Algebraic logic
Encyclopedia
In mathematical logic
, algebraic logic is the study of logic presented in an algebraic style.
What is now usually called classical algebraic logic focuses on the identification and algebraic description of models
appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics
for these deductive system
s) and connected problems like representation
and duality. Well known results like the representation theorem for Boolean algebras and Stone duality
fall under the umbrella of classical algebraic logic.
Work in the more recent abstract algebraic logic
(AAL) focuses on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator.
s, often bounded lattices
, as model
s (interpretations) of certain logic
s, making logic a branch of order theory
.
In algebraic logic:
In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure
which are its models
are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal
and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators."
Algebraic formalisms going beyond first-order logic
in at least some respects include:
, see Brady (2000) and Grattan-Guinness (2000) and their ample references. For postwar history, see Maddux-1991 and Quine-1976.
Algebraic logic has at least two meanings:
Algebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memoranda Leibniz wrote in the 1680s, some of which were published in the 19th century and translated into English by Clarence Lewis in 1918. But nearly all of Leibniz's known work on algebraic logic was published only in 1903 after Louis Couturat
discovered it in Leibniz's Nachlass
. Parkinson (1966) and Loemker (1969) translated selections from Couturat's volume into English.
Brady (2000) discusses the rich historical connections between algebraic logic and model theory
. The founders of model theory, Ernst Schröder
and Leopold Loewenheim, were logicians in the algebraic tradition. Alfred Tarski
, the founder of set theoretic
model theory
as a major branch of contemporary mathematical logic
, also:
Modern mathematical logic
began in 1847, with two pamphlets whose respective authors were Augustus DeMorgan 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 of some writings by Leopold Loewenheim and Thoralf Skolem
, algebraic logic went into eclipse soon after the 1910-13 publication 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 the Parkinson and Loemker translations. Our present understanding of Leibniz as a logician stems mainly from the work of Wolfgang Lenzen, summarized in Lenzen (2004). To see how present-day work in logic
and metaphysics
can draw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000).
s but without much background in order theory
and/or universal algebra
; the book covers these prerequisites at length. This book however has been criticized for poor and sometimes incorrect presentation of AAL results. http://www.jstor.org/stable/3094793 draft
Historical perspective
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, algebraic logic is the study of logic presented in an algebraic style.
What is now usually called classical algebraic logic focuses on the identification and algebraic description of models
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics
Algebraic semantics
An programming language theory, the algebraic semantics of a programming language is a form of axiomatic semantics based on algebraic laws for describing and reasoning about program semantics in a formal manner....
for these deductive system
Deductive system
A deductive system consists of the axioms and rules of inference that can be used to derive the theorems of the system....
s) and connected problems like representation
Representation (mathematics)
In mathematics, representation is a very general relationship that expresses similarities between objects. Roughly speaking, a collection Y of mathematical objects may be said to represent another collection X of objects, provided that the properties and relationships existing among the...
and duality. Well known results like the representation theorem for Boolean algebras and Stone duality
Stone duality
In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they form a natural generalization of Stone's representation...
fall under the umbrella of classical algebraic logic.
Work in the more recent abstract algebraic logic
Abstract Algebraic Logic
In mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...
(AAL) focuses on the process of algebraization itself, like classifying various forms of algebraizability using the Leibniz operator.
Algebras as models of logics
Algebraic logic treats algebraic structureAlgebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s, often bounded lattices
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...
, as model
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
s (interpretations) of certain logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
s, making logic a branch of order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
.
