Thealgebra of logic, as anexplicit algebraicsystem showing the underlying mathematical structure of logic, wasintroduced by George Boole (1815–1864) in his bookTheMathematical Analysis of Logic (1847). It is therefore to bedistinguished from the more general approach ofalgebraiclogic. The methodology initiated by Boole was successfullycontinued in the 19^th century in the work of WilliamStanley Jevons (1835–1882), Charles Sanders Peirce(1839–1914), Ernst Schröder (1841–1902), among manyothers, thereby establishing a tradition in (mathematical) logic. FromBoole’s first book until the influence after WWI of the monumentalworkPrincipia Mathematica (1910–1913) by Alfred NorthWhitehead (1861–1947) and Bertrand Russell (1872–1970),versions of the algebra of logic were the most developed form ofmathematical logic above all as presented in Schröder’s threevolumesVorlesungen über die Algebra der Logik(1890–1905). Furthermore, this tradition motivated theinvestigations of Leopold Löwenheim (1878–1957) thateventually gave rise to model theory. In addition, in 1941, AlfredTarski (1901–1983) in his paper “On the calculus ofrelations” returned to Peirce’s relation algebra as presented inSchröder’sAlgebra der Logik. The tradition of thealgebra of logic played a key role in the notion ofLogic asCalculus as opposed to the notion ofLogic as UniversalLanguage. Beyond Tarski’s algebra of relations, the influence ofthe algebraic tradition in logic can be found in other mathematicaltheories, such as category theory. However this influence lies outsidethe scope of this entry, which is divided into 10 sections.

• 1. Introduction
• 2. 1847—The Beginnings of the Modern Versions of the Algebra of Logic
• 3. 1854—Boole’s Final Presentation of his Algebra of Logic
• 4. Jevons: An Algebra of Logic Based on Total Operations
• 5. Peirce: Basing the Algebra of Logic on Subsumption
• 6. De Morgan and Peirce: Relations and Quantifiers in the Algebra of Logic
• 7. Schröder’s systematization of the algebra of logic
• 8. Huntington: Axiomatic Investigations of the Algebra of Logic
• 9. Stone: Models for the Algebra of Logic
• 10. Skolem: Quantifier Elimination and Decidability
• 11. Tarski and the Revival of Algebraic Logic
• Bibliography
  • Primary Sources
  • Secondary Sources
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

Boole’sThe Mathematical Analysis of Logic presents manyinteresting logic novelties: It was the beginning ofnineteenth-century mathematization of logic and provided analgorithmic alternative (via a slight modification ofordinary algebra) to thecatalog approach used in traditionallogic (even if reduction procedures were developed in the latter).Instead of a list of valid forms of argument, the validity ofarguments were determined on the basis of general principles andrules. Furthermore, it provided an effective method for provinglogical laws on the basis of a system of postulates. As Boole wrotelater, it was a proper “science of reasoning”, and not a“mnemonic art” like traditional Syllogistics (Boole 1997:136). Three-quarters of the way through this book, after finishinghis discussion of syllogistic logic, Boole started to develop thegeneral tools that would be used in hisLaws of Thought(1854) to greatly extend traditional logic by permitting an argumentto have many premises and to involve many classes. To handle theinfinitely many possible logical arguments of this expanded logic, hepresented theorems that provided key tools for an algorithmic analysis(a catalog was no longer feasible).

Boole’s ideas were conceived independently of earlier anticipations,like those developed by G.W. Leibniz. They emerged from the particularcontexts of English mathematics (see Peckhaus 2009). According toVíctor Sánchez Valencia, the tradition that originatedwith Boole came to be known as thealgebra of logic since thepublication in 1879 ofPrinciples of the Algebra of Logic byAlexander MacFarlane (see Sánchez Valencia 2004: 389).MacFarlane considered “the analytical method of reasoning aboutQuality proposed by Boole” as an algebra (see MacFarlane 1879:3).

This approach differs from what is usually calledalgebraiclogic; though there is some overlap, the historical developmentof the two areas are different. Algebraic logic is understood as:

a style [of logic]in which concepts and relationsare expressed by mathematical symbols [\(\ldots\)] so thatmathematical techniques can be applied. Here mathematics shall meanmostly algebra, i.e., the part of mathematics concerned with finitaryoperations on some set. (Hailperin 2004: 323)

Algebraic logic can be already found in the work of Leibniz, JacobBernoulli and other modern thinkers, and it undoubtedly constitutes animportant antecedent of Boole’s approach. In a broader perspective,both are part of the tradition ofsymbolic knowledge in theformal sciences, as first conceived by Leibniz (see Esquisabel 2012).This idea of algebraic logic was continued to some extent in theFrench Enlightenment in the work of Condillac and Condorcet (seeGrattan-Guinness 2000: 14 ff.)

Boole’s methodology for dealing with logical problems can be describedas follows:

1. Translate the logical data into suitable equations;
2. Apply algebraic techniques to solve these equations;
3. Translate this solution, if possible, back into the originallanguage.

