The Algebra of Logic Tradition Stanford Encyclopedia of Philosophy

Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. Elementary algebra, on the other hand, uses arithmetic operators such as addition, multiplication, subtraction and division. Boolean algebra is therefore a formal way of describing logical operations, in the same way that elementary algebra describes numerical operations.

For infinite Boolean algebras the notion of ultrafilter becomes considerably more delicate. The elements greater than or equal to an atom always form an ultrafilter, but so do many other sets; for example, in the Boolean algebra of finite and cofinite sets of integers, the cofinite sets form an ultrafilter even though none of them are atoms. These form the basis for nonstandard analysis, providing representations for such classically inconsistent objects as infinitesimals and delta functions. The technique we just used to prove an identity of Boolean algebra can be generalized to all identities in a systematic way that can be taken as a sound and complete axiomatization of, or axiomatic system for, the equational laws of Boolean logic. The customary formulation of an axiom system consists of a set of axioms that "prime the pump" with some initial identities, along with a set of inference rules for inferring the remaining identities from the axioms and previously proved identities. Model theory can be regarded as the product of Hilbert’s methodology
of metamathematics and the algebra of logic tradition, represented
specifically by the results due to Löwenheim and Skolem.

Completeness

This is defined analogously to complete Boolean algebras, but with sups and infs limited to countable arity. That is, a sigma-algebra is a Boolean algebra with all countable sups and infs. Because the sups and infs are of bounded cardinality, unlike the situation with complete Boolean algebras, the Gaifman-Hales result does not apply and free sigma-algebras do exist. Unlike the situation with CABAs however, the free countably generated sigma algebra is not a power set algebra.

If \(f\) is a homomorphism from a BA \(A\) into a complete BA
\(B\), and if \(A\) is a subalgebra of \(C\), then \(f\) can be
extended to a homomorphism of \(C\) into \(B\). Another general algebraic notion
which applies to Boolean algebras is the notion of a free
algebra. Namely, the free
BA on \(\kappa\) is the BA of closed-open subsets of the two element
discrete space raised to the \(\kappa\) power.

The prototypical Boolean algebra

In everyday relaxed conversation, nuanced or complex answers such as "maybe" or "only on the weekend" are acceptable. In more focused situations such as a court of law or theorem-based mathematics however it is deemed advantageous to frame questions so as to admit a simple yes-or-no answer—is the defendant guilty or not guilty, is the proposition true or false—and to disallow any other answer. However much of a straitjacket this might prove in practice for the respondent, the principle of the simple yes-no question has become a central feature of both judicial and mathematical logic, making two-valued logic deserving of organization and study in its own right. Examples 2 and 3 are special cases of a general construct of algebra called direct product, applicable not just to Boolean algebras but all kinds of algebra including groups, rings, etc.

It is convenient when referring to generic propositions to use Greek letters Φ, Ψ,… As metavariables (variables outside the language of propositional calculus, used when talking about propositional calculus) to denote propositions. For the purposes of this definition it is irrelevant how the operations came to satisfy the laws, whether by fiat or proof. All concrete Boolean algebras satisfy the laws (by proof rather than fiat), whence every concrete Boolean algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set and certain operations satisfying certain laws or axioms by fiat is entirely analogous to the abstract definitions of group, ring, field etc. characteristic of modern or abstract algebra.

Axiomatizing Boolean algebras

Whereas the foregoing has addressed the subject of Boolean algebra, this section deals with mathematical objects called Boolean algebras, defined in full generality as any model of the Boolean laws. We begin with a special case of the notion definable without reference to the laws, namely concrete Boolean algebras, and then give the formal definition of the general notion. Boolean algebra treats the equational theory of the maximal two-element finitary algebra, called the Boolean prototype, and the models of that theory, called Boolean algebras.[3] These terms are defined as follows.

The NAND (dually NOR) operation lacks all these, thus forming a basis by itself. The Boolean prototype is a Boolean algebra, since trivially it satisfies its own laws. We did not call it that initially in order to avoid any appearance of circularity in the definition.

Axiomatizing Boolean algebra

Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of an infinite set cannot be defined within the system — let alone the cardinality of such as set. A basic result of Tarski is that the elementary theory of Boolean
algebras is decidable.

• The latter is because the conjunction of any nonzero periodic sequence x with a sequence of greater period is neither 0 nor x.
• However, the concept of an infinite set cannot be defined within the system — let alone the cardinality of such as set.
• Hence no smaller example is possible, other than the degenerate algebra obtained by taking X to be empty so as to make the empty set and X coincide.

In this pursuit of generalized syllogisms
he introduced various other operations on binary relations, including
the converse operation, and he developed a fragment of a calculus for
these operations. His main paper on this subject was axiomatic definition of boolean algebra the fourth in the
series, called “On the syllogism, No. By adopting De Morgan’s convention of using upper-case/lower-case
letters for complements, Jevons’ system was not suited to provide
equational axioms for modern Boolean algebra.

Boolean rings

The Boolean algebra of all 32-bit bit vectors is the two-element Boolean algebra raised to the 32nd power, or power set algebra of a 32-element set, denoted 232. All Boolean algebras we have exhibited thus far have been direct powers of the two-element Boolean algebra, justifying the name "power set algebra". Polyadic algebra is another approach to an
algebra of logic for first-order logic—it was created by Halmos
(1956c).

• The focus of work in these systems was again to see to what
  extent one could parallel the famous results of Stone for Boolean
  algebra from the 1930s.
• The laws of Boolean algebra are the equations in the language of Boolean algebra satisfied by the prototype.
• More generally, Boudet, Jouannaud, and Schmidt-Schauß (1989) gave an algorithm to solve equations between arbitrary Boolean-ring expressions.
• Unlike consistency, independence is not a necessary requirement for a functioning axiomatic system — though it is usually sought after to minimize the number of axioms in the system.
• Second, Boolean algebra uses logical operators such as conjunction (and) denoted as ∧, disjunction (or) denoted as ∨, and the negation (not) denoted as ¬.