What is the diference between boolean algebra and sentential calculus?

Question

In both you have 2 truth-values, you can built truth-tables...
So what is the diference between boolean algebra and classical sentential calculus?

Answer 1

In mathematical logic, a propositional calculus (sentential calculus) is a formal system that represents the materials and the principles of propositional logic (sentential logic). Propositional logic is a domain of formal subject matter that is, up to isomorphism, constituted by the structural relationships of mathematical objects called propositions.

In general terms, a calculus is a formal system that consists of a set of syntactic expressions (well-formed formulas or wffs), a distinguished subset of these expressions, plus a set of transformation rules that define a binary relation on the space of expressions.

When the expressions are interpreted for mathematical purposes, the transformation rules are typically intended to preserve some type of semantic equivalence relation among the expressions. In particular, when the expressions are interpreted as a logical system, the semantic equivalence is typically intended to be logical equivalence. In this setting, the transformation rules can be used to derive logically equivalent expressions from any given expression. These derivations include as special cases (1) the problem of simplifying expressions and (2) the problem of deciding whether a given expression is equivalent to an expression in the distinguished subset, typically interpreted as the subset of logical axioms.

The set of axioms may be empty, a nonempty finite set, a countably infinite set, or given by axiom schemata. A formal grammar recursively defines the expressions and well-formed formulas (wffs) of the language. In addition a semantics is given which defines truth and valuations (or interpretations). It allows us to determine which wffs are valid, that is, are theorems.

The language of a propositional calculus consists of (1) a set of primitive symbols, variously referred to as atomic formulas, placeholders, proposition letters, or variables, and (2) a set of operator symbols, variously interpreted as logical operators or logical connectives. A well-formed formula (wff) is any atomic formula or any formula that can be built up from atomic formulas by means of operator symbols.

The following outlines a standard propositional calculus. Many different formulations exist which are all more or less equivalent but differ in (1) their language, that is, the particular collection of primitive symbols and operator symbols, (2) the set of axioms, or distinguished formulas, and (3) the set of transformation rules that are available.


In abstract algebra, a Boolean algebra is an algebraic structure (a collection of elements and operations on them obeying defining axioms) that captures essential properties of both set operations and logic operations. Specifically, it deals with the set operations of intersection, union, complement; and the logic operations of AND, OR, NOT.

For example, the logical assertion that a statement a and its negation ¬a cannot both be true,

Boolean lattice of subsets
parallels the set-theory assertion that a subset A and its complement AC have empty intersection,


Because truth values can be represented as binary numbers or as voltage levels in logic circuits, the parallel extends to these as well. Thus the theory of Boolean algebras has many practical applications in electrical engineering and computer science, as well as in mathematical logic.

A Boolean algebra is also called a Boolean lattice. The connection to lattices (special partially ordered sets) is suggested by the parallel between set inclusion, A ⊆ B, and ordering, a ≤ b. Consider the lattice of all subsets of {x,y,z}, ordered by set inclusion. This Boolean lattice is a partially ordered set in which, say, {x} ≤ {x,y}. Any two lattice elements, say p = {x,y} and q = {y,z}, have a least upper bound, here {x,y,z}, and a greatest lower bound, here {y}. Suggestively, the least upper bound (or join or supremum) is denoted by the same symbol as logical OR, p∨q; and the greatest lower bound (or meet or infimum) is denoted by same symbol as logical AND, p∧q.

The lattice interpretation helps in generalizing to Heyting algebras, which are Boolean algebras freed from the restriction that either a statement or its negation must be true. Heyting algebras correspond to intuitionist (constructivist) logic just as Boolean algebras correspond to classical logic.

Answer 2

In formal logic, mathematics and computer science, Boolean algebras, or Boolean lattices, are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set-theoretic operations intersection, union and complement.

Calculus is branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit-the notion of tending toward, or approaching, an ultimate value. The English physicist Isaac Newton and the German mathematician GW Leibniz, working independently, developed the calculus during the 17th cent.

Answer 3

Good question, seems to me sentential calculus would be the operations used in boolean algebra.

Answer 4

Boolean algebra and Sentential calculus?
I dont have the slightest clue to what ur asking about....
But learning these two i hope can bring world peace and serenity in our lives....
Study hard and do put to good use everything u learn for the benefit of all...
A million or more kids dont even get to study the alphabet...
Goodluck...

Answer 5

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