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172-407: Kurt Friedrich Gödel ( / ˈ ɡ ɜːr d əl / GUR -dəl ; German: [kʊʁt ˈɡøːdl̩] ; April 28, 1906 – January 14, 1978) was a logician , mathematician , and philosopher . Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century (at

344-501: A r y ) ∧ Q ( J o h n ) ) {\displaystyle \exists Q(Q(Mary)\land Q(John))} " . In this case, the existential quantifier is applied to the predicate variable " Q {\displaystyle Q} " . The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories. But it has drawbacks in regard to its meta-logical properties and ontological implications, which

516-528: A cell-mediated immunity reaction as these lesions predominantly contain T-helper cells and macrophages . In rheumatic fever, these lesions can be found in any layer of the heart causing different types of carditis . The inflammation may cause a serofibrinous pericardial exudate described as "bread-and-butter" pericarditis , which usually resolves without sequelae. Involvement of the endocardium typically results in fibrinoid necrosis and wart formation along

688-505: A proof system . Logic plays a central role in many fields, such as philosophy , mathematics , computer science , and linguistics . Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to the conclusion "I don't have to work". Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions

860-500: A streptococcal throat infection . Signs and symptoms include fever , multiple painful joints , involuntary muscle movements , and occasionally a characteristic non- itchy rash known as erythema marginatum . The heart is involved in about half of the cases. Damage to the heart valves, known as rheumatic heart disease (RHD), usually occurs after repeated attacks but can sometimes occur after one. The damaged valves may result in heart failure , atrial fibrillation and infection of

1032-462: A throat infection . Symptoms include: fever, painful joints with those joints affected changing with time, involuntary muscle movements , and occasionally a characteristic non-itchy rash known as erythema marginatum . The heart is involved in about half of the cases. Damage to the heart valves usually occurs only after several attacks but may occasionally occur after a single case of RF. The damaged valves may result in heart failure and also increase

1204-793: A virulence factor that is highly antigenic . The antibodies which the immune system generates against the M protein may cross-react with heart muscle cell protein myosin , heart muscle glycogen and smooth muscle cells of arteries, inducing cytokine release and tissue destruction. However, the only proven cross-reaction is with perivascular connective tissue . This inflammation occurs through direct attachment of complement and Fc receptor -mediated recruitment of neutrophils and macrophages. Characteristic Aschoff bodies , composed of swollen eosinophilic collagen surrounded by lymphocytes and macrophages can be seen on light microscopy. The larger macrophages may become Anitschkow cells or Aschoff giant cells . Rheumatic valvular lesions may also involve

1376-445: A central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments

1548-480: A certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth. This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to non-monotonicity and defeasibility : it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn. Ampliative reasoning plays

1720-573: A complex argument to be successful, each link of the chain has to be successful. Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning . The strongest form of support corresponds to deductive reasoning . But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases,

1892-425: A conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion. These general characterizations apply to logic in the widest sense, i.e., to both formal and informal logic since they are both concerned with assessing the correctness of arguments. Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic. Formal logic

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2064-508: A dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion. Gödel became a permanent member of the Institute for Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became

2236-510: A formal language together with a set of axioms and a proof system used to draw inferences from these axioms. In logic, axioms are statements that are accepted without proof. They are used to justify other statements. Some theorists also include a semantics that specifies how the expressions of the formal language relate to real objects. Starting in the late 19th century, many new formal systems have been proposed. A formal language consists of an alphabet and syntactic rules. The alphabet

2408-686: A formal language while informal logic investigates them in their original form. On this view, the argument "Birds fly. Tweety is a bird. Therefore, Tweety flies." belongs to natural language and is examined by informal logic. But the formal translation "(1) ∀ x ( B i r d ( x ) → F l i e s ( x ) ) {\displaystyle \forall x(Bird(x)\to Flies(x))} ; (2) B i r d ( T w e e t y ) {\displaystyle Bird(Tweety)} ; (3) F l i e s ( T w e e t y ) {\displaystyle Flies(Tweety)} "

2580-453: A full professor at the Institute in 1953 and an emeritus professor in 1976. During his time at the institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed timelike curves , to Einstein's field equations in general relativity . He is said to have given this elaboration to Einstein as a present for his 70th birthday. His "rotating universes" would allow time travel to

2752-415: A given argument is valid. Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be translated into formal language before their validity can be assessed. The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning

2924-551: A given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In chess , for example, the definitory rules dictate that bishops may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's king . It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning. A formal system of logic consists of

3096-402: A great variety of topics. They include metaphysical theses about ontological categories and problems of scientific explanation. But in a more narrow sense, it is identical to term logic or syllogistics. A syllogism is a form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a subject , a predicate, and a copula connecting

3268-474: A lecture by David Hilbert in Bologna on completeness and consistency in mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik ( Principles of Mathematical Logic ), an introduction to first-order logic in which the problem of completeness was posed: "Are the axioms of a formal system sufficient to derive every statement that

3440-426: A life-threatening type II hypersensitivity reaction . Usually, self reactive B cells remain anergic in the periphery without T cell co-stimulation. During a streptococcal infection, mature antigen-presenting cells such as B cells present the bacterial antigen to CD4+T cells which differentiate into helper T 2 cells . Helper T 2 cells subsequently activate the B cells to become plasma cells and induce

3612-614: A logical connective like "and" to form a new complex proposition. In Aristotelian logic, the subject can be universal , particular , indefinite , or singular . For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates". Aristotelian logic only includes predicates for simple properties of entities. But it lacks predicates corresponding to relations between entities. The predicate can be linked to

