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Predicate 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. Predicate 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. From Propositional Logic to Predicate Logic Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, 1 the dog goes to school. No contractors are dependable. Some engineers are contractors. Therefore, 2 some engineers are not dependable. All actresses are graceful. Anushka is a dancer. Anushka is an actress. 3 Therefore, some dancers are graceful. Every passenger either travels in first class or second class. Each passenger 4 is in second class if and only if he or she is not wealthy. Some passengers are wealthy. Not all passengers are wealthy. Therefore, some passengers travel in second class. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 18

  3. From Propositional Logic to Predicate Logic Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, 1 the dog goes to school. No contractors are dependable. Some engineers are contractors. Therefore, 2 some engineers are not dependable. All actresses are graceful. Anushka is a dancer. Anushka is an actress. 3 Therefore, some dancers are graceful. Every passenger either travels in first class or second class. Each passenger 4 is in second class if and only if he or she is not wealthy. Some passengers are wealthy. Not all passengers are wealthy. Therefore, some passengers travel in second class. Propositional Logic Insufficiency Quantifications: ‘some’, ‘none’, ‘all’, ‘every’, ‘wherever’ etc. Associations: ‘x goes to some place y’, ‘z travels in first class’ etc. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 2 / 18

  4. Predicate Logic Argument Formulation Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  5. Predicate Logic Argument Formulation Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Formal Constructs and Fundamentals Following are the representational extensions made in First-Order Logic (Predicate Logic) over Propositional Logic constructs: Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  6. Predicate Logic Argument Formulation Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Formal Constructs and Fundamentals Following are the representational extensions made in First-Order Logic (Predicate Logic) over Propositional Logic constructs: New Additions: Variables (for e.g., x , y ) and Constants (for e.g., Ankush , Dog ) Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  7. Predicate Logic Argument Formulation Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Formal Constructs and Fundamentals Following are the representational extensions made in First-Order Logic (Predicate Logic) over Propositional Logic constructs: New Additions: Variables (for e.g., x , y ) and Constants (for e.g., Ankush , Dog ) Functional Symbols: Functional constructs returning Non-Boolean values (for e.g., Age ( x ) indicates ‘the age of x ’) Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  8. Predicate Logic Argument Formulation Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Formal Constructs and Fundamentals Following are the representational extensions made in First-Order Logic (Predicate Logic) over Propositional Logic constructs: New Additions: Variables (for e.g., x , y ) and Constants (for e.g., Ankush , Dog ) Functional Symbols: Functional constructs returning Non-Boolean values (for e.g., Age ( x ) indicates ‘the age of x ’) Predicate Symbols: Constructs indicating associations having Boolean outcomes (for e.g., goes ( x , y ) indicates ‘ x goes to the place y ’) Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  9. Predicate Logic Argument Formulation Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Formal Constructs and Fundamentals Following are the representational extensions made in First-Order Logic (Predicate Logic) over Propositional Logic constructs: New Additions: Variables (for e.g., x , y ) and Constants (for e.g., Ankush , Dog ) Functional Symbols: Functional constructs returning Non-Boolean values (for e.g., Age ( x ) indicates ‘the age of x ’) Predicate Symbols: Constructs indicating associations having Boolean outcomes (for e.g., goes ( x , y ) indicates ‘ x goes to the place y ’) Connectors: Well-defined connectors, such as, ¬ (negation), ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (if and only if) etc. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  10. Predicate Logic Argument Formulation Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Formal Constructs and Fundamentals Following are the representational extensions made in First-Order Logic (Predicate Logic) over Propositional Logic constructs: New Additions: Variables (for e.g., x , y ) and Constants (for e.g., Ankush , Dog ) Functional Symbols: Functional constructs returning Non-Boolean values (for e.g., Age ( x ) indicates ‘the age of x ’) Predicate Symbols: Constructs indicating associations having Boolean outcomes (for e.g., goes ( x , y ) indicates ‘ x goes to the place y ’) Connectors: Well-defined connectors, such as, ¬ (negation), ∧ (conjunction), ∨ (disjunction), → (implication), ↔ (if and only if) etc. Quantifiers: Existantial ( ∃ , i.e. there exists) and Universal ( ∀ , i.e. for all) Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 3 / 18

  11. Predicate Logic Argument Formulation: Example-1 Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 4 / 18

  12. Predicate Logic Argument Formulation: Example-1 Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Logical Formulation Variables: x and y Constants: Ankush , Dog and School Predicate: goes ( x , y ): x goes to y Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 4 / 18

  13. Predicate Logic Argument Formulation: Example-1 Example Wherever Ankush goes, so does the pet dog. Ankush goes to school. So, the dog goes to school. Logical Formulation Variables: x and y Constants: Ankush , Dog and School Predicate: goes ( x , y ): x goes to y Formula: F 1 : ∀ x ( goes ( Ankush , x ) → goes ( Dog , x )) F 2 : goes ( Ankush , School ) G : goes ( Dog , School ) Requirement: To prove whether ( F 1 ∧ F 2 ) → G is valid Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 4 / 18

  14. Predicate Logic Argument Formulation: Example-2 Example No contractors are dependable. Some engineers are contractors. Therefore, some engineers are not dependable. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 18

  15. Predicate Logic Argument Formulation: Example-2 Example No contractors are dependable. Some engineers are contractors. Therefore, some engineers are not dependable. Logical Formulation Predicates: Assuming the variable as x . contractor ( x ) : x is a contractor dependable ( x ) : x is dependable engineer ( x ) : x is an engineer Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 18

  16. Predicate Logic Argument Formulation: Example-2 Example No contractors are dependable. Some engineers are contractors. Therefore, some engineers are not dependable. Logical Formulation Predicates: Assuming the variable as x . contractor ( x ) : x is a contractor dependable ( x ) : x is dependable engineer ( x ) : x is an engineer Formula: F 1 : ∀ x ( contractor ( x ) → ¬ dependable ( x )) ( Alt . ) : ¬∃ x ( contractor ( x ) ∧ dependable ( x )) F 2 : ∃ x ( engineer ( x ) ∧ contractor ( x )) ( Alt . ) : ∃ x ( engineer ( x ) → contractor ( x )) ∧ ∃ x engineer ( x ) G : ∃ x ( engineer ( x ) ∧ ¬ dependable ( x )) Requirement: To prove whether ( F 1 ∧ F 2 ) → G is valid Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 5 / 18

  17. Predicate Logic Argument Formulation: Example-3 Example All actresses are graceful. Anushka is a dancer. Anushka is an actress. Therefore, some dancers are graceful. Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 18

  18. Predicate Logic Argument Formulation: Example-3 Example All actresses are graceful. Anushka is a dancer. Anushka is an actress. Therefore, some dancers are graceful. Logical Formulation Predicates: Assuming the variable as x . actress ( x ) : x is an actress graceful ( x ) : x is graceful dancer ( x ) : x is a dancer Aritra Hazra (CSE, IITKGP) CS21001 : Discrete Structures Autumn 2020 6 / 18

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