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Text of the page (random words):
vant 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 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 the following is therefore an equivalent definition a boolean algebra is a complemented distributive lattice the section on axiomatization lists other axiomatizations any of which can be made the basis of an equivalent definition representable boolean algebras edit although every concrete boolean algebra is a boolean algebra not every boolean algebra need be concrete let n be a square free positive integer one not divisible by the square of an integer for example 30 but not 12 the operations of greatest common divisor least common multiple and division into n that is x n x can be shown to satisfy all the boolean laws when their arguments range over the positive divisors of n hence those divisors form a boolean algebra these divisors are not subsets of a set making the divisors of n a boolean algebra that is not concrete according to our definitions however if each divisor of n is represented by the set of its prime factors this nonconcrete boolean algebra is isomorphic to the concrete boolean algebra consisting of all sets of prime factors of n with union corresponding to least common multiple intersection to greatest common divisor and complement to division into n so this example while not technically concrete is at least morally concrete via this representation called an isomorphism this example is an instance of the following notion a boolean algebra is called representable when it is isomorphic to a concrete boolean algebra the next question is answered positively as follows every boolean algebra is representable that is up to isomorphism abstract and concrete boolean algebras are the same thing this result depends on the boolean prime ideal theorem a choice principle slightly weaker than the axiom of choice this strong relationship implies a weaker result strengthening the observation in the previous subsection to the following easy consequence of representability the laws satisfied by all boolean algebras coincide with those satisfied by the prototypical boolean algebra it is weaker in the sense that it does not of itself imply representability boolean algebras are special here for example a relation algebra is a boolean algebra with additional structure but it is not the case that every relation algebra is representable in the sense appropriate to relation algebras axiomatizing boolean algebra edit main articles axiomatization of boolean algebras and boolean algebras canonically defined the above definition of an abstract boolean algebra as a set together with operations satisfying the boolean laws raises the question of what those laws are a simplistic answer is all boolean laws which can be defined as all equations that hold for the boolean algebra of 0 and 1 however since there are infinitely many such laws this is not a satisfactory answer in practice leading to the question of it suffices to require only finitely many laws to hold in the case of boolean algebras the answer is yes the finitely many equations listed above are sufficient thus boolean algebra is said to be finitely axiomatizable or finitely based moreover the number of equations needed can be further reduced to begin with some of the above laws are implied by some of the others a sufficient subset of the above laws consists of the pairs of associativity commutativity and absorption laws distributivity of over or the other distributivity law one suffices and the two complement laws in fact this is the traditional axiomatization of boolean algebra as a complemented distributive lattice by introducing additional laws not listed above it becomes possible to shorten the list of needed equations yet further for instance with the vertical bar representing the sheffer stroke operation the single axiom a b c a a c a c displaystyle a mid b mid c mid a mid a mid c mid a c is sufficient to completely axiomatize boolean algebra it is also possible to find longer single axioms using more conventional operations see minimal axioms for boolean algebra 32 propositional logic edit main article propositional calculus propositional logic is a logical system that is intimately connected to boolean algebra 5 many syntactic concepts of boolean algebra carry over to propositional logic with only minor changes in notation and terminology while the semantics of propositional logic are defined via boolean algebras in a way that the tautologies theorems of propositional logic correspond to equational theorems of boolean algebra syntactically every boolean term corresponds to a propositional formula of propositional logic in this translation between boolean algebra and propositional logic boolean variables x y become propositional variables or atoms p q boolean terms such as x y become propositional formulas p q 0 becomes false or and 1 becomes true or 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 the semantics of propositional logic rely on truth assignments the essential idea of a truth assignment is that the propositional variables are mapped to elements of a fixed boolean algebra and then the truth value of a propositional formula using these letters is the element of the boolean algebra that is obtained by computing the value of the boolean term corresponding to the formula 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 these semantics permit a translation between tautologies of propositional logic and equational theorems of boolean algebra every tautology φ of propositional logic can be expressed as the boolean equation φ 1 which will be a theorem of boolean algebra conversely every theorem φ ψ of boolean algebra corresponds to the tautologies φ ψ φ ψ and φ ψ φ ψ if is in the language these last tautologies can also be written as φ ψ ψ φ or as two separate theorems φ ψ and ψ φ if is available