# Truth Tables

Philosophy I · Logic · https://tryals.app/learn/philosophy-i/truth-tables

## Deciding by Exhaustion

A **truth table** lists every possible assignment of truth values to the atomic sentences and computes the value of the compound in each. Because there are no other possibilities, the table settles the matter completely, propositional logic is **decidable**, and the truth table is the decision procedure.

The size is fixed by the number of distinct atoms:

$$\text{rows} = 2^n$$

One atom gives 2 rows, two give 4, three give 8. The growth is the method's weakness: ten atoms need 1024 rows, and twenty need over a million. It always works and quickly stops being practical, which is precisely why natural deduction exists.

Building the table has a standard discipline. List the atoms, fill $2^n$ rows so every combination appears exactly once (halving the block size at each column keeps this systematic), then work outward from the smallest subformulas to the main connective. The column under the **main connective** is the sentence's own column.

The table classifies any sentence into one of three kinds:

| Classification | Column shows | Example |
|---|---|---|
| Tautology | True in every row | $P \vee \neg P$ |
| Contradiction | False in every row | $P \wedge \neg P$ |
| Contingency | True in some rows, false in others | $P \wedge Q$ |

Two sentences are **logically equivalent** when their columns match row for row. The **De Morgan** equivalences are the ones worth knowing by heart:

$$\neg(P \wedge Q) \equiv \neg P \vee \neg Q, \qquad \neg(P \vee Q) \equiv \neg P \wedge \neg Q$$

Negating a conjunction gives a disjunction, and vice versa. The connective flips.

Tables also test arguments. An argument is valid exactly when no row makes all premises true and the conclusion false, so you build the table and look for that one bad row. Finding none proves validity; finding one is a **counterexample** and disproves it outright.

> **Common pitfall:** counting rows from the number of *letters written* rather than the number of *distinct* atoms. $P \wedge (P \vee Q)$ contains three letter-occurrences but only two distinct atoms, so it needs 4 rows, not 8.

## Practice questions

8 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. How many rows does a truth table need for a sentence containing 5 distinct atomic propositions?

**Answer:** 32

**Why:** $2^5 = 32$ rows. The exponential growth is the whole reason natural deduction was developed, truth tables always work but stop being usable well before interesting arguments get large.

Page: https://tryals.app/practice/philosophy-i/truth-tables/how-many-rows-does-a-truth-table-need-for-a-sentence-containing-5

### 2. Why does the exponential growth of truth tables motivate natural deduction?

A. Tables remain correct but become impractically large for many atoms
B. Tables fail to decide sentences containing deeply nested connectives
C. Tables start yielding incorrect classifications once atoms exceed three
D. Natural deduction can validate complex sentences that tables cannot settle

**Answer:** A. Tables remain correct but become impractically large for many atoms

**Why:** Tables never become *wrong*, they become unusable. A twenty-atom sentence needs over a million rows, all correct and none of them worth writing. Natural deduction reaches the same verdicts by following rules instead of exhausting cases.

Page: https://tryals.app/practice/philosophy-i/truth-tables/why-does-the-exponential-growth-of-truth-tables-motivate-natural

### 3. Two sentences count as logically equivalent as soon as their truth-table columns agree in at least one row.

**Answer:** False

**Why:** False, the columns must match in **every** row. Almost any two sentences agree somewhere by coincidence, so agreement in one row establishes nothing at all. This is why De Morgan’s laws are checked row by row.

Page: https://tryals.app/practice/philosophy-i/truth-tables/two-sentences-count-as-logically-equivalent-as-soon-as-their

### 4. Arrange the steps of building a truth table in order.

**Answer:**

1. Identify the distinct atomic propositions
2. Write 2 to the power n rows covering every combination
3. Compute the columns for the smallest subformulas
4. Work outward to the main connective
5. Read the sentence’s value from the main connective column

**Why:** The order is forced by dependency: rows need the atom count, and each compound column needs its subformulas. The main connective column is the sentence’s own, reading the wrong column is a common slip.

Page: https://tryals.app/practice/philosophy-i/truth-tables/arrange-the-steps-of-building-a-truth-table-in-order

### 5. Propositional logic is termed decidable because truth tables offer an exhaustive decision procedure. What does this exhaustive quality guarantee when evaluating an argument?

A. A mechanical guarantee of reaching a definitive verdict
B. The ability to evaluate premises without checking atoms
C. A guarantee that valid arguments have no false premises
D. The derivation of conclusions purely by syntactic rules

**Answer:** A. A mechanical guarantee of reaching a definitive verdict

**Why:** Decidability ensures a definitive semantic result via finite valuation, not syntactic derivation or false-free premises. Finding invalidity requires only one counterexample row, while syntactic methods like natural deduction bypass truth tables entirely.

Page: https://tryals.app/practice/philosophy-i/truth-tables/propositional-logic-is-termed-decidable-because-truth-tables-offer-an

### 6. Which are correct statements of De Morgan’s laws?

A. not (P and Q) is equivalent to not-P or not-Q
B. Negating a conjunction produces a disjunction
C. not (P or Q) is equivalent to not-P and not-Q
D. not (P and Q) is equivalent to not-P and not-Q

**Answer:** A. not (P and Q) is equivalent to not-P or not-Q; B. Negating a conjunction produces a disjunction; C. not (P or Q) is equivalent to not-P and not-Q

**Why:** The connective **flips** when negation moves inside. the fourth option keeps the conjunction, which is the standard error: "not both" does not mean "neither".

Page: https://tryals.app/practice/philosophy-i/truth-tables/which-are-correct-statements-of-de-morgans-laws

### 7. Match each truth-table result to what it establishes about an argument.

**Answer:**

- No row has true premises and a false conclusion → The argument is valid
- Some row has true premises and a false conclusion → That row is a counterexample
- The conclusion is true in every row → The conclusion is a tautology
- Premises and conclusion columns match exactly → They are logically equivalent

**Why:** Validity is established by exhausting the possibilities and finding no counterexample. One bad row refutes; no bad rows prove. This asymmetry, one counterexample suffices, runs through all of logic.

Page: https://tryals.app/practice/philosophy-i/truth-tables/match-each-truth-table-result-to-what-it-establishes-about-an

### 8. A sentence contains 4 distinct atomic propositions. Set the number of rows its truth table requires.

**Answer:** 16 (within ±1.5)

**Why:** $2^4 = 16$ rows. Doubling per atom is easy to underestimate: going from 4 atoms to 8 does not double the work, it multiplies it by sixteen.

Page: https://tryals.app/practice/philosophy-i/truth-tables/a-sentence-contains-4-distinct-atomic-propositions-set-the-number-of