115-543: The Alfred Tarski Lectures are an annual distinction in mathematical logic and series of lectures held at the University of California, Berkeley . Established in tribute to Alfred Tarski on the fifth anniversary of his death, the award has been given every year since 1989. Following a 2-year hiatus after the 2020 lecture was not given due to the COVID-19 pandemic , the lectures resumed in 2023. The list of past Tarski lecturers
230-415: 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,
345-589: A civic art of rhetoric, combining the almost incompatible properties of techne and appropriateness to citizens." Each of Aristotle's divisions plays a role in civic life and can be used in a different way to affect the polis . Because rhetoric is a public art capable of shaping opinion, some of the ancients, including Plato found fault in it. They claimed that while it could be used to improve civic life, it could be used just as easily to deceive or manipulate. The masses were incapable of analyzing or deciding anything on their own and would therefore be swayed by
460-434: 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
575-594: A course of study has evolved since its ancient beginnings, and has adapted to the particular exigencies of various times, venues, and applications ranging from architecture to literature. Although the curriculum has transformed in a number of ways, it has generally emphasized the study of principles and rules of composition as a means for moving audiences. Rhetoric began as a civic art in Ancient Greece where students were trained to develop tactics of oratorical persuasion, especially in legal disputes. Rhetoric originated in
690-430: 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
805-400: 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
920-415: 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
1035-521: 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
1150-494: 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
1265-576: A good man, a person enlightened on a variety of civic topics. He describes the proper training of the orator in his major text on rhetoric, De Oratore , which he modeled on Plato's dialogues. Modern works continue to support the claims of the ancients that rhetoric is an art capable of influencing civic life. In Political Style , Robert Hariman claims that "questions of freedom, equality, and justice often are raised and addressed through performances ranging from debates to demonstrations without loss of moral content". James Boyd White argues that rhetoric
SECTION 10
#17328765900771380-473: A group. This definition of rhetoric as identification broadens the scope from strategic and overt political persuasion to the more implicit tactics of identification found in an immense range of sources . Among the many scholars who have since pursued Burke's line of thought, James Boyd White sees rhetoric as a broader domain of social experience in his notion of constitutive rhetoric . Influenced by theories of social construction , White argues that culture
1495-518: A guide to creating persuasive messages and arguments: Memory was added much later to the original four canons. During the Renaissance rhetoric enjoyed a resurgence, and as a result nearly every author who wrote about music before the Romantic era discussed rhetoric. Joachim Burmeister wrote in 1601, "there is only little difference between music and the nature of oration". Christoph Bernhard in
1610-518: A limited field, ignoring many critical applications of rhetorical theory, criticism, and practice. Simultaneously, the neo-Sophists threaten to expand rhetoric beyond a point of coherent theoretical value. In more recent years, people studying rhetoric have tended to enlarge its object domain beyond speech. Kenneth Burke asserted humans use rhetoric to resolve conflicts by identifying shared characteristics and interests in symbols. People engage in identification , either to assign themselves or another to
1725-421: 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
1840-561: 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
1955-403: 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
2070-485: 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
2185-470: A persuasive speech, were first codified in classical Rome: invention , arrangement , style , memory , and delivery . From Ancient Greece to the late 19th century, rhetoric played a central role in Western education in training orators , lawyers , counsellors, historians , statesmen , and poets . Scholars have debated the scope of rhetoric since ancient times. Although some have limited rhetoric to
2300-422: 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
2415-404: A positive image, potentially at the expense of suppressing dissent or criticism. An example of this is the government's actions in freezing bank accounts and regulating internet speech, ostensibly to protect the vulnerable and preserve freedom of expression, despite contradicting values and rights. The origins of the rhetoric language begin in Ancient Greece. It originally began by a group named
SECTION 20
