Algebraic logic

In mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.

What is now usually called classical algebraic logic focuses on the identification and algebraic description of

Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator (Czelakowski 2003).

A homogeneous

"The basic operations are set-theoretic union, intersection and complementation, the relative multiplication, and conversion."[1]

The conversion refers to the converse relation that always exists, contrary to function theory. A given relation may be represented by a logical matrix; then the converse relation is represented by the transpose matrix. A relation obtained as the composition of two others is then represented by the logical matrix obtained by matrix multiplication using Boolean arithmetic.

An example of calculus of relations arises in erotetics, the theory of questions. In the universe of utterances there are statements S and questions Q. There are two relations π and α from Q to S: q α a holds when a is a direct answer to question q. The other relation, q π p holds when p is a presupposition of question q. The converse relation π^T runs from S to Q so that the composition π^Tα is a homogeneous relation on S.^[2] The art of putting the right question to elicit a sufficient answer is recognized in Socratic method dialogue.

The description of the key binary relation properties has been formulated with the calculus of relations. The univalence property of functions describes a relation R that satisfies the formula ${\displaystyle R^{T}R\subseteq I,}$ where I is the identity relation on the range of R. The injective property corresponds to univalence of ${\displaystyle R^{T}}$, or the formula ${\displaystyle RR^{T}\subseteq I,}$ where this time I is the identity on the domain of R.

But a univalent relation is only a partial function, while a univalent total relation is a function. The formula for totality is ${\displaystyle I\subseteq RR^{T}.}$ Charles Loewner and Gunther Schmidt use the term mapping for a total, univalent relation.^[3][4]

The facility of

using ${\displaystyle {\bar {R}}}$ for the complement of relation R. These equivalences provide alternative formulas for univalent relations (${\displaystyle R{\bar {I}}\subseteq {\bar {R}}}$), and total relations (${\displaystyle {\bar {R}}\subseteq R{\bar {I}}}$). Therefore, mappings satisfy the formula ${\displaystyle {\bar {R}}=R{\bar {I}}.}$ Schmidt uses this principle as "slipping below negation from the left".

${\displaystyle f{\bar {A}}={\overline {fA}}.}$

The relation algebra structure, based in set theory, was transcended by Tarski with axioms describing it. Then he asked if every algebra satisfying the axioms could be represented by a set relation. The negative answer^[6] opened the frontier of abstract algebraic logic.^[7][8][9]

Algebraic logic treats algebraic structures, often bounded lattices, as models (interpretations) of certain logics, making logic a branch of order theory.

In algebraic logic:

• Variables are tacitly
  universe of discourse. There are no existentially quantified variables or open formulas
  ;
• Terms are built up from variables using primitive and defined operations. There are no connectives;
• Formulas, built from terms in the usual way, can be equated if they are logically equivalent. To express a tautology, equate a formula with a truth value;
• The rules of proof are the substitution of equals for equals^{[clarification needed]}, and uniform replacement. Modus ponens remains valid, but is seldom employed.

In the table below, the left column contains one or more

Algebraic formalisms going beyond first-order logic in at least some respects include:

• Combinatory logic, having the expressive power of set theory;
• ZFC
  .

Algebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memoranda

Some writings by Leopold Löwenheim and Thoralf Skolem on algebraic logic appeared after the 1910–13 publication of Principia Mathematica, and Tarski revived interest in relations with his 1941 essay "On the Calculus of Relations".^[9]

According to Helena Rasiowa, "The years 1920-40 saw, in particular in the Polish school of logic, researches on non-classical propositional calculi conducted by what is termed the logical matrix method. Since logical matrices are certain abstract algebras, this led to the use of an algebraic method in logic."^[17]

Brady (2000) discusses the rich historical connections between algebraic logic and model theory. The founders of model theory, Ernst Schröder and Leopold Loewenheim, were logicians in the algebraic tradition. Alfred Tarski, the founder of set theoretic model theory as a major branch of contemporary mathematical logic, also:

• Initiated abstract algebraic logic with relation algebras^[9]
• Invented cylindric algebra
• Co-discovered Lindenbaum–Tarski algebra.

In the practice of the calculus of relations,

Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before the Parkinson and Loemker translations. Our present understanding of Leibniz as a logician stems mainly from the work of Wolfgang Lenzen, summarized in Lenzen (2004). To see how present-day work in logic and metaphysics can draw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000).

• Boolean algebra
• Codd's theorem
• Computer algebra
• Universal algebra