Lesson 3: Grouping and precedence (beginner, 12 min)

Q: Where do -> and fit?

Below OR: A OR B -> C is (A OR B) -> C, and is lowest of all. Chains of -> associate to the right, and chains of must be bracketed.

Which operator applies first when there are no brackets, how the parser reads A OR B AND C, and why dropping the brackets from the entry rule lets a banned member in. Four steps, beginner level, about twelve minutes.

Level: beginner12 min11 claims in this lesson were confirmed by the exact checker at build time

Learn which operator binds first, read `A OR B AND C` the way the parser does, and use parentheses to say what you mean.

The order is fixed: parentheses and the postfix complement first, then NOT, AND, XOR, OR, -> and <->. The first step states it and reads A OR B AND C as A OR (B AND C). The second step puts the two readings, A OR (B AND C) and (A OR B) AND C, on one table and finds an assignment that tells them apart. The third does the same for NOT A AND B against NOT (A AND B), and the fourth removes the brackets from (member OR invited) AND NOT banned to show that member OR invited AND NOT banned admits a banned member.

Three of the four steps carry a verified claim that the paired formulas differ, and the exercises are two multiple-choice questions about how the parser groups an expression and one hunt for a distinguishing assignment. The precedence page in the operators section prints the fully bracketed form of anything you type, which is the habit the lesson is trying to build.

The lesson is short because the rule is short; the difficulty is entirely in sentences, where “A or B and C” has no precedence, and the lesson's advice is to bracket every mixed group rather than remember the order.

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Steps

1. AND binds before OR

Without parentheses, operators apply in a fixed order, highest first: parentheses (and the postfix complement A'), NOT, AND, XOR, OR, ->, <->.

So A OR B AND C means A OR (B AND C), just as 1 + 2 × 3 means 1 + (2 × 3). Tangibool can always show this fully grouped reading, and it adds a hint when AND and OR are mixed without parentheses.

f := A OR B AND C · output

AND binds before OR
#	A	B	C	f
0	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	FALSE
2	FALSE	TRUE	FALSE	FALSE
3	FALSE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	TRUE
5	TRUE	FALSE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE
7	TRUE	TRUE	TRUE	TRUE

Open the fully grouped reading of f and compare it with the text you typed.

2. Two groupings, two different rules

reading1 := A OR (B AND C) and reading2 := (A OR B) AND C use the same inputs and operators, yet they are different rules.

One assignment is enough to tell them apart: with A=TRUE, B=FALSE, C=FALSE, reading1 is TRUE and reading2 is FALSE. Such a distinguishing assignment settles any debate about which grouping was meant.

reading1 := A OR (B AND C) · outputreading2 := (A OR B) AND C · output

Two groupings, two different rules
#	A	B	C	reading1	reading2
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	FALSE	FALSE
2	FALSE	TRUE	FALSE	FALSE	FALSE
3	FALSE	TRUE	TRUE	TRUE	TRUE
4	TRUE	FALSE	FALSE	TRUE	FALSE
5	TRUE	FALSE	TRUE	TRUE	TRUE
6	TRUE	TRUE	FALSE	TRUE	FALSE
7	TRUE	TRUE	TRUE	TRUE	TRUE

Compare reading1 with reading2 and replay the counterexample the checker finds.

not-equivalent reading1 / reading2 · holds

3. NOT binds tightly

NOT A AND B means (NOT A) AND B: the NOT applies only to A. To negate the whole conjunction, write NOT (A AND B).

The two differ, for example, when both inputs are FALSE.

tight := NOT A AND B · outputwide := NOT (A AND B) · output

NOT binds tightly
#	A	B	tight	wide
0	FALSE	FALSE	FALSE	TRUE
1	FALSE	TRUE	TRUE	TRUE
2	TRUE	FALSE	FALSE	TRUE
3	TRUE	TRUE	FALSE	FALSE

Set both inputs to FALSE and compare tight with wide.

not-equivalent tight / wide · holds

4. Parentheses carry meaning

Drop the parentheses from the entry rule and its meaning changes: member OR invited AND NOT banned reads as member OR (invited AND NOT banned), so a banned member gets in.

When a rule mixes AND and OR, write the parentheses you mean, even where precedence would give the same result.

intended := (member OR invited) AND NOT banned · outputunbracketed := member OR invited AND NOT banned · output

member: the person is a member; invited: the person has an invitation; banned: the person is banned;

Parentheses carry meaning
#	member	invited	banned	intended	unbracketed
0	FALSE	FALSE	FALSE	FALSE	FALSE
1	FALSE	FALSE	TRUE	FALSE	FALSE
2	FALSE	TRUE	FALSE	TRUE	TRUE
3	FALSE	TRUE	TRUE	FALSE	FALSE
4	TRUE	FALSE	FALSE	TRUE	TRUE
5	TRUE	FALSE	TRUE	FALSE	TRUE
6	TRUE	TRUE	FALSE	TRUE	TRUE
7	TRUE	TRUE	TRUE	FALSE	TRUE

Set member=TRUE, invited=FALSE, banned=TRUE and compare intended with unbracketed.

not-equivalent intended / unbracketed · holds

Exercises

How does the parser group A OR B AND C? (choose-grouping)
Which reading matches NOT A AND B OR C? (choose-grouping)
Find an assignment on which reading1 and reading2 give different results. (find-assignment)

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Questions

Does the engine ever change my grouping?

No. It keeps the brackets you write and applies the fixed order where you write none; the fully bracketed form on every page shows what it read.

Where do -> and <-> fit?

Below OR: A OR B -> C is (A OR B) -> C, and <-> is lowest of all. Chains of -> associate to the right, and chains of <-> must be bracketed.

Laws

Associative law: (A AND B) AND C = A AND (B AND C)

Tools

Truth table generator: any Boolean expression, up to 12 variables, with intermediate columns

Lessons

Generated from the Tangibool engine, semantics tangibool-semantics/1, engine 1.0.0, build 2026-11-07. Every table, map, simplification and verdict on these pages was computed by the Tangibool engine when the site was built, and every claim was re-checked by the same exact checker the app uses. Limits are stated where they apply. © 2026 Foreranger LTD.