Propositional Logic

Negation

• The negation of a proposition A, written as ¬A, has the opposite truth value of A. If A is true, then ¬A is false, and vice versa.
• In simple terms, negation flips the truth value of a proposition.

Conjunction (AND)

• The truth value of a conjunction is true only if both propositions are true.
• If either proposition is false, the entire conjunction is false.

Proposition

• A proposition is a statement that can be either true or false.
• For instance, "The sky is blue" is a proposition because it is a declarative statement that can be evaluated as true or false.

'If-Then' Symbol

• The symbol for 'if-then' in logical operators is →.

Truth Table

• A truth table systematically lists all possible truth value combinations of propositions and the corresponding truth values of compound propositions formed from them.
• Used to determine if a statement is true or false for all possible combinations of inputs.

Compound Proposition

• A compound proposition is formed by combining existing propositions using logical operators like conjunction (AND), disjunction (OR), negation (NOT), implication (IF-THEN), and equivalence (IF AND ONLY IF).

A ∨ B (OR)

• The result of the proposition A ∨ B is true if at least one of A or B is true.
• This is true even if both A and B are true.

A → B (Implication)

• A → B represents the statement "If A, then B," or "A implies B."
• The only case where A→B is false is when A is true, and B is false.

A ⇔ B (Equivalence)

• The compound proposition A ⇔ B, commonly read as "A if and only if B," is true only when both A and B have the same truth value.
• This means that both A and B are true, or both A and B are false for the proposition to be true.

[(A→B)∧A]→B

• Represents a logical argument where the antecedent is the conjunction of two propositions: A implies B, and A.
• The consequent of the implication is B.
• This argument is typically used to demonstrate a proof by conditional proof, as the overall implication is true if the antecedent and consequent are true.

(~A ^ B) ∧ [B (~D)]

• Given the values A=T, B=F, C=T, and D=F, the proposition evaluates to:
  • (~T ^ F) ∧ [F (~F)]
  • (F ^ F) ∧ [F (T)]
  • F ∧ F
  • F
• Therefore, due to the conjunction, the result of the entire proposition is false.

Role of Conclusion

• The conclusion determines the validity of an argument.
• If, within an argument, the conclusion can be derived from the premises, using logical reasoning (often using truth tables or other logical methods), then the argument is valid.
• A valid argument with true premises guarantees a true conclusion.