#17328765900772530-473: A revival with the rise of democratic institutions during the late 18th and early 19th centuries. Hugh Blair was a key early leader of this movement. In his most famous work, Lectures on Rhetoric and Belles Lettres , he advocates rhetorical study for common citizens as a resource for social success. Many American colleges and secondary schools used Blair's text throughout the 19th century to train students of rhetoric. Political rhetoric also underwent renewal in
2645-478: 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
2760-521: A school of pre-Socratic philosophers known as the Sophists c. 600 BCE . Demosthenes and Lysias emerged as major orators during this period, and Isocrates and Gorgias as prominent teachers. Modern teachings continue to reference these rhetoricians and their work in discussions of classical rhetoric and persuasion. Rhetoric was taught in universities during the Middle Ages as one of
2875-485: 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
2990-423: 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
3105-404: 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
3220-457: 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,
3335-571: 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
3450-402: 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
3565-443: A tool to influence communities from local to national levels. Political parties employ "manipulative rhetoric" to advance their party-line goals and lobbyist agendas. They use it to portray themselves as champions of compassion, freedom, and culture, all while implementing policies that appear to contradict these claims. It serves as a form of political propaganda, presented to sway and maintain public opinion in their favor, and garner
Tarski Lectures - Misplaced Pages Continue
3680-500: 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
3795-581: 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
3910-425: Is "reconstituted" through language. Just as language influences people, people influence language. Language is socially constructed, and depends on the meanings people attach to it. Because language is not rigid and changes depending on the situation, the very usage of language is rhetorical. An author, White would say, is always trying to construct a new world and persuading his or her readers to share that world within
4025-438: Is Epistemic?". In it, he focuses on uncovering the most appropriate definitions for the terms "rhetoric", "knowledge", and "certainty". According to Harpine, certainty is either objective or subjective. Although both Scotts and Cherwitz and Hikins theories deal with some form of certainty, Harpine believes that knowledge is not required to be neither objectively nor subjectively certain. In terms of "rhetoric", Harpine argues that
4140-517: 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
4255-529: Is also known for describing her process of invention in "The Exaltation of Inanna," moving between first- and third-person address to relate her composing process in collaboration with the goddess Inanna, reflecting a mystical enthymeme in drawing upon a Cosmic audience. Later examples of early rhetoric can be found in the Neo-Assyrian Empire during the time of Sennacherib (704–681 BCE ). In ancient Egypt , rhetoric had existed since at least
4370-464: Is an overwhelming majority that does support the concept of certainty as a requirement for knowledge , but it is at the definition of certainty where parties begin to diverge. One definition maintains that certainty is subjective and feeling-based, the other that it is a byproduct of justification . The more commonly accepted definition of rhetoric claims it is synonymous with persuasion . For rhetorical purposes, this definition, like many others,
4485-468: Is capable not only of addressing issues of political interest but that it can influence culture as a whole. In his book, When Words Lose Their Meaning , he argues that words of persuasion and identification define community and civic life. He states that words produce "the methods by which culture is maintained, criticized, and transformed". Rhetoric remains relevant as a civic art. In speeches, as well as in non-verbal forms, rhetoric continues to be used as
4600-449: 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
4715-516: Is difficult to define. Political discourse remains the paradigmatic example for studying and theorizing specific techniques and conceptions of persuasion or rhetoric. Throughout European History , rhetoric meant persuasion in public and political settings such as assemblies and courts. Because of its associations with democratic institutions, rhetoric is commonly said to flourish in open and democratic societies with rights of free speech , free assembly, and political enfranchisement for some portion of
Tarski Lectures - Misplaced Pages Continue
4830-496: Is especially used by the fields of marketing, politics, and literature. Another area of rhetoric is the study of cultural rhetorics, which is the communication that occurs between cultures and the study of the way members of a culture communicate with each other. These ideas can then be studied and understood by other cultures, in order to bridge gaps in modes of communication and help different cultures communicate effectively with each other. James Zappen defines cultural rhetorics as
4945-468: 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