In algebraic logic:
- Variables are tacitly universally quantifiedUniversal quantificationIn predicate logic, universal quantification formalizes the notion that something is true for everything, or every relevant thing....
over some universe of discourse. There are no existentially quantified variablesExistential quantificationIn predicate logic, an existential quantification is the predication of a property or relation to at least one member of the domain. It is denoted by the logical operator symbol ∃ , which is called the existential quantifier...
or open formulaSentence (mathematical logic)In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...
s; - TermsTerm (mathematics)A term is a mathematical expression which may form a separable part of an equation, a series, or another expression.-Definition:In elementary mathematics, a term is either a single number or variable, or the product of several numbers or variables separated from another term by a + or - sign in an...
are built up from variables using primitive and defined operationsOperation (mathematics)The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
. There are no connectiveLogical connectiveIn logic, a logical connective is a symbol or word used to connect two or more sentences in a grammatically valid way, such that the compound sentence produced has a truth value dependent on the respective truth values of the original sentences.Each logical connective can be expressed as a...
s; - FormulaFormulaIn mathematics, a formula is an entity constructed using the symbols and formation rules of a given logical language....
s, built from terms in the usual way, can be equated if they are logically equivalentLogical equivalenceIn logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
. To express a tautologyTautology (logic)In logic, a tautology is a formula which is true in every possible interpretation. Philosopher Ludwig Wittgenstein first applied the term to redundancies of propositional logic in 1921; it had been used earlier to refer to rhetorical tautologies, and continues to be used in that alternate sense...
, equate a formula with a truth value; - The rules of proof are the substitution of equals for equals, and uniform replacement. Modus ponensModus ponensIn classical logic, modus ponendo ponens or implication elimination is a valid, simple argument form. It is related to another valid form of argument, modus tollens. Both Modus Ponens and Modus Tollens can be mistakenly used when proving arguments...
remains valid, but is seldom employed.
In the table below, the left column contains one or more logical or mathematical systems, and the algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
which are its models
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
are shown on the right in the same row. Some of these structures are either Boolean algebras or proper extensions thereof. Modal
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
and other nonclassical logics are typically modeled by what are called "Boolean algebras with operators."
Algebraic formalisms going beyond first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
in at least some respects include:
- Combinatory logicCombinatory logicCombinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming...
, having the expressive power of set theorySet theorySet theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
; - Relation algebraRelation algebraIn mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
, arguably the paradigmatic algebraic logic, can express Peano arithmetic and most axiomatic set theories, including the canonical ZFC.
| logical system | its models |
| Classical sentential logic | Lindenbaum-Tarski algebra Two-element Boolean algebra |
| Intuitionistic Intuitionistic logic Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either... propositional logic |
Heyting algebra Heyting algebra In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b... |
| Łukasiewicz logic | MV-algebra MV-algebra In abstract algebra, a branch of pure mathematics, an MV-algebra is an algebraic structure with a binary operation \oplus, a unary operation \neg, and the constant 0, satisfying certain axioms... |
| Modal logic K Normal modal logic In logic, a normal modal logic is a set L of modal formulas such that L contains:* All propositional tautologies;* All instances of the Kripke schema: \Box\toand it is closed under:... |
Modal algebra Modal logic Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is... |
| Lewis Clarence Irving Lewis Clarence Irving Lewis , usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism. First a noted logician, he later branched into epistemology, and during the last 20 years of his life, he wrote much on ethics.-Early years:Lewis was born in... 's S4 Modal logic Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is... |
Interior algebra Interior algebra In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordinary propositional logic... |
| Lewis Clarence Irving Lewis Clarence Irving Lewis , usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism. First a noted logician, he later branched into epistemology, and during the last 20 years of his life, he wrote much on ethics.-Early years:Lewis was born in... 's S5 S5 (modal logic) In logic and philosophy, S5 is one of five systems of modal logic proposed byClarence Irving Lewis and Cooper Harold Langford in their 1932 book Symbolic Logic.It is a normal modal logic, and one of the oldest systems of modal logic of any kind.... ; Monadic predicate logic |
Monadic Boolean algebra Monadic Boolean algebra In abstract algebra, a monadic Boolean algebra is an algebraic structure with signaturewhere ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.The prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities:... |
| First-order logic First-order logic First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic... |
complete Boolean algebra Boolean-valued model In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take values in some fixed complete Boolean... Cylindric algebra Cylindric algebra The notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Indeed, cylindric algebras are Boolean algebras equipped with additional... Polyadic algebra Polyadic algebra Polyadic algebras are algebraic structures introduced by Paul Halmos. They are related to first-order logic in a way analogous to the relationship between Boolean algebras and propositional logic .There are other ways to relate first-order logic to algebra, including Tarski's cylindric algebras... Predicate functor logic Predicate functor logic In mathematical logic, predicate functor logic is one of several ways to express first-order logic by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors that operate on terms to yield terms... |
| Set theory Set theory Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics... |
Combinatory logic Combinatory logic Combinatory logic is a notation introduced by Moses Schönfinkel and Haskell Curry to eliminate the need for variables in mathematical logic. It has more recently been used in computer science as a theoretical model of computation and also as a basis for the design of functional programming... Relation algebra Relation algebra In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation... |
History
For the history of algebraic logic before World War IIWorld War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, see Brady (2000) and Grattan-Guinness (2000) and their ample references. For postwar history, see Maddux-1991 and Quine-1976.