then the single tautology φ ψ can be used applications edit one motivating application of propositional calculus is the analysis of propositions and deductive arguments in natural language 33 whereas the proposition if x 3 then x 1 4 depends on the meanings of such symbols as and 1 the proposition if x 3 then x 3 does not it is true merely by virtue of its structure and remains true whether x 3 is replaced by x 4 or the moon is made of green cheese the generic or abstract form of this tautology is if p then p or in the language of boolean algebra p p citation needed replacing p by x 3 or any other proposition is called instantiation of p by that proposition the result of instantiating p in an abstract proposition is called an instance of the proposition thus x 3 x 3 is a tautology by virtue of being an instance of the abstract tautology p p all occurrences of the instantiated variable must be instantiated with the same proposition to avoid such nonsense as p x 3 or x 3 x 4 propositional calculus restricts attention to abstract propositions those built up from propositional variables using boolean operations instantiation is still possible within propositional calculus but only by instantiating propositional variables by abstract propositions such as instantiating q by q p in p q p to yield the instance p q p p the availability of instantiation as part of the machinery of propositional calculus avoids the need for metavariables within the language of propositional calculus since ordinary propositional variables can be considered within the language to denote arbitrary propositions the metavariables themselves are outside the reach of instantiation not being part of the language of propositional calculus but rather part of the same language for talking about it that this sentence is written in where there is a need to be able to distinguish propositional variables and their instantiations as being distinct syntactic entities deductive systems for propositional logic edit an axiomatization of propositional calculus is a set of tautologies called axioms and one or more inference rules for producing new tautologies from old a proof in an axiom system a is a finite nonempty sequence of propositions each of which is either an instance of an axiom of a or follows by some rule of a from propositions appearing earlier in the proof thereby disallowing circular reasoning the last proposition is the theorem proved by the proof every nonempty initial segment of a proof is itself a proof whence every proposition in a proof is itself a theorem an axiomatization is sound when every theorem is a tautology and complete when every tautology is a theorem 34 sequent calculus edit main article sequent calculus 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 a b 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 entailment differs from implication in that whereas the latter is a binary operation that returns a value in a boolean algebra the former is a binary relation which either holds or does not hold in this sense entailment is an external form of implication meaning external to the boolean algebra thinking of the reader of the sequent as also being external and interpreting and comparing antecedents and succedents in some boolean algebra the natural interpretation of is as in the partial order of the boolean algebra defined by x y just when x y y this ability to mix external implication and internal implication in the one logic is among the essential differences between sequent calculus and propositional calculus 35 applications edit boolean algebra as the calculus of two values is fundamental to computer circuits computer programming and mathematical logic and is also used in other areas of mathematics such as set theory and statistics 5 computers edit in the early 20th century several electrical engineers who 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 today all modern general purpose computers perform their functions using two value boolean logic that is their electrical circuits are a physical manifestation of two value boolean logic they achieve this in various ways as voltages on wires in high speed circuits and capacitive storage devices as orientations of a magnetic domain in ferromagnetic storage devices as holes in punched cards or paper tape and so on some early computers used decimal circuits or mechanisms instead of two valued logic circuits 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 computers use two value boolean circuits for the above reasons the most common computer architectures use ordered sequences of boolean values called bits of 32 or 64 values e g 01101000110101100101010101001011 when programming in machine code assembly language and certain other programming languages programmers work with the low level digital structure of the data registers these registers operate on voltages where zero volts represents boolean 0 and a reference voltage often 5 v 3 3 v or 1 8 v represents boolean 1 such languages support both numeric operations and logical operations in this context numeric means that the computer treats sequences of bits as binary numbers base two numbers and executes arithmetic operations like add subtract multiply or divide logical refers to the boolean logical operations of disjunction conjunction and negation between two sequences of bits in which each bit in one sequence is simply compared to its counterpart in the other sequence programmers therefore have the option of working in and applying the rules of either numeric algebra or boolean algebra as needed a core differentiating feature between these families of operations is the existence of the carry operation in the first but not the second two valued logic edit other areas where two values is a good choice are the law and mathematics 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 limiting 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 organiz...