Propositional calculus is commonly organized as a Hilbert system, whose operations are just those of Boolean algebra and whose theorems are Boolean tautologies, those Boolean terms equal to the Boolean constant 1. Another form is sequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists of propositions called sequents, such as A∨B, A∧C,… The two halves of a sequent are called the antecedent and the succedent respectively. The customary metavariable denoting an antecedent or part thereof is Γ, and for a succedent Δ; thus Γ,A ⊢ Δ would denote a sequent whose succedent is a list Δ and whose antecedent is a list Γ with an additional proposition A appended after it. The antecedent is interpreted as the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of the succedent by the antecedent. The Boolean algebras we have seen so far have all been concrete, consisting of bit vectors or equivalently of subsets of some set.
The shading indicates the value of the operation for each combination of regions, with dark denoting 1 and light 0 (some authors use the opposite convention). There is nothing magical about the choice of symbols for the values of Boolean algebra. We could rename 0 and 1 to say α and β, and as long as we did so consistently throughout it would still be Boolean algebra, albeit with some obvious cosmetic differences. The category Bool of Boolean algebras has as objects all Boolean algebras and as morphisms the Boolean homomorphisms between them.
After dealing with several special
cases, Schröder recommended this topic as an important research
area—the quest for an Elimination Theorem would be known as the
Elimination Problem. De Morgan wrote a series of six papers called “On the
Syllogism” in the years 1846 to 1863 (reprinted in De Morgan
1966). In his efforts to generalize the syllogism, De Morgan replaced
the copula “is” with a general binary relation in the
second paper of the series dating from 1850. By allowing different
binary relations in the two premises of a syllogism, he was led to
introduce the composition of the two binary relations to express the
conclusion of the syllogism.
Of course, it is possible to code more than two symbols in any given medium. For example, one might use respectively 0, 1, 2, and 3 volts to code a four-symbol alphabet on a wire, or holes of different sizes in a punched card. In practice, the tight constraints of high speed, small size, and low power combine to make noise a major factor. This makes it hard to distinguish between symbols when there are several possible symbols that could occur at a single site. Rather than attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages per wire, high and low.
De Morgan and Peirce: Relations and Quantifiers in the Algebra of Logic
Note, however, that the absorption law and even the associativity law can be excluded from the set of axioms as they can be derived from the other axioms (see Proven properties). ] intuitively recognized that Boolean algebra was analogous to the behavior of certain types of electrical circuits. Claude Shannon formally proved such behavior was logically equivalent to Boolean algebra in his 1937 master’s thesis, A Symbolic Analysis of Relay and Switching Circuits.
Such a Boolean algebra consists of a set and operations on that set which can be shown to satisfy the laws of Boolean algebra. Ultimately periodic sequences, sequences that become periodic after an initial finite bout of lawlessness. They constitute a proper extension of Example 5 (meaning that Example 5 is a proper subalgebra of Example 7) and also of Example 4, since constant sequences are periodic with period one. Sequences may vary as to when they settle down, but any finite set of sequences will all eventually settle down no later than their slowest-to-settle member, whence ultimately periodic sequences are closed under all Boolean operations and so form a Boolean algebra. This example has the same atoms and coatoms as Example 4, whence it is not atomless and therefore not isomorphic to Example 5/6. However it contains an infinite atomless subalgebra, namely Example 5, and so is not isomorphic to Example 4, every subalgebra of which must be a Boolean algebra of finite sets and their complements and therefore atomic.
Structure theory and cardinal functions on Boolean algebras
However this influence lies outside
the scope of this entry, which is divided into 10 sections. The advantage of Boolean algebra is that it is valid when truth-values—i.e., the truth or falsity of a given proposition or logical statement—are used as variables instead of the numeric quantities employed by ordinary algebra. It lends itself to manipulating propositions that are either true (with truth-value 1) or false (with truth-value axiomatic definition of boolean algebra 0). Two such propositions can be combined to form a compound proposition by use of the logical connectives, or operators, AND or OR. (The standard symbols for these connectives are ∧ and ∨, respectively.) The truth-value of the resulting proposition is dependent on the truth-values of the components and the connective employed. For example, the propositions a and b may be true or false, independently of one another.
Apart from model theory, Tarski revived the algebra of relations in
his 1941 paper “On the Calculus of Relations”. First he
outlined a formal logic based on allowing quantification over both
elements and relations, and then he turned to a more detailed study of
the quantifier-free formulas of this system that involved only
relation variables. After presenting a list of axioms that obviously
held in the algebra of relations as presented in Schröder’s third
volume he proved that these axioms allowed one to reduce
quantifier-free relation formulas to equations.
In classical semantics, only the two-element Boolean algebra is used, while in Boolean-valued semantics arbitrary Boolean algebras are considered. A tautology is a propositional formula that is assigned truth value 1 by every truth assignment of its propositional variables to an arbitrary Boolean algebra (or, equivalently, every truth assignment to the two element Boolean algebra). Given any complete axiomatization of Boolean algebra, such as the axioms for a complemented distributive lattice, a sufficient condition for an algebraic structure of this kind to satisfy all the Boolean laws is that it satisfy just those axioms. More generally one may complement any of the eight subsets of the three ports of either an AND or OR gate. The resulting sixteen possibilities give rise to only eight Boolean operations, namely those with an odd number of 1’s in their truth table. There are eight such because the “odd-bit-out” can be either 0 or 1 and can go in any of four positions in the truth table.
There being sixteen binary Boolean operations, this must leave eight operations with an even number of 1’s in their truth tables. Two of these are the constants 0 and 1 (as binary operations that ignore both their inputs); four are the operations that depend nontrivially on exactly one of their two inputs, namely x, y, ¬x, and ¬y; and the remaining two are x⊕y (XOR) and its complement x≡y. A sup of X is a least upper bound on X, namely an upper bound on X that is less or equal to every upper bound on X. The sup of x and y always exists in the underlying poset of a Boolean algebra, being x∨y, and likewise their inf exists, namely x∧y. Infinite subsets of a Boolean algebra may or may not have a sup and/or an inf; in a power set algebra they always do. When all the algebras being multiplied together in this way are the same algebra A we call the direct product a direct power of A.
Other definitions of Boolean algebra
In the case of a family of operations forming an algebra, the indices are called operation symbols, constituting the language of that algebra. The operation indexed by each symbol is called the denotation or interpretation of that symbol. Each operation symbol specifies the arity of its interpretation, whence all possible interpretations of a symbol have the same arity. In general it is possible for an algebra to interpret distinct symbols with the same operation, but this is not the case for the prototype, whose symbols are in one-one correspondence with its operations. The prototype therefore has 22n n-ary operation symbols, called the Boolean operation symbols and forming the language of Boolean algebra.
- For example, a number-theoretic statement might be expressible in the language of arithmetic (i.e. the language of the Peano axioms) and a proof might be given that appeals to topology or complex analysis.
- For example, small independence is the smallest size of an
infinite maximal independent set; and small cellularity is the
smallest size of an infinite partition of unity.
- Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion).
- A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values.
- Infinite subsets of a Boolean algebra may or may not have a sup and/or an inf; in a power set algebra they always do.
- It can be shown that all countably infinite atomless Boolean algebras are isomorphic, that is, up to isomorphism there is only one such algebra.
An example of such a body of propositions is the theory of the natural numbers, which is only partially axiomatized by the Peano axioms (described below). This is a proper subalgebra of Example 5 (a proper subalgebra equals the intersection of itself with its algebra). These can be understood as the finitary operations, with the first period of such a sequence giving the truth table of the operation it represents.