Algebraic logic has at least two meanings:
- The study of Boolean algebra begun by George BooleGeorge BooleGeorge Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
, and of relation algebraRelation algebraIn mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
begun by Augustus DeMorgan, extended by Charles Sanders Peirce, and taking definitive form in the work of Ernst SchröderErnst SchröderErnst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
; - Abstract algebraic logicAbstract Algebraic LogicIn mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...
, a branch of contemporary mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
.
Algebraic logic is, perhaps, the oldest approach to formal logic, arguably beginning with a number of memoranda Leibniz wrote in the 1680s, some of which were published in the 19th century and translated into English by Clarence Lewis in 1918. But nearly all of Leibniz's known work on algebraic logic was published only in 1903 after Louis Couturat
Louis Couturat
Louis Couturat was a French logician, mathematician, philosopher, and linguist.-Life:Born in Ris-Orangis, Essonne, France, he was educated in philosophy and mathematics at the École Normale Supérieure...
discovered it in Leibniz's Nachlass
Nachlass
Nachlass is a German word, used in academia to describe the collection of manuscripts, notes, correspondence, and so on left behind when a scholar dies. The word is a compound in German: nach means 'after', and the verb lassen means 'leave'. The plural can be either Nachlasse or Nachlässe...
. Parkinson (1966) and Loemker (1969) translated selections from Couturat's volume into English.
Brady (2000) discusses the rich historical connections between algebraic logic and model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
. The founders of model theory, Ernst Schröder
Ernst Schröder
Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
and Leopold Loewenheim, were logicians in the algebraic tradition. Alfred Tarski
Alfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
, the founder of set theoretic
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
as a major branch of contemporary mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, also:
- Co-discovered Lindenbaum-Tarski algebra;
- Invented cylindric algebraCylindric algebraThe notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Indeed, cylindric algebras are Boolean algebras equipped with additional...
; - Wrote the 1941 paper that revived relation algebraRelation algebraIn mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
, which can be viewed as the starting point of abstract algebraic logicAbstract Algebraic LogicIn mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...
.
Modern mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
began in 1847, with two pamphlets whose respective authors were Augustus DeMorgan and George Boole
George Boole
George Boole was an English mathematician and philosopher.As the inventor of Boolean logic—the basis of modern digital computer logic—Boole is regarded in hindsight as a founder of the field of computer science. Boole said,...
. They, and later C.S. Peirce, Hugh MacColl
Hugh MacColl
Hugh MacColl was a Scot who trained as a mathematician and became a logician. MacColl was the youngest son of a poor highland family which was at least in part Gaelic-speaking...
, Frege, Peano, Bertrand Russell
Bertrand Russell
Bertrand Arthur William Russell, 3rd Earl Russell, OM, FRS was a British philosopher, logician, mathematician, historian, and social critic. At various points in his life he considered himself a liberal, a socialist, and a pacifist, but he also admitted that he had never been any of these things...
, and A. N. Whitehead all shared Leibniz's dream of combining symbolic logic, mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, and philosophy
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
. Relation algebra
Relation algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
is arguably the culmination of Leibniz's approach to logic. With the exception of some writings by Leopold Loewenheim and Thoralf Skolem
Thoralf Skolem
Thoralf Albert Skolem was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.-Life:...
