💡Critical Thinking Unit 9 – Propositional Logic and Truth Tables

Key Concepts and Definitions

• Propositional logic deals with statements that can be either true or false, known as propositions
• Propositions are declarative sentences that make a claim about the world and can be assigned a truth value
• Logical connectives are symbols used to join propositions together to form compound statements
  • Common connectives include conjunction ($\land$), disjunction ($\lor$), negation ($\lnot$), implication ($\rightarrow$), and biconditional ($\leftrightarrow$)
• Truth tables are a systematic way to represent and evaluate the truth values of propositions and compound statements
• Tautology is a compound statement that is always true, regardless of the truth values of its individual propositions
• Contradiction is a compound statement that is always false, regardless of the truth values of its individual propositions
• Contingency is a compound statement that can be either true or false, depending on the truth values of its individual propositions

Propositional Logic Basics

• Propositional logic is a branch of logic that studies the relationships between propositions and their truth values
• Propositions are represented by letters ($p$, $q$, $r$, etc.) and can only have two possible truth values: true ($T$) or false ($F$)
• Compound statements are formed by connecting propositions using logical connectives
• The truth value of a compound statement depends on the truth values of its constituent propositions and the logical connectives used
• Parentheses are used to indicate the order of operations and to group propositions together
• The main goal of propositional logic is to determine the validity of arguments and to identify logical equivalences between statements
• Propositional logic forms the foundation for more advanced logical systems, such as first-order logic and modal logic

Logical Connectives

• Conjunction ($\land$) represents the logical "and" and is true only when both propositions are true
• Disjunction ($\lor$) represents the logical "or" and is true when at least one of the propositions is true
  • Inclusive disjunction allows for both propositions to be true, while exclusive disjunction ($\oplus$) requires exactly one proposition to be true
• Negation ($\lnot$) represents the logical "not" and reverses the truth value of a proposition
• Implication ($\rightarrow$) represents a conditional statement, where the truth of the first proposition (antecedent) implies the truth of the second proposition (consequent)
  • The implication is false only when the antecedent is true and the consequent is false
• Biconditional ($\leftrightarrow$) represents a logical equivalence between two propositions and is true when both propositions have the same truth value
• The order of precedence for logical connectives is: negation, conjunction, disjunction, implication, and biconditional
• Logical connectives allow for the creation of complex compound statements that can be analyzed using truth tables

Truth Tables: Structure and Purpose

• Truth tables are a tabular representation of the truth values of propositions and compound statements
• Each row in a truth table represents a unique combination of truth values for the individual propositions
• The number of rows in a truth table is determined by the number of propositions involved ($2^n$ rows for $n$ propositions)
• The columns in a truth table represent the individual propositions and the compound statement being evaluated
• Truth tables allow for the systematic evaluation of the truth value of a compound statement for all possible combinations of truth values of its constituent propositions
• By analyzing truth tables, one can identify tautologies, contradictions, and contingencies
• Truth tables are also used to determine the logical equivalence between different compound statements

Constructing Truth Tables

• Begin by identifying the propositions involved in the compound statement and assigning them letters ($p$, $q$, $r$, etc.)
• Determine the number of rows needed based on the number of propositions ($2^n$ rows for $n$ propositions)
• Create columns for each proposition and list all possible combinations of truth values in the rows
  • For two propositions, the combinations are: $TT$, $TF$, $FT$, and $FF$
• Add columns for each logical connective used in the compound statement, following the order of precedence
• Evaluate the truth value of the compound statement for each row using the truth tables for the logical connectives
• The final column will represent the truth value of the entire compound statement for each combination of truth values of the propositions
• Analyze the truth table to identify tautologies (all $T$s in the final column), contradictions (all $F$s), and contingencies (a mix of $T$s and $F$s)

Evaluating Arguments with Truth Tables

• An argument in propositional logic consists of premises and a conclusion
• To evaluate the validity of an argument, construct a truth table that includes columns for the premises, the conclusion, and the argument as a whole
• The argument is valid if and only if the truth value of the conclusion is true whenever the truth values of all the premises are true
  • In other words, there should be no row in the truth table where the premises are all true and the conclusion is false
• If the argument is valid, the truth table will show a tautology in the column representing the argument as a whole
• If the argument is invalid, there will be at least one row where the premises are true and the conclusion is false, indicating a counterexample to the argument
• Truth tables provide a systematic way to evaluate the validity of arguments in propositional logic, helping to identify sound and unsound arguments

Common Fallacies and Pitfalls

• The fallacy of affirming the consequent occurs when one incorrectly concludes that the antecedent must be true because the consequent is true
  • Example: "If it is raining, the ground is wet. The ground is wet, therefore it is raining." (The ground could be wet due to other reasons)
• The fallacy of denying the antecedent occurs when one incorrectly concludes that the consequent must be false because the antecedent is false
  • Example: "If it is raining, 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)
• The fallacy of the converse error occurs when one assumes that the converse of a true implication is also true
  • Example: "If a shape is a square, then it is a rectangle. Therefore, if a shape is a rectangle, then it is a square." (Not all rectangles are squares)
• The fallacy of the inverse error occurs when one assumes that the inverse of a true implication is also true
  • Example: "If a shape is a square, then it is a rectangle. Therefore, if a shape is not a square, then it is not a rectangle." (A shape can be a rectangle without being a square)
• It is important to be aware of these fallacies and to carefully analyze the structure of arguments to avoid drawing incorrect conclusions

Real-World Applications

• Propositional logic is used in computer science and programming to design logical circuits and to develop algorithms
  • Logical connectives are represented using Boolean operators (AND, OR, NOT) in programming languages
• In artificial intelligence and expert systems, propositional logic is used to represent and reason about knowledge and to make inferences based on available information
• Propositional logic is applied in philosophical arguments and debates to analyze the validity of claims and to identify logical inconsistencies
• In mathematics, propositional logic serves as a foundation for more advanced logical systems, such as first-order logic and set theory
• Legal reasoning often relies on propositional logic to evaluate the consistency and validity of arguments presented in court cases
• Propositional logic is used in linguistics to study the logical structure of sentences and to analyze the relationships between different parts of speech
• In decision-making and problem-solving, propositional logic helps to break down complex situations into simpler, more manageable components and to evaluate the consequences of different choices