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Propositional Logic Aritra Hazra Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, Paschim Medinipur, West Bengal, India - 721302. Email: aritrah@cse.iitkgp.ac.in Autumn 2020 Aritra Hazra (CSE, IITKGP)

  1. Propositional Logic Aritra Hazra Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, Paschim Medinipur, West Bengal, India - 721302. Email: aritrah@cse.iitkgp.ac.in Autumn 2020 Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 1 / 18

  2. Introduction to Logic History and Genesis Indic: Geometry and Calculations, Nyaya and Vaisisekha, Argumentation Theory, Sanskrit and Binary Arguments, Chatustoki (Logical Argumentation), Philosophers, Vedanta, Formal Systems China: Confucious, Mozi, Master Mo (Mohist School), Basic Formal Systems, Buddhist Systems from India Greek: Thales and Pythagoras (Postulates, Geometry), Heraclitus and Permenides (Logos), Plato (Logic beyond Geometry), Aristotle (Syllogism, Syntax), Stoics Middle-East: Egyptian logic, Arabic (Avisennian logic), Inductive logic Medieval-Europe: Post Aristotle, Precursor to First-Order logic Today: Propositional, Predicate, Higher-Order, Psychology, Philosophy Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 18

  3. Introduction to Logic History and Genesis Indic: Geometry and Calculations, Nyaya and Vaisisekha, Argumentation Theory, Sanskrit and Binary Arguments, Chatustoki (Logical Argumentation), Philosophers, Vedanta, Formal Systems China: Confucious, Mozi, Master Mo (Mohist School), Basic Formal Systems, Buddhist Systems from India Greek: Thales and Pythagoras (Postulates, Geometry), Heraclitus and Permenides (Logos), Plato (Logic beyond Geometry), Aristotle (Syllogism, Syntax), Stoics Middle-East: Egyptian logic, Arabic (Avisennian logic), Inductive logic Medieval-Europe: Post Aristotle, Precursor to First-Order logic Today: Propositional, Predicate, Higher-Order, Psychology, Philosophy Applications Problem Solving using Logical Arguments Automated Reasoning and Artificial Intelligence Automated Learning and Deduction / Derivation Circuit Behaviour and Program Verification Cognition Models and Neural Network Analysis Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 18

  4. Some Example Arguments Example-1 If I am the VP of Gymkhana, then I am well-known in IIT. I am the VP of a Gymkhana. Therefore, I am well-known in IIT. If Ninaad is the VP of Gymkhana, then Ninaad is well-known in IIT. Ninaad b is the VP of Gymkhana. Therefore, Ninaad is well-known in IIT. If Neha is the VP of Gymkhana, then Neha is well-known in IIT. Neha is c NOT the VP of Gymkhana. Therefore, Neha is NOT well-known in IIT. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  5. Some Example Arguments Example-1 If I am the VP of Gymkhana, then I am well-known in IIT. I am the VP of a Gymkhana. Therefore, I am well-known in IIT. If Ninaad is the VP of Gymkhana, then Ninaad is well-known in IIT. Ninaad b is the VP of Gymkhana. Therefore, Ninaad is well-known in IIT. If Neha is the VP of Gymkhana, then Neha is well-known in IIT. Neha is c NOT the VP of Gymkhana. Therefore, Neha is NOT well-known in IIT. Example-2 If Ninaad is elected as the VP of Gymkhana, then Ayushi is chosen as a a G-Sec AND Devang is chosen as a Treasurer. Ayushi is NOT chosen as a G-Sec. Therefore, Ninaad is NOT elected as VP of Gymkhana. If Ninaad is elected as the VP of Gymkhana, then Ayushi is chosen as a b G-Sec AND Devang is chosen as a Treasurer. Devang is chosen as a Treasurer. Therefore, Ninaad is elected as VP of Gymkhana. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  6. Representation and Deduction using Propositional Logic Formal Representation Propositions: Choice of Boolean variables with true or false values. Connectors: Well-defined connectors, such as, ¬ (negation), ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (if and only if) etc. The meaning (semantics) is given by their Truth-tables. Codification: Boolean Formulas constructed from the statements in arguments. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 4 / 18

