1 / 22

CT214 – Logical Foundations of Computing Lecture 4 Propositional Calculus

CT214 – Logical Foundations of Computing Lecture 4 Propositional Calculus. Proof Methods. Deduction Theorem Also known as “Conditional proof” Used to deduce proofs in a given theory Reductio Ad Absurdum Means “Reduction to the absurd”

jared
Download Presentation

CT214 – Logical Foundations of Computing Lecture 4 Propositional Calculus

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CT214 – Logical Foundations of Computing Lecture 4 Propositional Calculus

  2. Proof Methods • Deduction Theorem Also known as “Conditional proof” Used to deduce proofs in a given theory • Reductio Ad Absurdum Means “Reduction to the absurd” Also known as “Indirect proof” or “Proof by contradiction”

  3. Deduction Theorem Definition: If the logical expression B can be derived from the premises P1, P2, P3, …, PN, then the logical expression PN -> B can be derived from P1, P2, P3, …, PN-1 If P1, P2, P3, …, PN B Then P1, P2, P3, …, PN-1 PN -> B

  4. Deduction Theorem If P1, P2, P3, …, PN B Then P1, P2, P3, …, PN-1 PN -> B To prove PN -> B, assume PN and prove B.

  5. Deduction Theorem Example: (1) P -> Q (2) R -> S ¬Q -> S (3) P v R To prove ¬Q -> S, assume ¬Q and prove S. (1) P -> Q (2) R -> S S (3) P v R (4) ¬Q

  6. Deduction Theorem Example: (1) P -> Q (2) R -> S S (3) P v R (4) ¬Q Answer: P -> Q, ¬Q (1 + 4) ¬P Modus Tollens (5) P v R, ¬P (3 + 5)

  7. Deduction Theorem P v R, ¬P (3 + 5) R Disjunctive Syllogism (6) R -> S, R (2 + 6) S Modus Ponens

  8. Reductio Ad Absurdum Definition: If the logical expression B can be derived from the premises P1, P2, P3, …, PN, then the compliment of B together with the premises P1, P2, P3, …, PN can be used to prove a contradiction. If P1, P2, P3, …, PN, ¬B S And P1, P2, P3, …, PN, ¬B¬S Then P1, P2, P3, …, PN B

  9. Reductio Ad Absurdum If P1, P2, P3, …, PN, ¬B S And P1, P2, P3, …, PN, ¬B¬S Then P1, P2, P3, …, PN B To prove B, assume ¬B and prove a contradiction.

  10. Reductio Ad Absurdum Example: (1) P -> Q (2) R -> S ¬Q -> S (3) P v R To prove ¬Q -> S, assume ¬(¬Q -> S) and prove contradiction. (1) P -> Q (2) R -> S False (3) P v R (4) ¬(¬Q -> S)

  11. Reductio Ad Absurdum Example: (1) P -> Q (2) R -> S False (3) P v R (4) ¬(¬Q -> S) Answer: ¬(¬Q -> S) (4) ¬(¬¬Q v S) Definition ¬(Q v S) Double Negative

  12. Reductio Ad Absurdum ¬(Q v S) Double Negative ¬Q ^ ¬S De Morgan (5) ¬Q Simplification (6) ¬Q ^ ¬S (5) ¬S Simplification (7) P -> Q, ¬Q (1 + 6) ¬P Modus Tollens (8) P v R, ¬P (3 + 8)

  13. Reductio Ad Absurdum P v R, ¬P (3 + 8) R Disjunctive Syllogism (9) R -> S, R (2 + 9) S Modus Ponens (10) S, ¬S (10 + 7) S ^ ¬S Conjunction F Compliment

  14. Natural Language Proofs Prove the validity of a statement made in natural language by converting the natural language statements to logical expressions. For example: Determine the validity of: “Liverpool will win the premiership if Torres doesn’t get injured. If Ronaldo gets injured, Man United will finish third. Torres and Ronaldo both get injured. Therefore, Liverpool don’t win the premiership and Man United finish third.”

  15. Natural Language Proofs “Liverpool will win the premiership if Torres doesn’t get injured. If Ronaldo gets injured, Man United will finish third. Torres and Ronaldo both get injured. Therefore, Liverpool don’t win the premiership and Man United finish third.” Liverpool will win the premiership L Torres doesn’t get injured ¬T Ronaldo gets injured R Man united finish third M

  16. Natural Language Proofs “Liverpool will win the premiership if Torres doesn’t get injured. If Ronaldo gets injured, Man United will finish third. Torres and Ronaldo both get injured. Therefore, Liverpool don’t win the premiership and Man United finish third.” (1) L -> ¬T (2) R -> M ¬L ^ M (3) T ^ R

  17. Natural Language Proofs (1) L -> ¬T (2) R -> M ¬L ^ M (3) T ^ R Answer: T ^ R (3) T Simplification (4) T ^ R (3) R Simplification (5)

  18. Natural Language Proofs L -> ¬T, T (1 + 4) ¬L Modus Tollens (6) R -> M, R (2 + 5) M Modus Ponens (7) ¬L, M (6 + 7) ¬L ^ M Conjunction

  19. Natural Language Proofs (1) L -> ¬T (2) R -> M ¬L ^ M (3) T ^ R Assume ¬(¬L ^ M) and prove a contradiction (1) L -> ¬T (2) R -> M False (3) T ^ R (4) ¬(¬L ^ M)

  20. Natural Language Proofs (1) L -> ¬T (2) R -> M False (3) T ^ R (4) ¬(¬L ^ M) Answer: ¬(¬L ^ M) (4) ¬¬L v ¬M De Morgan L v ¬M Double Negative (5)

  21. Natural Language Proofs T ^ R (3) T Simplification (6) T ^ R (3) R Simplification (7) L -> ¬T, T (1 + 6) ¬L Modus Tollens (8) R -> M, R (2 + 7) M Modus Ponens (9)

  22. Natural Language Proofs L v ¬M, ¬L (5 + 8) ¬M Disjunctive Syllogism (10) ¬M, M (10 + 9) ¬M ^ M Conjunction F Compliment

More Related