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3784-477: A major textile firm, and Marianne Gödel ( née Handschuh, 1879–1966). At the time of his birth the city had a German-speaking majority which included his parents. His father was Catholic and his mother was Protestant and the children were raised as Protestants. The ancestors of Kurt Gödel were often active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer in his time and for some years

3956-689: A member of the Brünner Männergesangverein (Men's Choral Union of Brünn). Gödel automatically became a citizen of Czechoslovakia at age 12 when the Austro-Hungarian Empire collapsed following its defeat in the First World War . According to his classmate Klepetař , like many residents of the predominantly German Sudetenländer , "Gödel considered himself always Austrian and an exile in Czechoslovakia". In February 1929, he

4128-463: A moderate or high risk. Minor criteria Rheumatic fever can be prevented by effectively and promptly treating strep throat with antibiotics. Globally, rheumatic fever is seen in populations that are socioeconomically disadvantaged and with limited access to health care. Overcrowding and exposure to domestic air pollution have been cited as associated risk factors. In those who have previously had rheumatic fever, antibiotics may be used in

4300-582: A preceding streptococcal infection include: recent scarlet fever , raised antistreptolysin O or other streptococcal antibody titre, or positive throat culture. The last revision of 2015 suggested variable diagnostic criteria in low-risk and high-risk populations to avoid overdiagnosis in the first category and underdiagnosis in the last one. Low-risk populations were defined as those with acute rheumatic fever annual incidence ≤2 per 100 000 school-aged children or all-age rheumatic heart disease prevalence of ≤1 per 1000. All other populations were categorised as having

4472-640: A preventative manner as secondary prophylaxis . Antibiotic prophylaxis after an episode of acute rheumatic fever is recommended owing to the high likelihood of recurrence. Streptococcal pharyngitis may occur asymptomatically and rheumatic fever may recur even after a treated infection. The American Heart Association recommends, based on low quality evidence but with high predicted efficacy, that people with mitral stenosis due to rheumatic heart disease receive prophylactic antibiotics for 10 years or until age 40, whichever would be longer. The AHA also supports good dental hygiene in people with RHD, and antibiotics for

4644-537: A proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem. On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in

4816-664: A sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, p {\displaystyle p} ("yesterday was Sunday") and q {\displaystyle q} ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are ¬ {\displaystyle \lnot } ( not ), ∨ {\displaystyle \lor } ( or ), → {\displaystyle \to } ( if...then ), and ↑ {\displaystyle \uparrow } ( Sheffer stroke ). Given

4988-410: A subsequent untreated infection is substantially greater (about 50%). The rate of development is far lower in individuals who have received antibiotic treatment. Persons who have had a case of rheumatic fever have a tendency to develop flare-ups with repeated strep infections. The recurrence of rheumatic fever is relatively common in the absence of maintenance of low dose antibiotics, especially during

5160-400: A time when Bertrand Russell , Alfred North Whitehead , and David Hilbert were using logic and set theory to investigate the foundations of mathematics ), building on earlier work by Frege, Richard Dedekind , and Georg Cantor . Gödel's discoveries in the foundations of mathematics led to the proof of his completeness theorem in 1929 as part of his dissertation to earn a doctorate at

5332-420: Is sound when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by

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5504-498: Is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife]." In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic , not pantheistic , following Leibniz rather than Spinoza ." Of religion(s) in general, he said: "Religions are for

5676-402: Is a formula that is true of arithmetic, but not provable in that system. To make this precise, Gödel had to produce a method to encode (as natural numbers) statements, proofs, and the concept of provability; he did this by a process known as Gödel numbering . In his two-page paper Zum intuitionistischen Aussagenkalkül (1932), Gödel refuted the finite-valuedness of intuitionistic logic . In

5848-404: Is a priority to prevent complications and progression. However, spontaneous regression is more likely in borderline RHD than in definite cases, and its natural history may vary between populations. Echocardiographic screening among children and timely initiation of secondary antibiotic prophylaxis in children with evidence of early stages of rheumatic heart disease may be effective to reduce

6020-471: Is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers . This means that they have a truth value : they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences . Propositions are the denotations of sentences and are usually seen as abstract objects . For example, the English sentence "the tree

6192-441: Is a restricted version of classical logic. It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that A {\displaystyle A} follows from ¬ ¬ A {\displaystyle \lnot \lnot A} . This is a valid rule of inference in classical logic but it

6364-416: Is also known as symbolic logic and is widely used in mathematical logic . It uses a formal approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the logical form of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content. Formal logic

6536-453: Is an example of the existential quantifier " ∃ {\displaystyle \exists } " applied to the individual variable " x {\displaystyle x} " . In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula " ∃ Q ( Q ( M

6708-415: Is blurry in some cases, such as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between. The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term " induction " to cover all forms of non-deductive arguments. But in a more narrow sense, induction

6880-481: Is caused by its destructive effects on cardiac valve tissue. The complicated pathogenesis of RHD is not fully understood, though it has been observed to use molecular mimicry via group A streptococci carbohydrates and genetic predisposition involving HLA Class II genes that trigger autoimmune reactions . Molecular mimicry occurs when epitopes are shared between host antigens and Streptococcus antigens. This causes an autoimmune reaction against native tissues in

7052-421: Is commonly defined in terms of arguments or inferences as the study of their correctness. An argument is a set of premises together with a conclusion. An inference is the process of reasoning from these premises to the conclusion. But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on