In other words, symbolic formulation of logical problems and solutionof logical equations constitutes Boole’s method (see SánchezValencia 2004: 389).

Later, in hisPure Logic from 1864, Jevons changed Boole’spartial operation of union of disjoint sets to the modern unrestrictedunion and eliminated Boole’s questionable use of uninterpretable terms(see Jevons 1890). Peirce (1880) eliminated explicitly theAristotelian derivation of particular statements from universalstatements by giving the modern meaning for “All \(A\) is\(B\)”. In addition, he extended the algebra of logic forclasses to the algebra of logic for binary relations and introducedgeneral sums and products to handle quantification. ErnstSchröder, taking inspiration from previous work by Hermann(1809–1877) and Robert Grassmann (1815–1901) and using theframework developed by Peirce, developed and systematized the19^th Century achievements in the algebra of logic in histhree-volume workVorlesungen über die Algebra der Logik(1890–1910).

The contributions of Gottlob Frege (1848–1925) to logic from theperiod 1879–1903, based on anaxiomatic approach tologic, had very little influence at the time (and the same can be saidof thediagrammatic systems of C.S. Peirce developed at theturn of the century). Whitehead and Russell rejected the algebra oflogic approach, with its predominantly equational formulations andalgebraic symbolism, in favor of an approach strongly inspired by theaxiomatic system of Frege, and using the notation developed byGiuseppe Peano, namely to use logical connectives, relation symbolsand quantifiers.

During the first two decades of the twentieth century, the algebra oflogic was further developed in the works of Platon SergeevichPoretzsky (1846–1907), Louis Couturat (1868–1914), LeopoldLöwenheim (1878–1957), and Heinrich Behmann(1891–1970) (see Styazhkin 1969). In particular, eliminationtheorems in the algebra of logic influenced decision procedures forfragments of first-order and second-order logic (see Mancosu, Zach,Badesa 2009).

After WWI David Hilbert (1862–1943), who had at first adoptedthe algebraic approach, picked up on the approach ofPrincipia, and the algebra of logic fell out of favor.However, in 1941, Tarski treated relation algebras as an equationallydefined class. Such a class has many models besides thecollectionof all binary relations on a given universe that was consideredin the 1800s, just as there are many Boolean algebras besides thepower set Boolean algebras studied in the 1800s. In the years1948–1952 Tarski, along with his students Chin and Thompson,created cylindric algebras as an algebraic logic companion tofirst-order logic, and in 1956 Paul Halmos (1916–2006)introduced polyadic algebras for the same purpose. As Halmos (1956 b,c and d) noted, these new algebraic logics tended to focus on studyingthe extent to which they captured first-order logic and on theiruniversal algebraic aspects such as axiomatizations and structuretheorems, but offered little insight into the nature of thefirst-order logic which inspired their creation.

2. 1847—The Beginnings of the Modern Versions of the Algebra of Logic

In late 1847, Boole and Augustus De Morgan (1806–1871) eachpublished a book on logic—Boole’sMathematical Analysis ofLogic (1847) and De Morgan’sFormal Logic (1847). DeMorgan’s approach was to dissect every aspect of traditional deductivelogic (usually called ‘Aristotelian logic’) into itsminutest components, to consider ways to generalize these components,and then, in some cases, undertake to build a logical system usingthese components. Unfortunately, he was never able to incorporate hisbest ideas into a significant system. His omission of a symbol forequality made it impossible to develop an equational algebra of logic.It seems that synthesis was not De Morgan’s strong suit.

De Morgan’s book of 1847 was part of a revival in logic studies thatoriginated at the beginning of 19th Century with Joseph Diez Gergonne(1771–1859) in France, and Bernhard Bolzano (1781–1848) inBohemia, among others. George Bentham and William Hamilton in theUnited Kingdom were also part of this revival and their studiesfocused on the nature of variations of the categorical sentences intraditional syllogistic, including what was called“quantification of the predicate”; for example, “All\(A\) are some \(B\)” or “Some \(A\) are all\(B\)”. It was thought that this problem required an extensionof the syllogistic logic of Aristotle and that some form of symbolicmethod was needed both to handle such statements and provide aclassification of their different types (see Heinemann 2015 chapters 2and 3).

Boole approached logic from a completely different perspective, namelyhow to cast Aristotelian logic in the garb of symbolical algebra.Using symbolical algebra was a theme with which he was well-acquaintedfrom his work in differential equations, and from the various papersof his young friend and mentor Duncan Farquharson Gregory(1813–1844), who made attempts to cast other subjects such asgeometry into the language of symbolical algebra. Since theapplication of symbolical algebra to differential equations hadproceeded through the introduction of differential operators, it musthave been natural for Boole to look for operators that applied in thearea of Aristotelian logic. He readily came up with the idea of using“selection” operators, for example, a selection operatorfor the color red would select the red members from a class. In his1854 book, Boole realized that it was simpler to omit selectionoperators and work directly with classes. (However he kept theselection operators to justify his claim that his laws of logic werenot ultimately based on observations concerning the use of language,but were actually deeply rooted in the processes of the human mind.)From now on in this article, when discussing Boole’s 1847 book, theselection operators have been replaced with the simpler directformulation using classes.