5060-638: Is maintained by UC Berkeley. Mathematical logic Mathematical logic 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
5175-655: Is merely a form of flattery and functions similarly to culinary arts , which mask the undesirability of unhealthy food by making it taste good. Plato considered any speech of lengthy prose aimed at flattery as within the scope of rhetoric. Some scholars, however, contest the idea that Plato despised rhetoric and instead view his dialogues as a dramatization of complex rhetorical principles. Aristotle both redeemed rhetoric from his teacher and narrowed its focus by defining three genres of rhetoric— deliberative , forensic or judicial, and epideictic . Yet, even as he provided order to existing rhetorical theories, Aristotle generalized
5290-542: Is no institution devised by man which the power of speech has not helped us to establish." With this statement he argues that rhetoric is a fundamental part of civic life in every society and that it has been necessary in the foundation of all aspects of society. He further argues in Against the Sophists that rhetoric, although it cannot be taught to just anyone, is capable of shaping the character of man. He writes, "I do think that
5405-641: Is one of many counterintuitive results of the axiom of choice. Rhetoric Rhetoric ( / ˈ r ɛ t ə r ɪ k / ) is the art of persuasion . It is one of the three ancient arts of discourse ( trivium ) along with grammar and logic / dialectic . As an academic discipline within the humanities , rhetoric aims to study the techniques that speakers or writers use to inform, persuade, and motivate their audiences . Rhetoric also provides heuristics for understanding, discovering, and developing arguments for particular situations. Aristotle defined rhetoric as "the faculty of observing in any given case
5520-446: 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
5635-521: Is still associated with its political origins. However, even the original instructors of Western speech—the Sophists —disputed this limited view of rhetoric. According to Sophists like Gorgias , a successful rhetorician could speak convincingly on a topic in any field, regardless of his experience in that field. This suggested rhetoric could be a means of communicating any expertise, not just politics. In his Encomium to Helen , Gorgias even applied rhetoric to fiction by seeking, for his amusement, to prove
5750-642: 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
5865-401: 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
SECTION 50
#17328765900775980-521: Is too broad. The same issue presents itself with definitions that are too narrow. Rhetoricians in support of the epistemic view of rhetoric have yet to agree in this regard. Philosophical teachings refer to knowledge as a justified true belief . However, the Gettier Problem explores the room for fallacy in this concept. Therefore, the Gettier Problem impedes the effectivity of the argument of Richard A. Cherwitz and James A. Hikins, who employ
6095-529: 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
6210-500: 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
6325-551: The Middle Kingdom period ( c. 2080–1640 BCE ). The five canons of eloquence in ancient Egyptian rhetoric were silence, timing, restraint, fluency, and truthfulness. The Egyptians held eloquent speaking in high esteem. Egyptian rules of rhetoric specified that "knowing when not to speak is essential, and very respected, rhetorical knowledge", making rhetoric a "balance between eloquence and wise silence". They also emphasized "adherence to social behaviors that support
6440-402: The justified true belief standpoint in their argument for rhetoric as epistemic . Celeste Condit Railsback takes a different approach, drawing from Ray E. McKerrow's system of belief based on validity rather than certainty . William D. Harpine refers to the issue of unclear definitions that occurs in the theories of "rhetoric is epistemic" in his 2004 article "What Do You Mean, Rhetoric
6555-421: 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
6670-403: 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
6785-538: The Athenians did, indeed rely on persuasive speech, more during public speak, and four new political processes, also increasing the sophists trainings leading too many victories for legal cases, public debate, and even a simple persuasive speech. This ultimately led to concerns rising on falsehood over truth, with highly trained, persuasive speakers, knowingly, misinforming. Rhetoric has its origins in Mesopotamia . Some of
6900-464: The Athenians persuasive speech, with the goal of navigating the courts and senate. The sophists became speech teachers known as Sophia; Greek for "wisdom" and root for philosophy, or " love of wisdom" – the sophists came to be common term for someone who sold wisdom for money. Although there is no clear understanding why the Sicilians engaged to educating the Athenians persuasive speech. It is known that
7015-408: The Sophists, who wanted to teach the Athenians to speak persuasively in order to be able to navigate themselves in the court and senate. What inspired this form of persuasive speech came about through a new form of government, known as democracy, that was being experimented with. Consequently people began to fear that persuasive speech would overpower truth. Aristotle however believed that this technique
