Notes on Propositional Logic

Propositional Logic

Introduction to Propositional Logic

• Valid Argument
  • A valid argument is characterized by the following:
  • If all premises are true, the conclusion must also necessarily be true.
  • There is no possible scenario where the conclusion could be false if the premises are true.
  • Validity concerns the structure and relation between premises and conclusions rather than the actual truth of content.
    • Implication: This allows for symbolization of statements.

Comparison of Mathematics and Logic

• Mathematical Statements vs. Logical Statements
  • Example:
  • 2 apples + 3 apples = 5 apples
  • 2x + 3x = 5x
  • Logical Examples:
  • P1: Calgary is in Canada.
  • P2: If Calgary is in Canada, then it has cold winters.
  • C: Therefore, Calgary has cold winters.
  • More Examples:
  • P1: Calgary is in Mexico.
  • P2: If Calgary is in Mexico, then it has warm winters.
  • C: Therefore, Calgary has warm winters.

Formal Logic Overview

• Logic
  • The study of correct reasoning.
• Formal Logic
  • The science dedicated to understanding deductively valid inferences.
  • Evaluates how conclusions legitimately follow from premises, independent of their context or content.
• Categories of Logic
  • Categorical Logic
  • Propositional Logic

Propositional Logic

• Definition
  • Propositional logic examines the logical relationships among statements (also referred to as claims or propositions).
• Statements
  • Simple Statement: An assertion that does not contain other statements.
  • Examples:
    • "The moon is made of cheese."
    • "Calgary is in Canada."
  • Compound Statement: Composed of at least two simple statements.
  • Examples:
    • “If the moon is made of cheese, then Italians love it.”
    • “If Calgary is in Canada, then it has cold winters.”

Symbolization in Propositional Logic

• Logical Connectives
  • Use of symbols to express relationships between statements:
  • Conjunction (and): $p ext{ and } q$
  • Disjunction (or): $p ext{ or } q$
  • Negation (not): $<br />\neg p$
  • Conditional (if-then): $p o q$
  • Example:
    • “If I have a time machine, I will change history.”
    • Symbolically, $p o q$.

Logical Connectives and Examples

• Connective Meanings:
  • Conjunction (and): p ext{ & } q
  • Example: "Alice rode her bike, and John walked."
  • Disjunction (or): $p ext{ v } q$
  • Example: "Either Alice rode her bike or John walked."
  • Negation (not): $<br />\neg p$
  • Example: "It is not the case that Alice rode her bike."
  • Conditional (if-then): $p o q$
  • Example: "If Alice rode her bike, then John walked."

Evaluating Validity Using Truth Tables

• Truth Tables:
  • A method used to determine the validity of deductive arguments by laying out the possible truth values of each proposition and the overall truth value of the compound statement.

Components of Propositional Logic

• Conjunction:
  • Formed by joining two simple statements.
  • Examples:
  • "Calgary is a large city, and it is in Japan."
  • Common Signal Words: but, and, also.
• Disjunction:
  • Examples:
  • "You hate pineapple pizza or you are weird."
• Negation:
  • Denies a statement.
  • Examples:
  • "Calgary is not in Japan."
• Conditional:
  • Statements based on conditions, using "if…then" structure.
  • Examples:
  • "If it snows, I will stay home."

Necessary vs. Sufficient Conditions

• Sufficient Condition: An event that guarantees a particular event's occurrence if satisfied.
  • Example: "If I eat a burger, then I will be full."
• Necessary Condition: A requirement that must be met for an event to occur.
  • Example: "Having a valid passport is a necessary condition for traveling internationally."
• Usage in Logic: Distinguishing between conditions that are sufficient or necessary is crucial for proper formulation of arguments.

Practice Exercises and Translation into Symbolic Form

• Exercises: Construct sentences using logical connectives and translate into symbolic form.
• Examples to Translate:
  • "People die, but ideas live forever."
  • "If we don’t start preserving the rain forests, many of the species that live there will not survive."
  • "As long as he returns the money, Ron will not be prosecuted."
• Practice Scenarios:
  • Use provided statements to construct logical expressions using symbols (e.g., $p, q$).

Conclusion

• Propositional logic allows for the systematic analysis of arguments through the use of logical connectives, truth tables, and symbolic representation. Understanding these concepts and practicing translation and evaluation will strengthen logical reasoning skills.