182-420: Ludwig Josef Johann Wittgenstein ( / ˈ v ɪ t ɡ ən ʃ t aɪ n , - s t aɪ n / VIT -gən-s(h)tyne , Austrian German: [ˈluːdvɪk ˈjoːsɛf ˈjoːhan ˈvɪtɡn̩ʃtaɪn] ; 26 April 1889 – 29 April 1951) was an Austrian philosopher who worked primarily in logic , the philosophy of mathematics , the philosophy of mind , and the philosophy of language . From 1929 to 1947, Wittgenstein taught at
364-501: 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
546-480: A "reproductive" rather than a "productive" thinker, he attributed this to his Jewish sense of identity. He wrote: 'The saint is the only Jewish "genius". Even the greatest Jewish thinker is no more than talented. (Myself for instance).' There is much discussion around the extent to which Wittgenstein and his siblings, who were of 3/4 Jewish descent, saw themselves as Jews. The issue has arisen in particular regarding Wittgenstein's schooldays, because Adolf Hitler was, for
728-422: A British engineer who was his friend and classmate at the time, reported that Logic 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 the structure of arguments alone, independent of their topic and content. Informal logic
910-594: A book review; and a children's dictionary. His voluminous manuscripts were edited and published posthumously. The first and best-known of this posthumous series is the 1953 book Philosophical Investigations . A 1999 survey among American university and college teachers ranked the Investigations as the most important book of 20th-century philosophy , standing out as "the one crossover masterpiece in twentieth-century philosophy, appealing across diverse specializations and philosophical orientations". His philosophy
1092-445: 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
1274-480: 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
1456-573: 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,
1638-425: 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
1820-510: 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
2002-686: 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)} "
SECTION 10
#17327660841292184-496: A fourth daughter Dora who died as a baby; and five boys: Johannes (Hans), Kurt, Rudolf (Rudi), Paul – who became a concert pianist despite losing an arm in World War I – and Ludwig, who was the youngest of the family. The children were baptized as Catholics, received formal Catholic instruction, and were raised in an exceptionally intense environment. The family was at the centre of Vienna's cultural life; Bruno Walter described
2366-415: 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
2548-551: 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
2730-402: 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
2912-691: A house of study) in Leipzig to provide different forms of studies to Jewish students who were prohibited from attending any institutions in Germany. Jewish studies were emphasized and much of the Jewish community of Leipzig became involved. Like all other cities claimed by the Nazis, Leipzig was subject to aryanisation . Beginning in 1933 and increasing in 1939, Jewish business owners were forced to give up their possessions and stores. This eventually intensified to
3094-474: A letter from an old friend in Austria, a priest. In it he says that he hopes my work will go well, if it should be God's will. Now that is all I want: if it should be God's will. In Culture and Value , Wittgenstein writes: Is what I am doing [my work in philosophy] really worth the effort? Yes, but only if a light shines on it from above. His close friend Norman Malcolm wrote: Wittgenstein's mature life
3276-614: 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
3458-545: A religious point of view." Wittgenstein referred to Augustine of Hippo in his Philosophical Investigations . Philosophically, Wittgenstein's thought shows alignment with religious discourse. For example, he would become one of the century's fiercest critics of scientism . Wittgenstein's religious belief emerged during his service for the Austrian army in World War I, and he was a devoted reader of Dostoevsky's and Tolstoy's religious writings. He viewed his wartime experiences as
3640-664: 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
3822-665: 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 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
SECTION 20
#17327660841294004-590: A significant role in precipitating the fall of communism in Central and Eastern Europe, mainly through demonstrations starting from St. Nicholas Church . The immediate effects of the reunification of Germany included the collapse of the local economy (which had come to depend on highly polluting heavy industry ), severe unemployment, and urban blight. By the early 2000s the trend had reversed, and since then Leipzig has undergone some significant changes, including urban and economic rejuvenation, and modernisation of
