Lesson 6.2 — The Symbols

Nine symbols cover all of sentential logic. Each is a label for an idea you already know.

The Roster

That's the lot. Nine symbols. You already know what each one means.

Variable Names

In symbol-world, sentences are usually represented by capital letters: P, Q, R, S, etc. Each letter stands in for some sentence.

"It is raining" → R
"The street is wet" → W

Then:

"IF it is raining, THEN the street is wet" → R → W

The letters are just names. You pick them when you set up the problem.

How Symbols Stack

The same parentheses rules from Lesson 2.1 apply. Parens tell you which piece to evaluate first.

"(IF P THEN Q) AND (R OR NOT S)"  →  (P → Q) ∧ (R ∨ ¬S)

Each connector becomes its symbol. The structure stays the same.

Symbol Order of Operations

When parens are missing, there's a default order:

1. ¬ (NOT) binds tightest.
2. ∧ (AND) next.
3. ∨ (OR) next.
4. → (IF) next.
5. ↔ (IFF) loosest.

So ¬P ∧ Q ∨ R → S reads as ((¬P ∧ Q) ∨ R) → S.

But honestly, just use parens. The defaults are confusing.

Reading Symbols Out Loud

The trick to fluency: read every symbol as its English name.

• P ∧ Q → "P and Q"
• ¬R → "not R"
• (P ∨ Q) → R → "if (P or Q) then R"
• P ↔ ¬Q → "P if and only if not Q"

After enough practice, you start reading symbols the way you read regular text.

Famous Examples in Symbols

Modus Ponens: P, P → Q ∴ Q   (Read: "P, plus P implies Q, therefore Q.")

Modus Tollens: P → Q, ¬Q ∴ ¬P

De Morgan's Laws:

• ¬(P ∧ Q) ↔ (¬P ∨ ¬Q)
• ¬(P ∨ Q) ↔ (¬P ∧ ¬Q)

You already knew all of these in English. Now they're three or four symbols long.

Name That Symbol

Quick recognition drill. Click an option to check.

Recap

• Nine symbols cover all of sentential logic. ∧ ∨ ¬ → ↔ ⊕ ⊤ ⊥ ∴
• Variables are capital letters: P, Q, R…
• Parens still apply. Use them when in doubt.
• Read symbols out loud in English to build fluency.