Boolean algebra vs propositional logic

Boolean algebra and propositional logic are two vocabularies for the same two-valued system: one speaks of 0, 1, sums and products, the other of propositions, connectives and validity. The formula (p -> q) AND p below is modus ponens as an algebra problem, and it reduces to p AND q.

Propositional logic starts from propositions (p, q) and connectives (NOT, AND, OR, implication, biconditional) and asks about truth, validity and consequence: from p -> q and p, does q follow? Boolean algebra starts from the two-element algebra with its laws, identity, complement, distribution, and asks about functions and their forms. The engine does not choose: it prints every formula in a words, programming, mathematical and algebra notation, and both laws and argument checks run on the same two-valued semantics.

The formula on this page makes the identification concrete. (p -> q) AND p, the conjunction of modus ponens's premises, is TRUE on one row of four, Σm(3), p and q both TRUE. Its DNF is p AND q (implication elimination, distribution, complement), its CNF is p AND (NOT p OR q), and the Karnaugh map derives p AND q. In logic's terms: whenever both premises hold, q holds, so the argument is valid. In algebra's terms: the product of the premises is the product p·q, which is at most q.

The vocabularies diverge in what they emphasise. Algebra has no notion of an argument, only of functions; logic has no Karnaugh maps, only proofs and tables. Philosophy courses teach the logic vocabulary with letters p, q, r and arrows; digital design teaches the algebra vocabulary with A, B, C, overbars and dots. Both are served by the same reference, and the glossary gives each term in both tongues.

Example

f := (p -> q) AND p · output

Exact checks

Always true (tautology)? No Counterexample: p = F, q = F · Exhaustively checked and solver cross-checked · 1 rows

Always false (contradiction)? No Counterexample: p = T, q = T · Exhaustively checked and solver cross-checked · 4 rows

Can be true? Yes Example assignment: p = T, q = T · Exhaustively checked and solver cross-checked · 4 rows

Truth table (4 rows)

Boolean algebra vs propositional logic
#	p	q	f
0	FALSE	FALSE	FALSE
1	FALSE	TRUE	FALSE
2	TRUE	FALSE	FALSE
3	TRUE	TRUE	TRUE

Logic diagram

Karnaugh map

p \ q

0m0

0m1

0m2

1m3

cells m3p AND q

p AND q · Derived p AND q. Exactly equivalent on every assignment.

Minterms and maxterms

f(p, q) = Σm(3)f(p, q) = ΠM(0, 1, 2)

Variable order: p, q · 4 rows

Normal forms

Conjunctive normal form (CNF)

p AND (NOT p OR q)

2 clauses, 3 literals · complete · Implication elimination ×1

Disjunctive normal form (DNF)

p AND q

1 terms, 2 literals · complete · Implication elimination ×1, Distribution ×1, Complement ×1

NAND-only and NOR-only

NAND-only form

NAND(NAND(NAND(p, NAND(q, q)), p), NAND(NAND(p, NAND(q, q)), p))

7 gates · verified equivalent on every row

NOR-only form

NOR(NOR(NOR(p, p), q), NOR(p, p))

4 gates · verified equivalent on every row

The common slip

Thinking that p -> q is an algebra operation that implication “really is”. It is NOT p OR q, p' + q in algebra notation, and the two names describe one truth table; a validity question and a function question about it have the same answer.

Questions

Is a tautology the same as the constant 1?

Yes: a tautology is a formula TRUE on every row, which as a function is the constant 1. The simplifier reduces p OR NOT p to TRUE, and the tautology check reports it holds on every row.

Why does the engine have both argument mode and decision mode?

Because the two vocabularies ask different questions of the same formulas: argument mode asks whether a conclusion follows from premises, decision mode asks what a rule outputs. The semantics underneath are identical.

Guides

Compare

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.