4186-409: A sincere and devoted sympathy, changed over time, much like his philosophical ideas. In 1912, Wittgenstein wrote to Russell saying that Mozart and Beethoven were the actual sons of God. However, Wittgenstein resisted formal religion, saying it was hard for him to "bend the knee", though his grandfather's beliefs continued to influence Wittgenstein – as he said, "I cannot help seeing every problem from
4368-497: A story about Paul practising on one of the pianos in the Wittgensteins' main family mansion, when he suddenly shouted at Ludwig in the next room: I cannot play when you are in the house, as I feel your skepticism seeping towards me from under the door! The family palace housed seven grand pianos and each of the siblings pursued music "with an enthusiasm that, at times, bordered on the pathological". The eldest brother, Hans,
4550-870: A student at the Realschule , Wittgenstein was influenced by Austrian philosopher Otto Weininger 's 1903 book Geschlecht und Charakter ( Sex and Character ). Weininger (1880–1903), who was Jewish, argued that the concepts of male and female exist only as Platonic forms , and that Jews tend to embody the Platonic femininity. Whereas men are basically rational, women operate only at the level of their emotions and sexual organs. Jews, Weininger argued, are similar, saturated with femininity, with no sense of right and wrong, and no soul. Weininger argues that man must choose between his masculine and feminine sides, consciousness and unconsciousness, platonic love and sexuality. Love and sexual desire stand in contradiction, and love between
4732-453: A top mark (5) in religious studies; a 2 for conduct and English, 3 for French, geography, history, mathematics and physics, and 4 for German, chemistry, geometry and freehand drawing. He had particular difficulty with spelling and failed his written German exam because of it. He wrote in 1931: My bad spelling in youth, up to the age of about 18 or 19, is connected with the whole of the rest of my character (my weakness in study). Wittgenstein
4914-570: A total of 63 Ortsteile (localities). Some of these correspond to outlying villages which have been annexed by Leipzig. Like many cities in Eastern Germany, Leipzig has an oceanic climate ( Köppen : Cfb ), with significant continental influences due to its inland location. Winters are cold, with an average temperature of around 1 °C (34 °F). Summers are generally warm, averaging at 19 °C (66 °F) with daytime temperatures of 24 °C (75 °F). Precipitation in winter
5096-911: A trade city since at least the time of the Holy Roman Empire . The city sits at the intersection of the Via Regia and the Via Imperii , two important medieval trade routes. Leipzig's trade fair dates back to 1190. Between 1764 and 1945, the city was a centre of publishing. After the Second World War and during the period of the German Democratic Republic (East Germany) Leipzig remained a major urban centre in East Germany, but its cultural and economic importance declined. Events in Leipzig in 1989 played
5278-416: A trial in which he strove to conform to the will of God, and in a journal entry from 29 April 1915, he writes: Perhaps the nearness of death will bring me the light of life. May God enlighten me. I am a worm, but through God I become a man. God be with me. Amen. Around this time, Wittgenstein wrote that "Christianity is indeed the only sure way to happiness", but he rejected the idea that religious belief
5460-488: A while, at the same school at the same time. Laurence Goldstein argues that it is "overwhelmingly probable" that the boys met each other and that Hitler would have disliked Wittgenstein, a "stammering, precocious, precious, aristocratic upstart ..."; Strathern flatly states they never met. Other commentators have dismissed as irresponsible and uninformed any suggestion that Wittgenstein's wealth and unusual personality might have fed Hitler's antisemitism, in part because there
5642-625: A woman and a man is therefore doomed to misery or immorality. The only life worth living is the spiritual one – to live as a woman or a Jew means one has no right to live at all; the choice is genius or death. Weininger committed suicide, shooting himself in 1903, shortly after publishing the book. Wittgenstein, then 14, attended Weininger's funeral. Many years later, as a professor at the University of Cambridge , Wittgenstein distributed copies of Weininger's book to his bemused academic colleagues. He said that Weininger's arguments were wrong, but that it
Ludwig Wittgenstein - Misplaced Pages Continue
5824-440: A year. Historian Brigitte Hamann writes that he stood out from the other boys: he spoke an unusually pure form of High German with a stutter, dressed elegantly, and was sensitive and unsociable. Monk writes that the other boys made fun of him, singing after him: "Wittgenstein wandelt wehmütig widriger Winde wegen Wienwärts" ("Wittgenstein wanders wistfully Vienna-wards (in) worsening winds"). In his leaving certificate, he received
6006-420: 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
6188-476: 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
6370-441: 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
