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Algebraic Logic andAlgebraic Mathematics

ContentsArticlesAlgebraic Logic 1

Boolean algebra 1Algebraic logic 16ukasiewicz logic 19Intuitionistic logic 21Mathematical logic 27Heyting arithmetic 41Metatheory 42Metalogic 43

Quantum Logics and Quantum Computers 46Many-valued logic 46Quantum logic 50Quantum computer 57

Abstract Algebra 68Abstract algebra 68Universal algebra 72Heyting algebra 77MV-algebra 87Group algebra 89Lie algebra 92Affine Lie algebra 99Lie group 101Algebroid 111

Quantum Algebra and Geometry 113Quantum affine algebra 113Clifford algebra 114Von Neumann algebra 126C*-algebra 136KacMoody algebra 141Hopf algebra 143Quantum group 150

Group representation 158Unitary representation 161Representation theory of the Lorentz group 163Stonevon Neumann theorem 172PeterWeyl theorem 177Quasi-Hopf algebra 180Quasitriangular Hopf algebra 181Ribbon Hopf algebra 182Quasi-triangular Quasi-Hopf algebra 183Quantum inverse scattering method 184Yangian 185Exterior algebra 187Superalgebra 202Supergroup 205Noncommutative quantum field theory 206Standard Model 208Noncommutative standard model 218Noncommutative geometry 220

Algebraic Geometry and Analytic Geometry 224Algebraic geometry 224List of algebraic geometry topics 236Duality 241Universal algebraic geometry 248Grothendieck topology 249GrothendieckHirzebruchRiemannRoch theorem 256Algebraic geometry and analytic geometry 258Differential geometry 260

Algebraic Topology, Group Theory and Groupoids 266Algebraic topology 266Groupoid 270Group theory 276Abelian group 284Galois group 290Grothendieck group 291Esquisse d'un Programme 296Galois theory 299

Grothendieck's Galois theory 305Galois cohomology 306Homological algebra 307Homology theory 311Homotopical algebra 314De Rham cohomology 315Crystalline cohomology 318Cohomology 322K-theory 326Algebraic K-theory 328Topological K-theory 337

Category Theory and Categorical Logic 339Category theory 339Category 347Glossary of category theory 352Dual 354Abelian category 355Functor 358Yoneda lemma 362Limit 365Adjoint functors 374Natural transformations 388Variety 393Domain theory 395Enriched category 400Topos 404Descent 410Stack 412Categorical logic 417Timeline of category theory and related mathematics 420List of important publications in mathematics 437

Higher Dimensional Algebras 462Higher-dimensional algebra 462Higher category theory 465Duality 467

ReferencesArticle Sources and Contributors 476Image Sources, Licenses and Contributors 483

Article LicensesLicense 485

1Algebraic Logic

Boolean algebraIn mathematics and mathematical logic, Boolean algebra is the subarea of algebra in which the values of thevariables are the truth values true and false, usually denoted 1 and 0 respectively. Instead of elementary algebrawhere the values of the variables are numbers, and the main operations are addition and multiplication, the mainoperations of Boolean algebra are the conjunction and, denoted , the disjunction or, denoted , and the negationnot, denoted .Boolean algebra was introduced in 1854 by George Boole in his book An Investigation of the Laws of Thought.According to Huntington the term "Boolean algebra" was first suggested by Sheffer in 1913.[1]

Boolean algebra has been fundamental in the development of computer science and digital logic. It is also used in settheory and statistics.

HistoryBoole's algebra predated the modern developments in abstract algebra and mathematical logic; it is however seen asconnected to the origins of both fields. In an abstract setting, Boolean algebra was perfected in the late 19th centuryby Jevons, Schrder, Huntington, and others until it reached the modern conception of an (abstract) mathematicalstructure. For example, the empirical observation that one can manipulate expressions in the algebra of sets bytranslating them into expressions in Boole's algebra is explained in modern terms by saying that the algebra of sets isa Boolean algebra (note the indefinite article). In fact, M. H. Stone proved in 1936 that every Boolean algebra isisomorphic to a field of sets.In the 1930s, while studying switching circuits, Claude Shannon observed that one could also apply the rules ofBoole's algebra in this setting, and he introduced switching algebra as a way to analyze and design circuits byalgebraic means in terms of logic gates. Shannon already had at his disposal the abstract mathematical apparatus,thus he cast his switching algebra as the two-element Boolean algebra. In circuit engineering settings today, there islittle need to consider other Boolean algebras, thus "switching algebra" and "Boolean algebra" are often usedinterchangeably.[2] Efficient implementation of Boolean functions is a fundamental problem in the design ofcombinatorial logic circuits. Modern electronic design automation tools for VLSI circuits often rely on an efficientrepresentation of Boolean functions known as (reduced ordered) binary decision diagrams (BDD) for logic synthesisand formal verification.Logic sentences that can be expressed in classical propositional calculus have an equivalent expression in Booleanalgebra. Thus, Boolean logic is sometimes used to denote propositional calculus performed in this way. Booleanalgebra is not sufficient to capture logic formulas using quantifiers, like those from first order logic. Although thedevelopment of mathematical logic did not follow Boole's program, the connection between his algebra and logicwas later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many otherlogics. The problem of determining whether the variables of a given Boolean (propositional) formula can be assignedin such a way as to make the formula evaluate to true is called the Boolean satisfiability problem (SAT), and is ofimportance to theoretical computer science, being the first problem shown to be NP-complete. The closely relatedmodel of computation known as a Boolean circuit relates time complexity (of an algorithm) to circuit complexity.

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ValuesWhereas in elementary algebra expressions denote mainly numbers, in Boolean algebra they denote the truth valuesfalse and true. These values are represented with the bits (or binary digits) being 0 and 1. They do not behave likethe integers 0 and 1, for which 1 + 1 = 2, but may be identified with the elements of the two-element field GF(2), forwhich 1 + 1 = 0 with + serving as the Boolean operation XOR.Boolean algebra also deals with functions which have their values in the set {0, 1}. A sequence of bits is acommonly used such function. Another common example is the subsets of a set E: to a subset F of E is associatedthe indicator function that takes the value 1 on F and 0 outside F.As with elementary algebra, the purely equational part of the theory may be developed without considering explicitvalues for the variables.[3]

Operations

Basic operationsThe basic operations of Boolean algebra are the following ones: And (conjunction), denoted xy (sometimes x AND y or Kxy), satisfies xy = 1 if x = y = 1 and xy = 0 otherwise. Or (disjunction), denoted xy (sometimes x OR y or Axy), satisfies xy = 0 if x = y = 0 and xy = 1 otherwise. Not (negation), denoted x (sometimes NOT x, Nx or !x), satisfies x = 0 if x = 1 and x = 1 if x = 0.If the truth values 0 and 1 are interpreted as integers, these operation may be expressed with the ordinary operationsof the arithmetic:

xy = xy,xy = x + y - xy,x = 1 - x.

Alternatively the values of xy, xy, and x can be expressed by tabulating their values with truth tables as follows.

x y xy xy

0 0 0 0

1 0 0 1

0 1 0 1

1 1 1 1

+ Figure 1. Truth tables

One may consider that only the negation and one of the two other operations are basic, because of the followingidentities that allow to define the conjunction in terms of the negation and the disjunction, and vice versa:

x y = (x y)x y = (x y)

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Derived operationsWe have so far seen three Boolean operations. We referred to these as basic, meaning that they can be taken as abasis for other Boolean operations that can be built up from them by composition, the manner in which operationsare combined or compounded. Here are some examples of operations composed from the basic operations.

xy = (xy)

xy = (xy)(xy)

xy = (xy)

These definitions give rise to the following truth tables giving the values of these operations for all four possibleinputs.

x y xy xy xy

0 0 1 0 1

1 0 0 1 0

0 1 1 1 0

1 1 1 0 1

The first operation, xy, or Cxy, is called material implication. If x is true then the value of xy is taken to bethat of y. But if x is false then we ignore the value of y; however we must return some truth value and there are onlytwo choices, so we choose the value that entails less, namely true. (Relevance logic addresses this by viewing animplication with a false premise as something other than either true or false.)The second operation, xy, or Jxy, is called exclusive or to distinguish it from disjunction as the inclusive kind. Itexcludes the possibility of both x and y. Defined in terms of arithmetic it is addition mod2 where 1+1 =0.The third operation, the complement of exclusive or, is equivalence or Boolean equality: xy, or Exy, is true justwhen x and y have the same value. Hence xy as its complement can be understood as xy, being true just when xand y are different. Its counterpart in arithmetic mod 2 is x + y + 1.

LawsA law of Boolean algebra is an identity such as x(yz) = (xy)z between two Boolean terms, where a Booleanterm is defined as an expression built up from variables and the constants 0 and 1 using the operations , , and .The concept can be extended to terms involving other Boolean operations such as , , and , but such extensionsare unnecessary for the purposes to which the laws are put. Such purposes include the definition of a Booleanalgebra as any model of the Boolean laws, and as a means for deriving new laws from old as in the derivation ofx(yz) = x(zy) from yz = zy as treated in the section on axiomatization.

