A syllogism ( Ancient Greek : συλλογισμός , syllogismos , 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
91-517: The Gödel Lecture is an honor in mathematical logic given by the Association for Symbolic Logic , associated with an annual lecture at the association's general meeting. The award is named after Kurt Gödel and has been given annually since 1990. The list of award winners and lecture titles is maintained online by the Association for Symbolic Logic. Mathematical logic Mathematical logic
182-415: A cumulative hierarchy of sets. New Foundations takes a different approach; it allows objects such as the set of all sets at the cost of restrictions on its set-existence axioms. The system of Kripke–Platek set theory is closely related to generalized recursion theory. Two famous statements in set theory are the axiom of choice and the continuum hypothesis . The axiom of choice, first stated by Zermelo,
273-529: A calculus, a method of representing categorical statements (and statements that are not provided for in syllogism as well) by the use of quantifiers and variables. A noteworthy exception is the logic developed in Bernard Bolzano 's work Wissenschaftslehre ( Theory of Science , 1837), the principles of which were applied as a direct critique of Kant, in the posthumously published work New Anti-Kant (1850). The work of Bolzano had been largely overlooked until
364-434: A correspondence between syntax and semantics in first-order logic. Gödel used the completeness theorem to prove the compactness theorem , demonstrating the finitary nature of first-order logical consequence . These results helped establish first-order logic as the dominant logic used by mathematicians. In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems , which proved
455-430: A definition of the real numbers in terms of Dedekind cuts of rational numbers, a definition still employed in contemporary texts. Georg Cantor developed the fundamental concepts of infinite set theory. His early results developed the theory of cardinality and proved that the reals and the natural numbers have different cardinalities. Over the next twenty years, Cantor developed a theory of transfinite numbers in
546-400: A finitistic system together with a principle of transfinite induction . Gentzen's result introduced the ideas of cut elimination and proof-theoretic ordinals , which became key tools in proof theory. Gödel gave a different consistency proof, which reduces the consistency of classical arithmetic to that of intuitionistic arithmetic in higher types. The first textbook on symbolic logic for
637-415: A formalized mathematical statement, whether the statement is true or false. Ernst Zermelo gave a proof that every set could be well-ordered , a result Georg Cantor had been unable to obtain. To achieve the proof, Zermelo introduced the axiom of choice , which drew heated debate and research among mathematicians and the pioneers of set theory. The immediate criticism of the method led Zermelo to publish
728-521: A foundational theory for mathematics. Fraenkel proved that the axiom of choice cannot be proved from the axioms of Zermelo's set theory with urelements . Later work by Paul Cohen showed that the addition of urelements is not needed, and the axiom of choice is unprovable in ZF. Cohen's proof developed the method of forcing , which is now an important tool for establishing independence results in set theory. Leopold Löwenheim and Thoralf Skolem obtained
819-494: A function as a rule for computation, or a smooth graph, were no longer adequate. Weierstrass began to advocate the arithmetization of analysis , which sought to axiomatize analysis using properties of the natural numbers. The modern (ε, δ)-definition of limit and continuous functions was already developed by Bolzano in 1817, but remained relatively unknown. Cauchy in 1821 defined continuity in terms of infinitesimals (see Cours d'Analyse, page 34). In 1858, Dedekind proposed
910-756: A method of valid logical reasoning, will always be useful in most circumstances, and for general-audience introductions to logic and clear-thinking. In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism. Aristotle defines the syllogism as "a discourse in which certain (specific) things having been supposed, something different from the things supposed results of necessity because these things are so." Despite this very general definition, in Prior Analytics Aristotle limits himself to categorical syllogisms that consist of three categorical propositions , including categorical modal syllogisms. The use of syllogisms as
1001-421: A milestone in recursion theory and proof theory, but has also led to Löb's theorem in modal logic. The method of forcing is employed in set theory, model theory, and recursion 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 of categorical logic , but category theory is not ordinarily considered
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#17328770259781092-561: A model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory , and they are a key reason for the prominence of first-order logic in mathematics. Gödel's incompleteness theorems establish additional limits on first-order axiomatizations. The first incompleteness theorem states that for any consistent, effectively given (defined below) logical system that