6552-556: Is about half that of the summer. The amount of sunshine differs significantly between winter and summer, with an average of around 51 hours of sunshine in December (1.7 hours per day) compared with 229 hours of sunshine in July (7.4 hours per day). Leipzig was first documented in 1015 in the chronicles of Bishop Thietmar of Merseburg as urbs Libzi ( Chronicon , VII, 25) and endowed with city and market privileges in 1165 by Otto
6734-416: 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
6916-453: 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
7098-495: 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 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
7280-415: 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
7462-421: 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
Ludwig Wittgenstein - Misplaced Pages Continue
7644-407: 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
7826-453: 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
8008-438: 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. Leipzig Leipzig ( / ˈ l aɪ p s ɪ ɡ , - s ɪ x / LYPE -sig, -sikh , German: [ˈlaɪptsɪç] ; Upper Saxon : Leibz'sch ; Upper Sorbian : Lipsk )
8190-610: 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
8372-432: 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
8554-415: 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
8736-447: 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,
8918-688: 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
9100-518: 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
9282-589: Is no evidence they had anything to do with each other. Several commentators have argued that a school photograph of Hitler may show Wittgenstein in the lower left corner, While Wittgenstein would later claim that "[m]y thoughts are 100% Hebraic", as Hans Sluga has argued, if so, His was a self-doubting Judaism, which had always the possibility of collapsing into a destructive self-hatred (as it did in Weininger's case) but which also held an immense promise of innovation and genius. By Hebraic, he meant to include
SECTION 50
#17327660841299464-414: Is no indication that Hitler would have seen Wittgenstein as Jewish. Wittgenstein and Hitler were born just six days apart, though Hitler had to re-sit his mathematics exam before being allowed into a higher class, while Wittgenstein was moved forward by one, so they ended up two grades apart at the Realschule . Monk estimates that they were both at the school during the 1904–1905 school year, but says there
9646-407: 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
9828-442: Is now part of the city, Friedrich Schiller wrote his poem " Ode to Joy ". An older spelling of Leipzig in English is Leipsic . The Latin name Lipsia was also used. The name Leipzig is commonly held to derive from lipa , the common Slavic designation for linden trees , making the city's name etymologically related to Lipetsk, Russia and Liepaja, Latvia . Based on medieval attestations like Lipzk (c. 1190),
10010-578: Is often divided into an early period, exemplified by the Tractatus , and a later period, articulated primarily in the Philosophical Investigations . The "early Wittgenstein" was concerned with the logical relationship between propositions and the world, and he believed that by providing an account of the logic underlying this relationship, he had solved all philosophical problems. The "later Wittgenstein", however, rejected many of
10192-467: Is one of the oldest symphony orchestras in the world. Several well-known composers lived and worked in Leipzig, including Johann Sebastian Bach (1723 to 1750) and Felix Mendelssohn (1835 to 1847). The University of Music and Theatre "Felix Mendelssohn Bartholdy" was founded in 1843. The Oper Leipzig , one of the most prominent opera houses in Germany, was founded in 1693. During a stay in Gohlis , which
10374-541: 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
10556-430: 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
10738-542: Is the most populous city in the German state of Saxony . The city has a population of 628,718 inhabitants as of 2023. It is the eighth-largest city in Germany and is part of the Central German Metropolitan Region . The name of the city is usually interpreted as a Slavic term meaning place of linden trees , in line with many other Slavic placenames in the region. Leipzig is located about 150 km (90 mi) southwest of Berlin, in
10920-925: Is the part of the North European Plain in Germany. The city sits on the White Elster , a river that rises in the Czech Republic and flows into the Saale south of Halle. The Pleiße and the Parthe join the White Elster in Leipzig, and the large inland delta-like landscape the three rivers form is called Leipziger Gewässerknoten . The site is characterized by swampy areas such as the Leipzig Riparian Forest ( Leipziger Auenwald ), though there are also some limestone areas to
11102-410: 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
SECTION 60
#173276608412911284-432: 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