Monotone lawsBoolean algebra satisfies many of the same laws as ordinary algebra when we match up with addition and withmultiplication. In particular the following laws are common to both kinds of algebra:

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(Associativity of ) x(yz) = (xy)z

(Associativity of ) x(yz) = (xy)z

(Commutativity of ) xy = yx

(Commutativity of ) xy = yx

(Distributivity of over ) x(yz) = (xy)(xz)

(Identity for ) x0 = x

(Identity for ) x1 = x

(Annihilator for ) x0 = 0

Boolean algebra however obeys some additional laws, in particular the following:

(Idempotence of ) xx = x

(Idempotence of ) xx = x

(Absorption 1) x(xy) = x

(Absorption 2) x(xy) = x

(Distributivity of over ) x(yz) = (xy)(xz)

(Annihilator for ) x1 = 1

A consequence of the first of these laws is 11 = 1, which is false in ordinary algebra, where 1+1 = 2. Taking x = 2in the second law shows that it is not an ordinary algebra law either, since 22 = 4. The remaining four laws can befalsified in ordinary algebra by taking all variables to be 1, for example in Absorption Law 1 the left hand side is1(1+1) = 2 while the right hand side is 1, and so on.All of the laws treated so far have been for conjunction and disjunction. These operations have the property thatchanging either argument either leaves the output unchanged or the output changes in the same way as the input.Equivalently, changing any variable from 0 to 1 never results in the output changing from 1 to 0. Operations withthis property are said to be monotone. Thus the axioms so far have all been for monotonic Boolean logic.Nonmonotonicity enters via complement as follows.

Nonmonotone lawsThe complement operation is defined by the following two laws.

(Complementation 1) xx = 0

(Complementation 2) xx = 1.

All properties of negation including the laws below follow from the above two laws alone.In both ordinary and Boolean algebra, negation works by exchanging pairs of elements, whence in both algebras itsatisfies the double negation law (also called involution law)

(Double negation) x = x.

But whereas ordinary algebra satisfies the two laws

(x)(y) = xy

(x) + (y) = (x + y),

Boolean algebra satisfies De Morgan's laws,

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(De Morgan 1) (x)(y) = (xy)

(De Morgan 2) (x)(y) = (xy).

CompletenessAt this point we can now claim to have defined Boolean algebra, in the sense that the laws we have listed up to nowentail the rest of the subject. The laws Complementation 1 and 2, together with the monotone laws, suffice for thispurpose and can therefore be taken as one possible complete set of laws or axiomatization of Boolean algebra. Everylaw of Boolean algebra follows logically from these axioms. Furthermore Boolean algebras can then be defined asthe models of these axioms as treated in the section thereon.To clarify, writing down further laws of Boolean algebra cannot give rise to any new consequences of these axioms,nor can it rule out any model of them. Had we stopped listing laws too soon, there would have been Boolean lawsthat did not follow from those on our list, and moreover there would have been models of the listed laws that werenot Boolean algebras.This axiomatization is by no means the only one, or even necessarily the most natural given that we did not payattention to whether some of the axioms followed from others but simply chose to stop when we noticed we hadenough laws, treated further in the section on axiomatizations. Or the intermediate notion of axiom can besidestepped altogether by defining a Boolean law directly as any tautology, understood as an equation that holds forall values of its variables over 0 and 1. All these definitions of Boolean algebra can be shown to be equivalent.Boolean algebra has the interesting property that x = y can be proved from any non-tautology. This is because thesubstitution instance of any non-tautology obtained by instantiating its variables with constants 0 or 1 so as towitness its non-tautologyhood reduces by equational reasoning to 0 = 1. For example the non-tautologyhood of xy= x is witnessed by x = 1 and y = 0 and so taking this as an axiom would allow us to infer 10 = 1 as a substitutioninstance of the axiom and hence 0 = 1. We can then show x = y by the reasoning x = x1 = x0 = 0 = 1 = y1 = y0= y.

Duality principleThere is nothing magical about the choice of symbols for the values of Boolean algebra. We could rename 0 and 1 tosay and , and as long as we did so consistently throughout it would still be Boolean algebra, albeit with someobvious cosmetic differences.But suppose we rename 0 and 1 to 1 and 0 respectively. Then it would still be Boolean algebra, and moreoveroperating on the same values. However it would not be identical to our original Boolean algebra because now wefind behaving the way used to do and vice versa. So there are still some cosmetic differences to show that we'vebeen fiddling with the notation, despite the fact that we're still using 0s and 1s.But if in addition to interchanging the names of the values we also interchange the names of the two binaryoperations, now there is no trace of what we have done. The end product is completely indistinguishable from whatwe started with. We might notice that the columns for xy and xy in the truth tables had changed places, but thatswitch is immaterial.When values and operations can be paired up in a way that leaves everything important unchanged when all pairs areswitched simultaneously, we call the members of each pair dual to each other. Thus 0 and 1 are dual, and and are dual. The Duality Principle, also called De Morgan duality, asserts that Boolean algebra is unchanged when alldual pairs are interchanged.One change we did not need to make as part of this interchange was to complement. We say that complement is a self-dual operation. The identity or do-nothing operation x (copy the input to the output) is also self-dual. A more complicated example of a self-dual operation is (xy) (yz) (zx). It can be shown that self-dual operations must

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take an odd number of arguments; thus there can be no self-dual binary operation.The principle of duality can be explained from a group theory perspective by fact that there are exactly fourfunctions that are one-to-one mappings (automorphisms) of the set of Boolean polynomials back to itself: theidentity function, the complement function, the dual function and the contradual function (complemented dual).These four functions form a group under function composition, isomorphic to the Klein four-group, acting on the setof Boolean polynomials.

Diagrammatic representations

Venn diagramsA Venn diagram[4] is a representation of a Boolean operation using shaded overlapping regions. There is one regionfor each variable, all circular in the examples here. The interior and exterior of region x corresponds respectively tothe values 1 (true) and 0 (false) for variable x. The shading indicates the value of the operation for each combinationof regions, with dark denoting 1 and light 0 (some authors use the opposite convention).The three Venn diagrams in the figure below represent respectively conjunction xy, disjunction xy, andcomplement x.

Figure 2. Venn diagrams for conjunction, disjunction, and complement

For conjunction, the region inside both circles is shaded to indicate that xy is 1 when both variables are 1. The otherregions are left unshaded to indicate that xy is 0 for the other three combinations.The second diagram represents disjunction xy by shading those regions that lie inside either or both circles. Thethird diagram represents complement x by shading the region not inside the circle.While we have not shown the Venn diagrams for the constants 0 and 1, they are trivial, being respectively a whitebox and a dark box, neither one containing a circle. However we could put a circle for x in those boxes, in whichcase each would denote a function of one argument, x, which returns the same value independently of x, called aconstant function. As far as their outputs are concerned, constants and constant functions are indistinguishable; thedifference is that a constant takes no arguments, called a zeroary or nullary operation, while a constant functiontakes one argument, which it ignores, and is a unary operation.Venn diagrams are helpful in visualizing laws. The commutativity laws for and can be seen from the symmetryof the diagrams: a binary operation that was not commutative would not have a symmetric diagram becauseinterchanging x and y would have the effect of reflecting the diagram horizontally and any failure of commutativitywould then appear as a failure of symmetry.Idempotence of and can be visualized by sliding the two circles together and noting that the shaded area thenbecomes the whole circle, for both and .

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To see the first absorption law, x(xy) = x, start with the diagram in the middle for xy and note that the portion ofthe shaded area in common with the x circle is the whole of the x circle. For the second absorption law, x(xy) = x,start with the left diagram for xy and note that shading the whole of the x circle results in just the x circle beingshaded, since the previous shading was inside the x circle.The double negation law can be seen by complementing the shading in the third diagram for x, which shades the xcircle.To visualize the first De Morgan's law, (x)(y) = (xy), start with the middle diagram for xy and complementits shading so that only the region outside both circles is shaded, which is what the right hand side of the lawdescribes. The result is the same as if we shaded that region which is both outside the x circle and outside the ycircle, i.e. the conjunction of their exteriors, which is what the left hand side of the law describes.The second De Morgan's law, (x)(y) = (xy), works the same way with the two diagrams interchanged.The first complement law, xx = 0, says that the interior and exterior of the x circle have no overlap. The secondcomplement law, xx = 1, says that everything is either inside or outside the x circle.

Digital logic gatesDigital logic is the application of the Boolean algebra of 0 and 1 to electronic hardware consisting of logic gatesconnected to form a circuit diagram. Each gate implements a Boolean operation, and is depicted schematically by ashape indicating the operation. The shapes associated with the gates for conjunction (AND-gates), disjunction(OR-gates), and complement (inverters) are as follows.

