205 lines
5.0 KiB
Markdown
205 lines
5.0 KiB
Markdown
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**Slide 3: Recap on Logical Implication (Entailment) |-|=-**
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- Entailment notation: p |=q if and only if the implication p$\implies$q is a tautology.
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- Example:
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- p $\land$ q |=q
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- Truth table for p $\implies$ q:
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| p | q | p $\land$ q | p $\implies$ q |
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|---|---|------|--------|
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| T | T | T | T |
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| T | F | F | F |
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| F | T | F | T |
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| F | F | F | T |
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**Slide 4: (r $\implies$ s) $\land$ (r $\implies$ $\lnot$s) |-|=-**
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- Intuitively, if r implies both s and $\lnot$s, then r must be false.
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- Truth table for (r $\implies$ s) $\land$ (r $\implies$ $\lnot$s):
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| r | s | $\lnot$s | (r $\implies$ s) $\land$ (r $\implies$ $\lnot$s) |
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|---|---|---|------------------------|
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| T | T | F | F |
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| T | F | F | F |
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| F | T | T | T |
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| F | F | T | T |
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**Slide 5: p $\vdash$ q**
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- Notation: p $\vdash$ q means q is provable from p using inference rules.
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- Example:
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- A $\implies$ B, $\lnot$A, therefore $\lnot$B
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**Slide 6: Differences Between |-|=- and $\vdash$**
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- |= indicates semantic entailment (truth conditions).
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- $\vdash$ represents syntactic derivation (inference rules).
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**Slide 7: Recap on Inference Rules**
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- Example inference rules:
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- Modus Ponens ($\implies$Elim):
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p $\implies$ q, p $\vdash$ q
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- Conjunction Introduction ($\land$Intro):
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p $\vdash$ q, p $\vdash$ r $\vdash$ p $\land$ q
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- Conditional Proof ($\implies$Intro):
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p $\vdash$ r, p $\vdash$ s $\vdash$ p $\implies$ (r $\land$ s)
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**Slide 8: Layout of an Inference Rule**
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- Premises above the line, conclusion below the line.
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- Example inference rule ($\implies$Intro):
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p $\vdash$ r, p $\vdash$ s
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p $\implies$ (r $\land$ s)
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**Slide 9: Presentation of Proofs**
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- Steps:
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- Number each step.
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- Justify each step with previous line(s) and inference rule used.
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**Slide 10: Deriving $\lnot$p $\implies$ r From (p $\land$ q) $\lor$ r**
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- Example proof:
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(p $\land$ q) $\lor$ r, $\lnot$E
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…
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$\lnot$p $\implies$ r
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**Slide 11: Two Special Inference Rules**
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- Deductive Theorem ($\implies$Intro):
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p $\vdash$ r, p $\vdash$ s
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p $\implies$ (r $\land$ s)
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- Reductio ad absurdum ($\lnot$Intro):
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p $\vdash$ r, p $\vdash$ $\lnot$s
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p $\vdash$ $\lnot$r
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**Slide 12: Conditional Proofs**
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- Strategy: Assume p, deduce q if possible, discharge assumption.
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- Example:
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(p $\land$ q) $\lor$ r
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…
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$\lnot$p $\implies$ r
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**Slide 13: Indirect Proofs**
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- Strategy: Assume negation of goal, deduce contradiction.
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- Example:
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(p $\land$ q) $\lor$ r
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…
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$\lnot$p $\implies$ r
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**Slide 14: Solution to Exercise**
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Given argument:
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A (You eat carefully) ⇒ B (You have a healthy digestive system)
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C (You exercise regularly) ⇒ D (You are very fit)
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B ∨ D ⇒ E (You live to a ripe old age)
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¬E
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Therefore, ¬A ∧ ¬C
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**Proof:**
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| Line | Formula | Justification |
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| ---- | --------- | -------------------- |
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| 1 | A ⇒ B | Premise |
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| 2 | C ⇒ D | Premise |
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| 3 | B ∨ D ⇒ E | Premise |
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| 4 | ¬E | Premise |
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| 5 | ¬(B ∨ D) | Modus Tollens (3, 4) |
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| 6 | ¬B ∧ ¬D | De Morgan's Law (5) |
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| 7 | ¬B | ∧Elim (6) |
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| 8 | ¬A | Modus Tollens (1, 7) |
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| 9 | ¬D | ∧Elim (6) |
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| 10 | ¬C | Modus Tollens (2, 9) |
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| 11 | ¬A ∧ ¬C | ∧Intro (8, 10) |
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**Conclusion:**
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We have proven that ¬A ∧ ¬C, i.e., you did not eat carefully and you did not exercise regularly.
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**Slide 15: Two Special Inference Rules (continued)**
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- Deductive Theorem:
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p $\vdash$ r, p $\vdash$ s
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p $\implies$ (r $\land$ s)
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- Reductio ad absurdum:
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p $\vdash$ r, p $\vdash$ $\lnot$s
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p $\vdash$ $\lnot$r
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**Slide 16: Soundness and Completeness**
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- Sound: Valid argument with true premises.
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- Complete: Derives any sentence entailed by premises.
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**Slide 17: Formal Proofs of Natural Language Arguments**
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- Steps:
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- Identify atomic propositions.
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- Formalize argument in logic.
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- Check for invalidity.
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- Attempt proof.
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**Slide 18: Example - Travel**
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- Argument:
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…
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Therefore, if my neighbours claim to be impressed then they are just pretending.
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**Slide 19: Example - Travel (continued)**
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- Formalize argument:
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p $\implies$ q, $\lnot$p $\implies$ $\lnot$r, $\lnot$q
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…
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$\lnot$r
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- Proof:
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…
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$\lnot$p $\implies$ r
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**Slide 20: Example - Nutrition**
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- Argument:
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…
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Therefore, you did not eat carefully and you did not exercise regularly.
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**Slide 21: Example - Nutrition (continued)**
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- Formalize argument:
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A $\implies$ B, C $\implies$ D, B $\lor$ D $\implies$ E, $\lnot$E
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…
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$\lnot$A $\land$ $\lnot$C
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- Proof:
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…
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$\lnot$A $\land$ $\lnot$C
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**Slide 22: Application to Software Engineering**
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- Questions about software specifications and claims are arguments.
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**Slide 23: Reading and References**
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- Russell and Norvig, Artificial Intelligence (4th Edition)
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- Nissanke, Introductory Logic and Sets for Computer Scientists
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- Gray, Logic, Algebra and Databases
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