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7224-878: Is controversial and based on dated literature. Corticosteroids may be considered, especially in people with allergies to NSAIDs or severe disease, although use of steroids may cause tissue atrophy, which could present challenges during future cardiac surgery for valve repair. Some patients develop significant carditis which manifests as congestive heart failure . This requires the usual treatment for heart failure: ACE inhibitors , diuretics , beta blockers , and digoxin . Unlike typical heart failure, rheumatic heart failure responds well to corticosteroids. About 33 million people are affected by rheumatic heart disease with an additional 47 million having asymptomatic damage to their heart valves. As of 2010 globally it resulted in 345,000 deaths, down from 463,000 in 1990. In Western countries, rheumatic fever has become fairly rare since

7396-407: Is controversial because it belongs to the field of psychology , not logic, and because appearances may be different for different people. Fallacies are usually divided into formal and informal fallacies. For formal fallacies, the source of the error is found in the form of the argument. For example, denying the antecedent is one type of formal fallacy, as in "if Othello is a bachelor, then he

7568-453: Is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises. According to an influential view by Alfred Tarski , deductive arguments have three essential features: (1) they are formal, i.e. they depend only on

7740-423: Is established by verification using a proof. Intuitionistic logic is especially prominent in the field of constructive mathematics , which emphasizes the need to find or construct a specific example to prove its existence. Rheumatic fever Rheumatic fever ( RF ) is an inflammatory disease that can involve the heart , joints , skin , and brain . The disease typically develops two to four weeks after

7912-575: Is given annually to an outstanding paper in theoretical computer science. Gödel's philosophical notebooks are being edited at the Kurt Gödel Research Centre at the Berlin-Brandenburg Academy of Sciences and Humanities . Five volumes of Gödel's collected works have been published. The first two include his publications; the third includes unpublished manuscripts from his Nachlass , and the final two include correspondence. In

8084-606: Is green" is different from the German sentence "der Baum ist grün" but both express the same proposition. Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions. These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like

8256-432: Is interested in deductively valid arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false. For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a rule of inference . For example, modus ponens is a rule of inference according to which all arguments of

8428-415: Is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the law of excluded middle . It states that for every sentence, either it or its negation is true. This means that every proposition of the form A ∨ ¬ A {\displaystyle A\lor \lnot A} is true. These deviations from classical logic are based on the idea that truth

8600-447: Is male; Othello is not a bachelor; therefore Othello is not male". But most fallacies fall into the category of informal fallacies, of which a great variety is discussed in the academic literature. The source of their error is usually found in the content or the context of the argument. Informal fallacies are sometimes categorized as fallacies of ambiguity, fallacies of presumption, or fallacies of relevance. For fallacies of ambiguity,

8772-529: Is most often associated with RHD, and its combination with certain DQ alleles is seemingly associated with the development of valvular lesions. The mechanism by which MHC class II molecules increase a host's susceptibility to autoimmune reactions in RHD is unknown, but it is likely related to the role HLA molecules play in presenting antigens to T cell receptors, thus triggering an immune response. Also found on human chromosome 6

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8944-688: Is necessary, then it is also possible. This means that ◊ A {\displaystyle \Diamond A} follows from ◻ A {\displaystyle \Box A} . Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that ◻ A {\displaystyle \Box A} is equivalent to ¬ ◊ ¬ A {\displaystyle \lnot \Diamond \lnot A} . Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, deontic logic concerns

9116-518: Is necessary. For example, if the formula B ( s ) {\displaystyle B(s)} stands for the sentence "Socrates is a banker" then the formula ◊ B ( s ) {\displaystyle \Diamond B(s)} articulates the sentence "It is possible that Socrates is a banker". To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something

9288-407: Is not the best or most likely explanation. Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as fallacies . Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion. So the argument "it is sunny today; therefore spiders have eight legs" is fallacious even though

9460-446: Is not the case for ampliative arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into inductive and abductive arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens. Abductive arguments are inferences to

9632-839: Is now known as Gödel's ontological proof . Gödel was awarded (with Julian Schwinger ) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science , in 1974. Gödel was elected a resident member of the American Philosophical Society in 1961 and a Foreign Member of the Royal Society (ForMemRS) in 1968 . He was a Plenary Speaker of the ICM in 1950 in Cambridge, Massachusetts. Later in his life, Gödel suffered periods of mental instability and illness. Following

9804-541: Is only one type of ampliative argument alongside abductive arguments . Some philosophers, like Leo Groarke, also allow conductive arguments as another type. In this narrow sense, induction is often defined as a form of statistical generalization. In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains. In this sense, one may infer that "all elephants are gray" based on one's past observations of

9976-522: Is powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that: These theorems ended a half-century of attempts, beginning with the work of Gottlob Frege and culminating in Principia Mathematica and Hilbert's program , to find a non- relatively consistent axiomatization sufficient for number theory (that

10148-405: Is required. Otherwise complications are treated as usual. Rheumatic fever occurs in about 325,000 children each year and about 33.4 million people currently have rheumatic heart disease. Those who develop RF are most often between the ages of 5 and 14, with 20% of first-time attacks occurring in adults. The disease is most common in the developing world and among indigenous peoples in

10320-666: Is singular and monumental—indeed it is more than a monument, it is a landmark which will remain visible far in space and time. ... The subject of logic has certainly completely changed its nature and possibilities with Gödel's achievement. In 1930 Gödel attended the Second Conference on the Epistemology of the Exact Sciences , held in Königsberg , 5–7 September. There, he presented his completeness theorem of first-order logic, and, at

10492-430: Is studied by formal logic. The study of natural language arguments comes with various difficulties. For example, natural language expressions are often ambiguous, vague, and context-dependent. Another approach defines informal logic in a wide sense as the normative study of the standards, criteria, and procedures of argumentation. In this sense, it includes questions about the role of rationality , critical thinking , and