The lines on the left of each gate represent input wires or ports. The value of the input is represented by a voltage onthe lead. For so-called "active-high" logic 0 is represented by a voltage close to zero or "ground" while 1 isrepresented by a voltage close to the supply voltage; active-low reverses this. The line on the right of each gaterepresents the output port, which normally follows the same voltage conventions as the input ports.Complement is implemented with an inverter gate. The triangle denotes the operation that simply copies the input tothe output; the small circle on the output denotes the actual inversion complementing the input. The convention ofputting such a circle on any port means that the signal passing through this port is complemented on the waythrough, whether it is an input or output port.There being eight ways of labeling the three ports of an AND-gate or OR-gate with inverters, this convention gives awide range of possible Boolean operations realized as such gates so decorated. Not all combinations are distincthowever: any labeling of AND-gate ports with inverters realizes the same Boolean operation as the opposite labelingof OR-gate ports (a given port of the AND-gate is labeled with an inverter if and only if the corresponding port ofthe OR-gate is not so labeled). This follows from De Morgan's laws.If we complement all ports on every gate, and interchange AND-gates and OR-gates, as in Figure 4 below, we endup with the same operations as we started with, illustrating both De Morgan's laws and the Duality Principle. Notethat we did not need to change the triangle part of the inverter, illustrating self-duality for complement.

Boolean algebra 8

Because of the pairwise identification of gates via the Duality Principle, even though 16 schematic symbols can bemanufactured from the two basic binary gates AND and OR by furnishing their ports with inverters (circles), theyonly represent eight Boolean operations, namely those operations with an odd number of ones in their truth table.Altogether there are 16 binary Boolean operations, the other eight being those with an even number of ones in theirtruth table, namely the following. The constant 0, viewed as a binary operation that ignores both its inputs, has noones, the six operations x, y, x, y (as binary operations that ignore one input), xy, and xy have two ones, and theconstant 1 has four ones.

Boolean algebrasThe term "algebra" denotes both a subject, namely the subject of algebra, and an object, namely an algebraicstructure. Whereas the foregoing has addressed the subject of Boolean algebra, this section deals with mathematicalobjects called Boolean algebras, defined in full generality as any model of the Boolean laws. We begin with a specialcase of the notion definable without reference to the laws, namely concrete Boolean algebras, and then give theformal definition of the general notion.

Concrete Boolean algebrasA concrete Boolean algebra or field of sets is any nonempty set of subsets of a given set X closed under the setoperations of union, intersection, and complement relative to X.(As an aside, historically X itself was required to be nonempty as well to exclude the degenerate or one-elementBoolean algebra, which is the one exception to the rule that all Boolean algebras satisfy the same equations since thedegenerate algebra satisfies every equation. However this exclusion conflicts with the preferred purely equationaldefinition of "Boolean algebra," there being no way to rule out the one-element algebra using only equations01does not count, being a negated equation. Hence modern authors allow the degenerate Boolean algebra and let X beempty.)Example 1. The power set 2X of X, consisting of all subsets of X. Here X may be any set: empty, finite, infinite, oreven uncountable.Example 2. The empty set and X. This two-element algebra shows that a concrete Boolean algebra can be finite evenwhen it consists of subsets of an infinite set. It can be seen that every field of subsets of X must contain the empty setand X. Hence no smaller example is possible, other than the degenerate algebra obtained by taking X to be empty soas to make the empty set and X coincide.Example 3. The set of finite and cofinite sets of integers, where a cofinite set is one omitting only finitely manyintegers. This is clearly closed under complement, and is closed under union because the union of a cofinite set withany set is cofinite, while the union of two finite sets is finite. Intersection behaves like union with "finite" and"cofinite" interchanged.Example 4. For a less trivial example of the point made by Example 2, consider a Venn diagram formed by n closed curves partitioning the diagram into 2n regions, and let X be the (infinite) set of all points in the plane not on any curve but somewhere within the diagram. The interior of each region is thus an infinite subset of X, and every point in X is in exactly one region. Then the set of all 22n possible unions of regions (including the empty set obtained as the union of the empty set of regions and X obtained as the union of all 2n regions) is closed under union,

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intersection, and complement relative to X and therefore forms a concrete Boolean algebra. Again we have finitelymany subsets of an infinite set forming a concrete Boolean algebra, with Example 2 arising as the case n = 0 of nocurves.

Subsets as bit vectorsA subset Y of X can be identified with an indexed family of bits with index set X, with the bit indexed by x X being1 or 0 according to whether or not x Y. (This is the so-called characteristic function notion of a subset.) Forexample a 32-bit computer word consists of 32 bits indexed by the set {0,1,2,,31}, with 0 and 31 indexing the lowand high order bits respectively. For a smaller example, if X = {a,b,c} where a, b, c are viewed as bit positions inthat order from left to right, the eight subsets {}, {c}, {b}, {b,c}, {a}, {a,c}, {a,b}, and {a,b,c} of X can be identifiedwith the respective bit vectors 000, 001, 010, 011, 100, 101, 110, and 111. Bit vectors indexed by the set of naturalnumbers are infinite sequences of bits, while those indexed by the reals in the unit interval [0,1] are packed toodensely to be able to write conventionally but nonetheless form well-defined indexed families (imagine coloringevery point of the interval [0,1] either black or white independently; the black points then form an arbitrary subset of[0,1]).From this bit vector viewpoint, a concrete Boolean algebra can be defined equivalently as a nonempty set of bitvectors all of the same length (more generally, indexed by the same set) and closed under the bit vector operations ofbitwise , , and , as in 10100110 = 0010, 10100110 = 1110, and 1010 = 0101, the bit vector realizations ofintersection, union, and complement respectively.

The prototypical Boolean algebraThe set {0,1} and its Boolean operations as treated above can be understood as the special case of bit vectors oflength one, which by the identification of bit vectors with subsets can also be understood as the two subsets of aone-element set. We call this the prototypical Boolean algebra, justified by the following observation.

The laws satisfied by all nondegenerate concrete Boolean algebras coincide with those satisfied by theprototypical Boolean algebra.

This observation is easily proved as follows. Certainly any law satisfied by all concrete Boolean algebras is satisfiedby the prototypical one since it is concrete. Conversely any law that fails for some concrete Boolean algebra musthave failed at a particular bit position, in which case that position by itself furnishes a one-bit counterexample to thatlaw. Nondegeneracy ensures the existence of at least one bit position because there is only one empty bit vector.The final goal of the next section can be understood as eliminating "concrete" from the above observation. We shallhowever reach that goal via the surprisingly stronger observation that, up to isomorphism, all Boolean algebras areconcrete.

Boolean algebras: the definitionThe Boolean algebras we have seen so far have all been concrete, consisting of bit vectors or equivalently of subsetsof some set. Such a Boolean algebra consists of a set and operations on that set which can be shown to satisfy thelaws of Boolean algebra.Instead of showing that the Boolean laws are satisfied, we can instead postulate a set X, two binary operations on X,and one unary operation, and require that those operations satisfy the laws of Boolean algebra. The elements of Xneed not be bit vectors or subsets but can be anything at all. This leads to the more general abstract definition.

A Boolean algebra is any set with binary operations and and a unary operation thereon satisfying theBoolean laws.

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

Boolean algebra 10

algebra is a Boolean algebra according to our definitions. This axiomatic definition of a Boolean algebra as a set andcertain 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, asufficient condition for an algebraic structure of this kind to satisfy all the Boolean laws is that it satisfy just thoseaxioms. 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 equivalentdefinition.

Representable Boolean algebrasAlthough every concrete Boolean algebra is a Boolean algebra, not every Boolean algebra need be concrete. Let n bea square-free positive integer, one not divisible by the square of an integer, for example 30 but not 12. Theoperations of greatest common divisor, least common multiple, and division into n (that is, x = n/x), can be shownto satisfy all the Boolean laws when their arguments range over the positive divisors of n. Hence those divisors forma Boolean algebra. These divisors are not subsets of a set, making the divisors of n a Boolean algebra that is notconcrete according to our definitions.However if we represent each divisor of n by the set of its prime factors, we find that this nonconcrete Booleanalgebra is isomorphic to the concrete Boolean algebra consisting of all sets of prime factors of n, with unioncorresponding 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 anisomorphism. 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 obvious 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 quite nontrivial resultdepends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of choice, and istreated in more detail in the article Stone's representation theorem for Boolean algebras. This strong relationshipimplies a weaker result strengthening the observation in the previous subsection to the following easy consequenceof 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, forexample a relation algebra is a Boolean algebra with additional structure but it is not the case that every relationalgebra is representable in the sense appropriate to relation algebras.

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Axiomatizing Boolean algebraThe above definition of an abstract Boolean algebra as a set and operations satisfying "the" Boolean laws raises thequestion, what are those laws? A simple-minded answer is "all Boolean laws," which can be defined as all equationsthat hold for the Boolean algebra of 0 and 1. Since there are infinitely many such laws this is not a terriblysatisfactory answer in practice, leading to the next question: does it suffice to require only finitely many laws tohold?In the case of Boolean algebras the answer is yes. In particular the finitely many equations we have listed abovesuffice. We say that Boolean algebra is finitely axiomatizable or finitely based.Can this list be made shorter yet? Again the answer is yes. To begin with, some of the above laws are implied bysome of the others. A sufficient subset of the above laws consists of the pairs of associativity, commutativity, andabsorption laws, distributivity of over (or the other distributivity lawone suffices), and the two complementlaws. 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 yet further. In 1933 EdwardHuntington showed that if the basic operations are taken to be xy and x, with xy considered a derived operation(e.g. via De Morgan's law in the form xy = (xy)), then the equation (xy)(xy) = x along with the twoequations expressing associativity and commutativity of completely axiomatized Boolean algebra. When the onlybasic operation is the binary NAND operation (xy), Stephen Wolfram has proposed in his book A New Kind ofScience the single axiom (((xy)z)(x((xz)x))) = z as a one-equation axiomatization of Boolean algebra, where forconvenience here xy denotes the NAND rather than the AND of x and y.

