👁️‍🗨️Formal Logic I Unit 2 – Propositional Logic: Symbols and Language

What's This All About?

• Propositional logic is a branch of logic that studies the relationships between propositions or statements
• Focuses on the logical connectives that join simple propositions to form more complex propositions
• Propositions are declarative sentences that are either true or false, but not both simultaneously
• Propositional logic provides a formal system for analyzing the truth values of propositions and their combinations
• Helps in understanding the logical structure of arguments and determining their validity
• Fundamental to fields such as mathematics, computer science, and philosophy
• Lays the foundation for more advanced topics in logic, such as predicate logic and modal logic

Key Terms and Symbols

• Proposition: a declarative sentence that is either true or false, denoted by lowercase letters (p, q, r)
• Logical connectives: symbols used to join propositions, creating compound propositions
  • Negation (¬): "not"; reverses the truth value of a proposition
  • Conjunction (∧): "and"; true only when both propositions are true
  • Disjunction (∨): "or"; true when at least one proposition is true
  • Implication (→): "if...then"; true except when the antecedent is true and the consequent is false
  • Biconditional (↔): "if and only if"; true when both propositions have the same truth value
• Tautology: a compound proposition that is always true, regardless of the truth values of its components
• Contradiction: a compound proposition that is always false, regardless of the truth values of its components
• Contingency: a compound proposition that can be either true or false, depending on the truth values of its components

Building Blocks of Propositional Logic

• Atomic propositions: the simplest propositions that cannot be broken down further, represented by lowercase letters (p, q, r)
• Compound propositions: formed by combining atomic propositions using logical connectives
• Truth values: the possible values a proposition can have, either true (T) or false (F)
• Truth tables: a method for determining the truth value of a compound proposition based on the truth values of its components
• Logical equivalence: two propositions are logically equivalent if they have the same truth value for all possible combinations of truth values of their components
• Logical validity: an argument is valid if the conclusion necessarily follows from the premises, regardless of their truth values

Putting It All Together: Forming Statements

• Identify the atomic propositions in a given statement
• Represent the statement using logical connectives and symbols
• Examples:
  • "If it rains, then the ground is wet" can be represented as $p \rightarrow q$, where $p$ is "it rains" and $q$ is "the ground is wet"
  • "Either the cake is vanilla, or it is chocolate" can be represented as $p \vee q$, where $p$ is "the cake is vanilla" and $q$ is "the cake is chocolate"
• Determine the truth value of the compound proposition based on the truth values of its components
• Analyze the logical structure of arguments and determine their validity using propositional logic

Truth Tables: The Logic Cheat Sheet

• Truth tables are a systematic way to determine the truth value of a compound proposition for all possible combinations of truth values of its components
• Each row in a truth table represents a unique combination of truth values for the atomic propositions
• The number of rows in a truth table is determined by the number of atomic propositions ($2^n$, where $n$ is the number of atomic propositions)
• To fill out a truth table:
  1. List all the atomic propositions in the leftmost columns
  2. Fill in all possible combinations of truth values for the atomic propositions
  3. Evaluate the truth value of the compound proposition for each row using the definitions of the logical connectives
• Truth tables can be used to:
  • Determine the logical equivalence of two propositions
  • Identify tautologies, contradictions, and contingencies
  • Analyze the validity of arguments

Common Pitfalls and How to Avoid Them

• Confusing the inclusive "or" (disjunction) with the exclusive "or" (XOR)
  • In propositional logic, the disjunction ($\vee$) is inclusive, meaning it is true when at least one of the propositions is true
  • The exclusive "or" (XOR) is true only when exactly one of the propositions is true, and it is not a standard connective in propositional logic
• Misinterpreting the implication ($\rightarrow$) as a biconditional ($\leftrightarrow$)
  • The implication is true except when the antecedent (left side) is true and the consequent (right side) is false
  • The biconditional is true only when both propositions have the same truth value
• Forgetting to use parentheses to clarify the order of operations
  • Logical connectives have a specific order of precedence: negation, conjunction, disjunction, implication, biconditional
  • Use parentheses to override the default order of operations and ensure the intended meaning is conveyed
• Neglecting to consider all possible combinations of truth values when creating truth tables
  • Make sure to include all possible combinations of truth values for the atomic propositions to avoid incomplete or incorrect truth tables

Real-World Applications

• Computer programming: propositional logic is used in boolean algebra, which is fundamental to programming languages and digital circuits
• Artificial intelligence: propositional logic is used in knowledge representation and reasoning systems
• Debating and argumentation: understanding propositional logic can help in constructing valid arguments and identifying logical fallacies
• Legal reasoning: propositional logic is used to analyze the logical structure of legal arguments and contracts
• Mathematical proofs: propositional logic is used as a foundation for more advanced mathematical logic and proof techniques
• Philosophical arguments: propositional logic is used to analyze and evaluate philosophical arguments and theories

Practice Makes Perfect: Example Problems

1. Represent the following statement using propositional logic symbols: "If I study hard, then I will pass the exam."

   • Let $p$ be "I study hard" and $q$ be "I will pass the exam"
   • The statement can be represented as $p \rightarrow q$

2. Create a truth table for the compound proposition $(p \rightarrow q) \wedge (\neg q \rightarrow \neg p)$.

   • The truth table will have 4 rows (2^2 combinations)
   • Fill in the truth values for $p$, $q$, $p \rightarrow q$, $\neg q$, $\neg p$, $\neg q \rightarrow \neg p$, and the final compound proposition

3. Determine whether the following argument is valid using a truth table:

   • Premise 1: If it rains, then the ground is wet.
   • Premise 2: The ground is not wet.
   • Conclusion: Therefore, it did not rain.
   • Let $p$ be "it rains" and $q$ be "the ground is wet"
   • Represent the premises and conclusion using propositional logic symbols
   • Create a truth table to analyze the validity of the argument

4. Prove that $(p \rightarrow q) \wedge (q \rightarrow r)$ is logically equivalent to $(p \wedge q) \rightarrow r$ using a truth table.

   • Create a truth table with 8 rows (2^3 combinations)
   • Fill in the truth values for $p$, $q$, $r$, and the two compound propositions
   • Compare the truth values of the two compound propositions for each row to determine their logical equivalence