---

#### **Slide 5: Exercise**
**Example**:
-$\frac{9}{z} = 3$
-$\text{even}(n) \lor \text{odd}(n)$

| **Variable** | **Status**               | **Justification**                   |
|--------------|-------------------------|-----------------------------------|
|$z$      | Unknown variable        |$\exists z \cdot \frac{9}{z} = 3$|
|$n$      | Genuine variable        |$\forall n \cdot (\text{even}(n) \lor \text{odd}(n))$|

---

#### **Slide 6: Extra Inference Rules for Predicate Logic**
| **Quantifier** | **Introduction Rule**                     | **Elimination Rule**                                  |
|-----------------|----------------------------------------|---------------------------------------------------|
|$\forall$(for all) |$\forall \text{Intro}: \varphi(x), x \neq y \vdash \varphi(x)$|$\forall \text{Elim}: \forall x \cdot \varphi(x), \varphi(x) \vdash \varnothing$|
|$\exists$(there exists) |$\exists \text{Intro}: \varphi(x), x \neq y \vdash \exists x \cdot \varphi(x)$|$\exists \text{Elim}: \exists x \cdot \varphi(x), \varphi(x) \vdash \varnothing$|

---

#### **Slide 7: Freeing Variables from$\forall$**
- Example:
  -$\forall x \cdot \text{lecturer}(x), \text{lecturer}(bryant)$
  -$\forall x \cdot \text{lecturer}(x), \text{lecturer}(y)$
  -$y$is a genuine variable.

---

#### **Slide 8: Freeing Variables from$\exists$**
- Example:
  -$\exists x \cdot \text{student}(x), \text{student}(y)$
  -$y$is an unknown variable.

---

#### **Slide 9: Introduction of$\forall$**
- Constraint:
  - Do not introduce$\forall$on the basis of an individual (a constant).

---

#### **Slide 10: Constraint 1**
- When eliminating$\exists$, do not instantiate its variable with a constant.

---

#### **Slide 11: Constraint 2**
- When eliminating$\exists$, do not instantiate its variable with an existing free variable.

---

#### **Slide 12: Constraint 3**
- Do not introduce$\forall$on the basis of an individual (a constant).

---

#### **Slide 13: Constraint 4**
- Do not introduce$\forall$on the basis of an unknown.

---

#### **Slide 14: Constraint 5**
- Do not quantify a variable by introducing$\forall$in a formula if the formula contains another var obtained by eliminating$\exists$.

---

#### **Slide 15: Constraint 6**
- Within the scope of an assumption, do not introduce$\forall$on the basis of a variable appearing in the assumption.

---

#### **Slide 16: Constraint 7**
- Every instantiation, whether following the elimination of a$\forall$or$\exists$, must always be done with a free var, rather than a bound var.

---

#### **Slide 17: Constraint 8**
- Beware of binding any newly quantified var by an unintended quantifier.

---

#### **Slide 18: Summary**
- Use inference rules to eliminate and introduce logical connectives.
- Handle quantifiers carefully, following constraints when eliminating and introducing them.

---

#### **Slide 19: Summary of Constraints**
1. Do not assume a property holds for a particular individual based on it holding for at least one individual.
2. When eliminating$\exists$, do not instantiate its var with a constant or existing free var.
3. Do not introduce$\forall$on the basis of a constant or unknown.
4. Do not quantify a variable by introducing$\forall$if the formula contains another var from$\exists$elimination.
5. Within an assumption, do not introduce$\forall$based on a var in the assumption.
6. Instantiate variables only with free vars, not bound vars.
7. Avoid binding newly quantified vars by unintended quantifiers.

---

#### **Slide 20: Deducing a Conclusion**
Example:
1.$\forall x \cdot (\text{master}(x) \lor \text{slave}(x)) \Rightarrow \text{adult}(x) \land \text{man}(x)$
2.$\neg \forall x \cdot (\text{adult}(x) \land \text{man}(x))$
3.$\exists x \cdot (\neg (\text{adult}(x) \land \text{man}(x)))$
4.$\neg (\text{master}(x) \lor \text{slave}(x))$
5.$\neg \text{master}(x) \land \neg \text{slave}(x)$
6.$\neg \text{master}(x)$
7.$\exists y \cdot (\neg \text{master}(y))$

---