Propositional logicPropositional logic is a logical system that is intimately connected to Boolean algebra. Many syntactic concepts ofBoolean algebra carry over to propositional logic with only minor changes in notation and terminology, while thesemantics of propositional logic are defined via Boolean algebras in a way that the tautologies (theorems) ofpropositional logic correspond to equational theorems of Boolean algebra.Syntactically, every Boolean term corresponds to a propositional formula of propositional logic. In this translationbetween Boolean algebra and propositional logic, Boolean variables x,y become propositional variables (oratoms) P,Q,, Boolean terms such as xy become propositional formulas PQ, 0 becomes false or , and 1becomes true or T. It is convenient when referring to generic propositions to use Greek letters , , asmetavariables (variables outside the language of propositional calculus, used when talking about propositionalcalculus) to denote propositions.The semantics of propositional logic rely on truth assignments. The essential idea of a truth assignment is that thepropositional variables are mapped to elements of a fixed Boolean algebra, and then the truth value of apropositional formula using these letters is the element of the Boolean algebra that is obtained by computing thevalue of the Boolean term corresponding to the formula. In classical semantics, only the two-element Booleanalgebra is used, while in Boolean-valued semantics arbitrary Boolean algebras are considered. A tautology is apropositional formula that is assigned truth value 1 by every truth assignment of its propositional variables to anarbitrary 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 Booleanalgebra. Every tautology of propositional logic can be expressed as the Boolean equation = 1, which will be atheorem of Boolean algebra. Conversely every theorem = of Boolean algebra corresponds to the tautologies() () and () (). If is in the language these last tautologies can also be written as() (), or as two separate theorems and ; if is available then the single tautology can be used.

Boolean algebra 12

ApplicationsOne motivating application of propositional calculus is the analysis of propositions and deductive arguments innatural language. Whereas the proposition "if x = 3 then x+1 = 4" depends on the meanings of such symbols as + and1, 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 tautologyis "if P then P", or in the language of Boolean algebra, "P P".Replacing P by x = 3 or any other proposition is called instantiation of P by that proposition. The result ofinstantiating P in an abstract proposition is called an instance of the proposition. Thus "x = 3 x = 3" is a tautologyby virtue of being an instance of the abstract tautology "P P". All occurrences of the instantiated variable must beinstantiated 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 usingBoolean operations. Instantiation is still possible within propositional calculus, but only by instantiatingpropositional variables by abstract propositions, such as instantiating Q by QP in P(QP) to yield the instanceP((QP)P).(The availability of instantiation as part of the machinery of propositional calculus avoids the need for metavariableswithin the language of propositional calculus, since ordinary propositional variables can be considered within thelanguage to denote arbitrary propositions. The metavariables themselves are outside the reach of instantiation, notbeing part of the language of propositional calculus but rather part of the same language for talking about it that thissentence is written in, where we need to be able to distinguish propositional variables and their instantiations asbeing distinct syntactic entities.)

Deductive systems for propositional logicAn axiomatization of propositional calculus is a set of tautologies called axioms and one or more inference rules forproducing new tautologies from old. A proof in an axiom system A is a finite nonempty sequence of propositionseach of which is either an instance of an axiom of A or follows by some rule of A from propositions appearing earlierin the proof (thereby disallowing circular reasoning). The last proposition is the theorem proved by the proof. Everynonempty initial segment of a proof is itself a proof, whence every proposition in a proof is itself a theorem. Anaxiomatization is sound when every theorem is a tautology, and complete when every tautology is a theorem.

Sequent calculus

Propositional calculus is commonly organized as a Hilbert system, whose operations are just those of Booleanalgebra and whose theorems are Boolean tautologies, those Boolean terms equal to the Boolean constant 1. Anotherform is sequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists ofpropositions called sequents, such as AB, AC, A, BC,. The two halves of a sequent are called theantecedent and the succedent respectively. The customary metavariable denoting an antecedent or part thereof is ,and for a succedent ; thus ,A would denote a sequent whose succedent is a list and whose antecedent is alist with an additional proposition A appended after it. The antecedent is interpreted as the conjunction of itspropositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of thesuccedent by the antecedent.Entailment differs from implication in that whereas the latter is a binary operation that returns a value in a Booleanalgebra, the former is a binary relation which either holds or does not hold. In this sense entailment is an externalform of implication, meaning external to the Boolean algebra, thinking of the reader of the sequent as also beingexternal and interpreting and comparing antecedents and succedents in some Boolean algebra. The naturalinterpretation of is as in the partial order of the Boolean algebra defined by x y just when xy = y. This abilityto mix external implication and internal implication in the one logic is among the essential differencesbetween sequent calculus and propositional calculus.

Boolean algebra 13

Applications

Two-valued logicBoolean algebra as the calculus of two values is fundamental to digital logic, computer programming, andmathematical logic, and is also used in other areas of mathematics such as set theory and statistics.Digital logic codes its symbols in various ways: as voltages on wires in high-speed circuits and capacitive storagedevices, as orientations of a magnetic domain in ferromagnetic storage devices, as holes in punched cards or papertape, and so on. Now it is possible to code more than two symbols in any given medium. For example one might userespectively 0, 1, 2, and 3 volts to code a four-symbol alphabet on a wire, or holes of different sizes in a punchedcard. In practice however the tight constraints of high speed, small size, and low power combine to make noise amajor factor. This makes it hard to distinguish between symbols when there are many of them at a single site. Ratherthan attempting to distinguish between four voltages on one wire, digital designers have settled on two voltages perwire, high and low. To obtain four symbols one uses two wires, and so on.Programmers programming in machine code, assembly language, and other programming languages that expose thelow-level digital structure of the data registers operate on whatever symbols were chosen for the hardware,invariably bit vectors in modern computers for the above reasons. Such languages support both the numericoperations of addition, multiplication, etc. performed on words interpreted as integers, as well as the logicaloperations of disjunction, conjunction, etc. performed bit-wise on words interpreted as bit vectors. Programmerstherefore have the option of working in and applying the laws of either numeric algebra or Boolean algebra asneeded. A core differentiating feature is carry propagation with the former but not the latter.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 situationssuch as a court of law or theorem-based mathematics however it is deemed advantageous to frame questions so as toadmit a simple yes-or-no answeris the defendant guilty or not guilty, is the proposition true or falseand todisallow any other answer. However much of a straitjacket this might prove in practice for the respondent, theprinciple of the simple yes-no question has become a central feature of both judicial and mathematical logic, makingtwo-valued logic deserving of organization and study in its own right.A central concept of set theory is membership. Now an organization may permit multiple degrees of membership,such as novice, associate, and full. With sets however an element is either in or out. The candidates for membershipin a set work just like the wires in a digital computer: each candidate is either a member or a nonmember, just aseach wire is either high or low.Algebra being a fundamental tool in any area amenable to mathematical treatment, these considerations combine tomake the algebra of two values of fundamental importance to computer hardware, mathematical logic, and settheory.Two-valued logic can be extended to multi-valued logic, notably by replacing the Boolean domain {0,1} with theunit interval [0,1], in which case rather than only taking values 0 or 1, any value between and including 0 and 1 canbe assumed. Algebraically, negation (NOT) is replaced with 1x, conjunction (AND) is replaced withmultiplication ( ), and disjunction (OR) is defined via De Morgan's law. Interpreting these values as logical truthvalues yields a multi-valued logic, which forms the basis for fuzzy logic and probabilistic logic. In theseinterpretations, a value is interpreted as the "degree" of truth to what extent a proposition is true, or the probabilitythat the proposition is true.

