Five rules of inference, and every semantic consequence becomes a derivation. Equational logic is complete, and the proof is the free algebra construction read backwards.
Engineering · Mathematics11 min readKV-MATH-0222
Learning objectives
List the rules of equational logic and apply them to derive an equation.
Distinguish ⊢ from ⊨ and state what completeness asserts.
Prove soundness by induction on derivations.
Outline the completeness proof via the free algebra.
Explain why the deduction theorem fails for equational logic.
Recognise the limits: derivability is not always decidable.
01The formal system
Equational logic derives equations from equations. A derivation is a finite sequence of equations, each an axiom of Σ or obtained from earlier ones by a rule.
The rules of inference
Rule
From
Infer
Reflexivity
—
p ≈ p
Symmetry
p ≈ q
q ≈ p
Transitivity
p ≈ q and q ≈ r
p ≈ r
Congruence (replacement)
p₁ ≈ q₁, …, pₙ ≈ qₙ
f(p₁,…,pₙ) ≈ f(q₁,…,qₙ)
Substitution
p ≈ q
p(σ) ≈ q(σ) for any substitution σ
The first three make derivability an equivalence relation on terms; the congruence rule makes it compatible with the operations, hence a congruence on the term algebra; and substitution makes it fully invariant. Those three observations are the entire structure of the completeness proof.
02Soundness
Every derivable equation is semantically valid: if Σ ⊢ p ≈ q then Σ ⊨ p ≈ q.
ProcedureSoundness by induction on the derivation
in: Σ ⊢ p ≈ q → out: Σ ⊨ p ≈ q
input: derivation of p ≈ q from Σ; algebra A with A ⊨ Σ
base: each axiom of Σ holds in A by hypothesis
reflexivity: p^A = p^A trivially
symmetry and transitivity: equality of functions is an equivalence relation
congruence: if pᵢ^A = qᵢ^A then f^A∘(p⃗^A) = f^A∘(q⃗^A) pointwise
substitution: p^A = q^A as functions means they agree on ALL assignments,
in particular on those factoring through σ
conclude: every line of the derivation holds in A, so A ⊨ p ≈ q
The substitution case is where the implicit universal quantification of identities is used; an identity is a statement about all assignments, not one. Caveat: soundness is the easy half — completeness is the theorem.
Note that substitution is sound precisely because identities are universally quantified. If p ≈ q were an assertion about particular elements, substituting terms for variables would not be justified.
03Completeness
The converse holds: every semantically valid consequence is derivable.
Key resultBirkhoff's completeness theorem for equational logic
Σ ⊢ p ≈ q if and only if Σ ⊨ p ≈ q. Syntactic derivability and semantic consequence coincide exactly for equational reasoning.
Form the derivability relation
Let θ = { ⟨p, q⟩ : Σ ⊢ p ≈ q }. By the first four rules θ is a congruence on the term algebra T(X); by substitution it is fully invariant.
Take the quotient
A := T(X)/θ is an algebra of the type, and by construction A ⊨ Σ, since each axiom of Σ is derivable from Σ.
Test the target equation
Since A ⊨ Σ and Σ ⊨ p ≈ q by hypothesis, A ⊨ p ≈ q. Evaluating at the assignment sending each variable to its own class gives p/θ = q/θ.
Read off derivability
p/θ = q/θ means ⟨p, q⟩ ∈ θ, which is precisely Σ ⊢ p ≈ q.
The quotient T(X)/θ is exactly the free algebra for the variety M(Σ). So the completeness theorem and the free algebra construction are two readings of the same object, and Birkhoff's two theorems are more closely related than their statements suggest.
04Why there is no deduction theorem
Propositional logic has a deduction theorem: if Γ, φ ⊢ ψ then Γ ⊢ φ → ψ. Equational logic has no such thing, because there is no implication in the language.
CautionAdding a hypothesis is not the same as forming an implication
The equational language contains only equations, so an implication between equations is not itself an equation. Statements of the form 'if p ≈ q then r ≈ s' are quasi-identities, and they define quasivarieties rather than varieties. Any argument that silently converts a hypothesis into an implication has left equational logic.
