**Slide 3: Recap on Logical Implication (Entailment) |-|=-**

- Entailment notation: p |=q if and only if the implication p$\implies$q is a tautology.
- Example:
  - p $\land$ q |=q
  - Truth table for p $\implies$ q:

| p | q | p $\land$ q | p $\implies$ q |
|---|---|------|--------|
| T | T | T | T |
| T | F | F | F |
| F | T | F | T |
| F | F | F | T |

**Slide 4: (r $\implies$ s) $\land$ (r $\implies$ $\lnot$s) |-|=-**

- Intuitively, if r implies both s and $\lnot$s, then r must be false.
- Truth table for (r $\implies$ s) $\land$ (r $\implies$ $\lnot$s):

| r | s | $\lnot$s | (r $\implies$ s) $\land$ (r $\implies$ $\lnot$s) |
|---|---|---|------------------------|
| T | T | F | F |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |

**Slide 5: p $\vdash$ q**

- Notation: p $\vdash$ q means q is provable from p using inference rules.
- Example:
  - A $\implies$ B, $\lnot$A, therefore $\lnot$B

**Slide 6: Differences Between |-|=- and $\vdash$**

- |= indicates semantic entailment (truth conditions).
- $\vdash$ represents syntactic derivation (inference rules).

**Slide 7: Recap on Inference Rules**

- Example inference rules:
  - Modus Ponens ($\implies$Elim):

    p $\implies$ q, p $\vdash$ q

  - Conjunction Introduction ($\land$Intro):

    p $\vdash$ q, p $\vdash$ r $\vdash$ p $\land$ q

  - Conditional Proof ($\implies$Intro):

    p $\vdash$ r, p $\vdash$ s $\vdash$ p $\implies$ (r $\land$ s)

**Slide 8: Layout of an Inference Rule**

- Premises above the line, conclusion below the line.
- Example inference rule ($\implies$Intro):

    p $\vdash$ r, p $\vdash$ s
      p $\implies$ (r $\land$ s)

**Slide 9: Presentation of Proofs**

- Steps:
  - Number each step.
  - Justify each step with previous line(s) and inference rule used.

**Slide 10: Deriving $\lnot$p $\implies$ r From (p $\land$ q) $\lor$ r**

- Example proof:

    (p $\land$ q) $\lor$ r, $\lnot$E
     …
      $\lnot$p $\implies$ r

**Slide 11: Two Special Inference Rules**

- Deductive Theorem ($\implies$Intro):

    p $\vdash$ r, p $\vdash$ s
      p $\implies$ (r $\land$ s)

- Reductio ad absurdum ($\lnot$Intro):

    p $\vdash$ r, p $\vdash$ $\lnot$s
      p $\vdash$ $\lnot$r

**Slide 12: Conditional Proofs**

- Strategy: Assume p, deduce q if possible, discharge assumption.
- Example:

    (p $\land$ q) $\lor$ r
     …
      $\lnot$p $\implies$ r

**Slide 13: Indirect Proofs**

- Strategy: Assume negation of goal, deduce contradiction.
- Example:

    (p $\land$ q) $\lor$ r
     …
      $\lnot$p $\implies$ r

**Slide 14: Solution to Exercise**

Given argument:
A (You eat carefully) ⇒ B (You have a healthy digestive system)
C (You exercise regularly) ⇒ D (You are very fit)
B ∨ D ⇒ E (You live to a ripe old age)
¬E
Therefore, ¬A ∧ ¬C

**Proof:**

| Line | Formula   | Justification        |
| ---- | --------- | -------------------- |
| 1    | A ⇒ B     | Premise              |
| 2    | C ⇒ D     | Premise              |
| 3    | B ∨ D ⇒ E | Premise              |
| 4    | ¬E        | Premise              |
| 5    | ¬(B ∨ D)  | Modus Tollens (3, 4) |
| 6    | ¬B ∧ ¬D   | De Morgan's Law (5)  |
| 7    | ¬B        | ∧Elim (6)            |
| 8    | ¬A        | Modus Tollens (1, 7) |
| 9    | ¬D        | ∧Elim (6)            |
| 10   | ¬C        | Modus Tollens (2, 9) |
| 11   | ¬A ∧ ¬C   | ∧Intro (8, 10)       |

**Conclusion:**
We have proven that ¬A ∧ ¬C, i.e., you did not eat carefully and you did not exercise regularly.

**Slide 15: Two Special Inference Rules (continued)**

- Deductive Theorem:

    p $\vdash$ r, p $\vdash$ s
      p $\implies$ (r $\land$ s)

- Reductio ad absurdum:

    p $\vdash$ r, p $\vdash$ $\lnot$s
      p $\vdash$ $\lnot$r

**Slide 16: Soundness and Completeness**

- Sound: Valid argument with true premises.
- Complete: Derives any sentence entailed by premises.

**Slide 17: Formal Proofs of Natural Language Arguments**

- Steps:
  - Identify atomic propositions.
  - Formalize argument in logic.
  - Check for invalidity.
  - Attempt proof.

**Slide 18: Example - Travel**

- Argument:

   …
      Therefore, if my neighbours claim to be impressed then they are just pretending.

**Slide 19: Example - Travel (continued)**

- Formalize argument:

    p $\implies$ q, $\lnot$p $\implies$ $\lnot$r, $\lnot$q
     …
      $\lnot$r

- Proof:

   …
      $\lnot$p $\implies$ r

**Slide 20: Example - Nutrition**

- Argument:

   …
      Therefore, you did not eat carefully and you did not exercise regularly.

**Slide 21: Example - Nutrition (continued)**

- Formalize argument:
    A $\implies$ B, C $\implies$ D, B $\lor$ D $\implies$ E, $\lnot$E
     …
      $\lnot$A $\land$ $\lnot$C
- Proof:
   …
      $\lnot$A $\land$ $\lnot$C

**Slide 22: Application to Software Engineering**

- Questions about software specifications and claims are arguments.

**Slide 23: Reading and References**

- Russell and Norvig, Artificial Intelligence (4th Edition)
- Nissanke, Introductory Logic and Sets for Computer Scientists
- Gray, Logic, Algebra and Databases