Boolean algebra 14

Boolean operationsThe original application for Boolean operations was mathematical logic, where it combines the truth values, true orfalse, of individual formulas.Natural languages such as English have words for several Boolean operations, in particular conjunction (and),disjunction (or), negation (not), and implication (implies). But not is synonymous with and not. When used tocombine situational assertions such as "the block is on the table" and "cats drink milk," which naively are either trueor false, the meanings of these logical connectives often have the meaning of their logical counterparts. Howeverwith descriptions of behavior such as "Jim walked through the door", one starts to notice differences such as failureof commutativity, for example the conjunction of "Jim opened the door" with "Jim walked through the door" in thatorder is not equivalent to their conjunction in the other order, since and usually means and then in such cases.Questions can be similar: the order "Is the sky blue, and why is the sky blue?" makes more sense than the reverseorder. Conjunctive commands about behavior are like behavioral assertions, as in get dressed and go to school.Disjunctive commands such love me or leave me or fish or cut bait tend to be asymmetric via the implication thatone alternative is less preferable. Conjoined nouns such as tea and milk generally describe aggregation as with setunion while tea or milk is a choice. However context can reverse these senses, as in your choices are coffee and teawhich usually means the same as your choices are coffee or tea (alternatives). Double negation as in "I don't not likemilk" rarely means literally "I do like milk" but rather conveys some sort of hedging, as though to imply that there isa third possibility. "Not not P" can be loosely interpreted as "surely P", and although P necessarily implies "not notP" the converse is suspect in English, much as with intuitionistic logic. In view of the highly idiosyncratic usage ofconjunctions in natural languages, Boolean algebra cannot be considered a reliable framework for interpreting them.Boolean operations are used in digital logic to combine the bits carried on individual wires, thereby interpreting themover {0,1}. When a vector of n identical binary gates are used to combine two bit vectors each of n bits, theindividual bit operations can be understood collectively as a single operation on values from a Boolean algebra with2n elements.Naive set theory interprets Boolean operations as acting on subsets of a given set X. As we saw earlier this behaviorexactly parallels the coordinate-wise combinations of bit vectors, with the union of two sets corresponding to thedisjunction of two bit vectors and so on.The 256-element free Boolean algebra on three generators is deployed in computer displays based on raster graphics,which use bit blit to manipulate whole regions consisting of pixels, relying on Boolean operations to specify how thesource region should be combined with the destination, typically with the help of a third region called the mask.Modern video cards offer all 223=256 ternary operations for this purpose, with the choice of operation being aone-byte (8-bit) parameter. The constants SRC = 0xaa or 10101010, DST = 0xcc or 11001100, and MSK = 0xf0 or11110000 allow Boolean operations such as (SRC^DST)&MSK (meaning XOR the source and destination and thenAND the result with the mask) to be written directly as a constant denoting a byte calculated at compile time, 0x60in the (SRC^DST)&MSK example, 0x66 if just SRC^DST, etc. At run time the video card interprets the byte as theraster operation indicated by the original expression in a uniform way that requires remarkably little hardware andwhich takes time completely independent of the complexity of the expression.Solid modeling systems for computer aided design offer a variety of methods for building objects from other objects, combination by Boolean operations being one of them. In this method the space in which objects exist is understood as a set S of voxels (the three-dimensional analogue of pixels in two-dimensional graphics) and shapes are defined as subsets of S, allowing objects to be combined as sets via union, intersection, etc. One obvious use is in building a complex shape from simple shapes simply as the union of the latter. Another use is in sculpting understood as removal of material: any grinding, milling, routing, or drilling operation that can be performed with physical machinery on physical materials can be simulated on the computer with the Boolean operation xy or xy, which in set theory is set difference, remove the elements of y from those of x. Thus given two shapes one to be machined and the other the material to be removed, the result of machining the former to remove the latter is described simply

Boolean algebra 15

as their set difference.

Boolean searches

Search engine queries also employ Boolean logic. For this application, each web page on the Internet may beconsidered to be an "element" of a "set". The following examples use a syntax supported by Google.[5]

Doublequotes are used to combine whitespace-separated words into a single search term.[6]

Whitespace is used to specify logical AND, as it is the default operator for joining search terms:

"Search term 1" "Search term 2"

The OR keyword is used for logical OR:

"Search term 1" OR "Search term 2"

The minus sign is used for logical NOT (AND NOT):

"Search term 1" "Search term 2"

References[1] cf footnote on page 278: "* The name Boolean algebra (or Boolean "algebras") for the calculus originated by Boole, extended by Schrder,

and perfected by Whitehead seems to have been first suggested by Sheffer, in 1913" quoted from E. V. Huntington January 1933, "NEWSETS OF INDEPENDENT POSTULATES FOR THE ALGEBRA OF LOGIC, WITH SPECIAL REFERENCE TO WHITEHEAD ANDRUSSELL'S PRINCIPIA MATHEMATICA", http:/ / www. ams. org/ journals/ tran/ 1933-035-01/ S0002-9947-1933-1501684-X/S0002-9947-1933-1501684-X. pdf

[2] , online sample (http:/ / www. wiley. com/ college/ engin/ balabanian293512/ pdf/ ch02. pdf)[3][3] Halmos, Paul (1963). Lectures on Boolean Algebras. van Nostrand.[4] J. Venn, On the Diagrammatic and Mechanical Representation of Propositions and Reasonings, Philosophical Magazine and Journal of

Science, Series 5, vol. 10, No. 59, July 1880.[5] Not all search engines support the same query syntax. Additionally, some organizations (such as Google) provide "specialized" search

engines that support alternate or extended syntax. (See e.g., Syntax cheatsheet (http:/ / www. google. com/ help/ cheatsheet. html), Googlecodesearch supports regular expressions (http:/ / www. google. com/ intl/ en/ help/ faq_codesearch. html#regexp)).

[6][6] Doublequote-delimited search terms are called "exact phrase" searches in the Google documentation.

Mano, Morris; Ciletti, Michael D. (2013). Digital Design. Pearson. ISBN978-0-13-277420-8.

Further reading J. Eldon Whitesitt (1995). Boolean algebra and its applications. Courier Dover Publications.

ISBN978-0-486-68483-3. Suitable introduction for students in applied fields. Dwinger, Philip (1971). Introduction to Boolean algebras. Wrzburg: Physica Verlag. Sikorski, Roman (1969). Boolean Algebras (3/e ed.). Berlin: Springer-Verlag. ISBN978-0-387-04469-9. Bocheski, Jzef Maria (1959). A Prcis of Mathematical Logic. Translated from the French and German editions

by Otto Bird. Dordrecht, South Holland: D. Reidel.Historical perspective

George Boole (1848). " The Calculus of Logic, (http:/ / www. maths. tcd. ie/ pub/ HistMath/ People/ Boole/CalcLogic/ CalcLogic. html)" Cambridge and Dublin Mathematical Journal III: 18398.

Theodore Hailperin (1986). Boole's logic and probability: a critical exposition from the standpoint ofcontemporary algebra, logic, and probability theory (2nd ed.). Elsevier. ISBN978-0-444-87952-3.

Dov M. Gabbay, John Woods, ed. (2004). The rise of modern logic: from Leibniz to Frege. Handbook of theHistory of Logic 3. Elsevier. ISBN978-0-444-51611-4., several relevant chapters by Hailperin, Valencia, andGrattan-Guinesss

Calixto Badesa (2004). The birth of model theory: Lwenheim's theorem in the frame of the theory of relatives. Princeton University Press. ISBN978-0-691-05853-5., chapter 1, "Algebra of Classes and Propositional

Boolean algebra 16

Calculus" Burris, Stanley, 2009. The Algebra of Logic Tradition (http:/ / plato. stanford. edu/ entries/

algebra-logic-tradition/ ). Stanford Encyclopedia of Philosophy. Radomir S. Stankovic; Jaakko Astola (2011). From Boolean Logic to Switching Circuits and Automata: Towards

Modern Information Technology (http:/ / books. google. com/ books?id=uagvEc2jGTIC). Springer.ISBN978-3-642-11681-0.

External links How Stuff Works Boolean Logic (http:/ / computer. howstuffworks. com/ boolean. htm) Science and Technology - Boolean Algebra (http:/ / oscience. info/ mathematics/ boolean-algebra-2/ ) contains a

list and proof of Boolean theorems and laws.

Algebraic logicIn mathematical logic, algebraic logic is the reasoning obtained by manipulating equations with free variables.What is now usually called classical algebraic logic focuses on the identification and algebraic description of modelsappropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semanticsfor these deductive systems) and connected problems like representation and duality. Well known results like therepresentation theorem for Boolean algebras and Stone duality fall under the umbrella of classical algebraic logic.Works in the more recent abstract algebraic logic (AAL) focus on the process of algebraization itself, like classifyingvarious forms of algebraizability using the Leibniz operator.

Algebras as models of logicsAlgebraic logic treats algebraic structures, often bounded lattices, as models (interpretations) of certain logics,making logic a branch of the order theory.In algebraic logic: Variables are tacitly universally quantified over some universe of discourse. There are no existentially quantified

variables or open formulas; Terms are built up from variables using primitive and defined operations. There are no connectives; Formulas, built from terms in the usual way, can be equated if they are logically equivalent. To express a

tautology, equate a formula with a truth value; The rules of proof are the substitution of equals for equals, and uniform replacement. Modus ponens remains

valid, but is seldom employed.In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structurewhich are its models are shown on the right in the same row. Some of these structures are either Boolean algebras orproper extensions thereof. Modal and other nonclassical logics are typically modeled by what are called "Booleanalgebras with operators."Algebraic formalisms going beyond first-order logic in at least some respects include: Combinatory logic, having the expressive power of set theory; Relation algebra, arguably the paradigmatic algebraic logic, can express Peano arithmetic and most axiomatic set

theories, including the canonical ZFC.