1183-742: A more inductive approach to the observation of nature, which involves experimentation, and leads to discovering and building on axioms to create a more general conclusion. Yet, a full method of drawing conclusions in nature is not the scope of logic or syllogism, and the inductive method was covered in Aristotle's subsequent treatise, the Posterior Analytics . In the 19th century, modifications to syllogism were incorporated to deal with disjunctive ("A or B") and conditional ("if A then B") statements. Immanuel Kant famously claimed, in Logic (1800), that logic
1274-403: A new concept – the computable function – had been discovered, and that this definition was robust enough to admit numerous independent characterizations. In his work on the incompleteness theorems in 1931, Gödel lacked a rigorous concept of an effective formal system; he immediately realized that the new definitions of computability could be used for this purpose, allowing him to state
1365-485: A particular sentence is true in every model that satisfies a particular set of axioms, then there must be a finite deduction of the sentence from the axioms. The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has
1456-422: A portion of set theory directly in their semantics. The most well studied infinitary logic is L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} . In this logic, quantifiers may only be nested to finite depths, as in first-order logic, but formulas may have finite or countably infinite conjunctions and disjunctions within them. Thus, for example, it
1547-436: A premise containing the modal words necessarily , possibly , or contingently . Aristotle's terminology in this aspect of his theory was deemed vague, and in many cases unclear, even contradicting some of his statements from On Interpretation . His original assertions on this specific component of the theory were left up to a considerable amount of conversation, resulting in a wide array of solutions put forth by commentators of
1638-439: A rectangle is a square that is a quadrangle." A categorical syllogism consists of three parts: Each part is a categorical proposition , and each categorical proposition contains two categorical terms. In Aristotle, each of the premises is in the form "All S are P," "Some S are P", "No S are P" or "Some S are not P", where "S" is the subject-term and "P" is the predicate-term: More modern logicians allow some variation. Each of
1729-478: A rough division of contemporary mathematical logic into four areas: Additionally, sometimes the field of computational complexity theory is also included as part of mathematical logic. Each area has a distinct focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic and other fields of mathematics, are not always sharp. Gödel's incompleteness theorem marks not only
1820-485: A second exposition of his result, directly addressing criticisms of his proof. This paper led to the general acceptance of the axiom of choice in the mathematics community. Skepticism about the axiom of choice was reinforced by recently discovered paradoxes in naive set theory . Cesare Burali-Forti was the first to state a paradox: the Burali-Forti paradox shows that the collection of all ordinal numbers cannot form
1911-423: A separate domain for each higher-type quantifier to range over, the quantifiers instead range over all objects of the appropriate type. The logics studied before the development of first-order logic, for example Frege's logic, had similar set-theoretic aspects. Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of
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#17328770259782002-404: A series of publications. In 1891, he published a new proof of the uncountability of the real numbers that introduced the diagonal argument , and used this method to prove Cantor's theorem that no set can have the same cardinality as its powerset . Cantor believed that every set could be well-ordered , but was unable to produce a proof for this result, leaving it as an open problem in 1895. In
2093-457: A set. Very soon thereafter, Bertrand Russell discovered Russell's paradox in 1901, and Jules Richard discovered Richard's paradox . Zermelo provided the first set of axioms for set theory. These axioms, together with the additional axiom of replacement proposed by Abraham Fraenkel , are now called Zermelo–Fraenkel set theory (ZF). Zermelo's axioms incorporated the principle of limitation of size to avoid Russell's paradox. In 1910,
2184-571: A stronger limitation than the one established by the Löwenheim–Skolem theorem. The second incompleteness theorem states that no sufficiently strong, consistent, effective axiom system for arithmetic can prove its own consistency, which has been interpreted to show that Hilbert's program cannot be reached. Many logics besides first-order logic are studied. These include infinitary logics , which allow for formulas to provide an infinite amount of information, and higher-order logics , which include
2275-402: A subfield of mathematical logic. Because of its applicability in diverse fields of mathematics, mathematicians including Saunders Mac Lane have proposed category theory as a foundational system for mathematics, independent of set theory. These foundations use toposes , which resemble generalized models of set theory that may employ classical or nonclassical logic. Mathematical logic emerged in
2366-409: A tool for understanding can be dated back to the logical reasoning discussions of Aristotle . Before the mid-12th century, medieval logicians were only familiar with a portion of Aristotle's works, including such titles as Categories and On Interpretation , works that contributed heavily to the prevailing Old Logic, or logica vetus . The onset of a New Logic, or logica nova , arose alongside