11466-540: 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
11648-599: 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)
11830-431: 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
12012-518: The Berlin Academy , the third eldest brother, Rudi, committed suicide in a Berlin bar. He had asked the pianist to play Thomas Koschat 's " Verlassen, verlassen, verlassen bin ich " ("Forsaken, forsaken, forsaken am I"), before mixing himself a drink of milk and potassium cyanide . He had left several suicide notes, one to his parents that said he was grieving over the death of a friend, and another that referred to his "perverted disposition". It
12194-497: The East German government . The reunification of Germany , however, was at first not good for Leipzig. The centrally planned heavy industry that had become the city's specialty was, in terms of the advanced economy of reunited Germany, almost completely unviable, and closed. Within only six years, 90% of jobs in industry had vanished. As unemployment rocketed, the population fell dramatically; some 100,000 people left Leipzig in
12376-819: The Haidbauer incident ), especially during mathematics classes; working during World War II as a hospital porter in London; and working as a hospital laboratory technician at the Royal Victoria Infirmary in Newcastle upon Tyne . According to a family tree prepared in Jerusalem after World War II, Wittgenstein's paternal great-great-grandfather was Moses Meier, an Ashkenazi Jewish land agent who lived with his wife, Brendel Simon, in Bad Laasphe in
12558-638: The Nazi government officially renamed the city Reichsmessestadt Leipzig (Reich Trade Fair City Leipzig). In 1989 Leipzig was dubbed a Hero City ( Heldenstadt ), alluding to the honorary title awarded in the former Soviet Union to certain cities that played a key role in the victory of the Allies during the Second World War, in recognition of the role that the Monday demonstrations there played in
12740-805: The Nuremberg Laws . However, due to the Leipzig Trade Fair and the international attention it garnered, Leipzig was especially cautious about its public image. Despite this, the Leipzig authorities were not afraid to strictly apply and enforce anti-semitic measures. On 20 December 1937, after the Nazis took control of the city, they renamed it Reichsmessestadt Leipzig, meaning the "Imperial Trade Fair City Leipzig". In early 1938, Leipzig saw an increase in Zionism through Jewish citizens. Many of these Zionists attempted to flee before deportations began. On 28 October 1938, Heinrich Himmler ordered
12922-734: The Potsdam Agreement . In the mid-20th century, the city's trade fair assumed renewed importance as a point of contact with the Comecon Eastern Europe economic bloc, of which East Germany was a member. At this time, trade fairs were held at a site in the south of the city, near the Monument to the Battle of the Nations . The planned economy of the German Democratic Republic, however, was not kind to Leipzig. Before
13104-636: The Principality of Wittgenstein , Westphalia . In July 1808, Napoleon issued a decree that everyone, including Jews, must adopt an inheritable family surname, so Meier's son, also Moses, took the name of his employers, the Sayn-Wittgensteins , and became Moses Meier Wittgenstein. His son, Hermann Christian Wittgenstein — who took the middle name "Christian" to distance himself from his Jewish background — married Fanny Figdor, also Jewish, who converted to Protestantism just before they married, and
13286-500: The Royal Meteorological Society researched and investigated the ionization of the upper atmosphere, by suspending instruments on balloons or kites. At Glossop, Wittgenstein worked under Professor of Physics Sir Arthur Schuster . He also worked on the design of a propeller with small jet ( Tip jet ) engines on the end of its blades, something he patented in 1911, and that earned him a research studentship from
13468-473: The University of Cambridge . Despite his position, only one book of his philosophy was published during his entire life: the 75-page Logisch-Philosophische Abhandlung ( Logical-Philosophical Treatise , 1921), which appeared, together with an English translation, in 1922 under the Latin title Tractatus Logico-Philosophicus . His only other published works were an article, " Some Remarks on Logical Form " (1929);
13650-494: The American Consulate was closed, and its building became a temporary place of stay for Americans and Allied refugees from Serbia , Romania and Japan . During the 1930s and 1940s, music was prominent throughout Leipzig. Many students attended Felix Mendelssohn Bartholdy College of Music and Theatre (then named Landeskonservatorium.) However, in 1944, it was closed due to World War II . It re-opened soon after
13832-671: The Christian tradition, in contradistinction to the Greek tradition, holding that good and evil could not be reconciled. He began his studies in mechanical engineering at the Technische Hochschule Berlin in Charlottenburg , Berlin, on 23 October 1906, lodging with the family of Professor Jolles. He attended for three semesters, and was awarded a diploma ( Abgangzeugnis ) on 5 May 1908. During his time at