SECTION 60
#17328765900777130-427: The assembly decides about future events, a juryman about past events: while those who merely decide on the orator's skill are observers. From this it follows that there are three divisions of oratory—(1) political, (2) forensic, and (3) the ceremonial oratory of display". Eugene Garver, in his critique of Aristotle's Rhetoric , confirms that Aristotle viewed rhetoric as a civic art. Garver writes, " Rhetoric articulates
7245-437: The available means of persuasion", and since mastery of the art was necessary for victory in a case at law, for passage of proposals in the assembly, or for fame as a speaker in civic ceremonies, he called it "a combination of the science of logic and of the ethical branch of politics". Aristotle also identified three persuasive audience appeals: logos , pathos , and ethos . The five canons of rhetoric , or phases of developing
7360-529: The blamelessness of the mythical Helen of Troy in starting the Trojan War . Plato defined the scope of rhetoric according to his negative opinions of the art. He criticized the Sophists for using rhetoric to deceive rather than to discover truth. In Gorgias , one of his Socratic Dialogues , Plato defines rhetoric as the persuasion of ignorant masses within the courts and assemblies. Rhetoric, in Plato's opinion,
7475-471: The city area – the citizens of Athens formed institutions to the red processes: are the Senate, jury trials, and forms of public discussions, but people needed to learn how to navigate these new institutions. With no forms of passing on the information, other than word of mouth the Athenians needed an effective strategy to inform the people. A group of wandering Sicilian's later known as the Sophists , began teaching
7590-469: 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 ,
7705-608: 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
7820-412: 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
7935-679: 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
8050-539: The definition of rhetoric as "the art of persuasion" is the best choice in the context of this theoretical approach of rhetoric as epistemic. Harpine then proceeds to present two methods of approaching the idea of rhetoric as epistemic based on the definitions presented. One centers on Alston's view that one's beliefs are justified if formed by one's normal doxastic while the other focuses on the causal theory of knowledge. Both approaches manage to avoid Gettier's problems and do not rely on unclear conceptions of certainty. In
8165-432: The definition of rhetoric to be the ability to identify the appropriate means of persuasion in a given situation based upon the art of rhetoric ( technê ). This made rhetoric applicable to all fields, not just politics. Aristotle viewed the enthymeme based upon logic (especially, based upon the syllogism) as the basis of rhetoric. Aristotle also outlined generic constraints that focused the rhetorical art squarely within
8280-414: The discourses of a wide variety of domains, including the natural and social sciences, fine art, religion, journalism, digital media, fiction, history, cartography , and architecture, along with the more traditional domains of politics and the law. Because the ancient Greeks valued public political participation, rhetoric emerged as an important curriculum for those desiring to influence politics. Rhetoric
8395-418: The discussion of rhetoric and epistemology , comes the question of ethics . Is it ethical for rhetoric to present itself in the branch of knowledge ? Scott rears this question, addressing the issue, not with ambiguity in the definitions of other terms, but against subjectivity regarding certainty . Ultimately, according to Thomas O. Sloane, rhetoric and epistemology exist as counterparts, working towards
8510-423: The division between the Sophists and Aristotle. Neo-Aristotelians generally study rhetoric as political discourse, while the neo-Sophistic view contends that rhetoric cannot be so limited. Rhetorical scholar Michael Leff characterizes the conflict between these positions as viewing rhetoric as a "thing contained" versus a "container". The neo-Aristotelian view threatens the study of rhetoric by restraining it to such
8625-405: The domain of public political practice. He restricted rhetoric to the domain of the contingent or probable: those matters that admit multiple legitimate opinions or arguments. Since the time of Aristotle, logic has changed. For example, modal logic has undergone a major development that also modifies rhetoric. The contemporary neo-Aristotelian and neo-Sophistic positions on rhetoric mirror
8740-553: The earliest examples of rhetoric can be found in the Akkadian writings of the princess and priestess Enheduanna ( c. 2285–2250 BCE ). As the first named author in history, Enheduanna's writing exhibits numerous rhetorical features that would later become canon in Ancient Greece. Enheduanna's "The Exaltation of Inanna ," includes an exordium , argument , and peroration , as well as elements of ethos , pathos , and logos , and repetition and metonymy . She