Since symbolical algebra was just the syntactic side of ordinaryalgebra, Boole needed ways to interpret the usual operations andconstants of algebra to create his algebra of logic for classes.Multiplication was interpreted as intersection, leading to his one newlaw, the idempotent law \(XX = X\) for multiplication, rediscovering alogical law already formulated by Leibniz. Addition was defined asunion, provided one was dealing withdisjoint classes; andsubtraction as class difference, provided one was subtracting asubclass from a class. In other cases, the addition and subtractionoperations were simply undefined, or as Boole wrote,uninterpretable. The usual laws of arithmetic told Boole that1 must be the universe and \(1 - X\) must be the complement of\(X\).

The next step in Boole’s system was to translate the four kinds ofcategorical propositions into equations, for example “All \(X\)is \(Y\)” becomes \(X = XY\), and “Some \(X\) is\(Y\)” becomes \(V = XY\), where \(V\) is a new symbol. Toeliminate the middle term in a syllogism Boole borrowed an eliminationtheorem from ordinary algebra, but it was too weak for his algebra oflogic. This would be remedied in his 1854 book. Boole found that hecould not always derive the desired conclusions with the abovetranslation of particular propositions (i.e., those with existentialimport), so he added the variants \(X = VY\), \(Y = VX\), and \(VX =VY\) (see the entry onBoole).

The symbolic algebra of the 1800s included much more than just thealgebra of polynomials, and Boole experimented to see which resultsand tools might apply to the algebra of logic. For example, he provedone of his results by using an infinite series expansion. Hisfascination with the possibilities of ordinary algebra led him toconsider questions such as: What would logic be like if the idempotentlaw were replaced by the law \(X^3 = X\)? His successors, especiallyJevons, would soon narrow the operations on classes to the ones thatwe use today, namely union, intersection and complement.

As mentioned earlier, three-quarters of the way through his brief bookof 1847, after finishing derivations of the traditional Aristoteliansyllogisms in his system, Boole announced that his algebra of logicwas capable of far more general applications. Then he proceeded to addgeneral theorems on developing (expanding) terms, providinginterpretations of equations, and using long division to express oneclass in an equation in terms of the other classes (with sideconditions added).

Boole’s theorems, completed and perfected in 1854, gave algorithms foranalyzing infinitely many argument forms. This opened a new andfruitful perspective, deviating from the traditional approach tologic, where for centuries scholars had struggled to come up withclever mnemonics to memorize a very small catalog of valid conversionsand syllogisms and their various interrelations.

De Morgan’sFormal Logic did not gain significantrecognition, primarily because it was a large collection of smallfacts without a significant synthesis. Boole’sThe MathematicalAnalysis of Logic had powerful methods that caught the attentionof a few scholars such as De Morgan and Arthur Cayley(1821–1895); but immediately there were serious questions aboutthe workings of Boole’s algebra of logic: Just how closely was it tiedto ordinary algebra? How could Boole justify the procedures of hisalgebra of logic? In retrospect it seems quite certain that Boole didnot know why his system worked. His claim, following Gregory, that inorder to justify using ordinary algebra it was enough to check thecommutative law \(XY = YX\) for multiplication and the distributivelaw \(X(Y + Z) = XY + XZ\), is clearly false. Nonetheless it is alsolikely that he had checked his results in a sufficient number of casesto give substance to his belief that his system was correct.

3. 1854—Boole’s Final Presentation of his Algebra of Logic

In his second book,The Laws of Thought, Boole not onlyapplied algebraic methods to traditional logic but also attempted somereforms to logic. He started by augmenting the laws of his 1847algebra of logic (without explicitly saying that his previous list ofthree axioms was inadequate), and made some comments on the rule ofinference (performing the same operation on both sides of anequation). But then he casually stated that the foundation of hissystem actually rested on a single (new) principle, namely it sufficedto check an argument by seeing if it was correct when the classsymbols took on only the values 0 and 1, and the operations were theusual arithmetical operations. Let us call thisBoole’s Rule of 0and 1. No meaningful justification was given for Boole’s adoptionof this new foundation, it was not given a special name, and the scantreferences to it in the rest of the book were usually rather clumsilystated. For a modern analysis of this Rule of 0 and 1 see Burris &Sankappanavar 2013.

The development of the algebra of logic in theLaws ofThought proceeded much as in his 1847 book, with minor changes tohis translation scheme, and with the selection operators replaced byclasses. There is a new and very important theorem (correcting the onehe had used in 1847), the Elimination Theorem, which says thefollowing: given an equation \(F(x,y, z, \ldots) = 0\) in the classsymbols \(x, y, z\), etc., the most general conclusion that followsfrom eliminating certain of the class symbols is obtained by (1)substituting 0s and 1s into \(F(x, y, z, \ldots)\) for the symbols tobe eliminated, in all possible ways, then (2) multiplying thesevarious substitution instances together and setting the product equalto 0. Thus eliminating \(y\) and \(z\) from \(F(x, y, z) = 0\) gives\(F(x, 0, 0)F(x, 0, 1)F(x, 1, 0)F(x, 1, 1) = 0\). This theorem alsoplayed an important role in Boole’s interpretation of Aristotle’ssyllogistic.

