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AI & Data Mining/Week 23/Week 23 - Deductive Proofs.md

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+**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