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10664-724: Is the cytokine TNF-α which is also associated with RHD. High expression levels of TNF-α may exacerbate valvular tissue inflammation, because as this cytokine circulates in the bloodstream, it triggers the activation of multiple pathways that stimulate further pro-inflammatory cytokine secretion. Mannose-binding lectin (MBL) is an inflammatory protein involved in pathogen recognition. Different variants of MBL2 gene regions are associated with RHD. RHD-induced mitral valve stenosis has been associated with MBL2 alleles encoding for high production of MBL. Aortic valve regurgitation in RHD patients has been associated with different MBL2 alleles that encode for low production of MBL. In addition,

10836-410: Is the set of basic symbols used in expressions . The syntactic rules determine how these symbols may be arranged to result in well-formed formulas. For instance, the syntactic rules of propositional logic determine that " P ∧ Q {\displaystyle P\land Q} " is a well-formed formula but " ∧ Q {\displaystyle \land Q} " is not since

11008-399: Is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ∧ {\displaystyle \land } ( and ) or → {\displaystyle \to } ( if...then ). Simple propositions also have parts, like "Sunday" or "work" in the example. The truth of a proposition usually depends on

11180-432: Is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference . They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach

11352-524: Is true in all models of the system?" This problem became the topic that Gödel chose for his doctoral work. In 1929, aged 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established his eponymous completeness theorem regarding first-order logic . He was awarded his doctorate in 1930, and his thesis (accompanied by additional work) was published by the Vienna Academy of Science . Kurt Gödel's achievement in modern logic

11524-540: Is unable to address. Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies. Many characterizations of informal logic have been suggested but there is no general agreement on its precise definition. The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments. On this view, informal logic studies arguments that are in informal or natural language. Formal logic can only examine them indirectly by translating them first into

11696-599: Is used to represent the ideas of knowing something in contrast to merely believing it to be the case. Higher-order logics extend classical logic not by using modal operators but by introducing new forms of quantification. Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula " ∃ x ( A p p l e ( x ) ∧ S w e e t ( x ) ) {\displaystyle \exists x(Apple(x)\land Sweet(x))} " ( some apples are sweet)

11868-431: Is why first-order logic is still more commonly used. Deviant logics are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue. Intuitionistic logic

12040-643: The American Heart Association in collaboration with other groups. According to revised Jones criteria, the diagnosis of rheumatic fever can be made when two of the major criteria, or one major criterion plus two minor criteria, are present along with evidence of streptococcal infection: elevated or rising antistreptolysin O titre or anti-DNase B . A recurrent episode can be diagnosed when three minor criteria are present. Exceptions are chorea and indolent carditis , each of which by itself can indicate rheumatic fever. An April 2013 review article in

12212-808: The American Mathematical Society . During this year, Gödel also developed the ideas of computability and recursive functions to the point where he was able to present a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering . In 1934, Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey , titled On undecidable propositions of formal mathematical systems . Stephen Kleene , who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published. Gödel visited

12384-557: The Indian Journal of Medical Research stated that echocardiographic and Doppler (E & D) studies, despite some reservations about their utility, have identified a massive burden of rheumatic heart disease, which suggests the inadequacy of the 1992 Jones' criteria. E & D studies have identified subclinical carditis in patients with rheumatic fever, as well as in follow-ups of rheumatic heart disease patients who initially presented as having isolated cases of Sydenham's chorea. Signs of

12556-461: The Indian subcontinent , and North Africa . Rheumatic fever primarily affects children between ages 5 and 17 years and occurs approximately 20 days after strep throat. In up to a third of cases, the underlying strep infection may not have caused any symptoms. The rate of development of rheumatic fever in individuals with untreated strep infection is estimated to be 3%. The incidence of recurrence with

12728-445: The U.S. Constitution that could allow the U.S. to become a dictatorship; this has since been dubbed Gödel's Loophole . Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his application. The judge turned out to be Phillip Forman , who knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought

12900-464: The University of Vienna , and the publication of Gödel's incompleteness theorems two years later, in 1931. The first incompleteness theorem states that for any ω-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example, Peano arithmetic ), there are true propositions about the natural numbers that can be neither proved nor disproved from

13072-560: The University of Vienna . During his teens, Gödel studied Gabelsberger shorthand , and criticisms of Isaac Newton , and the writings of Immanuel Kant . At the age of 18, Gödel joined his brother at the University of Vienna . He had already mastered university-level mathematics. Although initially intending to study theoretical physics , he also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism . He read Kant 's Metaphysische Anfangsgründe der Naturwissenschaft , and participated in

13244-508: The Vienna Circle with Moritz Schlick , Hans Hahn , and Rudolf Carnap . Gödel then studied number theory , but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell 's book Introduction to Mathematical Philosophy , he became interested in mathematical logic . According to Gödel, mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences." Attending

13416-479: The developed world . In 2015 it resulted in 319,400 deaths down from 374,000 deaths in 1990. Most deaths occur in the developing world where as many as 12.5% of people affected may die each year. Descriptions of the condition are believed to date back to at least the 5th century BCE in the writings of Hippocrates . The disease is so named because its symptoms are similar to those of some rheumatic disorders . The disease typically develops two to four weeks after

13588-613: The 1960s, probably due to the widespread use of antibiotics to treat streptococcus infections. While it has been far less common in the United States since the beginning of the 20th century, there have been a few outbreaks since the 1980s. The disease is most common among Indigenous Australians (particularly in central and northern Australia), Māori , and Pacific Islanders , and is also common in Sub-Saharan Africa , Latin America ,

13760-497: The 1994 film I.Q. , Lou Jacobi portrays Gödel. In the 2023 movie Oppenheimer , Gödel, played by James Urbaniak , briefly appears walking with Einstein in the gardens of Princeton. In German: In English: In English translation: Logician Logic is the study of correct reasoning . It includes both formal and informal logic . Formal logic is the study of deductively valid inferences or logical truths . It examines how conclusions follow from premises based on