2457-500: A variation of the logical system of Boole and Schröder but adding quantifiers. Peano was unaware of Frege's work at the time. Around the same time Richard Dedekind showed that the natural numbers are uniquely characterized by their induction properties. Dedekind 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
2548-581: Is computable ; this is not true in classical theories of arithmetic such as Peano arithmetic . Algebraic logic uses the methods of abstract algebra to study the semantics of formal logics. A fundamental example is the use of Boolean algebras to represent truth values in classical propositional logic, and the use of Heyting algebras to represent truth values in intuitionistic propositional logic. Stronger logics, such as first-order logic and higher-order logic, are studied using more complicated algebraic structures such as cylindric algebras . Set theory
2639-498: Is a man (minor premise), we may validly conclude that Socrates is mortal. Syllogistic arguments are usually represented in a three-line form: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. In antiquity, two rival syllogistic theories existed: Aristotelian syllogism and Stoic syllogism . From the Middle Ages onwards, categorical syllogism and syllogism were usually used interchangeably. This article
2730-517: Is a particular formal system of logic . Its syntax involves only finite expressions as well-formed formulas , while its semantics are characterized by the limitation of all quantifiers to a fixed domain of discourse . Early results from formal logic established limitations of first-order logic. The Löwenheim–Skolem theorem (1919) showed that if a set of sentences in a countable first-order language has an infinite model then it has at least one model of each infinite cardinality. This shows that it
2821-438: Is about drawing valid conclusions from assumptions ( axioms ), rather than about verifying the assumptions. However, people over time focused on the logic aspect, forgetting the importance of verifying the assumptions. In the 17th century, Francis Bacon emphasized that experimental verification of axioms must be carried out rigorously, and cannot take syllogism itself as the best way to draw conclusions in nature. Bacon proposed
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2912-449: Is capable of interpreting arithmetic, there exists a statement that is true (in the sense that it holds for the natural numbers) but not provable within that logical system (and which indeed may fail in some non-standard models of arithmetic which may be consistent with the logical system). For example, in every logical system capable of expressing the Peano axioms , the Gödel sentence holds for
3003-480: Is concerned only with this historical use. The syllogism was at the core of historical deductive reasoning, whereby facts are determined by combining existing statements, in contrast to inductive reasoning , in which facts are predicted by repeated observations. Within some academic contexts, syllogism has been superseded by first-order predicate logic following the work of Gottlob Frege , in particular his Begriffsschrift ( Concept Script ; 1879). Syllogism, being
3094-521: Is emphasized by the historian of logic John Corcoran in an accessible introduction to Laws of Thought . Corcoran also wrote a point-by-point comparison of Prior Analytics and Laws of Thought . According to Corcoran, Boole fully accepted and endorsed Aristotle's logic. Boole's goals were "to go under, over, and beyond" Aristotle's logic by: More specifically, Boole agreed with what Aristotle said; Boole's 'disagreements', if they might be called that, concern what Aristotle did not say. First, in
3185-737: Is explicated in modern fora of academia primarily in introductory material and historical study. One notable exception to this modern relegation is the continued application of Aristotelian logic by officials of the Congregation for the Doctrine of the Faith , and the Apostolic Tribunal of the Roman Rota , which still requires that any arguments crafted by Advocates be presented in syllogistic format. George Boole 's unwavering acceptance of Aristotle's logic
3276-468: Is impossible for a set of first-order axioms to characterize the natural numbers, the real numbers, or any other infinite structure up to isomorphism . As the goal of early foundational studies was to produce axiomatic theories for all parts of mathematics, this limitation was particularly stark. Gödel's completeness theorem established the equivalence between semantic and syntactic definitions of logical consequence in first-order logic. It shows that if
3367-427: Is known as the middle term ; in this example, humans . Both of the premises are universal, as is the conclusion. Here, the major term is die , the minor term is men , and the middle term is mortals . Again, both premises are universal, hence so is the conclusion. A polysyllogism, or a sorites , is a form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms
3458-417: Is one of many counterintuitive results of the axiom of choice. Syllogism In its earliest form (defined by Aristotle in his 350 BC book Prior Analytics ), a deductive syllogism arises when two true premises (propositions or statements) validly imply a conclusion, or the main point that the argument aims to get across. For example, knowing that all men are mortal (major premise), and that Socrates
