7.3 Logical Statements, Connectives, Conditionals & Truth Tables
Key Takeaways
- A proposition is a declarative statement that is definitively True ($T$) or False ($F$), but not both; questions, commands, opinions, and paradoxes are not propositions.
- A conjunction ($p \land q$) is True only when both variables are True; a disjunction ($p \lor q$) is False only when both variables are False.
- A conditional statement ($p \rightarrow q$) is False in only one case: when the hypothesis $p$ is True and conclusion $q$ is False ($T \rightarrow F$ is $F$); if $p$ is False, the conditional is vacuously True.
- A biconditional ($p \leftrightarrow q$) is True when both components share the identical truth value (both True or both False).
- Compound statements are classified by their truth table outputs: a Tautology is always True ($T$), a Contradiction is always False ($F$), and a Contingency contains both $T$ and $F$ values.
7.3 Logical Statements, Connectives, Conditionals & Truth Tables
Symbolic logic is the formal mathematical framework used to analyze statements, evaluate truth conditions, and assess deductive validity. On the CLEP College Mathematics exam, logic questions frequently test truth table evaluation, compound connectives, conditional implication behavior, and tautology identification. Mastering truth tables and formal logical definitions guarantees consistent exam accuracy.
1. Propositions vs. Non-Propositions
A proposition (or statement) is a declarative sentence that is definitively True ($T$) or False ($F$), but not both simultaneously.
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| PROPOSITION IDENTIFICATION GUIDE |
| |
| Sentence Proposition? Truth Value |
| -------- ------------ ----------- |
| "Austin is the capital of Texas." YES True (T) |
| "8 + 5 = 12" YES False (F) |
| "Every prime number is odd." YES False (F, 2) |
| "Please submit your exam." (Command) NO N/A |
| "What time does the lecture begin?" NO N/A |
| "x + 7 = 15" (Open algebraic sentence) NO N/A (x unknown) |
| "This sentence is false." (Paradox) NO N/A |
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2. Fundamental Logical Connectives
Compound statements are constructed by joining simple propositions using logical connectives.
| Connective | Name | Symbolic Form | Key Truth Condition |
|---|---|---|---|
| Negation | NOT | $\neg p$ or $\sim p$ | Inverts the truth value of $p$ |
| Conjunction | AND | $p \land q$ | True ONLY if both $p$ and $q$ are True |
| Disjunction | OR (Inclusive) | $p \lor q$ | False ONLY if both $p$ and $q$ are False |
| Exclusive OR | XOR | $p \oplus q$ | True if exactly one of $p$ or $q$ is True |
| Conditional | IF... THEN | $p \rightarrow q$ | False ONLY when $p$ is True and $q$ is False |
| Biconditional | IF AND ONLY IF | $p \leftrightarrow q$ | True when $p$ and $q$ have identical truth values |
Master Truth Table for Basic Connectives
| $p$ | $q$ | $\neg p$ | $p \land q$ | $p \lor q$ | $p \oplus q$ | $p \rightarrow q$ | $p \leftrightarrow q$ |
|---|---|---|---|---|---|---|---|
| T | T | F | T | T | F | T | T |
| T | F | F | F | T | T | F | F |
| F | T | T | F | T | T | T | F |
| F | F | T | F | F | F | T | T |
3. In-Depth Analysis of Conditional Statements ($p \rightarrow q$)
A conditional statement (implication) $p \rightarrow q$ links an antecedent (hypothesis $p$) to a consequent (conclusion $q$).
Verbal Phrasings for $p \rightarrow q$
- "If $p$, then $q$"
- "$p$ implies $q$"
- "$p$ only if $q$"
- "$q$ if $p$"
- "$p$ is sufficient for $q$"
- "$q$ is necessary for $p$"
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| THE CONDITIONAL CONTRACT ANALOGY |
| |
| Promise: "If you wash the car (p), I will pay you $20 (q)." |
| |
| Row 1: Wash car (T), Paid $20 (T) --> Promise Kept (TRUE) |
| Row 2: Wash car (T), NOT paid (F) --> Promise BROKEN (FALSE) |
| Row 3: Didn't wash (F), Paid $20 (T) --> Generosity (TRUE) |
| Row 4: Didn't wash (F), NOT paid (F) --> No breach (TRUE) |
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The Principle of Vacuous Truth
If the hypothesis $p$ is False, the conditional statement $p \rightarrow q$ is automatically True, regardless of whether the conclusion $q$ is True or False.