13932-465: The Greek word "logos", which has a variety of translations, such as reason , discourse , or language . Logic is traditionally defined as the study of the laws of thought or correct reasoning , and is usually understood in terms of inferences or arguments . Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences. An argument is a set of premises together with

14104-670: The IAS again in the autumn of 1935. The travelling and the hard work had exhausted him and the next year he took a break to recover from a depressive episode. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis ; he went on to show that these hypotheses cannot be disproved from the common system of axioms of set theory. He married Adele Nimbursky  [ es ; ast ] (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Gödel's parents had opposed their relationship because she

14276-532: The Institute merely ... to have the privilege of walking home with Gödel". Gödel and his wife, Adele, spent the summer of 1942 in Blue Hill, Maine , at the Blue Hill Inn at the top of the bay. Gödel was not merely vacationing but had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson Jr. conjectures that Gödel discovered

14448-471: The age of six or seven, Kurt suffered from rheumatic fever ; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage. Beginning at age four, Gödel suffered from "frequent episodes of poor health", which would continue for his entire life. Gödel attended the Evangelische Volksschule , a Lutheran school in Brünn from 1912 to 1916, and

14620-485: The allele IGHV4-61, located on chromosome 14, which helps code for the immunoglobulin heavy chain (IgH) is linked to greater susceptibility to RHD because it may affect protein structure of the IgH. Other genes are also being investigated to better understand the complexity of autoimmune reactions that occur in RHD. The original method of diagnosing rheumatic heart disease was through heart auscultation, specifically listening for

14792-418: The ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark". Fallacies of presumption have a wrong or unjustified premise but may be valid otherwise. In the case of fallacies of relevance, the premises do not support the conclusion because they are not relevant to it. The main focus of most logicians

14964-476: The assassination of his close friend Moritz Schlick , Gödel developed an obsessive fear of being poisoned , and would eat only food prepared by his wife Adele. Adele was hospitalized beginning in late 1977, and in her absence Gödel refused to eat; he weighed 29 kilograms (65 lb) when he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978. He

15136-399: The assessment of arguments. Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion. For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars

15308-551: The axiom of choice when proving the Hahn–Banach theorem . Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory. Gödel spent the spring of 1939 at the University of Notre Dame . After the Anschluss on 12 March 1938, Austria had become a part of Nazi Germany . Germany abolished

15480-477: The axioms. To prove this, Gödel developed a technique now known as Gödel numbering , which codes formal expressions as natural numbers. The second incompleteness theorem, which follows from the first, states that the system cannot prove its own consistency. Gödel also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted Zermelo–Fraenkel set theory , assuming that its axioms are consistent. The former result opened

15652-448: The bacteria than others. Other risk factors include malnutrition and poverty. Diagnosis of RF is often based on the presence of signs and symptoms in combination with evidence of a recent streptococcal infection. Treating people who have strep throat with antibiotics , such as penicillin , decreases the risk of developing rheumatic fever. In order to avoid antibiotic misuse this often involves testing people with sore throats for

15824-495: The basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like metaphysics , ethics , and epistemology . Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: ◊ {\displaystyle \Diamond } expresses that something is possible while ◻ {\displaystyle \Box } expresses that something

15996-487: The best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer. Arguments that fall short of the standards of correct reasoning often embody fallacies . Systems of logic are theoretical frameworks for assessing the correctness of arguments. Logic has been studied since antiquity . Early approaches include Aristotelian logic , Stoic logic , Nyaya , and Mohism . Aristotelian logic focuses on reasoning in

16168-645: The claim "either it is raining, or it is not". These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q " is a logical truth. Formal logic uses formal languages to express and analyze arguments. They normally have a very limited vocabulary and exact syntactic rules . These rules specify how their symbols can be combined to construct sentences, so-called well-formed formulas . This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether

16340-495: The color of elephants. A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray. Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference. Abductive inference may or may not take statistical observations into consideration. In either case,

16512-511: The conclusion "all ravens are black". A further approach is to define informal logic as the study of informal fallacies . Informal fallacies are incorrect arguments in which errors are present in the content and the context of the argument. A false dilemma , for example, involves an error of content by excluding viable options. This is the case in the fallacy "you are either with us or against us; you are not with us; therefore, you are against us". Some theorists state that formal logic studies

16684-458: The conclusion is true. Some theorists, like John Stuart Mill , give a more restrictive definition of fallacies by additionally requiring that they appear to be correct. This way, genuine fallacies can be distinguished from mere mistakes of reasoning due to carelessness. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them. However, this reference to appearances

16856-591: The conditional proposition p → q {\displaystyle p\to q} , one can form truth tables of its converse q → p {\displaystyle q\to p} , its inverse ( ¬ p → ¬ q {\displaystyle \lnot p\to \lnot q} ) , and its contrapositive ( ¬ q → ¬ p {\displaystyle \lnot q\to \lnot p} ) . Truth tables can also be defined for more complex expressions that use several propositional connectives. Logic

17028-438: The contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future. Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like ethics or epistemology that lie beyond

17200-451: The depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs. Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with

17372-718: The development of rheumatic fever. Some suggest the use of benzathine benzylpenicillin . Monthly injections of long-acting penicillin must be given for a period of five years in patients having one attack of rheumatic fever. If there is evidence of carditis, the length of therapy may be up to 40 years. Another important cornerstone in treating rheumatic fever includes the continual use of low-dose antibiotics (such as penicillin , sulfadiazine , or erythromycin ) to prevent recurrence. Aspirin at high doses has historically been used for treatment of rheumatic fever. However, due to side effects like gastritis and salicylate poisoning , necessitating serum monitoring of salicylate levels, and