3549-446: Is possible to say that an object is a whole number using a formula of L ω 1 , ω {\displaystyle L_{\omega _{1},\omega }} such as Higher-order logics allow for quantification not only of elements of the domain of discourse , but subsets of the domain of discourse, sets of such subsets, and other objects of higher type. The semantics are defined so that, rather than having
3640-556: Is the study of formal logic within mathematics . Major subareas include model theory , proof theory , set theory , and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics . Since its inception, mathematical logic has both contributed to and been motivated by
3731-642: Is the study of sets , which are abstract collections of objects. Many of the basic notions, such as ordinal and cardinal numbers, were developed informally by Cantor before formal axiomatizations of set theory were developed. The first such axiomatization , due to Zermelo, was extended slightly to become Zermelo–Fraenkel set theory (ZF), which is now the most widely used foundational theory for mathematics. Other formalizations of set theory have been proposed, including von Neumann–Bernays–Gödel set theory (NBG), Morse–Kelley set theory (MK), and New Foundations (NF). Of these, ZF, NBG, and MK are similar in describing
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3822-430: Is the subject of the conclusion, P is the predicate of the conclusion, and M is the middle term. The major premise links M with P and the minor premise links M with S. However, the middle term can be either the subject or the predicate of each premise where it appears. The differing positions of the major, minor, and middle terms gives rise to another classification of syllogisms known as the figure . Given that in each case
3913-401: Is the theorem that a line contains at least two points, or that circles of the same radius whose centers are separated by that radius must intersect. Hilbert developed a complete set of axioms for geometry , building on previous work by Pasch. The success in axiomatizing geometry motivated Hilbert to seek complete axiomatizations of other areas of mathematics, such as the natural numbers and
4004-404: Is traditional to use is rather than are as the copula , hence All A is B rather than All As are Bs . It is traditional and convenient practice to use a, e, i, o as infix operators so the categorical statements can be written succinctly. The following table shows the longer form, the succinct shorthand, and equivalent expressions in predicate logic: The convention here is that the letter S
4095-529: The Organon , found wide application and acceptance in Western science and mathematics for millennia. The Stoics , especially Chrysippus , began the development of predicate logic . In 18th-century Europe, attempts to treat the operations of formal logic in a symbolic or algebraic way had been made by philosophical mathematicians including Leibniz and Lambert , but their labors remained isolated and little known. In
4186-500: The Löwenheim–Skolem theorem , which says that first-order logic cannot control the cardinalities of infinite structures. Skolem realized that this theorem would apply to first-order formalizations of set theory, and that it implies any such formalization has a countable model . This counterintuitive fact became known as Skolem's paradox . In his doctoral thesis, Kurt Gödel proved the completeness theorem , which establishes
4277-421: The real line . This would prove to be a major area of research in the first half of the 20th century. The 19th century saw great advances in the theory of real analysis , including theories of convergence of functions and Fourier series . Mathematicians such as Karl Weierstrass began to construct functions that stretched intuition, such as nowhere-differentiable continuous functions . Previous conceptions of
4368-403: The 19th century. Concerns that mathematics had not been built on a proper foundation led to the development of axiomatic systems for 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 published a set of axioms for arithmetic that came to bear his name ( Peano axioms ), using
4459-561: The Venn diagrams, the black areas indicate no elements, and the red areas indicate at least one element. In the predicate logic expressions, a horizontal bar over an expression means to negate ("logical not") the result of that expression. It is also possible to use graphs (consisting of vertices and edges) to evaluate syllogisms. Similar: Cesare (EAE-2) Camestres is essentially like Celarent with S and P exchanged. Similar: Calemes (AEE-4) Similar: Datisi (AII-3) Disamis
4550-520: The categorical syllogism were central to expanding the syllogistic discussion. Rather than in any additions that he personally made to the field, Boethius' logical legacy lies in his effective transmission of prior theories to later logicians, as well as his clear and primarily accurate presentations of Aristotle's contributions. Another of medieval logic's first contributors from the Latin West, Peter Abelard (1079–1142), gave his own thorough evaluation of
4641-469: The collection is nonempty, the lack of a general, concrete rule by which the choice can be made renders the axiom nonconstructive. Stefan Banach and Alfred Tarski showed that the axiom of choice can be used to decompose a solid ball into a finite number of pieces which can then be rearranged, with no scaling, to make two solid balls of the original size. This theorem, known as the Banach–Tarski paradox ,