From an algebra of logic point of view, the 1854 treatment at timesseems less elegant than that in the 1847 book, but it gives a muchricher insight into how Boole thinks about the foundations for hisalgebra of logic. The final chapter on logic, Chapter XV, was anattempt to give a uniform proof of the Aristotelian conversions andsyllogisms. (It is curious that prior to Chapter XV Boole did notpresent any examples of arguments involving particular propositions.)The details of Chapter XV are quite involved, mainly because of theincrease in size of expressions when the Elimination and DevelopmentTheorems are applied. Boole simply left most of the work to thereader. Later commentators would gloss over this chapter, and no oneseems to have worked through its details.

Aside from the Rule of 0 and 1 and the Elimination Theorem, the 1854presentation is mainly interesting for Boole’s attempts to justify hisalgebra of logic. He argued that in symbolical algebra it was quiteacceptable to carry out equational deductions with partial operations,just as one would when the operations were total, as long as the termsin the premises and the conclusion were interpretable. He said thiswas the way ordinary algebra worked with the uninterpretable\(\sqrt{-1}\), the square root of −1. (The geometricinterpretation of complex numbers was recognized early on by Wessel,Argand, and Gauss, but it was only with the publications of Gauss andHamilton in the 1830s that doubts about the acceptability of complexnumbers in the larger mathematical community were overcome. It iscurious that in 1854 Boole regarded \(\sqrt{-1}\) asuninterpretable.)

There were a number of concerns regarding Boole’s approach to thealgebra of logic:

1. Was there a meaningful tie between his algebra of logic and thealgebra of numbers, or was it just an accident that they were sosimilar?
2. Could one handle particular propositions in an algebraic logicthat focused on equations?
3. Was it really acceptable to work with uninterpretable terms inequational derivations?
4. Was Boole using “Aristotelian” semantics (thesemantics presupposed in traditional logic, where the extension of aterm is nonempty)?

4. Jevons: An Algebra of Logic Based on Total Operations

Jevons, who had studied with De Morgan, was the first to offer analternative to Boole’s system. In 1863 he wrote to Boole that surelyBoole’s operation of addition should be replaced by the more natural‘inclusive or’ (or ‘union’), leading to thelaw \(X + X = X\). Boole completely rejected this suggestion (it wouldhave destroyed his system based on ordinary algebra) and broke off thecorrespondence. Jevons published his system in his 1864 book,PureLogic (reprinted in Jevons 1890). By ‘pure’ he meantthat he was casting off any dependence on the algebra ofnumbers—instead of classes, which are associated with quantity,he would use predicates, which are associated with quality, and hislaws would be derived directly from the (total) fundamental operationsof inclusive disjunction and conjunction. But he kept Boole’s use ofequations as the fundamental form of statements in his algebra oflogic.

By adopting De Morgan’s convention of using upper-case/lower-caseletters for complements, Jevons’ system was not suited to provideequational axioms for modern Boolean algebra. However, he refined hissystem of axioms and rules of inference until the result wasessentially the modern system of Boolean algebra forgroundterms, that is, terms where the class symbols are to be thoughtof as constants, not as variables.

It must be noticed that modern equational logic deals withuniversally quantified equations (which would have beencalledlaws in the 1800s). In the 19^th centuryalgebra of logic one could translate “All \(X\) is \(Y\)”as the equation \(X = XY\). This isnot to be viewed as theuniversally quantified expression \((\forall X)(\forall Y)(X = XY)\).\(X\) and \(Y\) are to be treated as constants (or schematic letters).Terms that only have constants (no variables) are calledground terms.

By carrying out this analysis in the special setting of an algebra ofpredicates (or equivalently, in an algebra of classes) Jevons playedan important role in the development of modern equational logic. Asmentioned earlier, Boole gave inadequate sets of equational axioms forhis system, originally starting with the two laws due to Gregory plushis idempotent law; these were accompanied by De Morgan’s inferencerule that one could carry out the same operation (Boole’s fundamentaloperations in his algebra of logic were addition, subtraction andmultiplication) on equals and obtain equals. Boole then switched tothe simple and powerful (but unexplained) Rule of 0 and 1.

Having replaced Boole’s fundamental operations with total operations,Jevons proceeded, over a period of many years, to work on the axiomsand rules for his system. Some elements of equational logic that wenow take for granted required a considerable number of years forJevons to resolve:

The Reflexive Law (\(A=A\)). In 1864 Jevons listed this as apostulate (1890, p. 11) and then in §24 he referred to \(A = A\)as a “useless Identical proposition”. In his 1869 paper onsubstitution it became the “Law of Identity”. In thePrinciples of Science (1874) it was one of the three“Fundamental Laws of Thought”.