Algebraic logic 17

Logical system Its models

Classical sentential logic Lindenbaum-Tarski algebra Two-element Boolean algebra

Intuitionistic propositional logic Heyting algebra

ukasiewicz logic MV-algebra

Modal logic K Modal algebra

Lewis's S4 Interior algebra

Lewis's S5; Monadic predicate logic Monadic Boolean algebra

First-order logic complete Boolean algebra Cylindric algebraPolyadic algebra

Predicate functor logic

Set theory Combinatory logic Relation algebra

HistoryAlgebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memorandaLeibniz wrote in the 1680s, some of which were published in the 19th century and translated into English byClarence Lewis in 1918. But nearly all of Leibniz's known work on algebraic logic was published only in 1903 afterLouis Couturat discovered it in Leibniz's Nachlass. Parkinson (1966) and Loemker (1969) translated selections fromCouturat's volume into English.Brady (2000) discusses the rich historical connections between algebraic logic and model theory. The founders ofmodel theory, Ernst Schrder and Leopold Loewenheim, were logicians in the algebraic tradition. Alfred Tarski, thefounder of set theoretic model theory as a major branch of contemporary mathematical logic, also: Co-discovered Lindenbaum-Tarski algebra; Invented cylindric algebra; Wrote the 1941 paper that revived relation algebra, which can be viewed as the starting point of abstract algebraic

logic.Modern mathematical logic began in 1847, with two pamphlets whose respective authors were AugustusDeMorganWikipedia:Disputed statement and George Boole. They, and later C.S. Peirce, Hugh MacColl, Frege,Peano, Bertrand Russell, and A. N. Whitehead all shared Leibniz's dream of combining symbolic logic, mathematics,and philosophy. Relation algebra is arguably the culmination of Leibniz's approach to logic. With the exception ofsome writings by Leopold Loewenheim and Thoralf Skolem, algebraic logic went into eclipse soon after the 1910-13publication of Principia Mathematica, not to be revived until Tarski's 1940 re-exposition of relation algebra.Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before theParkinson and Loemker translations. Our present understanding of Leibniz as a logician stems mainly from the workof Wolfgang Lenzen, summarized in Lenzen (2004). [1] To see how present-day work in logic and metaphysics candraw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000). [2]

Algebraic logic 18

References[1] http:/ / www. philosophie. uni-osnabrueck. de/ Publikationen%20Lenzen/ Lenzen%20Leibniz%20Logic. pdf[2] http:/ / mally. stanford. edu/ Papers/ leibniz. pdf

Further reading J. Michael Dunn; Gary M. Hardegree (2001). Algebraic methods in philosophical logic. Oxford University Press.

ISBN978-0-19-853192-0. Good introduction for readers with prior exposure to non-classical logics but withoutmuch background in order theory and/or universal algebra; the book covers these prerequisites at length. Thisbook however has been criticized for poor and sometimes incorrect presentation of AAL results. (http:/ / www.jstor. org/ stable/ 3094793)

Hajnal Andrka, Istvn Nmeti and Ildik Sain (2001). "Algebraic logic". In Dov M. Gabbay, Franz Guenthner.Handbook of philosophical logic, vol 2 (2nd ed.). Springer. ISBN978-0-7923-7126-7. draft (http:/ / www.math-inst. hu/ pub/ algebraic-logic/ handbook. pdf)

Willard Quine, 1976, "Algebraic Logic and Predicate Functors" in The Ways of Paradox. Harvard Univ. Press:283-307.

Historical perspective

Burris, Stanley, 2009. The Algebra of Logic Tradition (http:/ / plato. stanford. edu/ entries/algebra-logic-tradition/ ). Stanford Encyclopedia of Philosophy.

Brady, Geraldine, 2000. From Peirce to Skolem: A neglected chapter in the history of logic.North-Holland/Elsevier Science BV: catalog page (http:/ / www. elsevier. com/ wps/ find/ bookdescription.cws_home/ 621535/ description), Amsterdam, Netherlands, 625 pages.

Lenzen, Wolfgang, 2004, " Leibnizs Logic (http:/ / www. philosophie. uni-osnabrueck. de/ PublikationenLenzen/ Lenzen Leibniz Logic. pdf)" in Gabbay, D., and Woods, J., eds., Handbook of the History of Logic, Vol.3: The Rise of Modern Logic from Leibniz to Frege. North-Holland: 1-84.

Roger Maddux, 1991, "The Origin of Relation Algebras in the Development and Axiomatization of the Calculusof Relations," Studia Logica 50: 421-55.

Parkinson, G.H.R., 1966. Leibniz: Logical Papers. Oxford Uni. Press. Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots. Princeton Univ. Press. Loemker, Leroy (1969 (1956)), Leibniz: Philosophical Papers and Letters, Reidel. Zalta, E. N., 2000, " A (Leibnizian) Theory of Concepts (http:/ / mally. stanford. edu/ leibniz. pdf),"

Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3: 137-183.

External links Stanford Encyclopedia of Philosophy: " Propositional Consequence Relations and Algebraic Logic (http:/ / plato.

stanford. edu/ entries/ consequence-algebraic/ )" -- by Ramon Jansana. (mainly about abstract algebraic logic)

ukasiewicz logic 19

ukasiewicz logicIn mathematics, ukasiewicz logic (/lukvt/; Polish pronunciation:[wukavit]) is a non-classical, many valuedlogic. It was originally defined in the early 20th-century by Jan ukasiewicz as a three-valued logic;[1] it was latergeneralized to n-valued (for all finite n) as well as infinitely-many-valued variants, both propositional andfirst-order.[2] It belongs to the classes of t-norm fuzzy logics[3] and substructural logics.[4]

This article presents the ukasiewicz logic in its full generality, i.e. as an infinite-valued logic. For an elementaryintroduction to the three-valued instantiation 3, see three-valued logic.

LanguageThe propositional connectives of ukasiewicz logic are implication , negation , equivalence , weakconjunction , strong conjunction , weak disjunction , strong disjunction , and propositional constants

and . The presence of weak and strong conjunction and disjunction is a common feature of substructural logicswithout the rule of contraction, to which ukasiewicz logic belongs.

AxiomsThe original system of axioms for propositional infinite-valued ukasiewicz logic used implication and negation asthe primitive connectives:

Propositional infinite-valued ukasiewicz logic can also be axiomatized by adding the following axioms to theaxiomatic system of monoidal t-norm logic:

Divisibility: Double negation: That is, infinite-valued ukasiewicz logic arises by adding the axiom of double negation to basic t-norm logic BL, orby adding the axiom of divisibility to the logic IMTL.Finite-valued ukasiewicz logics require additional axioms.

Real-valued semanticsInfinite-valued ukasiewicz logic is a real-valued logic in which sentences from sentential calculus may be assigneda truth value of not only zero or one but also any real number in between (e.g. 0.25). Valuations have a recursivedefinition where:

for a binary connective and and where the definitions of the operations hold as follows:

Implication: Equivalence: Negation: Weak Conjunction: Weak Disjunction:

ukasiewicz logic 20

Strong Conjunction: Strong Disjunction: The truth function of strong conjunction is the ukasiewicz t-norm and the truth function of strongdisjunction is its dual t-conorm. The truth function is the residuum of the ukasiewicz t-norm. All truthfunctions of the basic connectives are continuous.By definition, a formula is a tautology of infinite-valued ukasiewicz logic if it evaluates to 1 under any valuation ofpropositional variables by real numbers in the interval [0,1].

Finite-valued and countable-valued semanticsUsing exactly the same valuation formulas as for real-valued semantics ukasiewicz (1922) also defined (up toisomorphism) semantics over any finite set of cardinality n 2 by choosing the domain as { 0, 1/(n 1), 2/(n 1), ..., 1 } any countable set by choosing the domain as { p/q | 0 p q where p is a non-negative integer and q is a positive

integer }.

General algebraic semanticsThe standard real-valued semantics determined by the ukasiewicz t-norm is not the only possible semantics ofukasiewicz logic. General algebraic semantics of propositional infinite-valued ukasiewicz logic is formed by theclass of all MV-algebras. The standard real-valued semantics is a special MV-algebra, called the standardMV-algebra.Like other t-norm fuzzy logics, propositional infinite-valued ukasiewicz logic enjoys completeness with respect tothe class of all algebras for which the logic is sound (that is, MV-algebras) as well as with respect to only linearones. This is expressed by the general, linear, and standard completeness theorems:

The following conditions are equivalent: is provable in propositional infinite-valued ukasiewicz logic is valid in all MV-algebras (general completeness) is valid in all linearly ordered MV-algebras (linear completeness) is valid in the standard MV-algebra (standard completeness).

References[1] ukasiewicz J., 1920, O logice trjwartociowej (in Polish). Ruch filozoficzny 5:170171. English translation: On three-valued logic, in L.

Borkowski (ed.), Selected works by Jan ukasiewicz, NorthHolland, Amsterdam, 1970, pp. 8788. ISBN 0-7204-2252-3[2] Hay, L.S., 1963, Axiomatization of the infinite-valued predicate calculus. Journal of Symbolic Logic 28:7786.[3] Hjek P., 1998, Metamathematics of Fuzzy Logic. Dordrecht: Kluwer.[4] Ono, H., 2003, "Substructural logics and residuated lattices an introduction". In F.V. Hendricks, J. Malinowski (eds.): Trends in Logic: 50

Years of Studia Logica, Trends in Logic 20: 177212.