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#17328770259784732-608: The completeness and compactness theorems from first-order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are fixed-point logic s that allow inductive definitions , like one writes for primitive recursive functions . One can formally define an extension of first-order logic — a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic . Lindström's theorem implies that
4823-403: The concept over time. This theory of the syllogism would not enter the context of the more comprehensive logic of consequence until logic began to be reworked in general in the mid-14th century by the likes of John Buridan . Aristotle's Prior Analytics did not, however, incorporate such a comprehensive theory on the modal syllogism—a syllogism that has at least one modalized premise, that is,
4914-426: The conclusion can be of type A, E, I or O, and the syllogism can be any of the four figures. A syllogism can be described briefly by giving the letters for the premises and conclusion followed by the number for the figure. For example, the syllogism BARBARA below is AAA-1, or "A-A-A in the first figure". The vast majority of the 256 possible forms of syllogism are invalid (the conclusion does not follow logically from
5005-431: The conclusion is S-P, the four figures are: (Note, however, that, following Aristotle's treatment of the figures, some logicians—e.g., Peter Abelard and Jean Buridan —reject the fourth figure as a figure distinct from the first.) Putting it all together, there are 256 possible types of syllogisms (or 512 if the order of the major and minor premises is changed, though this makes no difference logically). Each premise and
5096-412: The consistency of elementary arithmetic, respectively; the tenth was to produce a method that could decide whether a multivariate polynomial equation over the integers has a solution. Subsequent work to resolve these problems shaped the direction of mathematical logic, as did the effort to resolve Hilbert's Entscheidungsproblem , posed in 1928. This problem asked for a procedure that would decide, given
5187-679: The context of proof theory. At its core, mathematical logic deals with mathematical concepts expressed using formal logical systems . These systems, though they differ in many details, share the common property of considering only expressions in a fixed formal language . The systems of propositional logic and first-order logic are the most widely studied today, because of their applicability to foundations of mathematics and because of their desirable proof-theoretic properties. Stronger classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic . First-order logic
5278-406: The day. The system for modal syllogisms laid forth by Aristotle would ultimately be deemed unfit for practical use, and would be replaced by new distinctions and new theories altogether. Boethius (c. 475–526) contributed an effort to make the ancient Aristotelian logic more accessible. While his Latin translation of Prior Analytics went primarily unused before the 12th century, his textbooks on
5369-407: The early decades of the 20th century, the main areas of study were set theory and formal logic. The discovery of paradoxes in informal set theory caused some to wonder whether mathematics itself is inconsistent, and to look for proofs of consistency. In 1900, Hilbert posed a famous list of 23 problems for the next century. The first two of these were to resolve the continuum hypothesis and prove
5460-450: The first volume of Principia Mathematica by Russell and Alfred North Whitehead was published. This seminal work developed the theory of functions and cardinality in a completely formal framework of type theory , which Russell and Whitehead developed in an effort to avoid the paradoxes. Principia Mathematica is considered one of the most influential works of the 20th century, although the framework of type theory did not prove popular as
5551-405: The foremost logician of the later Middle Ages, contributed two significant works: Treatise on Consequence and Summulae de Dialectica , in which he discussed the concept of the syllogism, its components and distinctions, and ways to use the tool to expand its logical capability. For 200 years after Buridan's discussions, little was said about syllogistic logic. Historians of logic have assessed that
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#17328770259785642-422: The form (note: M – Middle, S – subject, P – predicate.): The premises and conclusion of a syllogism can be any of four types, which are labeled by letters as follows. The meaning of the letters is given by the table: In Prior Analytics , Aristotle uses mostly the letters A, B, and C (Greek letters alpha , beta , and gamma ) as term place holders, rather than giving concrete examples. It
5733-423: The importance of the incompleteness theorem for some time. Gödel's theorem shows that a consistency proof of any sufficiently strong, effective axiom system cannot be obtained in the system itself, if the system is consistent, nor in any weaker system. This leaves open the possibility of consistency proofs that cannot be formalized within the system they consider. Gentzen proved the consistency of arithmetic using
5824-443: The incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem , establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program. It showed the impossibility of providing a consistency proof of arithmetic within any formal theory of arithmetic. Hilbert, however, did not acknowledge