, algebraic logic went into eclipse soon after the 1910-13 publication of Principia Mathematica
Principia Mathematica
The Principia Mathematica is a three-volume work on the foundations of mathematics, written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, and 1913...
, not to be revived until Tarski's 1940 re-exposition of relation algebra
Relation algebra
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
.
Leibniz had no influence on the rise of algebraic logic because his logical writings were little studied before the Parkinson and Loemker translations. Our present understanding of Leibniz as a logician stems mainly from the work of Wolfgang Lenzen, summarized in Lenzen (2004). To see how present-day work in logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
and metaphysics
Metaphysics
Metaphysics is a branch of philosophy concerned with explaining the fundamental nature of being and the world, although the term is not easily defined. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:...
can draw inspiration from, and shed light on, Leibniz's thought, see Zalta (2000).
See also
- Abstract algebraic logicAbstract Algebraic LogicIn mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...
- Algebraic structureAlgebraic structureIn abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
- Boolean algebra (logic)
- Boolean algebra (structure)
- Cylindric algebraCylindric algebraThe notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Indeed, cylindric algebras are Boolean algebras equipped with additional...
- Lindenbaum-Tarski algebra
- Mathematical logicMathematical logicMathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
- Model theoryModel theoryIn mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
- Monadic Boolean algebraMonadic Boolean algebraIn abstract algebra, a monadic Boolean algebra is an algebraic structure with signaturewhere ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.The prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities:...
- Predicate functor logicPredicate functor logicIn mathematical logic, predicate functor logic is one of several ways to express first-order logic by purely algebraic means, i.e., without quantified variables. PFL employs a small number of algebraic devices called predicate functors that operate on terms to yield terms...
- Relation algebraRelation algebraIn mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation...
- Universal algebraUniversal algebraUniversal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
Further reading
Good introduction for readers with prior exposure to non-classical logicNon-classical logic
Non-classical logics is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations...
s but without much background in order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
and/or universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
; the book covers these prerequisites at length. This book however has been criticized for poor and sometimes incorrect presentation of AAL results. http://www.jstor.org/stable/3094793 draft
- 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. Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
. - Brady, Geraldine, 2000. From Peirce to Skolem: A neglected chapter in the history of logic. North-Holland/Elsevier Science BV: catalog page, Amsterdam, Netherlands, 625 pages.
- Lenzen, Wolfgang, 2004, "Leibniz’s Logic" 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 MadduxRoger MadduxRoger Maddux is an American mathematician specializing in algebraic logic.He completed his B.A. at Pomona College in 1969, and his Ph.D. in mathematics at the University of California in 1978, where he was one of Alfred Tarski's last students...
, 1991, "The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations," Studia Logica 50: 421-55. - Parkinson, G.H.R., 1966. Leibniz: Logical Papers. Oxford Uni. Press.
- Ivor Grattan-GuinnessIvor Grattan-GuinnessIvor Grattan-Guinness, born 23 June 1941, in Bakewell, in England, is a historian of mathematics and logic.He gained his Bachelor degree as a Mathematics Scholar at Wadham College, Oxford, got an M.Sc in Mathematical Logic and the Philosophy of Science at the London School of Economics in 1966...
, 2000. The Search for Mathematical Roots. Princeton Univ. Press. - Zalta, E. N., 2000, "A (Leibnizian) Theory of Concepts," Philosophiegeschichte und logische Analyse / Logical Analysis and History of Philosophy 3: 137-183.
External links
- Stanford Encyclopedia of PhilosophyStanford Encyclopedia of PhilosophyThe Stanford Encyclopedia of Philosophy is a freely-accessible online encyclopedia of philosophy maintained by Stanford University. Each entry is written and maintained by an expert in the field, including professors from over 65 academic institutions worldwide...
: "Propositional Consequence Relations and Algebraic Logic" -- by Ramon Jansana. (mainly about abstract algebraic logicAbstract Algebraic LogicIn mathematical logic, abstract algebraic logic is the study of the algebraization of deductive systemsarising as an abstraction of the well-known Lindenbaum-Tarski algebra, and how the resulting algebras are related to logical systems.-Overview:...
)