➕Logic and Formal Reasoning Unit 2 – Propositional Logic: Basics & Truth Tables

What's Propositional Logic?

• Propositional logic is a branch of logic that studies the relationships between propositions and their logical connectives
• Focuses on the truth values of propositions (statements that can be either true or false) and how they interact with each other
• Uses logical connectives (such as "and", "or", "not", "if-then", "if and only if") to combine propositions into more complex statements
• Provides a formal framework for analyzing the validity of arguments based on the truth values of their constituent propositions
  • For example, the argument "If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet." can be analyzed using propositional logic
• Serves as a foundation for more advanced logical systems, such as first-order logic and modal logic
• Has applications in various fields, including mathematics, computer science, and philosophy

Key Terms and Symbols

• Proposition: a declarative sentence that can be either true or false, but not both simultaneously
  • Examples: "The sky is blue", "2 + 2 = 5"
• Logical connectives: symbols or words used to combine propositions into more complex statements
  • Negation (¬ or ~): represents the opposite truth value of a proposition
  • Conjunction (∧ or &): represents "and"; true only when both propositions are true
  • Disjunction (∨ or |): represents "or"; true when at least one proposition is true
  • Implication (→ or ⇒): represents "if-then"; true except when the antecedent is true and the consequent is false
  • Biconditional (↔ or ≡): represents "if and only if"; true when both propositions have the same truth value
• Truth value: the property of a proposition being either true (T) or false (F)
• Tautology: a statement that is always true, regardless of the truth values of its constituent propositions
• Contradiction: a statement that is always false, regardless of the truth values of its constituent propositions
• Contingency: a statement that can be either true or false, depending on the truth values of its constituent propositions

Building Propositions

• Propositions are the building blocks of propositional logic and can be represented by letters (p, q, r, etc.)
• Simple propositions are single declarative sentences that can be either true or false
  • Examples: "The Earth is round" (p), "Cats can fly" (q)
• Compound propositions are formed by combining simple propositions using logical connectives
  • Example: "The Earth is round and cats can fly" (p ∧ q)
• The truth value of a compound proposition depends on the truth values of its constituent propositions and the logical connectives used
• Parentheses are used to clarify the order of operations when multiple logical connectives are present
  • Example: (p ∧ q) ∨ r means "either both p and q are true, or r is true"
• Propositions can be negated using the negation symbol (¬ or ~) to represent their opposite truth value
  • Example: If p is "The sky is blue", then ¬p is "The sky is not blue"

Logical Connectives

• Logical connectives are symbols or words used to combine propositions into more complex statements
• Negation (¬ or ~): represents the opposite truth value of a proposition
  • Example: If p is true, then ¬p is false
• Conjunction (∧ or &): represents "and"; true only when both propositions are true
  • Example: p ∧ q is true only if both p and q are true
• Disjunction (∨ or |): represents "or"; true when at least one proposition is true
  • Example: p ∨ q is true if either p or q (or both) are true
• Implication (→ or ⇒): represents "if-then"; true except when the antecedent is true and the consequent is false
  • Example: p → q is false only when p is true and q is false
• Biconditional (↔ or ≡): represents "if and only if"; true when both propositions have the same truth value
  • Example: p ↔ q is true when p and q are both true or both false
• The order of precedence for logical connectives is: negation, conjunction, disjunction, implication, biconditional
  • Parentheses can be used to override this order and clarify the intended meaning

Truth Tables Explained

• Truth tables are a method for systematically evaluating the truth values of compound propositions based on the truth values of their constituent propositions
• Each row in a truth table represents a possible combination of truth values for the constituent propositions
• The number of rows in a truth table is determined by the number of unique propositions (n) and is equal to 2^n
• The columns in a truth table represent the constituent propositions and the compound propositions formed using logical connectives
• To fill out a truth table:
  1. List all possible combinations of truth values for the constituent propositions
  2. Evaluate the truth values of the compound propositions for each row using the definitions of the logical connectives
• Truth tables can be used to determine the logical equivalence of different compound propositions
  • Two propositions are logically equivalent if they have the same truth value for all possible combinations of truth values of their constituent propositions
• Truth tables can also be used to identify tautologies, contradictions, and contingencies

Evaluating Complex Statements

• Complex statements in propositional logic are compound propositions that involve multiple logical connectives
• To evaluate the truth value of a complex statement, break it down into smaller parts and evaluate each part using the definitions of the logical connectives
• Use parentheses to clarify the order of operations and evaluate the innermost parentheses first
• Work outwards, applying the logical connectives in the order of precedence: negation, conjunction, disjunction, implication, biconditional
• Construct a truth table to systematically evaluate the truth value of the complex statement for all possible combinations of truth values of its constituent propositions
  • This can be helpful for verifying your work and identifying patterns
• Look for opportunities to simplify the complex statement by applying logical equivalences
  • For example, p ∧ (p ∨ q) is logically equivalent to p, so the complex statement can be simplified
• Remember that a tautology is always true, a contradiction is always false, and a contingency can be either true or false depending on the truth values of its constituent propositions

Common Fallacies

• Fallacies are errors in reasoning that can lead to invalid arguments or conclusions
• Affirming the consequent: concluding that the antecedent must be true because the consequent is true
  • Example: "If it is raining, then the ground is wet. The ground is wet, therefore it must be raining." (The ground could be wet for other reasons)
• Denying the antecedent: concluding that the consequent must be false because the antecedent is false
  • Example: "If it is raining, then the ground is wet. It is not raining, therefore the ground is not wet." (The ground could be wet even if it is not raining)
• Begging the question (circular reasoning): using the conclusion of an argument as a premise to support itself
  • Example: "God exists because the Bible says so, and the Bible is true because it is the word of God."
• False dilemma (false dichotomy): presenting a limited number of options as if they were the only possibilities, when in fact there are other alternatives
  • Example: "Either you're with us, or you're against us." (There could be neutral or nuanced positions)
• Equivocation: using a word or phrase with multiple meanings in different parts of an argument, leading to confusion or deception
  • Example: "A feather is light. What is light cannot be dark. Therefore, a feather cannot be dark." (The word "light" is used in two different senses)

Real-World Applications

• Propositional logic has numerous applications in various fields, including mathematics, computer science, and philosophy
• In mathematics, propositional logic is used to analyze and prove statements about numbers, sets, and other mathematical objects
  • For example, propositional logic can be used to prove that the square root of 2 is irrational
• In computer science, propositional logic is used to design and analyze digital circuits, as well as to develop algorithms and programming languages
  • Boolean algebra, which is based on propositional logic, is fundamental to the design of digital systems
• In artificial intelligence, propositional logic is used to represent and reason about knowledge in expert systems and other AI applications
  • Propositional logic can be used to encode rules and facts, and to make inferences based on those rules and facts
• In philosophy, propositional logic is used to analyze arguments and test their validity
  • Philosophers use propositional logic to clarify the structure of arguments and to identify fallacies and other errors in reasoning
• In linguistics, propositional logic is used to study the meaning and structure of sentences and to analyze the relationships between different parts of speech
  • Propositional logic can be used to represent the logical form of sentences and to study the entailment relations between sentences
• Propositional logic also has applications in fields such as law, economics, and social sciences, where it is used to analyze and reason about complex systems and decision-making processes