Propositional Logic Notes

Propositional Logic - Key Concepts

Symbolization of Natural Language Statements

• Learn how to represent natural language statements using symbols.
• Understand logical connectives and their function.
• Evaluate the validity of deductive arguments using truth tables.

Logical Connectives

• Conjunction (&): Represents 'and'.
  • Example: p & q ("Alice rode her bike and John walked.")
• Disjunction (v): Represents 'or'.
  • Example: p v q ("Either Alice rode her bike or John walked.")
• Negation (~): Represents 'not'.
  • Example: ~p ("It is not the case that Alice rode her bike.")
• Conditional (→): Represents 'if-then'.
  • Example: p → q ("If Alice rode her bike, then John walked.")

Truth Values

• Each statement can be classified as true or false.
• Truth Value of a Statement:
  • True statement: truth value is true.
  • False statement: truth value is false.
  • Example statements:
  • "Today is Friday."
  • "I am at SAIT."

Conjunction Truth Table

• Conjunction (p & q) is true only when both p and q are true.

• Conclusion: At least one false statement in a conjunction means the whole expression is false.

Disjunction

• A disjunction is true when at least one disjunct is true and false only when both are false.
• Two Interpretations:
  • Inclusive: Either one or the other or both (Default meaning in logic).
  • Example: "Either I am sick, or I am tired."
  • Exclusive: Either one or the other but not both.
  • Example: "You can have either the chicken or the fish as your in-flight meal."

Negation

• Negation reverses the truth value of a statement.
• Double Negation: A double negation equals no negation.
  • Example: "It is not the case that today is not Friday" simplifies to "Today is Friday."

Conditional Statements

• A conditional statement (if-then) is only false if the antecedent is true and the consequent is false.
• Example: "If today is rainy, I will stay home."

Evaluating Truth Values of Statements

• Evaluate the truth value of given statements based on knowledge of their components:
  • Example statements for evaluation
  • "It is hot outside."
  • "We are aliens & Dogs are animals."

Constructing Truth Tables

• Full truth tables can be created for multiple variables.
  • Example: For three variables (p, q) and resulting expressions
  • Example Table:

Practical Exercises

• Participants are encouraged to create their own truth tables for various expressions:
  • Expressions such as:
  • ~p, (p v q), ~(p & ~q), and others.

Conclusion

• Mastery of these concepts is crucial for understanding and evaluating logical arguments and truth statements in propositional logic.