Introduction to Propositional Logic

Introduction to Propositional Logic

• A statement is a declarative sentence that can be classified as true (1) or false (0).
• True and false correspond to 1 and 0 in Boolean logic, which is commonly used in philosophy.
• Examples of statements include:
  • "Milk is white" (generally true, but not for chocolate milk).
  • "The cardinality of the empty set is equal to zero" (true).
  • "Humans are just fish with legs" (false).

Non-Statements

• Questions cannot be classified as true or false, e.g., "Will you go to the store for me?"
• Imperatives (commands) also do not have truth values, e.g., "Kick me."

Propositions vs. Statements

• Statements are specific, while propositions capture the general idea.
• Propositions are typically denoted with capital letters (e.g., P, Q, R) for specific meanings.
• Lowercase letters (e.g., p, q, r) represent general propositions without specific context.

Well-Formed Formulas (WFF)

• A statement or proposition can be a well-formed formula (WFF) within propositional logic.
• "Not P" is a WFF representing negation.
• The logical conjunction "P and Q" connects two WFFs, represented as P ∧ Q.
• The logical disjunction "P or Q" is represented as P ∨ Q.
• Conditional statements "if P then Q" are denoted as P → Q.

Translating Well-Formed Formulas into English

• Translating logical expressions into plain English involves understanding the logical connectives.
• Example translation:
  • Given R (I write an exam), P (I cheat), Q (I will get caught), S (I will fail):
  • The formula "R and P → Q and S" translates to:
    • "If I write an exam and I cheat, then I will get caught and I will fail."

Key Connectives in Propositional Logic

• Negation: ¬P (not P)
• Conjunction: P ∧ Q (P and Q)
• Disjunction: P ∨ Q (P or Q)
• Conditional: P → Q (if P then Q)

Summary

• Understanding propositions, well-formed formulas, and connectives is fundamental to mastering propositional logic.
• Translating logical expressions properly helps clarify their meanings in natural language.