  7. Representation and Deduction using Propositional Logic Formal Representation Propositions: Choice of Boolean variables with true or false values. Connectors: Well-defined connectors, such as, ¬ (negation), ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (if and only if) etc. The meaning (semantics) is given by their Truth-tables. Codification: Boolean Formulas constructed from the statements in arguments. Deduction Process Obtaining truth of the combined formula expressing complete argument. Proving or Disproving the argument using truth-tables or formal deduction rules. Checking the Validity and Satisfiability of the formula analyzing its truth or falsification over various Interpretations. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 4 / 18

  8. Deduction using Propositional Logic: Example-1a Example If I am the VP of Gymkhana, then I am well-known in IIT. I am the VP of Gymkhana. Therefore, I am well-known in IIT. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 18

  9. Deduction using Propositional Logic: Example-1a Example If I am the VP of Gymkhana, then I am well-known in IIT. I am the VP of Gymkhana. Therefore, I am well-known in IIT. Argument Representation Propositions: v : I am the VP of Gymkhana, w : I am well-known in IIT. Codification: F 1 : v → w ≡ ( ¬ v ∨ w ), F 2 : v , G : w . Complete Formula for Deduction: ( F 1 ∧ F 2 ) → G ≡ (( v → w ) ∧ v ) → w Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 18

  10. Deduction using Propositional Logic: Example-1a Example If I am the VP of Gymkhana, then I am well-known in IIT. I am the VP of Gymkhana. Therefore, I am well-known in IIT. Argument Representation Propositions: v : I am the VP of Gymkhana, w : I am well-known in IIT. Codification: F 1 : v → w ≡ ( ¬ v ∨ w ), F 2 : v , G : w . Complete Formula for Deduction: ( F 1 ∧ F 2 ) → G ≡ (( v → w ) ∧ v ) → w Deduction Steps: Using Truth-tables ( v → w ) ∧ v (( v → w ) ∧ v ) → w v w v → w True True True True True True False False False True False True True False True False False True False True Tautology: Formula, ( F 1 ∧ F 2 ) → G , is Valid, i.e. True under all Interpretations. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 18

  11. Deduction using Propositional Logic: Example-1b Example If Ninaad is the VP of Gymkhana, then Ninaad is well-known in IIT. Ninaad is the VP of Gymkhana. Therefore, Ninaad is well-known in IIT. Argument Representation Propositions: v : Ninaad is the VP of Gymkhana, w : Ninaad is well-known in IIT. Codification: F 1 : v → w ≡ ( ¬ v ∨ w ), F 2 : v , G : w . Complete Formula for Deduction: ( F 1 ∧ F 2 ) → G ≡ (( v → w ) ∧ v ) → w Deduction Steps: Using Truth-tables ( v → w ) ∧ v (( v → w ) ∧ v ) → w v w v → w True True True True True True False False False True False True True False True False False True False True Tautology: Formula, ( F 1 ∧ F 2 ) → G , is Valid, i.e. True under all Interpretations. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 18

  12. Deduction using Propositional Logic: Example-1c Example If Neha is the VP of Gymkhana, then Neha is well-known in IIT. Neha is NOT the VP of Gymkhana. Therefore, Neha is NOT well-known in IIT. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 18

  13. Deduction using Propositional Logic: Example-1c Example If Neha is the VP of Gymkhana, then Neha is well-known in IIT. Neha is NOT the VP of Gymkhana. Therefore, Neha is NOT well-known in IIT. Deduction Process: Using Truth-tables Codification: F 1 : v → w , F 2 : ¬ v , G : ¬ w . Formula: ( F 1 ∧ F 2 ) → G ≡ (( v → w ) ∧ ¬ v ) → ¬ w ( v → w ) ∧ ¬ v (( v → w ) ∧ ¬ v ) → ¬ w v w v → w True True True False True Truth-table: True False False False True False True True True False False False True True True Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 18

  14. Deduction using Propositional Logic: Example-1c Example If Neha is the VP of Gymkhana, then Neha is well-known in IIT. Neha is NOT the VP of Gymkhana. Therefore, Neha is NOT well-known in IIT. Deduction Process: Using Truth-tables Codification: F 1 : v → w , F 2 : ¬ v , G : ¬ w . Formula: ( F 1 ∧ F 2 ) → G ≡ (( v → w ) ∧ ¬ v ) → ¬ w ( v → w ) ∧ ¬ v (( v → w ) ∧ ¬ v ) → ¬ w v w v → w True True True False True Truth-table: True False False False True False True True True False False False True True True Interpretations of a Complete Formula Valid? No! vs. Satisfiable? Yes! Invalid? Yes! vs. Unsatisfiable? No! Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 7 / 18

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