17544-481: The door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic , intuitionistic logic , and modal logic . Gödel was born April 28, 1906, in Brünn, Austria-Hungary (now Brno , Czech Republic ), into the German-speaking family of Rudolf Gödel (1874–1929), the managing director and part owner of

17716-458: The end of the talk, mentioned that this result does not generalise to higher-order logic, thus hinting at his incompleteness theorems . Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme (called in English " On Formally Undecidable Propositions of Principia Mathematica and Related Systems "). In that article, he proved for any computable axiomatic system that

17888-434: The field of ethics and introduces symbols to express the ideas of obligation and permission , i.e. to describe whether an agent has to perform a certain action or is allowed to perform it. The modal operators in temporal modal logic articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time. In epistemology, epistemic modal logic

18060-478: The first three to five years after the first episode. Recurrent bouts of rheumatic fever can lead to valvular heart disease . Heart complications may be long-term and severe, particularly if valves are involved. In countries in Southeast-Asia, sub-Saharan Africa, and Oceania, the percentage of people with rheumatic heart disease detected by listening to the heart was 2.9 per 1000 children and by echocardiography it

18232-485: The form "(1) p , (2) if p then q , (3) therefore q " are valid, independent of what the terms p and q stand for. In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of logical truths . A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all possible worlds and under all interpretations of its non-logical terms, like

18404-508: The form of syllogisms . It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as Gottlob Frege . Today, the most commonly used system is classical logic . It consists of propositional logic and first-order logic . Propositional logic only considers logical relations between full propositions. First-order logic also takes

18576-523: The form of the premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid. The modus ponens is a prominent rule of inference. It has the form " p ; if p , then q ; therefore q ". Knowing that it has just rained ( p {\displaystyle p} ) and that after rain the streets are wet ( p → q {\displaystyle p\to q} ), one can use modus ponens to deduce that

18748-419: The form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by logical necessity for the given propositions, independent of any other circumstances. Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference. Rules of inference specify

18920-421: The general form of arguments while informal logic studies particular instances of arguments. Another approach is to hold that formal logic only considers the role of logical constants for correct inferences while informal logic also takes the meaning of substantive concepts into account. Further approaches focus on the discussion of logical topics with or without formal devices and on the role of epistemology for

19092-636: The heart that are incorrectly recognized as "foreign" due to the cross-reactivity of antibodies generated as a result of epitope sharing. The valvular endothelium is a prominent site of lymphocyte-induced damage. CD4 + T cells are the major effectors of heart tissue autoimmune reactions in RHD. Normally, T cell activation is triggered by the presentation of bacterial antigens. In RHD, molecular mimicry results in incorrect T cell activation, and these T lymphocytes can go on to activate B cells , which will begin to produce self-antigen-specific antibodies. This leads to an immune response attack mounted against tissues in

19264-477: The heart that have been misidentified as pathogens. Rheumatic valves display increased expression of VCAM-1 , a protein that mediates the adhesion of lymphocytes. Self-antigen-specific antibodies generated via molecular mimicry between human proteins and streptococcal antigens up-regulate VCAM-1 after binding to the valvular endothelium. This leads to the inflammation and valve scarring observed in rheumatic valvulitis, mainly due to CD4+ T cell infiltration. While

19436-437: The infection; however, testing might not be available in the developing world . Other preventive measures include improved sanitation . In those with rheumatic fever and rheumatic heart disease, prolonged periods of antibiotics are sometimes recommended. Gradual return to normal activities may occur following an attack. Once RHD develops, treatment is more difficult. Occasionally valve replacement surgery or valve repair

19608-406: The internal parts of propositions into account, like predicates and quantifiers . Extended logics accept the basic intuitions behind classical logic and apply it to other fields, such as metaphysics , ethics , and epistemology . Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic. The word "logic" originates from

19780-407: The internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, predicates , which refer to properties and relations, and quantifiers, which treat notions like "some" and "all". For example, to express the proposition "this raven is black", one may use the predicate B {\displaystyle B} for the property "black" and

19952-418: The lines of closure of the left-sided heart valves. Warty projections arise from the deposition, while subendocardial lesions may induce irregular thickenings called MacCallum plaques . Chronic rheumatic heart disease (RHD) is characterized by repeated inflammation with fibrinous repair. The cardinal anatomic changes of the valve include leaflet thickening, commissural fusion, and shortening and thickening of

20124-522: The logical conjunction ∧ {\displaystyle \land } requires terms on both sides. A proof system is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content. For instance, the classical rule of conjunction introduction states that P ∧ Q {\displaystyle P\land Q} follows from

20296-454: The meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have the strongest form of support: if their premises are true then their conclusion must also be true. This

20468-527: The mechanisms of genetic predisposition remain unclear, a few genetic factors have been found to increase susceptibility to autoimmune reactions in RHD. The dominant contributors are a component of MHC class II molecules, found on lymphocytes and antigen-presenting cells, specifically the DR and DQ alleles on human chromosome 6 . Certain allele combinations appear to increase RHD autoimmune susceptibility. Human leukocyte antigen (HLA) class II allele DR7 ( HLA-DR7 )

20640-517: The most part bad, but not religion itself." According to his wife Adele, "Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning", while of Islam , he said, "I like Islam: it is a consistent [or consequential] idea of religion and open-minded." Douglas Hofstadter wrote the 1979 book Gödel, Escher, Bach to celebrate the work and ideas of Gödel, M. C. Escher and Johann Sebastian Bach . It partly explores