8855-407: 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
8970-503: The expanse of implications these words hold. Those who have identified this inconsistency maintain the idea that Scott's relation is important, but requires further study. The root of the issue lies in the ambiguous use of the term rhetoric itself, as well as the epistemological terms knowledge , certainty , and truth . Though counterintuitive and vague, Scott's claims are accepted by some academics, but are then used to draw different conclusions. Sonja K. Foss , for example, takes on
9085-455: The fall of the Roman republic, poetry became a tool for rhetorical training since there were fewer opportunities for political speech. Letter writing was the primary way business was conducted both in state and church, so it became an important aspect of rhetorical education. Rhetorical education became more restrained as style and substance separated in 16th-century France, and attention turned to
9200-450: 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
9315-619: The growth of the study of rhetoric in colleges across the United States. Harvard's rhetoric program drew inspiration from literary sources to guide organization and style, and studies the rhetoric used in political communication to illustrate how political figures persuade audiences. William G. Allen became the first American college professor of rhetoric, at New-York Central College , 1850–1853. Debate clubs and lyceums also developed as forums in which common citizens could hear speakers and sharpen debate skills. The American lyceum in particular
9430-448: The idea that rhetoric is concerned with negotiation and listening, not persuasion, which differs from ancient definitions. Some ancient rhetoric was disparaged because its persuasive techniques could be used to teach falsehoods. Communication as studied in cultural rhetorics is focused on listening and negotiation, and has little to do with persuasion. Rhetorical education focused on five canons . The Five Canons of Rhetoric serve as
9545-423: 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
9660-443: 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
9775-491: 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
9890-559: 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
10005-481: The latter half of the century said "...until the art of music has attained such a height in our own day, that it may indeed be compared to a rhetoric, in view of the multitude of figures" . Epistemology and rhetoric have been compared to one another for decades, but the specifications of their similarities have gone undefined. Since scholar Robert L. Scott stated that, "rhetoric is epistemic ," rhetoricians and philosophers alike have struggled to concretely define
10120-569: 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
10235-457: 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
10350-487: 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
10465-494: The modes of persuasion: ethos , pathos , and logos ) and trace rhetorical development through history. Rhetoric earned a more esteemed reputation as a field of study with the emergence of Communication Studies departments and of Rhetoric and Composition programs within English departments in universities, and in conjunction with the linguistic turn in Western philosophy . Rhetorical study has broadened in scope, and
10580-540: The most persuasive speeches. Thus, civic life could be controlled by whoever could deliver the best speech. Plato explores the problematic moral status of rhetoric twice: in Gorgias and in The Phaedrus , a dialogue best-known for its commentary on love. More trusting in the power of rhetoric to support a republic, the Roman orator Cicero argued that art required something more than eloquence. A good orator needed also to be
10695-482: 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 ,
10810-486: 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
10925-478: The population. Those who classify rhetoric as a civic art believe that rhetoric has the power to shape communities, form the character of citizens, and greatly affect civic life. Rhetoric was viewed as a civic art by several of the ancient philosophers. Aristotle and Isocrates were two of the first to see rhetoric in this light. In Antidosis , Isocrates states, "We have come together and founded cities and made laws and invented arts; and, generally speaking, there
11040-532: The same purpose of establishing knowledge , with the common enemy of subjective certainty . Rhetoric is a persuasive speech that holds people to a common purpose and therefore facilitates collective action. During the fifth century BCE, Athens had become active in metropolis and people all over there. During this time the Greek city state had been experimenting with a new form of government – democracy, demos , "the people". Political and cultural identity had been tied to
11155-639: The scientific method. Influential scholars like Peter Ramus argued that the processes of invention and arrangement should be elevated to the domain of philosophy, while rhetorical instruction should be chiefly concerned with the use of figures and other forms of the ornamentation of language. Scholars such as Francis Bacon developed the study of "scientific rhetoric" which rejected the elaborate style characteristic of classical oration. This plain language carried over to John Locke 's teaching, which emphasized concrete knowledge and steered away from ornamentation in speech, further alienating rhetorical instruction—which