The Symmetric Law (\(B = A\) follows from \(A = B\)). In 1864Jevons wrote “\(A = B\) and \(B = A\) are the samestatement”. This is a position he would maintain. In 1874 hewrote

I shall consider the two forms \(A = B\) and \(B = A\) to expressexactly the same identity written differently.

For a final form of his algebra of logic we turn to the laws which hehad scattered over 40 pages inPrinciples of Science (1874),having replaced his earlier use of + by \(\ORjev\), evidently to movefurther away from any appearance of a connection with the algebra ofnumbers:

Laws of Combination

Laws of Thought

For his single rule of inference Jevons chose his principle ofsubstitution—in modern terms this was essentially a combinationof ground replacement and transitivity. He showed how to derivetransitivity of equality from this; he could have derived symmetry aswell but did not. The associative law was missing—it wasimplicit in the lack of parentheses in his expressions.

It was only in hisStudies in Deductive Logic (1880) thatJevons mentioned McColl’s use of an accent to indicate negation. Afternoting that McColl’s accent allowed one to take the negation ofcomplex bracketed terms he went on to say that, for the most part, hefound the notation of De Morgan, the notation that he had always used,to be the more elegant.

5. Peirce: Basing the Algebra of Logic on Subsumption

Peirce started his research into the algebra of logic in the late1860s. In his paper “On an Improvement in Boole’s calculus ofLogic” (Peirce 1867), he arrived independently at the sameconclusion that Jevons had reached earlier, that one needed to replaceBoole’s partial operation of addition with the total operation ofunion (see CP 3.3.6). In his important 1880 paper, “On theAlgebra of Logic”, Peirce quietly broke with the traditionalextensional semantics and introduced a usual assumption of modernsemantics: the extension of a concept, understood as a class, could beempty (as well as the universe), and stated the truth values of thecategorical propositions that we use today. For example, he said theproposition “All \(A\) is \(B\)” is true if \(A\) and\(B\) are both the empty class. Conversion by Limitation, that is, theargument “All \(A\) is \(B\)” therefore “Some \(B\)is \(A\)”, was no longer a valid inference. Peirce said nothingabout the reasons for and merits of his departure from the traditionalsemantic assumption of existence.

Peirce also broke with Boole’s and Jevons’ use of equality as thefundamental primitive, using instead the relation of“subsumption” interpretable in different ways (subclassrelation, implication, etc.). He stated the partial order propertiesof subsumption and then proceeded to define the operations of + and× as least upper bounds and greatest lower bounds—heimplicitly assumed such bounds existed—and listed the keyequational properties of the algebras with two binary operations thatwe now call lattices. Then he claimed that the distributive lawfollowed, but said the proof was too tedious to include. Thefruitfulness of this perspective is evident in his seminal paper from1885. There Peirce introduced a system for propositional logic basedon five axioms for implication (represented by the sign‘\(-\kern-.4em<\)’), including what is now calledPeirce’s law. It certainly made the algebra of logic more elegant.

6. De Morgan and Peirce: Relations and Quantifiers in the Algebra of Logic

De Morgan wrote a series of six papers called “On theSyllogism” in the years 1846 to 1863 (reprinted in De Morgan1966). In his efforts to generalize the syllogism, De Morgan replacedthe copula “is” with a general binary relation in thesecond paper of the series dating from 1850. By allowing differentbinary relations in the two premises of a syllogism, he was led tointroduce the composition of the two binary relations to express theconclusion of the syllogism. In this pursuit of generalized syllogismshe introduced various other operations on binary relations, includingthe converse operation, and he developed a fragment of a calculus forthese operations. His main paper on this subject was the fourth in theseries, called “On the syllogism, No. IV, and on the logic ofrelations” published in 1859 (see De Morgan 1966).

Following De Morgan’s paper, Peirce, in his paper “Descriptionof a Notation for the Logic of Relatives, resulting from anAmplification of the Conceptions of Boole’s Calculus of Logic”from 1870, lifted Boole’s work to the setting of binaryrelations—with binary relations one had, in addition to union,intersection and complement, the natural operations of composition andconverse. A binary relation was characterized as a set of orderedpairs (see 3.328). He worked on this new calculus between 1870 and1883. Like De Morgan, Peirce also considered a number of other naturaloperations on relations. Peirce’s main paper on the subject was“On the Algebra of Logic” (1880). By employingunrestricted unions, denoted by Σ, and unrestrictedintersections, denoted by Π, Peirce thus introduced quantifiersinto his algebra of logic.

In a paper from 1882, “Brief Description of the Algebra ofRelatives”, reprinted in De Morgan 1966, he used thesequantifiers to define operations on relations by means of operationson certain kind of coefficients. De Morgan gets credit for introducingthe concept of relation, but Peirce is considered the true creator ofthe theory of relations (see, e.g., Tarski 1941: 73). However, Peircedid not develop this theory. As Calixto Badesa wrote, “thecalculus of relatives was never to Peirce’s liking” (Badesa2004: 32). He considered it too complicated because of the combinationof class operations with relational ones. Instead, he preferred from1885 onwards to develop a “general algebra” includingquantifiers but no operation on relations. In this way, he arrived atan elementary and informal presentation of what is now calledfirst-order logic (see Badesa 2004,loc. cit.).