14014-417: 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
14196-613: The Institute, Wittgenstein developed an interest in aeronautics . He arrived at the Victoria University of Manchester in the spring of 1908 to study for a doctorate, full of plans for aeronautical projects, including designing and flying his own plane. He conducted research into the behaviour of kites in the upper atmosphere, experimenting at a meteorological observation site near Glossop in Derbyshire . Specifically,
14378-591: The Leipzig-Thekla subcamp who were unable to march were either burned alive, shot or beaten to death by the Gestapo, SS , Volkssturm and German civilians in the Abtnaundorf massacre. Some were rescued by Polish forced laborers of another camp; at least 67 people survived. 84 victims were buried on 27 April 1945, however, the total number of victims remains unknown. During World War II, Leipzig
14560-704: The Polish Consulate and its library. In 1941, the American Consulate was also closed by order of the German authorities. During the war, Leipzig was the location of five subcamps of the Buchenwald concentration camp , in which over 8,000 men, women and children were imprisoned, mostly Polish, Jewish, Soviet and French, but also Italian, Czech and Belgian. In April 1945, most surviving prisoners were sent on death marches to various destinations in Saxony and German-occupied Czechoslovakia , whereas prisoners of
14742-697: The Rich . Leipzig Trade Fair , started in the Middle Ages , has become an event of international importance and is the oldest surviving trade fair in the world. This encouraged the grewing of the Leipzig merchant bourgeoisie . There are records of commercial fishing operations on the river Pleiße that, most likely, refer to Leipzig dating back to 1305, when the Margrave Dietrich the Younger granted
14924-459: The Second World War, Leipzig had developed a mixture of industry, creative business (notably publishing), and services (including legal services). During the period of the German Democratic Republic, services became the concern of the state, concentrated in East Berlin ; creative business moved to West Germany ; and Leipzig was left only with heavy industry. To make matters worse, this industry
15106-486: The Swedish-led side. On 24 December 1701, when Franz Conrad Romanus was mayor, an oil-fueled street lighting system was introduced. The city employed light guards who had to follow a specific schedule to ensure the punctual lighting of the 700 lanterns. The Leipzig region was the arena of the 1813 Battle of Leipzig between Napoleonic France and an allied coalition of Prussia , Russia , Austria and Sweden. It
15288-576: The US, the family was to an extent shielded from the hyperinflation that hit Austria in 1922 . However, their wealth diminished due to post-1918 hyperinflation and subsequently during the Great Depression , although even as late as 1938 they owned 13 mansions in Vienna alone. Wittgenstein was ethnically Jewish . His mother was Leopoldine Maria Josefa Kalmus, known among friends as "Poldi". Her father
15470-681: The Wittgensteins became the second wealthiest family in the Austro-Hungarian Empire , only the Rothschilds being wealthier. Karl Wittgenstein was viewed as the Austrian equivalent of Andrew Carnegie , with whom he was friends, and was one of the wealthiest men in the world by the 1890s. As a result of his decision in 1898 to invest substantially in the Netherlands and in Switzerland as well as overseas, particularly in
15652-523: The Wittgensteins considered themselves Jewish. This was evident during the Nazi era, when Ludwig's sister was assured by an official that they wouldn't be considered as Jews under the racial laws. Indignant at the state's attempt to dictate her identity, she demanded papers certifying their Jewish lineage. In his own writings, Wittgenstein frequently referred to himself as Jewish, often in a self-deprecating manner. For instance, while criticizing himself for being
15834-418: 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
16016-561: The architect Adolf Loos ." Later, in a period of severe personal depression after World War I, he gave away his remaining fortune to his brothers and sisters. Three of his four older brothers died by separate acts of suicide. Wittgenstein left academia several times: serving as an officer on the front line during World War I, where he was decorated a number of times for his courage; teaching in schools in remote Austrian villages, where he encountered controversy for using sometimes violent corporal punishment on both girls and boys (see, for example,
16198-399: 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
16380-473: The assumptions of the Tractatus , arguing that the meaning of words is best understood as their use within a given language game . Born in Vienna into one of Europe's richest families, he inherited a fortune from his father in 1913. Before World War I, he "made a very generous financial bequest to a group of poets and artists chosen by Ludwig von Ficker, the editor of Der Brenner , from artists in need. These included Trakl as well as Rainer Maria Rilke and
