First-order equational logic

Logic

In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...

 consists of quantifier-free terms of ordinary first-order logic

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

, with equality as the only predicate symbol. The model theory of this logic was developed into Universal algebra by Birkhoff et al. [Birkhoff, Gratzer, Cohn]. It was later made into a branch of category theory by Lawvere ("algebraic theories").

The terms of equational logic are built up from variables and constants using function symbols (or operations). Identities (equalities) of the form

Syllogism

Here are the four inference rules of logic E. (P[x:= E] denotes textual substitution of expression E for variable x in expression P):

Substitution

Substitution

Substitution may refer to:- Sciences :* Substitution , a syntactic transformation on strings of symbols of a formal language* Substitution of variables* Substitution cipher, a method of encryption...

: If P is a theorem, then so is P[x:= E].
|- P ---> |- P[x:= E]
Leibniz: If P = Q is a theorem, then so is E[x:= P] = E[x:= Q].
|- P = Q ---> |- E[x:= P] = E[x:= Q]
Transitivity

Transitivity

-In grammar:* Intransitive verb* Transitive verb, when a verb takes an object* Transitivity -In logic and mathematics:* Arc-transitive graph* Edge-transitive graph* Ergodic theory, a group action that is metrically transitive* Vertex-transitive graph...

: If P = Q and Q = R are theorems, then so is P = R.
|- P = Q, |- Q = R ---> |- P = R
Equanimity

Equanimity

Equanimity is a state of mental or emotional stability or composure arising from a deep awareness and acceptance of the present moment. Equanimity is promoted by several major religious groups.-Stoicism:...

: If P and P

Q are theorems, then so is Q.
|- P, |- P

Q ---> |- Q

History

Equational logic was developed over the years (beginning in the early 1980s) by researchers in the formal development of programs, who felt a need for an effective style of manipulation, of calculation. Involved were people like Roland Backhouse, Edsger W. Dijkstra, Wim H.J. Feijen, David Gries, Carel S. Scholten, and Netty van Gasteren. Wim Feijen is responsible for important details of the proof format.

The axioms are similar to those use by Dijkstra and Scholten in their monograph Predicate calculus and program semantics (Springer Verlag, 1990), but our order of presentation is slightly different.

In their monograph, Dijkstra and Scholten use the three inference rules Leibniz, Substitution, and Transitivity. However, Dijkstra/Scholten system is not a logic, as logicians use the word. Some of their manipulations are based on the meanings of the terms involved, and not on clearly presented syntactical rules of manipulation. The first attempt at making a real logic out of it appeared in A Logical Approach to Discrete Math. However, inference rule Equanimity is missing there, and the definition of theorem is contorted to account for it. The introduction of Equanimity and its use in the proof format is due to Gries and Schneider. It is used, for example, in the proofs of soundness and completeness, and it will appear in the second edition of A Logical Approach to Discrete Math.

Proof

We explain how the four inference rules are used in proofs, using the proof of ~p

p

false.

(0) ~p

p

false

(1) = < (3.9), ~(p

q)

~p

q, with q:= p >

(2) ~(p

p)

false

(3) = < Identity of

(3.9), with q:= p >

(4) ~true

false --(3.8)
First, lines (0)-(2) show a use of inference rule Leibniz:
(0) = (2)
is the conclusion of Leibniz, and its premise ~(p

p)

~p

p is given on line (1). In the same way, the equality on lines (2)-(4) are substantiated using Leibniz.
The "hint" on line (1) is supposed to give a premise of Leibniz, showing what substitution of equals for equals is being used. This premise is theorem (3.9) with the substitution p:= q, i.e.

(~(p

q)

~p

q)[p:=q]
This shows how inference rule Substitution is used within hints.
From (0) = (2) and (2) = (4), we conclude by inference rule Transitivity that (0) = (4). This shows how Transitivity is used.

Finally, note that line (4), ~true

false, is a theorem, as indicated by the hint to its right. Hence, by inference rule Equanimity, we conclude that line (0) is also a theorem. And (0) is what we wanted to prove.