Intuitionistic logic 21

Intuitionistic logicIntuitionistic logic, sometimes more generally called constructive logic, is a system of symbolic logic that differsfrom classical logic by replacing the traditional concept of truth with the concept of constructive provability. Forexample, in classical logic, propositional formulae are always assigned a truth value from the two element set oftrivial propositions ("true" and "false" respectively) regardless of whether we have direct evidence foreither case. In contrast, propositional formulae in intuitionistic logic are not assigned any definite truth value at alland instead only considered "true" when we have direct evidence, hence proof. (We can also say, instead of thepropositional formula being "true" due to direct evidence, that it is inhabited by a proof in the Curry-Howard sense.)Operations in intuitionistic logic therefore preserve justification, with respect to evidence and provability, rather thantruth-valuation. A consequence of this point of view is that intuitionistic logic is not a two-valued logic, nor even afinite-valued logic, in the familiar sense: although intuitionistic logic retains the trivial propositions fromclassical logic, each proof of a propositional formula is considered a valid propositional value, thus by Heyting'snotion of propositions-as-sets, propositional formulae are (potentially non-finite) sets of their proofs.Semantically, intuitionistic logic is a restriction of classical logic in which the law of excluded middle and doublenegation elimination are not admitted as axioms. Excluded middle and double negation elimination can still beproved for some propositions on a case by case basis, however, but do not hold universally as they do with classicallogic.Several semantics for intuitionistic logic have been studied. One semantics mirrors classical Boolean-valuedsemantics but uses Heyting algebras in place of Boolean algebras. Another semantics uses Kripke models.Intuitionistic logic is practically useful because its restrictions produce proofs that have the existence property,making it also suitable for other forms of mathematical constructivism. Informally, this means that if you have aconstructive proof that an object exists, you can turn that constructive proof into an algorithm for generating anexample of it.Formalized intuitionistic logic was originally developed by Arend Heyting to provide a formal basis for Brouwer'sprogramme of intuitionism.

Intuitionistic logic 22

Syntax

The RiegerNishimura lattice. Its nodes are the propositional formulas in onevariable up to intuitionistic logical equivalence, ordered by intuitionistic

logical implication.

The syntax of formulas of intuitionistic logic issimilar to propositional logic or first-order logic.However, intuitionistic connectives are notdefinable in terms of each other in the same wayas in classical logic, hence their choice matters.In intuitionistic propositional logic it iscustomary to use , , , as the basicconnectives, treating A as an abbreviation for(A ). In intuitionistic first-order logic bothquantifiers , are needed.

Many tautologies of classical logic can no longerbe proven within intuitionistic logic. Examplesinclude not only the law of excluded middle p p, but also Peirce's law ((p q) p) p,and even double negation elimination. Inclassical logic, both p p and also p pare theorems. In intuitionistic logic, only theformer is a theorem: double negation can beintroduced, but it cannot be eliminated. Rejecting p p may seem strange to those more familiar with classicallogic, but proving this propositional formula in intuitionistic logic would require producing a proof for the truth orfalsity of all possible propositional formulae, which is impossible for a variety of reasons.

Because many classically valid tautologies are not theorems of intuitionistic logic, but all theorems of intuitionisticlogic are valid classically, intuitionistic logic can be viewed as a weakening of classical logic, albeit one with manyuseful properties.

Sequent calculusGentzen discovered that a simple restriction of his system LK (his sequent calculus for classical logic) results in asystem which is sound and complete with respect to intuitionistic logic. He called this system LJ. In LK any numberof formulas is allowed to appear on the conclusion side of a sequent; in contrast LJ allows at most one formula inthis position.Other derivatives of LK are limited to intuitionisitic derivations but still allow multiple conclusions in a sequent. LJ'[1] is one example.

Hilbert-style calculusIntuitionistic logic can be defined using the following Hilbert-style calculus. This is similar to a way of axiomatizingclassical propositional logic.In propositional logic, the inference rule is modus ponens

MP: from and infer and the axioms are

THEN-1: THEN-2: AND-1: AND-2:

Intuitionistic logic 23

AND-3: OR-1: OR-2: OR-3: FALSE: To make this a system of first-order predicate logic, the generalization rules

-GEN: from infer , if is not free in -GEN: from infer , if is not free in are added, along with the axioms

PRED-1: , if the term t is free for substitution for the variable x in (i.e., if no occurrenceof any variable in t becomes bound in )

PRED-2: , with the same restriction as for PRED-1

Optional connectives

Negation

If one wishes to include a connective for negation rather than consider it an abbreviation for , it isenough to add: NOT-1': NOT-2': There are a number of alternatives available if one wishes to omit the connective (false). For example, one mayreplace the three axioms FALSE, NOT-1', and NOT-2' with the two axioms

NOT-1: NOT-2: as at Propositional calculus#Axioms. Alternatives to NOT-1 are or

.

Equivalence

The connective for equivalence may be treated as an abbreviation, with standing for. Alternatively, one may add the axioms

IFF-1: IFF-2: IFF-3: IFF-1 and IFF-2 can, if desired, be combined into a single axiom usingconjunction.

Intuitionistic logic 24

Relation to classical logic

The system of classical logic is obtained by adding any one of the following axioms:

(Law of the excluded middle. May also be formulated as .) (Double negation elimination) (Peirce's law)In general, one may take as the extra axiom any classical tautology that is not valid in the two-element Kripke frame

(in other words, that is not included in Smetanich's logic).Another relationship is given by the GdelGentzen negative translation, which provides an embedding of classicalfirst-order logic into intuitionistic logic: a first-order formula is provable in classical logic if and only if itsGdelGentzen translation is provable intuitionistically. Therefore intuitionistic logic can instead be seen as a meansof extending classical logic with constructive semantics.In 1932, Kurt Gdel defined a system of Gdel logics intermediate between classical and intuitionistic logic; suchlogics are known as intermediate logics.

Relation to many-valued logic

Kurt Gdel in 1932 showed that intuitionistic logic is not a finitely-many valued logic. (See the section titledHeyting algebra semantics below for a sort of "infinitely-many valued logic" interpretation of intuitionistic logic.)

Non-interdefinability of operatorsIn classical propositional logic, it is possible to take one of conjunction, disjunction, or implication as primitive, anddefine the other two in terms of it together with negation, such as in ukasiewicz's three axioms of propositionallogic. It is even possible to define all four in terms of a sole sufficient operator such as the Peirce arrow (NOR) orSheffer stroke (NAND). Similarly, in classical first-order logic, one of the quantifiers can be defined in terms of theother and negation.These are fundamentally consequences of the law of bivalence, which makes all such connectives merely Booleanfunctions. The law of bivalence does not hold in intuitionistic logic, only the law of non-contradiction. As a resultnone of the basic connectives can be dispensed with, and the above axioms are all necessary. Most of the classicalidentities are only theorems of intuitionistic logic in one direction, although some are theorems in both directions.They are as follows:Conjunction versus disjunction:

Conjunction versus implication:

Disjunction versus implication:

Universal versus existential quantification:

Intuitionistic logic 25

So, for example, "a or b" is a stronger propositional formula than "if not a, then b", whereas these are classicallyinterchangeable. On the other hand, "not (a or b)" is equivalent to "not a, and also not b".If we include equivalence in the list of connectives, some of the connectives become definable from others:

In particular, {, , } and {, , } are complete bases of intuitionistic connectives.As shown by Alexander Kuznetsov, either of the following connectives the first one ternary, the second onequinary is by itself functionally complete: either one can serve the role of a sole sufficient operator forintuitionistic propositional logic, thus forming an analog of the Sheffer stroke from classical propositional logic:[2]

SemanticsThe semantics are rather more complicated than for the classical case. A model theory can be given by Heytingalgebras or, equivalently, by Kripke semantics. Recently, a Tarski-like model theory was proved complete by BobConstable, but with a different notion of completeness than classically.