This is why quasivarieties form a genuinely different theory, closed under S, P and ultraproducts but not H, and why the corresponding completeness result requires model-theoretic machinery.
05Decidability of derivability
Completeness says derivability and validity coincide; it does not say either is decidable.
Semidecidable always
Enumerate derivations
Derivations are finite objects over a countable language, so the set of derivable equations is recursively enumerable for any recursively enumerable Σ.
Not decidable in general
Undecidable equational theories exist
There is no algorithm deciding which equations hold in all relation algebras (Tarski), nor for unary algebras (Mal'cev), nor for certain finitely based semigroup varieties.
So the situation mirrors first-order logic: complete, semidecidable, not decidable. Where decidability does hold — for finitely generated discriminator varieties, for instance — it is a substantial theorem rather than a consequence of completeness.
06Term rewriting as a decision procedure
When an equational theory can be presented as a confluent and terminating rewriting system, the word problem becomes decidable by normalisation: rewrite both sides to normal form and compare.
ProcedureDeciding p ≈ q by normalisation
in: p, q and a complete rewriting system → out: decision
input: rewriting system R that is terminating and confluent for the theory
reduce p by R until no rule applies → normal form n(p)
reduce q by R until no rule applies → normal form n(q)
if n(p) and n(q) are the same term: p ≈ q is derivable
otherwise: p ≈ q is not derivable
correctness requires BOTH termination and confluence
Knuth–Bendix completion attempts to turn an arbitrary equational axiom set into such a system, but may fail to terminate — consistent with the undecidability results above. Caveat: orienting equations into rules is a choice, and a bad orientation can destroy termination.
Knuth–Bendix completion and the automated theorem provers built on it are the practical face of equational logic. The sourcing policy page routes to current tools rather than reproducing tables of completed systems.
Frequently asked
Are the five rules independent?
Reflexivity is needed as a base case and cannot be derived. Symmetry and transitivity are independent of one another. The congruence rule is sometimes split into replacement-in-context form, and some presentations combine substitution with the axioms. Presentations differ; what matters is that the derivability relation is a fully invariant congruence on the term algebra.
Why 'fully invariant' rather than just 'congruence'?
Because the substitution rule adds closure under endomorphisms of the term algebra, which is exactly full invariance. A congruence on T(X) that is not fully invariant corresponds to a set of equations closed under the first four rules but not substitution, and does not define a variety.
Does completeness hold for infinitary equational logic?
Not in the same clean form. Allowing infinitely long terms breaks the finiteness of derivations that the enumeration argument relies on, and the correspondence with finitary closure operators and algebraic lattices fails too. Finitariness is load-bearing throughout.
Sources and further reading
S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.
Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.
Handbook application: from concept to controlled practice
Purpose. This expanded section turns the original page into a practical handbook. It preserves the supplied material and adds a repeatable way to apply, check and review Equational Logic and the Completeness Theorem. It does not replace a contract, legislation, a controlled standard, competent engineering judgement or specialist advice.
The operating aim is to turn a compact mathematical statement into a usable chain of definitions, claims, examples and checks. Read the original explanation first, then use the workflow and checks below to convert knowledge into evidence.
Treat Equational Logic and the Completeness Theorem as a network of definitions and implications, not as a list of formulas. The working vocabulary on this page—completeness, equational, logic, theorem, formal—should be made explicit before any proof or computation begins. Record the ambient set or structure, the permitted operations and the equality or equivalence relation in use. A compact theorem often changes meaning when the base field, finiteness condition, commutativity assumption or direction of an action changes.
For a proof, write the hypotheses as a checklist and mark the line at which each one is used. For a computation, state the representation of the input, the arithmetic model, the termination condition and the output invariant. For a classification problem, distinguish existence from uniqueness and distinguish an object from its representation. These separations prevent a correct local calculation from being mistaken for the general result.
A useful worked example should be small enough to inspect completely but rich enough to exercise the main mechanism. Compute the result in two ways where practical: symbolically and by substitution, structurally and numerically, or directly and through a normal form. Then include one near-miss example in which a hypothesis fails. The contrast explains why the theorem is shaped as it is and gives the reader a diagnostic pattern for later problems.