7. Schröder’s systematization of the algebra of logic

The German mathematician Ernst Schröder played a key role in thetradition of the algebra of logic. A good example was his challenge toPeirce to provide a proof of the distributive law, as one of the keyequational properties of the algebras with two binary operations.Peirce (1885) admitted that he could not provide a proof. Years laterHuntington (1904: 300–301) described part of the content of aletter he had received from Peirce in December 1903 that claimed toprovide the missing proof—evidently Peirce had stumbled acrossthe long lost pages after the death of Schröder in 1902. Peirceexplained to Huntington that he had originally assumed Schröder’schallenge was well-founded and that this apparent shortcoming of hispaper “was to be added to the list of blunders, due to thegrippe, with which that paper abounds, …”. ActuallyPeirce’s proof did not correct the error since the distributive lawdoes not hold in lattices in general; instead his proof brought in theoperation of complementation—he used the axiom

if \(a\) is not contained in the complement of \(b\) then \(a\) and\(b\) have a common lower bound.

On the basis of his previous algebraic work, Schröder wrote anencyclopedic three volume work at the end of the 19^thcentury calledVorlesungen über die Algebra der Logik(1890–1905), built on the subsumption framework with the modernsemantics of classes as presented by Peirce. This work was the resultof his research in algebra and revealed different influences.Schröder aimed at a general algebraic theory with applications inmany mathematical fields, where the algebra of logic was at the core.As Geraldine Brady pointed out, it offers the first exposition ofabstract lattice theory, the first exposition of Dedekind’s theory ofchains after Dedekind, the most comprehensive development of thecalculus of relations, and a treatment of the foundations ofmathematics on the basis of the relation calculus (see Brady 2000: 143f.)

The first volume concerned the equational logic of classes, the mainresult being Boole’s Elimination Theorem of 1854. Three rathercomplicated counter-examples to Peirce’s claim of distributivityappeared in an appendix to Vol. I, one of which involved nine-hundredand ninety identities for quasigroups. On the basis of this volume,Dedekind (1897) composed an elegant modern abstract presentation oflattices (which he calledDualgruppen); in this paper hepresented a five-element counter-example to Peirce’s claim of thedistributive law.

Volume II augments the algebra of logic for classes developed inVolume I so that it can handle existential statements. First, usingmodern semantics, Schröder proved that one cannot use equationsto express “Some \(X\) is \(Y\)”. However, he noted thatone can easily express it with a negated equation, namely \(XY \ne0\). Volume II, a study of the calculus of classes using bothequations and negated equations, attempted to cover the same topicscovered in Vol. I, in particular there was considerable effort devotedto finding an Elimination Theorem. After dealing with several specialcases, Schröder recommended this topic as an important researcharea—the quest for an Elimination Theorem would be known as theElimination Problem.

Inspired mainly by Peirce’s work, Schröder examined the algebraof logic for binary relations in Vol. III of hisVorlesungenüber die Algebra der Logik. As Tarski once noted, Peirce’swork was continued and extended in a very thorough and systematic wayby Schröder. One item of particular fascination for him was this:given an equation \(E(x, y, z, \ldots) = 0\) in this algebra, find thegeneral solution for one of the relation symbols, say for \(x\), interms of the other relation symbols. He managed, given a particularsolution \(x = x0\), to find a remarkable term \(S(t, y, z, \ldots)\)with the following properties: (1) \(x = S(t, y, z, \ldots)\) yields asolution to \(E = 0\) for any choice of relation \(t\), and (2) everysolution \(x\) of \(E = 0\) can be obtained in this manner by choosinga suitable \(t\). Peirce was not impressed by Schröder’spreoccupation with the problem of solving equations, and pointed outthat Schröder’s parametric solution was a bit of a hoax—theexpressive power of the algebra of logic for relations was so strongthat by evaluating the term \(S(t, y, z, \ldots)\) one essentiallycarried out the steps to check if \(E(t, y, z, \ldots) = 0\); if theanswer was yes then \(S(t, y, z, \ldots)\) returned the value \(t\),otherwise it would return the value \(x0\).

Summing up, Schröder constructed an algebraic version of modernpredicate logic and also a theory of relations. He applied it todifferent fields (e.g., Cantor’s set theory), and he considered hisalgebraic notation as a general or universal language(pasigraphy, see Peckhaus 2004 and Legris 2012). It is to benoted that Löwenheim in 1940 still thought it was as reasonableas set theory. According to him, Schröder’s idea of solving arelational equation was a precursor of Skolem functions, andSchröder inspired Löwenheim’s formulation and proof of thefamous theorem that every “arithmetical” sentence with aninfinite model has a countable model. Schröder’s calculus ofrelations was the basis for the doctoral dissertation of NorbertWiener (1894–1964) in Harvard (Wiener 1913). According to Brady,Wiener gave the first axiomatic treatment of the calculus ofrelations, preceding Tarski’s axiomatization by more than twenty years(see Brady 2000: 165).

