# The Language of Propositional Logic

Philosophy I · Logic · https://tryals.app/learn/philosophy-i/the-language-of-propositional-logic

## Building Sentences From Sentences

Propositional logic takes whole declarative sentences as its atoms — written $P$, $Q$, $R$ — and builds compounds from them with **connectives**. Each connective is **truth-functional**: the truth value of the compound is fixed entirely by the truth values of its parts, with nothing else relevant.

| Connective | Symbol | Read as | True when |
|---|---|---|---|
| Negation | $\neg P$ | not P | P is false |
| Conjunction | $P \wedge Q$ | P and Q | both are true |
| Disjunction | $P \vee Q$ | P or Q | at least one is true |
| Conditional | $P \rightarrow Q$ | if P then Q | except when P true and Q false |
| Biconditional | $P \leftrightarrow Q$ | P if and only if Q | both have the same value |

Two of these need care. **Disjunction** is *inclusive*: $P \vee Q$ is true when both disjuncts are true. Ordinary English often means the exclusive "or", "tea or coffee" usually excludes both, but the logical symbol does not.

The **conditional** is the one that surprises everyone. $P \rightarrow Q$ is defined as false in exactly one case: $P$ true and $Q$ false. So it comes out **true whenever $P$ is false**, regardless of $Q$. "If the moon is made of cheese then I am the Pope" is true, on this definition. This is the **material conditional**, and it captures only part of what "if" means in English, but it is the part that makes deduction work, and the surprise is a feature of the definition rather than a mistake.

**Brackets** matter as much as connectives. $\neg(P \wedge Q)$ and $\neg P \wedge Q$ are different sentences with different truth conditions, and the difference is exactly where ambiguity in ordinary language hides.

Translating from English is where the real work lies. "Unless" is usually $\vee$; "P only if Q" is $P \rightarrow Q$, **not** $Q \rightarrow P$; "neither P nor Q" is $\neg P \wedge \neg Q$.

> **Common pitfall:** reading "P only if Q" as "if P then Q" reversed. "You may vote only if you are registered" says registration is *necessary*, not sufficient, so it means voting $\rightarrow$ registered. The "only" marks the consequent, not the antecedent.

## Practice questions

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

### 1. A conditional fails only when it promises something and does not deliver. In how many of the four assignments of truth values to $P$ and $Q$ is $P \rightarrow Q$ false?

**Answer:** 1

**Why:** Exactly **one**: $P$ true with $Q$ false. In the other three rows the conditional is true, including both rows where $P$ is false, which is the result that surprises people.

Page: https://tryals.app/practice/philosophy-i/the-language-of-propositional-logic/a-conditional-fails-only-when-it-promises-something-and-does-not

### 2. In logic, "P or Q" is true when both P and Q are true.

**Answer:** True

**Why:** True, logical disjunction is **inclusive**. Ordinary English often means the exclusive sense ("tea or coffee" usually excludes both), which is why translations need care, but the symbol $\vee$ is always inclusive.

Page: https://tryals.app/practice/philosophy-i/the-language-of-propositional-logic/in-logic-p-or-q-is-true-when-both-p-and-q-are-true

### 3. Logical connectives are truth-functional, meaning the truth value of a compound depends solely on the truth values of its parts. What follows from this principle when assessing an argument framed in natural language?

A. We must interpret every natural disjunction as strictly inclusive
B. We can ignore context and meaning once truth values are fixed
C. Compound claims are only valid if their components are true facts
D. Every English sentence maps uniquely to one symbolic formulation

**Answer:** B. We can ignore context and meaning once truth values are fixed

**Why:** Truth-functionality abstracts away semantic nuance, causation, and context once inputs are set. Treating natural language this way enables rigorous formal deduction, though it deliberately excludes non-truth-functional aspects like temporal order.

Page: https://tryals.app/practice/philosophy-i/the-language-of-propositional-logic/logical-connectives-are-truth-functional-meaning-the-truth-value-of

### 4. Arrange these compounds so that each is true in strictly more of the four assignments than the one before it.

**Answer:**

1. P and not-P
2. P and Q
3. P or Q
4. if P then P

**Why:** True in 0, 1, 3 and 4 rows respectively. Counting satisfying rows orders sentences by logical strength: a contradiction excludes everything, a tautology excludes nothing, and the informative sentences lie in between.

Page: https://tryals.app/practice/philosophy-i/the-language-of-propositional-logic/arrange-these-compounds-so-that-each-is-true-in-strictly-more-of-the

### 5. Which translations are correct?

A. "P only if Q" becomes if P then Q
B. "P unless Q" becomes P or Q
C. "Neither P nor Q" becomes not-P and not-Q
D. "P only if Q" becomes if Q then P

**Answer:** A. "P only if Q" becomes if P then Q; B. "P unless Q" becomes P or Q; C. "Neither P nor Q" becomes not-P and not-Q

**Why:** "Only if" marks a *necessary* condition: "you may vote only if registered" means voting $\rightarrow$ registered. Reversing it would say registration guarantees voting, which is a quite different and much stronger claim.

Page: https://tryals.app/practice/philosophy-i/the-language-of-propositional-logic/which-translations-are-correct

### 6. Sort each sentence by whether the main connective is a conjunction, a disjunction or a conditional.

**Answer:**

- Main connective is conjunction: not-P and Q, P and (Q or R)
- Main connective is disjunction: (P and Q) or R, not-P or not-Q
- Main connective is conditional: if (P and Q) then R

**Why:** The main connective is whichever is applied last. "P and (Q or R)" is a conjunction while "(P and Q) or R" is a disjunction, even though both contain the same three letters and the same two connectives.

Page: https://tryals.app/practice/philosophy-i/the-language-of-propositional-logic/sort-each-sentence-by-whether-the-main-connective-is-a-conjunction-a

### 7. Complete the account of the material conditional.

**Answer:** The conditional is false in exactly **one** of the four assignments, namely when the antecedent is **true** and the consequent is false. It is therefore **true** whenever the antecedent is false, which is why it captures only **part** of what "if" means in ordinary English.

**Why:** One falsifying row, and therefore truth in the other three. The material conditional is a deliberate simplification of "if", it drops relevance and causation, keeping only what deduction needs.

Page: https://tryals.app/practice/philosophy-i/the-language-of-propositional-logic/complete-the-account-of-the-material-conditional