11270-454: The specific realm of political discourse , to many modern scholars it encompasses every aspect of culture. Contemporary studies of rhetoric address a much more diverse range of domains than was the case in ancient times. While classical rhetoric trained speakers to be effective persuaders in public forums and in institutions such as courtrooms and assemblies, contemporary rhetoric investigates human discourse writ large . Rhetoricians have studied
11385-410: 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
11500-553: The study of political discourse can help more than any other thing to stimulate and form such qualities of character." Aristotle, writing several years after Isocrates, supported many of his arguments and argued for rhetoric as a civic art. In the words of Aristotle, in the Rhetoric , rhetoric is "...the faculty of observing in any given case the available means of persuasion". According to Aristotle, this art of persuasion could be used in public settings in three different ways: "A member of
11615-453: The text. People engage in rhetoric any time they speak or produce meaning. Even in the field of science , via practices which were once viewed as being merely the objective testing and reporting of knowledge, scientists persuade their audience to accept their findings by sufficiently demonstrating that their study or experiment was conducted reliably and resulted in sufficient evidence to support their conclusions. The vast scope of rhetoric
11730-464: The three original liberal arts or trivium (along with logic and grammar ). During the medieval period, political rhetoric declined as republican oratory died out and the emperors of Rome garnered increasing authority. With the rise of European monarchs, rhetoric shifted into courtly and religious applications. Augustine exerted strong influence on Christian rhetoric in the Middle Ages, advocating
11845-402: 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
11960-553: 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
12075-416: The use of rhetoric to lead audiences to truth and understanding, especially in the church. The study of liberal arts, he believed, contributed to rhetorical study: "In the case of a keen and ardent nature, fine words will come more readily through reading and hearing the eloquent than by pursuing the rules of rhetoric." Poetry and letter writing became central to rhetorical study during the Middle Ages. After
12190-483: The view that, "rhetoric creates knowledge," whereas James Herrick writes that rhetoric assists in people's ability to form beliefs , which are defined as knowledge once they become widespread in a community. It is unclear whether Scott holds that certainty is an inherent part of establishing knowledge , his references to the term abstract. He is not the only one, as the debate's persistence in philosophical circles long predates his addition of rhetoric. There
12305-487: The wake of the U.S. and French revolutions. The rhetorical studies of ancient Greece and Rome were resurrected as speakers and teachers looked to Cicero and others to inspire defenses of the new republics. Leading rhetorical theorists included John Quincy Adams of Harvard , who advocated the democratic advancement of rhetorical art. Harvard's founding of the Boylston Professorship of Rhetoric and Oratory sparked
12420-463: 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
12535-430: 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
12650-445: Was an art, and that persuasive speech could have truth and logic embedded within it. In the end, rhetoric speech still remained popular and was used by many scholars and philosophers. The study of rhetoric trains students to speak and/or write effectively, and to critically understand and analyze discourse. It is concerned with how people use symbols, especially language, to reach agreement that permits coordinated effort. Rhetoric as
12765-486: 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
12880-460: Was identified wholly with such ornamentation—from the pursuit of knowledge. In the 18th century, rhetoric assumed a more social role, leading to the creation of new education systems (predominantly in England): " Elocution schools" in which girls and women analyzed classic literature, most notably the works of William Shakespeare , and discussed pronunciation tactics. The study of rhetoric underwent
12995-407: 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
13110-467: Was seen as both an educational and social institution, featuring group discussions and guest lecturers. These programs cultivated democratic values and promoted active participation in political analysis. Throughout the 20th century, rhetoric developed as a concentrated field of study, with the establishment of rhetorical courses in high schools and universities. Courses such as public speaking and speech analysis apply fundamental Greek theories (such as
13225-496: 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
#76923