8. Huntington: Axiomatic Investigations of the Algebra of Logic

At the turn of the 19^th Century, David Hilbert(1862–1943) presented, in hisGrundlagen der Geometrie,Euclidean geometry as an axiomatic subject that did not depend ondiagrams for its proofs (Hilbert 1899). This led to a wave of interestin studying axiom systems in mathematics; in particular one wanted toknow if the axioms were independent, and which primitives led to themost elegant systems. Edward Vermilye Huntington (1874–1952) wasone of the first to examine this issue for the algebra of logic. Hegave three axiomatizations of the algebra of logic, showed each set ofaxioms was independent, and that they were equivalent (see Huntington1904). In 1933 he returned to this topic with three new sets ofaxioms, one of which contained the following three equations (1933:280):

Shortly after this, Herbert Robbins (1915–2001) conjectured thatthe third equation could be replaced by the slightly simpler

Neither Huntington nor Robbins could prove this, and later itwithstood the efforts of many others, including even Tarski and histalented school at Berkeley. Building on partial results of Winker,the automated theorem prover EQP, designed by William McCune of theArgonne National Laboratory, found a proof of the Robbins Conjecturein 1996. This accomplishment was popularized in Kolata 2010.

According to Huntington (1933: 278), the term “Booleanalgebra” was introduced by Henry M. Sheffer (1882–1964) inthe paper where he showed that one could give a five-equationaxiomatization of Boolean algebra using the single fundamentaloperation of joint exclusion, now known as the Sheffer stroke (seeSheffer 1913). Whitehead and Russell claimed in the preface to thesecond edition ofPrincipia that the Sheffer stroke was thegreatest advance in logic since the publication ofPrincipia.(Hilbert and Ackermann (1928), by contrast, stated that the Shefferstroke was just a curiosity.) Neither realized that decades earlierSchröder had discovered that the dual of the Sheffer stroke wasalso such an operation—Schröder’s symbol for his operationwas that of a double-edged sword.

In the 1930s Garrett Birkhoff (1911–1996) established thefundamental results of equational logic, namely (1) equational classesof algebras are precisely the classes closed under homomorphisms,subalgebras and direct products, and (2) equational logic is based onfive rules: reflexivity, symmetry, transitivity, replacement, andsubstitution. In the 1940s, Tarski joined in this development ofequational logic; the subject progressed rapidly from the 1950s tillthe present time.

9. Stone: Models for the Algebra of Logic

Traditional logic studied certain simple relationships betweenclasses, namelybeing a subclass of andhaving a nonemptyintersection with. However, once one adopted an axiomaticapproach, the topic of possible models besides the obvious onessurfaced. Beltrami introduced models of non-Euclidean geometry in thelate 1860s. In the 1890s Schröder and Dedekind constructed modelsof the axioms of lattice theory to show that the distributive law didnot follow. But when it came to the algebra of classes, Schröderconsidered only the standard models, namely each was the collection ofall subclasses of a given class.

The study of general models of the axioms of Boolean algebra did notget underway until the late 1920s; it was soon brought to a very highlevel in the work of Marshall Harvey Stone (1903–1989) (see hispapers 1936, 1937). He was interested in the structure of rings oflinear operators and realized that the central idempotents, that is,the operators \(E\) that commuted with all other operators in the ringunder multiplication (that is, \(EL = LE\) for all \(L\) in the ring)and which were idempotent under multiplication (\(EE = E\)) played animportant role. In a natural way, the central idempotents formed aBoolean algebra.

Pursuing this direction of research led Stone to ask about thestructure of an arbitrary Boolean algebra, a question that he answeredby proving thatevery Boolean algebra is isomorphic to a Booleanalgebra of sets. In his work on Boolean algebras he noticed acertain analogy between kernels of homomorphisms and the idealsstudied in ring theory—this led him to give the name“ideal” to such kernels. Not long after this he discovereda translation between Boolean algebras and Boolean rings; under thistranslation the ideals of a Boolean algebra corresponded precisely tothe ideals of the associated Boolean ring. His next major contributionwas to establish a correspondence between Boolean algebras and certaintopological spaces now called Boolean spaces (or Stone spaces). Thiscorrespondence would later prove to be a valuable tool in theconstruction of exotic Boolean algebras. These results of Stone arestill a paradigm for developments in the algebra of logic.

Inspired by the rather brief treatment of first-order statements aboutrelations in Vol. III of theAlgebra der Logik,Löwenheim (1915) showed that if such a statement could besatisfied in an infinite domain then it could be satisfied in adenumerable domain. In 1920 Thoralf Skolem (1887–1963)simplified Löwenheim’s proof by introducing Skolem normal forms,and in 1928 Skolem replaced his use of normal forms with a simpleridea, namely to use what are now called Skolem functions. He usedthese functions to convert first-order sentences into universalsentences, that is to say, into sentences in prenex form with allquantifiers being universal (\(\forall\)).