- Example: "If $2 + 2 = 5$, then the moon is made of green cheese" is mathematically True because the antecedent ($2 + 2 = 5$) is False.
4. Biconditional Statements ($p \leftrightarrow q$)
A biconditional statement asserts that $p$ and $q$ imply each other: $(p \rightarrow q) \land (q \rightarrow p)$. It is written as "$p$ if and only if $q$" (abbreviated as "$p$ iff $q$").
- $p \leftrightarrow q$ is True when $p$ and $q$ are both True or both False.
- $p \leftrightarrow q$ is False when $p$ and $q$ have opposite truth values.
5. Constructing Multi-Column Truth Tables
For a compound expression with $n$ propositional variables, the truth table requires exactly $2^n$ rows ($2$ variables $= 4$ rows; $3$ variables $= 8$ rows).
Standard Operator Hierarchy (Order of Operations)
- Parentheses
(...) - Negation
~/¬ - Conjunction
^/∧and Disjunctionv/∨(left to right) - Conditional
→ - Biconditional
↔
Worked Example 1: Constructing a Truth Table for $\neg(p \land q) \lor (p \rightarrow q)$
| $p$ | $q$ | $p \land q$ | $\neg(p \land q)$ | $p \rightarrow q$ | $\neg(p \land q) \lor (p \rightarrow q)$ |
|---|---|---|---|---|---|
| T | T | T | F | T | T |
| T | F | F | T | F | T |
| F | T | F | T | T | T |
| F | F | F | T | T | T |
Result: The final column contains entirely True values. Thus, this statement is a tautology.
6. Classifications of Compound Statements
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| STATEMENT CLASSIFICATION BY TRUTH TABLE |
| |
| Classification Definition Example |
| -------------- ---------- ------- |
| Tautology True in EVERY row p v ~p |
| Contradiction False in EVERY row p ^ ~p |
| Contingency Contains at least one T p -> q |
| and at least one F p v q |
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- Tautology: A statement that is logically true under every possible interpretation (e.g., $p \lor \neg p$, $(p \land q) \rightarrow p$).
- Contradiction (Self-Contradiction): A statement that is logically false under every possible interpretation (e.g., $p \land \neg p$, $\neg(p \rightarrow p)$).
- Contingency: A statement whose truth value depends on the truth values of its constituent variables (neither a tautology nor a contradiction).
7. Common CLEP Traps & Strategic Checkpoints
- Trap 1: The $F \rightarrow F$ Case in Conditionals: Many students intuitively believe that if both $p$ and $q$ are False, $p \rightarrow q$ must be False. Remember: $F \rightarrow F$ is TRUE.
- Trap 2: "$p$ only if $q$" vs. "$p$ if $q$":
- "$p$ only if $q$" translates to $p \rightarrow q$ ($q$ is the consequent).
- "$p$ if $q$" translates to $q \rightarrow p$ ($q$ is the antecedent).
- Trap 3: Negation Precedence: In $\neg p \land q$, the negation applies only to $p$. If the entire conjunction is negated, parentheses are required: $\neg(p \land q)$.
- Trap 4: Truth Table Row Calculation: Ensure your truth table has $2^n$ rows. Missing rows guarantees an incorrect tautology/contingency classification.
Which of the following sentences represents a valid logical proposition?
Under which specific assignment of truth values is the conditional statement $p \rightarrow q$ evaluated as FALSE?
Which of the following compound propositional expressions is a TAUTOLOGY (true for every possible combination of truth values of its constituent variables)?