16562-495: 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
16744-487: 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
16926-572: The buildings hit, a firestorm occurred. This prompted firefighters to rush to the city; however, they were unable to control the fires. Unlike the firebombing of the neighbouring city of Dresden , this was a largely conventional bombing with high explosives rather than incendiaries. The resultant pattern of loss was a patchwork, rather than wholesale loss of its centre, but was nevertheless extensive. The Allied ground advance into Germany reached Leipzig in late April 1945. The U.S. 2nd Infantry Division and U.S. 69th Infantry Division fought their way into
17108-499: The centrepiece of the S-Bahn Mitteldeutschland ( S-Bahn Central Germany ) public transit system, Germany's largest S-Bahn network, with a system length of 802 km (498 mi). Leipzig has long been a major centre for music, including classical and modern dark wave . The Thomanerchor (English: St. Thomas Choir of Leipzig), a boys' choir, was founded in 1212. The Leipzig Gewandhaus Orchestra , established in 1743,
17290-458: The centrifugal force exerted by the revolving arms and ignited. Propellers of the time were typically wood, whereas modern blades are made from pressed steel laminates as separate halves, which are then welded together. This gives the blade a hollow interior and thereby creates an ideal pathway for the air and gas. Work on the jet-powered propeller proved frustrating for Wittgenstein, who had very little experience working with machinery. Jim Bamber,
17472-588: The city on 18 April and completed its capture after fierce urban action, in which fighting was often house-to-house and block-to-block, on 19 April 1945. In April 1945, the Mayor of Leipzig, SS- Gruppenführer Alfred Freyberg , his wife and daughter, together with Deputy Mayor and City Treasurer Ernest Kurt Lisso, his wife, daughter and Volkssturm Major and former Mayor Walter Dönicke, all committed suicide in Leipzig City Hall. The United States turned
17654-524: The city over to the Red Army as it pulled back from the line of contact with Soviet forces in July 1945 to the designated occupation zone boundaries. Leipzig became one of the major cities of the German Democratic Republic ( East Germany ). Following the end of World War II in 1945, Leipzig saw a slow return of Jews to the city. They were joined by large numbers of German refugees who had been expelled from Central and Eastern Europe in accordance with
17836-498: The city. Another, somewhat old-fashioned epithet is Pleiß-Athen ( Athens on the Pleiße River ), hinting at Leipzig's long academic and literary tradition, as the seat of one of the oldest German universities and a centre of the book trade. It is also referred to as "Little Paris" ( Klein-Paris ) after Johann Wolfgang Goethe 's Faust I , which is partly set in the famous Leipzig restaurant Auerbachs Keller . In 1937
18018-645: 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
18200-495: 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,
18382-511: 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
18564-458: 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
18746-591: 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
18928-438: 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
19110-467: The couple founded a successful business trading in wool in Leipzig . Ludwig's grandmother Fanny was a first cousin of the violinist Joseph Joachim . They had 11 children – among them Wittgenstein's father. Karl Otto Clemens Wittgenstein (1847–1913) became an industrial tycoon, and by the late 1880s was one of the richest men in Europe, with an effective monopoly on Austria's steel cartel. Thanks to Karl,
19292-589: The deportation of Polish Jews from Leipzig to Poland. The Polish Consulate sheltered 1,300 Polish Jews, preventing their deportation. On 9 November 1938, as part of Kristallnacht , in Gottschedstrasse , synagogues and businesses were set on fire. Only a couple of days later, on 11 November 1938, many Jews in the Leipzig area were deported to the Buchenwald Concentration Camp. As World War II came to an end, much of Leipzig
19474-451: 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
19656-638: The eastern one for the Royal Saxon State Railways and the western one for the Prussian state railways . In the 19th century, Leipzig was a centre of the German and Saxon liberal movements. The first German labor party , the General German Workers' Association ( Allgemeiner Deutscher Arbeiterverein , ADAV) was founded in Leipzig on 23 May 1863 by Ferdinand Lassalle ; about 600 workers from across Germany travelled to
19838-407: The end of World War I, when the Austrian troops he was commanding refused to obey his orders and deserted en masse . According to Gottlieb, Hermine had said Kurt seemed to carry "the germ of disgust for life within himself". Later, Ludwig wrote: I ought to have ... become a star in the sky. Instead of which I have remained stuck on earth. Wittgenstein was taught by private tutors at home until he