5915-491: The incompleteness theorems in generality that could only be implied in the original paper. Numerous results in recursion theory were obtained in the 1940s by Stephen Cole Kleene and Emil Leon Post . Kleene introduced the concepts of relative computability, foreshadowed by Turing, and the arithmetical hierarchy . Kleene later generalized recursion theory to higher-order functionals. Kleene and Georg Kreisel studied formal versions of intuitionistic mathematics, particularly in
6006-559: The issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing 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. The Handbook of Mathematical Logic in 1977 makes
6097-605: The late 20th century, among other reasons, because of the intellectual environment at the time in Bohemia , which was then part of the Austrian Empire . In the last 20 years, Bolzano's work has resurfaced and become subject of both translation and contemporary study. This led to the rapid development of sentential logic and first-order predicate logic , subsuming syllogistic reasoning, which was, therefore, after 2000 years, suddenly considered obsolete by many. The Aristotelian system
6188-569: The layman was written by Lewis Carroll , author of Alice's Adventures in Wonderland , in 1896. Alfred Tarski developed the basics of model theory . Beginning in 1935, a group of prominent mathematicians collaborated under the pseudonym Nicolas Bourbaki to publish Éléments de mathématique , a series of encyclopedic mathematics texts. These texts, written in an austere and axiomatic style, emphasized rigorous presentation and set-theoretic foundations. Terminology coined by these texts, such as
6279-457: The mid-19th century as a subfield of mathematics, reflecting the confluence of two traditions: formal philosophical logic and mathematics. Mathematical logic, also called 'logistic', 'symbolic logic', the ' algebra of logic ', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. Before this emergence, logic
6370-487: The middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by algebraists such as George Peacock , extended the traditional Aristotelian doctrine of logic into a sufficient framework for the study of foundations of mathematics . In 1847, Vatroslav Bertić made substantial work on algebraization of logic, independently from Boole. Charles Sanders Peirce later built upon
6461-482: The natural numbers but cannot be proved. Here a logical system is said to be effectively given if it is possible to decide, given any formula in the language of the system, whether the formula is an axiom, and one which can express the Peano axioms is called "sufficiently strong." When applied to first-order logic, the first incompleteness theorem implies that any sufficiently strong, consistent, effective first-order theory has models that are not elementarily equivalent ,
6552-486: The only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic. Modal logics include additional modal operators, such as an operator which states that a particular formula is not only true, but necessarily true. Although modal logic is not often used to axiomatize mathematics, it has been used to study the properties of first-order provability and set-theoretic forcing. Intuitionistic logic
6643-479: The premises has one term in common with the conclusion: in a major premise, this is the major term (i.e., the predicate of the conclusion); in a minor premise, this is the minor term (i.e., the subject of the conclusion). For example: Each of the three distinct terms represents a category. From the example above, humans , mortal , and Greeks : mortal is the major term, and Greeks the minor term. The premises also have one term in common with each other, which
6734-403: The premises). The table below shows the valid forms. Even some of these are sometimes considered to commit the existential fallacy , meaning they are invalid if they mention an empty category. These controversial patterns are marked in italics . All but four of the patterns in italics (felapton, darapti, fesapo and bamalip) are weakened moods, i.e. it is possible to draw a stronger conclusion from
6825-455: The premises. The letters A, E, I, and O have been used since the medieval Schools to form mnemonic names for the forms as follows: 'Barbara' stands for AAA, 'Celarent' for EAE, etc. Next to each premise and conclusion is a shorthand description of the sentence. So in AAI-3, the premise "All squares are rectangles" becomes "MaP"; the symbols mean that the first term ("square") is the middle term,
6916-425: The primary changes in the post-Middle Age era were changes in respect to the public's awareness of original sources, a lessening of appreciation for the logic's sophistication and complexity, and an increase in logical ignorance—so that logicians of the early 20th century came to view the whole system as ridiculous. The Aristotelian syllogism dominated Western philosophical thought for many centuries. Syllogism itself
7007-401: The realm of applications, Boole's system could handle multi-term propositions and arguments, whereas Aristotle could handle only two-termed subject-predicate propositions and arguments. For example, Aristotle's system could not deduce: "No quadrangle that is a square is a rectangle that is a rhombus" from "No square that is a quadrangle is a rhombus that is a rectangle" or from "No rhombus that is
7098-420: The realm of foundations, Boole reduced Aristotle's four propositional forms to one form, the form of equations, which by itself was a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving to logic—another revolutionary idea—involved Boole's doctrine that Aristotle's rules of inference (the "perfect syllogisms") must be supplemented by rules for equation solving. Third, in
