Lesson 2.1 — Combining

Stack the four words. Work a compound sentence down to a single truth value, step by step.

The Setup

Once you know the four words — AND, OR, NOT, IF — you can stack them. But order and grouping matter.

"I want pizza AND soda OR juice."

What does this mean?

• "Pizza AND (soda OR juice)" — pizza for sure, plus one drink.
• "(Pizza AND soda) OR juice" — either pizza+soda together, or just juice.

Different sentences. Same words.

Parentheses Save Us

In math, parens group things:

• 2 × (3 + 4) = 14
• (2 × 3) + 4 = 10

In logic, we do the same. Parens tell you which piece to evaluate first.

• pizza AND (soda OR juice) — group the OR first.
• (pizza AND soda) OR juice — group the AND first.

When you write or read a logical sentence, always use parens to make the grouping clear. English is sloppy. Logic is precise.

Working Out the Answer

To find the truth value of a compound sentence, follow one rule:

Innermost parens first, then work outward.

Each step, replace one sub-expression with its truth value. Keep going until the whole thing is a single TRUE or FALSE. It's the same idea as solving a math expression — you do the inside of the parens before the outside.

Try it with the widget below. Pick a TRUE or FALSE value for each of the three facts, and watch the compound sentence resolve into a single answer.

Each step takes the previous line and replaces one sub-expression with its value. By the last step, the whole sentence is a single TRUE or FALSE. That's the deconstruction skill: take a long compound, work the innermost piece, repeat until you're down to one word.

A Worked Example

Let's do one by hand. Suppose:

• A = "I cleaned my room" — TRUE
• B = "I did my homework" — TRUE
• C = "I went outside" — FALSE

Evaluate this compound:

(A AND B) OR C

Step 1 — substitute the values:

(TRUE AND TRUE) OR FALSE

Step 2 — innermost parens first. TRUE AND TRUE = TRUE.

TRUE OR FALSE

Step 3 — evaluate the OR. TRUE OR anything = TRUE.

TRUE

Done. The compound sentence is TRUE.

Tricky Bits

1. No parens? English defaults — but they're confusing.

Without parens, English usually treats AND as binding tighter than OR. So pizza AND soda OR juice defaults to (pizza AND soda) OR juice. But this is exactly the kind of default that gets readers confused. Best practice: always write the parens.

2. Stacking the same word.

"I cleaned my room AND I did my homework AND I walked the dog."

Three ANDs in a row — all three parts must be true. Order doesn't matter for stacked ANDs (or stacked ORs).

3. NOT binds tightest.

When a NOT sits next to a variable, it grabs only that variable. NOT A AND B means (NOT A) AND B, not NOT (A AND B). To negate a whole group, you need parens: NOT (A AND B).

Practice

Facts:

• A = I ate breakfast (TRUE)
• B = I brushed my teeth (TRUE)
• C = I went to school (FALSE — it's Saturday)

Work out each truth value:

1. (A AND B) OR C
2. A AND (B OR C)
3. (A OR C) AND B
4. (A AND C) OR (B AND C)
5. NOT (A AND C)
6. (NOT C) AND (A OR B)

Answers:

1. True — (TRUE AND TRUE) OR FALSE = TRUE OR FALSE = TRUE.
2. True — TRUE AND (TRUE OR FALSE) = TRUE AND TRUE = TRUE.
3. True — (TRUE OR FALSE) AND TRUE = TRUE AND TRUE = TRUE.
4. False — both AND groups contain C, and C is FALSE. Both groups false, OR of two falses is FALSE.
5. True — A AND C = TRUE AND FALSE = FALSE, then NOT FALSE = TRUE.
6. True — NOT C = TRUE, A OR B = TRUE, TRUE AND TRUE = TRUE.

Recap

• Parens tell you which piece to evaluate first.
• Innermost parens first, then work outward. Replace each sub-expression with its truth value, one step at a time.
• Without parens, English defaults to AND-binds-tighter-than-OR, NOT binds tightest of all. Don't rely on the defaults — write the parens.
• The whole sentence resolves into a single TRUE or FALSE. That's the deconstruction skill.