Truth Table Symbols

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Why This Matters

Truth table symbols are the alphabet of formal logic—without fluency in these symbols, you can't read, write, or evaluate logical arguments. You're being tested not just on recognizing what each symbol means, but on understanding how they transform truth values and when specific combinations yield true or false results. The connectives you'll learn here—conjunction, disjunction, negation, conditional, and biconditional—show up in every proof, every argument analysis, and every validity question you'll encounter.

Master these symbols and you'll be able to construct truth tables quickly, spot logical equivalences, and evaluate complex compound statements with confidence. Don't just memorize the symbols—know the truth conditions for each connective and understand why certain combinations produce the results they do. When you see $p \rightarrow q$, you should immediately think about what makes it false, not just what it "means" in English.

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Truth Values: The Building Blocks

Before you can evaluate any compound statement, you need the two fundamental truth values that every proposition must take. In classical logic, every statement is either true or false—never both, never neither.

True (T)

• Indicates a proposition is correct or satisfied—the baseline "yes" value in any logical evaluation
• Represented as T, 1, or ⊤ depending on notation style; all mean the same thing in truth tables
• Combines with connectives to determine the output of compound statements—knowing when T appears is half the battle

False (F)

• Indicates a proposition is incorrect or unsatisfied—the baseline "no" value in logical evaluation
• Represented as F, 0, or ⊥ depending on your textbook's notation conventions
• Critical for identifying invalidity—arguments fail when premises are true but conclusions are false

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Unary Connective: Negation

Negation is the only connective that operates on a single proposition. It flips the truth value—nothing more, nothing less.

Negation (¬)

• Inverts the truth value of any proposition—if $A$ is true, $\neg A$ is false; if $A$ is false, $\neg A$ is true
• Also written as ~, !, or NOT depending on notation; the tilde (~) is common in many textbooks
• Double negation cancels out—$\neg \neg A$ is logically equivalent to $A$, a key equivalence for simplification

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Conjunctive Connectives: Requiring Agreement

These connectives yield true only when their component propositions align in specific ways. Conjunction demands both be true; biconditional demands both match.

Conjunction (∧)

• True only when both propositions are true—this is the strictest binary connective, yielding T in just one of four rows
• Represents logical "AND"—think of it as a checklist where every item must be satisfied
• Symbol variations include & and •—but ∧ (wedge pointing up) is standard in formal logic

Biconditional (↔)

• True when both propositions share the same truth value—both T or both F yields T; mixed values yield F
• Represents "if and only if" (iff)—establishes logical equivalence between two statements
• Equivalent to $(A \rightarrow B) \land (B \rightarrow A)$—understanding this breakdown helps verify biconditional truth tables

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Disjunctive Connective: Requiring At Least One

Disjunction is the most permissive binary connective. It only fails when everything fails.

Disjunction (∨)

• True when at least one proposition is true—false only when both are false, making it the "easiest" connective to satisfy
• Represents inclusive "OR"—both can be true simultaneously (unlike exclusive or)
• Symbol points down (∨)—remember "∨" for "vel," the Latin word for "or"

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Conditional Connective: The Tricky One

The conditional causes more confusion than any other connective because its truth conditions don't match everyday "if-then" intuitions. Focus on the one case that makes it false.

Conditional (→)

• False only when the antecedent is true and consequent is false—this is the single F row; all other combinations yield T
• Represents "if...then" or implication—$A \rightarrow B$ means "A implies B" or "A is sufficient for B"
• Vacuously true when antecedent is false—if $A$ is false, $A \rightarrow B$ is automatically true regardless of $B$

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Quick Reference Table

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Self-Check Questions

1. Which two connectives yield true when both propositions are true AND when both are false? How do they differ in the mixed cases?

2. You're building a truth table for $\neg(p \lor q)$. How many rows do you need, and in which row(s) will the final column show T?

3. Compare the conditional (→) and biconditional (↔): if $p$ is false and $q$ is true, what truth value does each connective produce? Why?

4. A classmate claims that $p \rightarrow q$ and $q \rightarrow p$ are logically equivalent. Using truth table reasoning, explain why this is incorrect and identify what these two statements actually represent.

5. Which connective is the only unary operator, and how does this affect the structure of its truth table compared to binary connectives?