Outline
Propositional Logic - Basics
K. Subramani1
1Lane Department of Computer Science and Electrical Engineering West Virginia University
14 January and 16 January, 2013
Subramani Propositonal Logic Outline Outline
1 Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics
Subramani Propositonal Logic (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic?
Subramani Propositonal Logic (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law!
Subramani Propositonal Logic (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics.
Subramani Propositonal Logic (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics.
Subramani Propositonal Logic Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Subramani Propositonal Logic A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) -
Subramani Propositonal Logic Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Subramani Propositonal Logic (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example
Subramani Propositonal Logic (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black.
Subramani Propositonal Logic (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John?
Subramani Propositonal Logic (iv) This statement is false. (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese.
Subramani Propositonal Logic (Paradox).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false.
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Motivation
Why Logic? (i) The Law! (ii) Mathematics. (iii) Computer Science.
Definition Statement (or Atomic Proposition) - A sentence that is either true or false.
Example (i) The board is black. (ii) Are you John? (iii) The moon is made of green cheese. (iv) This statement is false. (Paradox).
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Outline
1 Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics
Subramani Propositonal Logic To make compound statements from simple ones. Basic Connectives are conjunction (AND) (∧), disjunction (OR) (∨), Negation (NOT) (0), implication (IF) (→) and Equivalence (IF AND ONLY IF) (↔). Two special symbols in propositional logic: > for true and ⊥ for false.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Boolean Connectives
Motivation
Subramani Propositonal Logic Basic Connectives are conjunction (AND) (∧), disjunction (OR) (∨), Negation (NOT) (0), implication (IF) (→) and Equivalence (IF AND ONLY IF) (↔). Two special symbols in propositional logic: > for true and ⊥ for false.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Boolean Connectives
Motivation To make compound statements from simple ones.
Subramani Propositonal Logic Two special symbols in propositional logic: > for true and ⊥ for false.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Boolean Connectives
Motivation To make compound statements from simple ones. Basic Connectives are conjunction (AND) (∧), disjunction (OR) (∨), Negation (NOT) (0), implication (IF) (→) and Equivalence (IF AND ONLY IF) (↔).
Subramani Propositonal Logic > for true and ⊥ for false.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Boolean Connectives
Motivation To make compound statements from simple ones. Basic Connectives are conjunction (AND) (∧), disjunction (OR) (∨), Negation (NOT) (0), implication (IF) (→) and Equivalence (IF AND ONLY IF) (↔). Two special symbols in propositional logic:
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Boolean Connectives
Motivation To make compound statements from simple ones. Basic Connectives are conjunction (AND) (∧), disjunction (OR) (∨), Negation (NOT) (0), implication (IF) (→) and Equivalence (IF AND ONLY IF) (↔). Two special symbols in propositional logic: > for true and ⊥ for false.
Subramani Propositonal Logic A propositional formula has a syntax and a semantics. Semantics refers to the meaning of a formula. Meaning is given by the truth values true and false, where true 6= false. An interpretation I assigns to every propositional variable, a single truth value.
Example I : {P → true, Q → false,...}.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Semantics
Definition
Subramani Propositonal Logic Semantics refers to the meaning of a formula. Meaning is given by the truth values true and false, where true 6= false. An interpretation I assigns to every propositional variable, a single truth value.
Example I : {P → true, Q → false,...}.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Semantics
Definition A propositional formula has a syntax and a semantics.
Subramani Propositonal Logic Meaning is given by the truth values true and false, where true 6= false. An interpretation I assigns to every propositional variable, a single truth value.
Example I : {P → true, Q → false,...}.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Semantics
Definition A propositional formula has a syntax and a semantics. Semantics refers to the meaning of a formula.
Subramani Propositonal Logic An interpretation I assigns to every propositional variable, a single truth value.
Example I : {P → true, Q → false,...}.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Semantics
Definition A propositional formula has a syntax and a semantics. Semantics refers to the meaning of a formula. Meaning is given by the truth values true and false, where true 6= false.
Subramani Propositonal Logic Example I : {P → true, Q → false,...}.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Semantics
Definition A propositional formula has a syntax and a semantics. Semantics refers to the meaning of a formula. Meaning is given by the truth values true and false, where true 6= false. An interpretation I assigns to every propositional variable, a single truth value.
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Semantics
Definition A propositional formula has a syntax and a semantics. Semantics refers to the meaning of a formula. Meaning is given by the truth values true and false, where true 6= false. An interpretation I assigns to every propositional variable, a single truth value.
Example I : {P → true, Q → false,...}.
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Conjunction
Semantics of Conjunction A B A ∧ B T T T T F F F T F F F F
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Disjunction
Semantics of Disjunction A B A ∨ B T T T T F T F T T F F F
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Negation
Semantics of Negation A A0 T F F T
Subramani Propositonal Logic Note Note that A → B is the same as A0 ∨ B. A is called the antecedent and B is the consequent of the implication.
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Implication
Semantics of Implication A B A → B T T T T F F F T T F F T
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Implication
Semantics of Implication A B A → B T T T T F F F T T F F T
Note Note that A → B is the same as A0 ∨ B. A is called the antecedent and B is the consequent of the implication.
Subramani Propositonal Logic Note Note that A ↔ B is the same as (A → B) ∧ (B → A).
Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Equivalence
Semantics of Equivalence A B A ↔ B T T T T F F F T F F F T
Subramani Propositonal Logic Statements, Symbolic Representations and Semantics Boolean Connectives and Semantics Equivalence
Semantics of Equivalence A B A ↔ B T T T T F F F T F F F T
Note Note that A ↔ B is the same as (A → B) ∧ (B → A).
Subramani Propositonal Logic