20812-465: The only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo–Fraenkel axioms for set theory (ZF). This result has had considerable consequences for working mathematicians, as it means they can assume

20984-409: The other hand, are true or false depending on whether they are in accord with reality. In formal logic, a sound argument is an argument that is both correct and has only true premises. Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For

21156-444: The other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like singular terms and predicates . For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by

21328-652: The past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric (an exact solution of the Einstein field equation ). He studied and admired the works of Gottfried Leibniz , but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed. To a lesser extent he studied Immanuel Kant and Edmund Husserl . In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury 's ontological proof of God's existence. This

21500-425: The possibility of Hitler making an atom bomb. Gödel never conveyed that letter to Einstein, although they did meet, because he was not convinced Hitler could achieve this feat. In any case, Leo Szilard had already conveyed the message to Einstein, and Einstein had already warned Roosevelt. In Princeton, Gödel accepted a position at the Institute for Advanced Study (IAS), which he had visited during 1933–34. Einstein

21672-485: The premises P {\displaystyle P} and Q {\displaystyle Q} . Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are different types of proof systems including natural deduction and sequent calculi . A semantics is a system for mapping expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance,

21844-413: The premises offer support for the conclusion because the conclusion is the best explanation of why the premises are true. In this sense, abduction is also called the inference to the best explanation . For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean

22016-470: The premises. But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on

22188-473: The prevention of infective endocarditis during dental procedures are recommended in high-risk patients. No vaccines are currently available to protect against S. pyogenes infection, although research is underway to develop one. Difficulties in developing a vaccine include the wide variety of strains of S. pyogenes present in the environment and the large amount of time and number of people that will be needed for appropriate trials for safety and efficacy of

22360-445: The production of antibodies against the cell wall of Streptococcus. However the antibodies may also react against the myocardium and joints, producing the symptoms of rheumatic fever. S. pyogenes is a species of aerobic , cocci , gram-positive bacteria that are non-motile, non- spore forming , and forms chains and large colonies . S. pyogenes has a cell wall composed of branched polymers which sometimes contain M protein ,

22532-629: The proof, he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic ). Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a Privatdozent (unpaid lecturer) there. In 1933 Adolf Hitler came to power in Germany, and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, Moritz Schlick , whose seminar had aroused Gödel's interest in logic,

22704-485: The propositional connective "and". Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using truth-functional propositional connectives, their truth only depends on the truth values of their parts. But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects. Whether

22876-406: The propositions are formed. For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid. Classical logic is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in

23048-417: The psychology of argumentation. Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic. Non-deductive arguments make their conclusion probable but do not ensure that it is true. An example is the inductive argument from the empirical observation that "all ravens I have seen so far are black" to

23220-471: The ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain . In 2005 John Dawson published a biography, Logical Dilemmas: The Life and Work of Kurt Gödel . Stephen Budiansky 's book about Gödel's life, Journey to the Edge of Reason: The Life of Kurt Gödel , was a New York Times Critics' Top Book of 2021. Gödel

23392-463: The risk of Reye syndrome , a serious and potentially deadly condition that may arise in children treated with aspirin or aspirin-containing products, alternatives to aspirin have been sought, especially in children. While evidence suggests that treatment of rheumatic fever–associated arthritis with naproxen may be equally effective as with aspirin, its role in managing carditis has not been established. Management of carditis in acute rheumatic fever

23564-402: The risk of atrial fibrillation and infection of the valves . Rheumatic fever is a systemic disease affecting the connective tissue around arterioles , and can occur after an untreated strep throat infection, specifically due to group A streptococcus (GAS), Streptococcus pyogenes . The similarity between antigens of Streptococcus pyogenes and multiple cardiac proteins can cause

23736-436: The rules of inference they accept as valid and the formal languages used to express them. Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic. For example, it has been suggested that only logically complete systems, like first-order logic , qualify as logics. For such reasons, some theorists deny that higher-order logics are logics in

23908-492: The scope of mathematics. Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives . For instance, propositional logic represents the conjunction of two atomic propositions P {\displaystyle P} and Q {\displaystyle Q} as the complex formula P ∧ Q {\displaystyle P\land Q} . Unlike predicate logic where terms and predicates are

24080-418: The semantics for classical propositional logic assigns the formula P ∧ Q {\displaystyle P\land Q} the denotation "true" whenever P {\displaystyle P} and Q {\displaystyle Q} are true. From the semantic point of view, a premise entails a conclusion if the conclusion is true whenever the premise is true. A system of logic

24252-604: The semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. For over two thousand years, Aristotelian logic was treated as the canon of logic in the Western world, but modern developments in this field have led to a vast proliferation of logical systems. One prominent categorization divides modern formal logical systems into classical logic , extended logics, and deviant logics . Aristotelian logic encompasses

24424-518: The sense that it is based on basic logical intuitions shared by most logicians. These intuitions include the law of excluded middle , the double negation elimination , the principle of explosion , and the bivalence of truth. It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are

24596-404: The simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true under all interpretations of its non-logical terms. In some modal logics , this means that the proposition is true in all possible worlds. Some theorists define logic as

24768-415: The simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by theories of reference . Some complex propositions are true independently of the substantive meanings of their parts. In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like

24940-870: The singular term r {\displaystyle r} referring to the raven to form the expression B ( r ) {\displaystyle B(r)} . To express that some objects are black, the existential quantifier ∃ {\displaystyle \exists } is combined with the variable x {\displaystyle x} to form the proposition ∃ x B ( x ) {\displaystyle \exists xB(x)} . First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer ∃ x B ( x ) {\displaystyle \exists xB(x)} from B ( r ) {\displaystyle B(r)} . Extended logics are logical systems that accept

