Propositional Logic Symbols

Why This Matters

Propositional logic symbols are the alphabet of formal reasoning—without fluency in these symbols, you can't read, write, or evaluate logical arguments. In Formal Logic II, you're being tested on your ability to translate natural language into symbolic form, construct truth tables, and identify valid argument structures. Every proof technique you'll encounter—from direct proof to reductio ad absurdum—depends on your command of these fundamental connectives.

Don't just memorize what each symbol looks like. You need to understand how each connective transforms truth values, when statements using them are true or false, and how they relate to one another through logical equivalences. The difference between acing an exam and struggling through it often comes down to whether you can instantly recognize that $p \rightarrow q$ is logically equivalent to $\neg p \lor q$, or spot that $p \leftrightarrow q$ is really just $(p \rightarrow q) \land (q \rightarrow p)$ in disguise.

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Building Blocks: Atomic Propositions and Constants

Before you can combine statements, you need the raw materials. These are the irreducible elements of propositional logic—the atoms from which all molecular statements are built.

Atomic Propositions (p, q, r)

• The simplest declarative statements—these cannot be broken down into smaller logical components
• Each atomic proposition has exactly one truth value—either true or false, never both or neither
• Serve as variables in logical formulas, allowing you to analyze argument structure independent of specific content

Tautology (⊤)

• A proposition that is always true—regardless of the truth values assigned to any component propositions
• Represents logical validity itself, appearing when a formula holds universally (e.g., $p \lor \neg p$)
• Critical in proofs—if you can derive ⊤, your reasoning is sound; if a premise set entails ⊤, it's trivially satisfiable

Contradiction (⊥)

• A proposition that is always false—no assignment of truth values can make it true
• Signals logical impossibility, such as $p \land \neg p$
• Central to proof by contradiction—deriving ⊥ from an assumption proves that assumption false

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Unary Operator: Negation

This is your only single-input connective—it operates on one proposition and flips its truth value.

Negation (¬)

• Reverses the truth value of any proposition—if $p$ is true, $\neg p$ is false, and vice versa
• Symbolically, $\neg p$ means "it is not the case that p"—equivalent to denial or contradiction of the original claim
• Foundation for De Morgan's Laws—understanding negation is essential for transforming $\neg(p \land q)$ into $\neg p \lor \neg q$

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Binary Connectives: Combining Propositions

These operators take two propositions and produce a new compound statement. Each has a distinct truth-functional definition that determines when the compound is true or false.

Conjunction (∧)

• True only when both conjuncts are true—this is the strictest binary connective
• Symbolically, $p \land q$ means "p and q"—both conditions must be satisfied simultaneously
• Models requirements and necessary conditions—use conjunction when multiple criteria must all hold

Disjunction (∨)

• True when at least one disjunct is true—this is inclusive or, meaning both can be true
• Symbolically, $p \lor q$ means "p or q (or both)"—only false when both disjuncts are false
• Appears in proof by cases—if you can prove a conclusion from $p$ and separately from $q$, and you know $p \lor q$, you've proven the conclusion

Exclusive OR (⊕)

• True when exactly one proposition is true—false when both are true or both are false
• Symbolically, $p \oplus q$ means "p or q, but not both"—captures mutual exclusivity
• Equivalent to $(p \lor q) \land \neg(p \land q)$—useful for translation problems asking you to express XOR using basic connectives

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Conditional Connectives: Implication and Equivalence

These connectives express logical relationships between propositions—they're the backbone of argument structure and logical inference.

Conditional (→)

• False only when the antecedent is true and the consequent is false—true in all other cases, including when the antecedent is false
• Symbolically, $p \rightarrow q$ means "if p, then q"—also read as "p implies q" or "p only if q"
• Logically equivalent to $\neg p \lor q$—this equivalence is heavily tested and essential for natural deduction

Biconditional (↔)

• True when both propositions share the same truth value—either both true or both false
• Symbolically, $p \leftrightarrow q$ means "p if and only if q"—often abbreviated "p iff q"
• Equivalent to $(p \rightarrow q) \land (q \rightarrow p)$—establishes that two propositions are logically interchangeable

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Structural Notation: Grouping and Precedence

Without clear structure, complex formulas become ambiguous. Parentheses impose order and eliminate interpretive confusion.

Parentheses ( )

• Group propositions to specify operation order—determines which connectives apply first in complex expressions
• Override default precedence rules—standard precedence is ¬, ∧, ∨, →, ↔, but parentheses take priority
• Essential for unambiguous translation—"p and q or r" could mean $(p \land q) \lor r$ or $p \land (q \lor r)$, which have different truth tables

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Quick Reference Table

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Self-Check Questions

1. Which two symbols represent propositions with fixed truth values regardless of their components, and how do their truth values differ?

2. Compare the truth conditions for ∨ and ⊕—in what specific case do they produce different truth values?

3. If $p \rightarrow q$ is true and $p$ is false, what can you conclude about $q$? Why does this sometimes seem counterintuitive?

4. How would you express $p \leftrightarrow q$ using only →, ∧, and the propositions p and q? What does this tell you about the relationship between conditionals and biconditionals?

5. Given the formula $\neg p \land q \lor r$, write two different parenthesized versions that would produce different truth values when $p$ is true, $q$ is false, and $r$ is true.