The terms of equational logic are built up from variables and constants using function symbols (or operations). Identities (equalities) of the form
 |
(1)
|
where 
and 
are terms, constitute the formal language of equational
logic. The syllogisms of equational logic are summarized
below.
1. Reflexivity:
 |
(2)
|
2. Symmetry:
 |
(3)
|
3. Transitivity:
 |
(4)
|
4. For 
a function symbol and 
,
 |
(5)
|
5. For 
a substitution (cf. unification),
 |
(6)
|
The above rules state that if the formula above the line is a theorem formally deducted from axioms by application of the syllogisms, then
the formula below the line is also a formal theorem. Usually, some finite set 
of identities is given as axiom
schemata.
Equational logic can be combined with first-order logic. In this case, the fourth rule is extended onto predicate symbols as well, and the fifth rule is omitted. These syllogisms can be turned into axiom schemata having the form of implications to which Modus Ponens can be applied. Major results of first-order logic hold in this extended theory.
If every identity in 
is viewed as two rewrite rules transforming the left-hand side into the right-hand
side and vice versa, then the respective term
rewriting system is equivalent to the equational logic defined by 
: The identity 
is deducible in the equational
logic iff 
in the term
rewriting system. This property is called logicality of term
rewriting systems.
Equational logic is complete, since if algebra 
is a model for 
, i.e., all identities from 
hold in algebra 
(cf. universal algebra),
then 
holds in 
iff it can be deduced in the equational logic defined by 
. This theorem is sometimes known as
Birkhoff's theorem.