25112-474: The smallest units, propositional logic takes full propositions with truth values as its most basic component. Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition. First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates

25284-453: The sound of blood regurgitation from possibly dysfunctional valves. However, studies have shown that echocardiography is much more efficient in detecting RHD due to its high sensitivity. An echocardiogram has the ability to detect signs of RHD before the development of more obvious symptoms such as tissue scarring and stenosis. Modified Jones criteria were first published in 1944 by T. Duckett Jones , MD. They have been periodically revised by

25456-418: The streets are wet ( q {\displaystyle q} ). The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false. Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in

25628-437: The strict sense. When understood in a wide sense, logic encompasses both formal and informal logic. Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse. Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments. In this regard, it considers problems that formal logic on its own

25800-403: The structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies , critical thinking , and argumentation theory . Informal logic examines arguments expressed in natural language whereas formal logic uses formal language . When used as a countable noun , the term "a logic" refers to a specific logical formal system that articulates

25972-438: The study of logical truths. Truth tables can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for

26144-700: The subject in two ways: either by affirming it or by denying it. For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one predicate in each case. Thus, these three propositions contain three predicates, referred to as major term , minor term , and middle term . The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how

26316-417: The subject to the predicate. For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is". The subject and the predicate are the terms of the proposition. Aristotelian logic does not contain complex propositions made up of simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using

26488-423: The symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted. Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism . It

26660-409: The table. This conclusion is justified because it is the best explanation of the current state of the kitchen. For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it

26832-563: The tendinous cords. It is caused by an autoimmune reaction to Group A β-hemolytic streptococci (GAS) that results in valvular damage. Fibrosis and scarring of valve leaflets, commissures and cusps leads to abnormalities that can result in valve stenosis or regurgitation. The inflammation caused by rheumatic fever, usually during childhood, is referred to as rheumatic valvulitis. About half of patients with rheumatic fever develop inflammation involving valvular endothelium . The majority of morbidity and mortality associated with rheumatic fever

27004-399: The term ampliative or inductive reasoning is used. Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic. A deductively valid argument is one whose premises guarantee the truth of its conclusion. For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs"

27176-507: The title Privatdozent , so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him. The University of Vienna turned his application down. His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year

27348-479: The truth values "true" and "false". The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression " p ∧ q {\displaystyle p\land q} " uses the logical connective ∧ {\displaystyle \land } ( and ). It could be used to express

27520-464: The vaccine. The management of rheumatic fever is directed toward the reduction of inflammation with anti-inflammatory medications such as aspirin or corticosteroids . Individuals with positive cultures for strep throat should also be treated with antibiotics . People with positive cultures for Streptococcus pyogenes should be treated with penicillin as long as allergy is not present. The use of antibiotics will not alter cardiac involvement in

27692-416: The valves . Rheumatic fever may occur following an infection of the throat by the bacterium Streptococcus pyogenes . If the infection is left untreated, rheumatic fever occurs in up to three percent of people. The underlying mechanism is believed to involve the production of antibodies against a person's own tissues. Due to their genetics, some people are more likely to get the disease when exposed to

27864-502: Was 12.9 per 1000 children. To assist in the identification of RHD in low resource settings and where prevalence of GAS infections is high, the World Heart Federation has developed criteria for RHD diagnosis using echocardiography, supported by clinical history if available. The WHF additionally defines criteria for use in people younger than age 20 to diagnose "borderline" RHD, as identification of cases of RHD among children

28036-460: Was a divorced dancer, six years older than he was. Subsequently, he left for another visit to the United States, spending the autumn of 1938 at the IAS and publishing Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, a classic of modern mathematics. In that work he introduced the constructible universe , a model of set theory in which

28208-463: Was also living in Princeton during this time. Gödel and Einstein developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to

28380-447: Was assassinated by one of his former students, Johann Nelböck . This triggered "a severe nervous crisis" in Gödel. He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases. In 1933, Gödel first traveled to the U.S., where he met Albert Einstein , who became a good friend. He delivered an address to the annual meeting of

28552-581: Was buried in Princeton Cemetery . Adele died in 1981. Gödel believed that God was personal, and called his philosophy "rationalistic, idealistic, optimistic, and theological". He formulated a formal proof for the existence of God known as Gödel's ontological proof . Gödel believed in an afterlife, saying, "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It

28724-405: Was discussed at length around the turn of the 20th century but it is not widely accepted today. Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex. A complex proposition has other propositions as its constituents, which are linked to each other through propositional connectives like "and" or "if...then". Simple propositions, on

28896-491: Was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Gödel had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna , where he attended medical school at

29068-476: Was granted release from his Czechoslovak citizenship and then, in April, granted Austrian citizenship. When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32. In 1948, after World War II , at the age of 42, he became an American citizen. In his family, the young Gödel was nicknamed Herr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at

29240-581: Was one of four mathematicians examined in David Malone 's 2008 BBC documentary Dangerous Knowledge . The Kurt Gödel Society , founded in 1987, is an international organization for the promotion of research in logic, philosophy, and the history of mathematics . The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic has held an annual Gödel Lecture since 1990. The Gödel Prize

29412-440: Was to serve as the foundation for other fields of mathematics). Gödel constructed a formula that claims it is unprovable in a given formal system. If it were provable, it would be false. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there

29584-653: Was up, Gödel and his wife left Vienna for Princeton . To avoid the difficulty of an Atlantic crossing, the Gödels took the Trans-Siberian Railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the US by train to Princeton. During this trip, Gödel was supposed to be carrying a secret letter from Viennese physicist Hans Thirring to Albert Einstein to alert Franklin D. Roosevelt of

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