10. Skolem: Quantifier Elimination and Decidability

Skolem was strongly influenced by Schröder’sAlgebra derLogik, starting with his PhD Thesis. Later he took a particularinterest in the quest for an Elimination Theorem in the calculus ofclasses. In his 1919 paper he established some results for lattices,in particular, he showed that one could decide the validity ofuniversal Horn sentences (i.e., universal sentences with a matrix thatis a disjunction of negated and unnegated atoms, with at most onepositive atom) by a procedure that we now recognize to be a polynomialtime algorithm. This algorithm was based on finding a least fixedpoint of a finite partial lattice under production rules derived fromuniversal Horn sentences. Although this result, which is equivalent tothe uniform word problem for lattices, was in the same paper asSkolem’s famous contribution to Löwenheim’s Theorem, it wasforgotten until a chance rediscovery in the early 1990s. (Whitman(1941) gave a different solution to the more limited equationaldecision problem for lattices; it became widely known as the solutionto the word problem in lattices.)

Skolem (1920) gave an elegant solution to the Elimination Problemposed by Schröder for the calculus of classes by showing that ifone added predicates to express “has at least \(n\)elements”, for each \(n = 1, 2, \ldots\), then there was asimple (but often lengthy) procedure to convert a first-order formulaabout classes into a quantifier-free formula. In particular thisshowed that the first-order theory of the calculus of classes wasdecidable. This quantifier-elimination result was used by Mostowski(1952) to analyze first-order properties of direct powers and directsums of single structures, and then by Feferman and Vaught (1959) todo the same for general direct sums and direct products ofstructures.

The elimination of quantifiers became a main method in mathematicallogic to prove decidability, and proving decidability was stated asthe main problem of mathematical logic in Hilbert and Ackermann(1928)—this goal was dropped in subsequent editions because ofthe famous undecidability result of Church and Turing.

11. Tarski and the Revival of Algebraic Logic

Model theory can be regarded as the product of Hilbert’s methodologyof metamathematics and the algebra of logic tradition, representedspecifically by the results due to Löwenheim and Skolem. But itwas Tarski who gave the discipline its classical foundation. Modeltheory is the study of the relations between a formal language and itsinterpretation in “realizations” (that is, a domain forthe variables of the language together with an interpretation for itsprimitive signs). If the interpretation happens to make a sentence ofthe language state something true, then the interpretation is amodel of the sentence (see the entry onmodel theory). Models consist basically of algebraic structures, and model theorybecame an autonomous mathematical discipline with its roots not onlyin the algebra of logic but in abstract algebra (see Sinaceur1999).

Apart from model theory, Tarski revived the algebra of relations inhis 1941 paper “On the Calculus of Relations”. First heoutlined a formal logic based on allowing quantification over bothelements and relations, and then he turned to a more detailed study ofthe quantifier-free formulas of this system that involved onlyrelation variables. After presenting a list of axioms that obviouslyheld in the algebra of relations as presented in Schröder’s thirdvolume he proved that these axioms allowed one to reducequantifier-free relation formulas to equations. Thus his calculus ofrelations became the study of a certain equational theory which henoted had the same relation to the study of all binary relations onsets as the equational theory of Boolean algebra had to the study ofall subsets of sets. This led to questions paralleling those alreadyposed and resolved for Boolean algebras, for example, was every modelof his axioms for relation algebras isomorphic to an algebra ofrelations on a set? One question had been answered by Arwin Korselt(1864–1947), namely there were first-order sentences in thetheory of binary relations that were not equivalent to an equation inthe calculus of relations—thus the calculus of relationsdefinitely had a weaker expressive power than the first-order theoryof relations. Actually the expressive power of relation algebra isexactly equivalent to first-order logic with just three variables.However, if in relation algebras (the calculus of relations) one wantsto formalize a set theory which has something such as the pair axiom,then one can reduce many variables to three variables, and so it ispossible to express any first-order statement of such a theory by anequation. Monk proved that, unlike the calculus of classes, there isno finite equational basis for the calculus of binary relations (seeMonk 1964). Tarski and Givant (1987) showed that the equational logicof relation algebras is so expressive that one can carry outfirst-order set theory in it.

Furthermore, cylindric algebras, essentially Boolean algebras equippedwith unary cylindric operations \(C_x\) which are intended to capturethe existential quantifiers (\(\exists x\)), were introduced in theyears 1948–1952 by Tarski, working with his students Louise Chinand Frederick Thompson (see Henkin & Tarski 1961), to create analgebra of logic that captured the expressive power of the first-ordertheory of binary relations. Polyadic algebra is another approach to analgebra of logic for first-order logic—it was created by Halmos(1956c). The focus of work in these systems was again to see to whatextent one could parallel the famous results of Stone for Booleanalgebra from the 1930s.

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