7189-447: The reappearance of Prior Analytics , the work in which Aristotle developed his theory of the syllogism. Prior Analytics , upon rediscovery, was instantly regarded by logicians as "a closed and complete body of doctrine", leaving very little for thinkers of the day to debate, and reorganize. Aristotle's theory on the syllogism for assertoric sentences was considered especially remarkable, with only small systematic changes occurring to
7280-514: The second term ("rectangle") is the predicate of the conclusion, and the relationship between the two terms is labeled "a" (All M are P). The following table shows all syllogisms that are essentially different. The similar syllogisms share the same premises, just written in a different way. For example "Some pets are kittens" (SiM in Darii ) could also be written as "Some kittens are pets" (MiS in Datisi). In
7371-410: The study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry , arithmetic , and analysis . In the early 20th century it was shaped by David Hilbert 's program to prove the consistency of foundational theories. Results of Kurt Gödel , Gerhard Gentzen , and others provided partial resolution to the program, and clarified
7462-471: The subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. For example, one might argue that all lions are big cats, all big cats are predators, and all predators are carnivores. To conclude that therefore all lions are carnivores is to construct a sorites argument. There are infinitely many possible syllogisms, but only 256 logically distinct types and only 24 valid types (enumerated below). A syllogism takes
7553-573: The syllogism concept, and accompanying theory in the Dialectica —a discussion of logic based on Boethius' commentaries and monographs. His perspective on syllogisms can be found in other works as well, such as Logica Ingredientibus . With the help of Abelard's distinction between de dicto modal sentences and de re modal sentences, medieval logicians began to shape a more coherent concept of Aristotle's modal syllogism model. The French philosopher Jean Buridan (c. 1300 – 1361), whom some consider
7644-402: The turn of the century. The two-dimensional notation Frege developed was never widely adopted and is unused in contemporary texts. From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes. This work summarized and extended the work of Boole, De Morgan, and Peirce, and was a comprehensive reference to symbolic logic as it was understood at the end of
7735-553: The uniqueness of the set of natural numbers (up to isomorphism) and the recursive definitions of addition and multiplication from the successor function and mathematical induction. In the mid-19th century, flaws in Euclid's axioms for geometry became known. In addition to the independence of the parallel postulate , established by Nikolai Lobachevsky in 1826, mathematicians discovered that certain theorems taken for granted by Euclid were not in fact provable from his axioms. Among these
7826-463: The words bijection , injection , and surjection , and the set-theoretic foundations the texts employed, were widely adopted throughout mathematics. The study of computability came to be known as recursion theory or computability theory , because early formalizations by Gödel and Kleene relied on recursive definitions of functions. When these definitions were shown equivalent to Turing's formalization involving Turing machines , it became clear that
7917-430: The work of Boole to develop a logical system for relations and quantifiers, which he published in several papers from 1870 to 1885. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift , published in 1879, a work generally considered as marking a turning point in the history of logic. Frege's work remained obscure, however, until Bertrand Russell began to promote it near
8008-486: Was developed by Heyting to study Brouwer's program of intuitionism, in which Brouwer himself avoided formalization. Intuitionistic logic specifically does not include the law of the excluded middle , which states that each sentence is either true or its negation is true. Kleene's work with the proof theory of intuitionistic logic showed that constructive information can be recovered from intuitionistic proofs. For example, any provably total function in intuitionistic arithmetic
8099-407: Was proved independent of ZF by Fraenkel, but has come to be widely accepted by mathematicians. It states that given a collection of nonempty sets there is a single set C that contains exactly one element from each set in the collection. The set C is said to "choose" one element from each set in the collection. While the ability to make such a choice is considered obvious by some, since each set in
8190-496: Was studied with rhetoric , with calculationes , through the syllogism , 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. Theories of logic were developed in many cultures in history, including China , India , Greece and the Islamic world . Greek methods, particularly Aristotelian logic (or term logic) as found in
8281-456: Was the one completed science, and that Aristotelian logic more or less included everything about logic that there was to know. (This work is not necessarily representative of Kant's mature philosophy, which is often regarded as an innovation to logic itself.) Kant's opinion stood unchallenged in the West until 1879, when Gottlob Frege published his Begriffsschrift ( Concept Script ). This introduced
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