Logic Basics for Beginners: Truth Tables & ArgumentsLogic is the study of correct reasoning. It gives us tools to represent claims clearly, test whether conclusions follow from premises, and spot hidden mistakes in arguments. This article introduces the core ideas a beginner needs: propositions, logical connectives, truth tables, formal arguments, validity, and common informal pitfalls. Practical examples and simple exercises are included so you can practice what you learn.
What is a proposition?
A proposition (or statement) is a sentence with a definite truth value: it is either true or false, not both. Examples:
- “Snow is white.” — a proposition.
- “Who is coming?” — not a proposition (it’s a question).
- “x > 2” — not a proposition unless x is given a value.
In symbolic logic we use capital letters like P, Q, R to stand for propositions.
Logical connectives (operators)
Connectives combine propositions to form more complex propositions. The main ones:
- Negation (NOT), symbol: ¬P
- True when P is false; false when P is true.
- Conjunction (AND), symbol: P ∧ Q
- True only when both P and Q are true.
- Disjunction (OR), symbol: P ∨ Q
- True when at least one of P or Q is true. (Inclusive OR)
- Conditional (IF…THEN), symbol: P → Q
- False only when P is true and Q is false; otherwise true.
- Biconditional (IF AND ONLY IF), symbol: P ↔ Q
- True when P and Q have the same truth value.
These operators let us build compound statements like (P ∧ Q) → R or ¬(P ∨ Q).
Truth tables: how they work
A truth table lists the truth value of a compound proposition for every possible combination of truth values of its component propositions. They are the primary tool for evaluating logical expressions and testing argument validity in propositional logic.
Example: truth table for P ∧ Q
| P | Q | P ∧ Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
Example: truth table for P → Q
| P | Q | P → Q |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
Note on the conditional: P → Q is true whenever P is false — this is called vacuous truth. It reflects the idea that a false hypothesis imposes no constraint on the conclusion.
Building truth tables: step-by-step
- List all component propositions (P, Q, R…).
- Determine the number of rows: 2^n for n propositions.
- Fill columns for each basic proposition with all combinations of T/F.
- Compute intermediate columns for sub-expressions.
- Compute the final column for the full expression.
Example: Evaluate (P ∨ Q) → R for P, Q, R.
- There are 8 rows (2^3).
- Compute P ∨ Q, then compute (P ∨ Q) → R for each row.
Logical equivalence and important identities
Two expressions are logically equivalent if they have identical truth values in every row of their truth tables. Common equivalences:
- De Morgan’s laws:
- ¬(P ∧ Q) ≡ ¬P ∨ ¬Q
- ¬(P ∨ Q) ≡ ¬P ∧ ¬Q
- Implication as disjunction:
- P → Q ≡ ¬P ∨ Q
- Double negation:
- ¬(¬P) ≡ P
- Distribution:
- P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
- P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)
Using these identities helps simplify expressions and design shorter truth tables.
Arguments, validity, and soundness
An argument consists of premises and a conclusion. Example:
- If it rains, the ground gets wet. (P → Q)
- It is raining. (P)
Therefore: The ground gets wet. (Q)
Validity: An argument is valid if whenever all premises are true, the conclusion must also be true. Validity depends only on form, not truth of the premises.
Soundness: An argument is sound if it is valid and its premises are actually true.
Testing validity with truth tables:
- Form the conjunction of all premises as one compound statement (Prem1 ∧ Prem2 ∧ …).
- Create an implication: (Prem1 ∧ Prem2 ∧ …) → Conclusion.
- If the implication is a tautology (true in every row), the argument is valid.
Example: Modus ponens Premises: P → Q, P Conclusion: Q Truth table shows that whenever P → Q and P are both true, Q must be true — so modus ponens is valid.
Common valid argument forms:
- Modus Ponens: P → Q, P ⊢ Q
- Modus Tollens: P → Q, ¬Q ⊢ ¬P
- Hypothetical Syllogism: P → Q, Q → R ⊢ P → R
- Disjunctive Syllogism: P ∨ Q, ¬P ⊢ Q
Invalid forms (fallacies) to watch for:
- Affirming the consequent: P → Q, Q ⊢ P (invalid)
- Denying the antecedent: P → Q, ¬P ⊢ ¬Q (invalid)
Tautologies, contradictions, and contingencies
- Tautology: true for every assignment of truth values (e.g., P ∨ ¬P).
- Contradiction: false for every assignment (e.g., P ∧ ¬P).
- Contingent statement: true for some assignments and false for others (most meaningful statements).
Identifying whether a formula is a tautology or contradiction is easy with truth tables.
Translating English to propositional logic
Tips:
- Identify simple, atomic claims and assign propositional letters.
- Map connectives: “and” → ∧, “or” → ∨, “if…then” → →, “if and only if” → ↔, “not” → ¬.
- Watch for ambiguity: “or” in English can be exclusive; clarify if needed.
- Parentheses matter: P ∧ Q ∨ R is ambiguous; prefer (P ∧ Q) ∨ R or P ∧ (Q ∨ R).
Example: “If Alice studies, she’ll pass. She studied. Therefore she passed.” Symbols: S → P, S ⊢ P (modus ponens).
Practice problems
- Build a truth table for (P → Q) ∧ (Q → R) → (P → R). Is it a tautology?
- Determine if the argument below is valid:
- If the alarm is set, the house is protected. The alarm is not set. Therefore the house is not protected.
- Show using truth tables that ¬(P ∨ Q) is equivalent to ¬P ∧ ¬Q.
Answers:
- The formula is a tautology (hypothetical syllogism as a tautological implication).
- Invalid (denying the antecedent).
- True by De Morgan’s law.
Common pitfalls and advice
- Confusing validity with truth: an argument can be valid with false premises.
- Misreading conditionals in everyday language — natural language conditionals often carry context not captured by material implication.
- Forgetting to use parentheses when building complex formulas.
- Overreliance on intuition; use truth tables for a definitive check.
Where to go next
After mastering truth tables and basic propositional argument forms, next steps:
- Predicate logic (quantifiers: ∀, ∃) — reasoning about properties and relations.
- Proof techniques (natural deduction) — derive conclusions step-by-step using rules.
- Formal methods and Boolean algebra — useful for computer science and circuit design.
Further practice: translate short English arguments into propositional form, construct truth tables, and check validity.
Leave a Reply