Boole's

But he kept Boole's use of equations as the fundamental form of statements in his algebra of logic.

The Algebra of Logic Tradition Burris, Stanley 2009

(Boole's fundamental operations in his algebra of logic were addition, subtraction and multiplication) on equals and obtain equals.

The Algebra of Logic Tradition Burris, Stanley 2009

No meaningful justification was given for Boole's adoption of this new foundation, it was not given a special name, and the scant references to it in the rest of the book were usually rather clumsily stated.

The Algebra of Logic Tradition Burris, Stanley 2009

The first volume concerned the equational logic of classes, the main result being Boole's Elimination Theorem of 1854.

The Algebra of Logic Tradition Burris, Stanley 2009

Jevons, who had studied with De Morgan, was the first to offer an alternative to Boole's system.

The Algebra of Logic Tradition Burris, Stanley 2009

Peirce also broke with Boole's and Jevons 'use of equality as the fundamental primitive, using instead

The Algebra of Logic Tradition Burris, Stanley 2009

Aside from the Rule of 0 and 1 and the Elimination Theorem, the 1854 presentation is mainly interesting for Boole's attempts to justify his algebra of logic.

The Algebra of Logic Tradition Burris, Stanley 2009

According to Boole's own evaluation, his main innovation was the Index Law

Leibniz's Influence on 19th Century Logic Peckhaus, Volker 2009

There were a number of concerns regarding Boole's approach to the algebra of logic:

The Algebra of Logic Tradition Burris, Stanley 2009

Boole's theorems, completed and perfected in 1854, gave algorithms for analyzing infinitely many argument forms.

The Algebra of Logic Tradition Burris, Stanley 2009