Truth tables are a fundamental tool in [[Truth-Functional Logic]] (TFL) used to determine the truth value of complex sentences based on the truth values of their component [[Atomic Sentences]]. ## Structure - Rows: Each row represents a possible combination of truth values for the [[Atomic Sentences]]. - Columns: Each column shows the truth value of a subformula or the entire formula. ## Construction 1. List all [[Atomic Sentences]] and create columns for each. 2. Create rows for all possible combinations of truth values (T/F) for these atoms. 3. Add columns for each subformula, building up to the main connective. 4. Fill in truth values based on the definitions of the logical [[Connectives]]. ## Uses 1. Determining [[Validity]]: An [[Argument]] is [[Valid]] if there's no row where all premises are true and the conclusion false. 2. Identifying [[Tautology|Tautologies]] and [[Contradiction|Contradictions]] 3. Checking for [[Logical Equivalence]] 4. Analyzing the truth-functional behavior of complex sentences ## Example For p → (q ∨ r): | p | q | r | q ∨ r | p → (q ∨ r) | |---|---|---|-------|--------------| | T | T | T | T | T | | T | T | F | T | T | | T | F | T | T | T | | T | F | F | F | F | | F | T | T | T | T | | F | T | F | T | T | | F | F | T | T | T | | F | F | F | F | T | Truth tables provide a systematic and exhaustive method for analyzing the logical properties of sentences in TFL, forming a cornerstone of truth-functional analysis.