Heyting algebra semanticsIn classical logic, we often discuss the truth values that a formula can take. The values are usually chosen as themembers of a Boolean algebra. The meet and join operations in the Boolean algebra are identified with the and logical connectives, so that the value of a formula of the form A B is the meet of the value of A and the value of Bin the Boolean algebra. Then we have the useful theorem that a formula is a valid proposition of classical logic if andonly if its value is 1 for every valuationthat is, for any assignment of values to its variables.A corresponding theorem is true for intuitionistic logic, but instead of assigning each formula a value from a Booleanalgebra, one uses values from a Heyting algebra, of which Boolean algebras are a special case. A formula is valid inintuitionistic logic if and only if it receives the value of the top element for any valuation on any Heyting algebra.It can be shown that to recognize valid formulas, it is sufficient to consider a single Heyting algebra whose elementsare the open subsets of the real line R. In this algebra, the and operations correspond to set intersection andunion, and the value assigned to a formula A B is int(AC B), the interior of the union of the value of B and thecomplement of the value of A. The bottom element is the empty set , and the top element is the entire line R. Thenegation A of a formula A is (as usual) defined to be A . The value of A then reduces to int(AC), the interiorof the complement of the value of A, also known as the exterior of A. With these assignments, intuitionistically validformulas are precisely those that are assigned the value of the entire line.For example, the formula (A A) is valid, because no matter what set X is chosen as the value of the formula A,the value of (A A) can be shown to be the entire line:

Value((A A)) =int((Value(A A))C) =int((Value(A) Value(A))C) =

Intuitionistic logic 26

int((X int((Value(A))C))C) =int((X int(XC))C)

A theorem of topology tells us that int(XC) is a subset of XC, so the intersection is empty, leaving:int(C) = int(R) = R

So the valuation of this formula is true, and indeed the formula is valid.But the law of the excluded middle, A A, can be shown to be invalid by letting the value of A be {y : y > 0 }.Then the value of A is the interior of {y : y 0 }, which is {y : y < 0 }, and the value of the formula is the union of{y : y > 0 } and {y : y < 0 }, which is {y : y 0 }, not the entire line.The interpretation of any intuitionistically valid formula in the infinite Heyting algebra described above results in thetop element, representing true, as the valuation of the formula, regardless of what values from the algebra areassigned to the variables of the formula. Conversely, for every invalid formula, there is an assignment of values tothe variables that yields a valuation that differs from the top element.[3] No finite Heyting algebra has both theseproperties.

Kripke semanticsBuilding upon his work on semantics of modal logic, Saul Kripke created another semantics for intuitionistic logic,known as Kripke semantics or relational semantics.[4]

Tarski-like semanticsIt was discovered that Tarski-like semantics for intuitionistic logic were not possible to prove complete. However,Robert Constable has shown that a weaker notion of completeness still holds for intuitionistic logic under aTarski-like model. In this notion of completeness we are concerned not with all of the statements that are true ofevery model, but with the statements that are true in the same way in every model. That is, a single proof that themodel judges a formula to be true must be valid for every model. In this case, there is not only a proof ofcompleteness, but one that is valid according to intuitionistic logic.[5]

Relation to other logicsIntuitionistic logic is related by duality to a paraconsistent logic known as Brazilian, anti-intuitionistic ordual-intuitionistic logic.The subsystem of intuitionistic logic with the FALSE axiom removed is known as minimal logic.

Notes[1][1] Proof Theory by G. Takeuti, ISBN 0-444-10492-5[2] Alexander Chagrov, Michael Zakharyaschev, Modal Logic, vol. 35 of Oxford Logic Guides, Oxford University Press, 1997, pp. 5859. ISBN

0-19-853779-4.[3] Alfred Tarski, Der Aussagenkalkl und die Topologie, Fundamenta Mathematicae 31 (1938), 103134. (http:/ / matwbn. icm. edu. pl/ tresc.

php?wyd=1& tom=31)[4] Intuitionistic Logic (http:/ / plato. stanford. edu/ entries/ logic-intuitionistic/ ). Written by Joan Moschovakis (http:/ / www. math. ucla. edu/

~joan/ ). Published in Stanford Encyclopedia of Philosophy.[5] R. Constable, M. Bickford, Intuitionistic completeness of first-order logic, Annals of Pure and Applied Logic, to appear, . Preprint on ArXiv

(http:/ / arxiv. org/ abs/ 1110. 1614).

Intuitionistic logic 27

References van Dalen, Dirk, 2001, "Intuitionistic Logic", in Goble, Lou, ed., The Blackwell Guide to Philosophical Logic.

Blackwell. Morten H. Srensen, Pawe Urzyczyn, 2006, Lectures on the Curry-Howard Isomorphism (chapter 2:

"Intuitionistic Logic"). Studies in Logic and the Foundations of Mathematics vol. 149, Elsevier. W. A. Carnielli (with A. B.M. Brunner). "Anti-intuitionism and paraconsistency" (http:/ / dx. doi. org/ 10. 1016/ j.

jal. 2004. 07. 016). Journal of Applied Logic Volume 3, Issue 1, March 2005, pages 161-184.

External links Stanford Encyclopedia of Philosophy: " Intuitionistic Logic (http:/ / plato. stanford. edu/ entries/

logic-intuitionistic/ )"by Joan Moschovakis. Intuitionistic Logic (http:/ / www. cs. le. ac. uk/ people/ nb118/ Publications/ ESSLLI'05. pdf) by Nick

Bezhanishvili and Dick de Jongh (from the Institute for Logic, Language and Computation at the University ofAmsterdam)

Semantical Analysis of Intuitionistic Logic I (https:/ / www. princeton. edu/ ~hhalvors/ restricted/kripke_intuitionism. pdf) by Saul A. Kripke from Harvard University, Cambridge, Mass., USA

Intuitionistic Logic (http:/ / www. phil. uu. nl/ ~dvdalen/ articles/ Blackwell(Dalen). pdf) by Dirk van Dalen The discovery of E.W. Beth's semantics for intuitionistic logic (http:/ / www. illc. uva. nl/ j50/ contribs/ troelstra/

troelstra. pdf) by A.S. Troelstra and P. van Ulsen Expressing Database Queries with Intuitionistic Logic (ftp:/ / ftp. cs. toronto. edu/ pub/ bonner/ papers/

hypotheticals/ naclp89. ps) (FTP one-click download) by Anthony J. Bonner. L. Thorne McCarty. KumarVadaparty. Rutgers University, Department of Computer Science.

Mathematical logicMathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics.Topically, mathematical logic bears close connections to metamathematics, the foundations of mathematics, andtheoretical computer science.[1] The unifying themes in mathematical logic include the study of the expressive powerof formal systems and the deductive power of formal proof systems.Mathematical logic is often divided into the fields of set theory, model theory, recursion theory, and proof theory.These areas share basic results on logic, particularly first-order logic, and definability. In computer science(particularly in the ACM Classification) mathematical logic encompasses additional topics not detailed in thisarticle; see logic in computer science for those.Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundationsof mathematics. This study began in the late 19th century with the development of axiomatic frameworks forgeometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilbert's program to prove theconsistency of foundational theories. Results of Kurt Gdel, Gerhard Gentzen, and others provided partial resolutionto the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost allordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven incommon axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses onestablishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics)rather than trying to find theories in which all of mathematics can be developed.

Mathematical logic 28

Subfields and scopeThe Handbook of Mathematical Logic makes a rough division of contemporary mathematical logic into four areas:1.1. set theory2.2. model theory3. recursion theory, and4. proof theory and constructive mathematics (considered as parts of a single area).Each area has a distinct focus, although many techniques and results are shared among multiple areas. Theborderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are notalways sharp. Gdel's incompleteness theorem marks not only a milestone in recursion theory and proof theory, buthas also led to Lb's theorem in modal logic. The method of forcing is employed in set theory, model theory, andrecursion theory, as well as in the study of intuitionistic mathematics.The mathematical field of category theory uses many formal axiomatic methods, and includes the study ofcategorical logic, but category theory is not ordinarily considered a subfield of mathematical logic. Because of itsapplicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed categorytheory as a foundational system for mathematics, independent of set theory. These foundations use toposes, whichresemble generalized models of set theory that may employ classical or nonclassical logic.

HistoryMathematical logic emerged in the mid-19th century as a subfield of mathematics independent of the traditionalstudy of logic (Ferreirs 2001, p.443). Before this emergence, logic was studied with rhetoric, through thesyllogism, and with philosophy. The first half of the 20th century saw an explosion of fundamental results,accompanied by vigorous debate over the foundations of mathematics.

Early historyTheories of logic were developed in many cultures in history, including China, India, Greece and the Islamic world.In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had beenmade by philosophical mathematicians including Leibniz and Lambert, but their labors remained isolated and littleknown.

19th centuryIn the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematicmathematical treatments of logic. Their work, building on work by algebraists such as George Peacock, extended thetraditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations ofmathematics(Katz 1998, p.686).Charles Sanders Peirce built upon the work of Boole to develop a logical system for relations and quantifiers, whichhe published in several papers from 1870 to 1885. Gottlob Frege presented an independent development of logicwith quantifiers in his Begriffsschrift, published in 1879, a work generally considered as marking a turning point inthe history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near theturn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused incontemporary texts.From 1890 to 1905, Ernst Schrder published Vorlesungen ber die Algebra der Logik in three volumes. This worksummarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference tosymbolic logic as it was understood at the end of the 19th century.

Mathematical logic 29

Foundational theories

Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systemsfor fundamental areas of mathematics such as arithmetic, analysis, and geometry.In logic, the term arithmetic refers to the theory of the natural numbers. Giuseppe Peano (1889) published a set ofaxioms for arithmetic that came to bear his name (Peano axioms), using a variation of the logical system of Booleand Schrder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time RichardDedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind (1888)proposed a different characterization, which lacked the formal logical character of Peano's axioms. Dedekind's work,however, proved theorems inaccessible in Peano's system, including the uniqueness of the set of natural numbers (upto isomorphism) and the recursive definitions of addition and multiplication fr