Verification is part of the mathematics. Check domains and codomains, substitute proposed solutions, test identity and zero cases, compare dimensions or cardinalities, and confirm that maps respect the required operations. In numerical work, report precision, conditioning and a residual rather than digits alone. In algorithmic work, separate mathematical correctness from implementation complexity and resource limits.
Step-by-step operating method
Fix the setting. State the objects, ambient structure, notation and assumptions before manipulating symbols.
Separate claims. Distinguish definitions, hypotheses, conclusions, equivalent conditions and consequences.
Choose a method. Select proof, construction, calculation or algorithm according to the question actually asked.
Work a small case. Use the smallest non-trivial example to expose the mechanism and test edge behaviour.
Verify independently. Substitute back, check invariants, test boundary cases or use an alternative derivation.
Worked-example protocol
Illustrative method—not a source theorem. Start with a small admissible input and list the definitions it must satisfy. Carry out each transformation on a separate line, citing the property that permits it. Preserve exact values until approximation is necessary. At the end, verify the output against the original definition and one invariant such as dimension, degree, determinant, order, norm or residual. Then alter one hypothesis and observe which step ceases to be valid. This protocol creates a reusable example without inventing a theorem-specific numerical answer.
Stage
Record
Quality check
Input
Objects, domain, notation, assumptions
Every symbol is defined
Method
Permitted operation or cited result at each step
All hypotheses hold
Output
Exact result and representation
Correct type, domain and form
Verification
Substitution, invariant or alternative derivation
Independent agreement
Boundary test
Zero, identity, degenerate or failed hypothesis
Scope is understood
Common failure modes and recovery actions
1. Watch for
Using a theorem without checking every hypothesis.
Recovery: Return to the governing definition or requirement and restate the decision in one sentence.
2. Watch for
Treating a suggestive example as a proof of the general case.
Recovery: Separate evidence from assumption, assign an owner and set a date for validation.
3. Watch for
Changing notation or conventions part-way through an argument.
Recovery: Run a small counterexample, boundary test, pilot or independent check before proceeding.
4. Watch for
Hiding a division-by-zero, convergence, finiteness or commutativity assumption.
Recovery: Record the consequence, decision and rationale, then update the controlled baseline.
5. Watch for
Reporting a computed result without a residual, substitution or structural check.
Recovery: Escalate when the issue affects safety, compliance, acceptance, material value or an agreed tolerance.
Review checklist
Can every symbol be traced to a definition or prior result?
Which hypothesis does each major step use?
Does the method cover zero, identity, degenerate and boundary cases?
Can the conclusion be checked by a second representation or calculation?
Are mandatory requirements distinguished from recommendations and illustrative values?
Are sources, assumptions, units, dates and versions recorded closely enough to reproduce the decision?
Have safety, legal, ethical, stakeholder and operational consequences been considered at the appropriate level?
Is there a named owner and a trigger for review, escalation, change or retirement?
Questions for deeper application
What is the most important distinction a practitioner must preserve when applying Equational Logic and the Completeness Theorem?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which assumption about completeness would change the result most if it proved false?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What evidence would allow an independent reviewer to reproduce or challenge the conclusion?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Which boundary, exception or failure case has not yet been tested?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
What must be handed over, monitored or reviewed after the immediate work is complete?
Answer with a fact or cited source where available. Where evidence is incomplete, record the assumption, consequence, responsible owner and next validation action.
Authoritative references and use notes
The sources below were selected as institutional or primary guidance for the broader practice. They support the handbook method; they do not imply that every statement or clause in a source applies to every project. Confirm the current edition, jurisdiction, contract and application before treating any requirement as mandatory.
MIT OpenCourseWare — Algebra I — Massachusetts Institute of Technology. Used for groups, vector spaces, linear transformations and linear groups. Accessed 2026-08-13.
The Stacks Project — table of contents — The Stacks Project. Used for commutative algebra, homological algebra, modules and derived categories. Accessed 2026-08-13.