20020-515: The exam for the more technically oriented k.u.k. Realschule in Linz , a small state school with 300 pupils. In 1903, when he was 14, he began his three years of formal schooling there, lodging nearby during the term with the family of Josef Strigl, a teacher at the local gymnasium, the family giving him the nickname Luki. On starting at the Realschule, Wittgenstein had been moved forward
20202-409: The example. The truth of a proposition usually depends on 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
20384-594: The fall of the East German regime. More recently, the city has sometimes been nicknamed Hypezig , the "Boomtown of eastern Germany", or "The better Berlin" ( Das bessere Berlin ) and is celebrated by the media as a hip urban centre for its vibrant lifestyle and creative scene with many startups . Leipzig is located in the Leipzig Bay , the southernmost part of the North German Plain , which
20566-569: The family's numerous music rooms. Wittgenstein, who valued precision and discipline, never considered contemporary classical music acceptable. He said to his friend Drury in 1930: Music came to a full stop with Brahms ; and even in Brahms I can begin to hear the noise of machinery. Ludwig Wittgenstein himself had absolute pitch , and his devotion to music remained vitally important to him throughout his life; he made frequent use of musical examples and metaphors in his philosophical writings, and he
20748-434: 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
20930-541: The fishing rights to the church and convent of St Thomas. There were a number of monasteries in and around the city, including a Franciscan monastery after which the Barfußgäßchen (Barefoot Alley) is named and a monastery of Irish monks ( Jacobskirche , destroyed in 1544) near the present day Ranstädter Steinweg (the old Via Regia ). The University of Leipzig was founded in 1409 and Leipzig developed into an important centre of German law and of
21112-468: The five brothers later committed suicide. Psychiatrist Michael Fitzgerald argues that Karl was a harsh perfectionist who lacked empathy, and that Wittgenstein's mother was anxious and insecure, unable to stand up to her husband. Johannes Brahms said of the family, whom he visited regularly: They seemed to act towards one another as if they were at court. The family appeared to have a strong streak of depression running through it. Anthony Gottlieb tells
21294-485: 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
21476-513: 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
21658-523: 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
21840-419: 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
22022-546: The foundation on the new railway. Leipzig expanded rapidly to more than 700,000 inhabitants. Huge Gründerzeit areas were built, which mostly survived both war and post-war demolition. With the opening of a fifth production hall in 1907, the Leipziger Baumwollspinnerei became the largest cotton mill company on the continent, housing over 240,000 spindles. Yearly production surpassed 5 million kilograms of yarn. During World War I , in 1917,
22204-421: 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
22386-516: The greatness lies in that with which we disagree. It is his enormous mistake which is great. I.e. roughly speaking if you just add a "~" to the whole book it says an important truth. In an unusual move, Wittgenstein took out a copy of Weininger's work on 1 June 1931 from the Special Order Books in the university library. He met Moore on 2 June, when he probably gave this copy to Moore. Despite their and their forebears' Christianization,
22568-465: The importance of the idea of confession . He wrote in his diaries about having made a major confession to his oldest sister, Hermine, while he was at the Realschule ; Monk speculates that it may have been about his loss of faith. He also discussed it with Gretl, his other sister, who directed him to Schopenhauer's The World as Will and Representation . As a teenager, Wittgenstein adopted Schopenhauer's epistemological idealism . However, after he studied
22750-406: 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
22932-407: 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
23114-524: The intersection of the ancient roads known as the Via Regia (King's highway), which traversed Germany in an east–west direction, and the Via Imperii (Imperial highway), a north–south road. Leipzig was a walled city in the Middle Ages and the current "ring" road around the historic centre of the city follows the line of the old city walls. Since 1992 Leipzig has been divided administratively into ten Stadtbezirke (boroughs), which in turn contain
23296-477: The life at the Wittgensteins' palace as an "all-pervading atmosphere of humanity and culture." Karl was a leading patron of the arts, commissioning works by Auguste Rodin and financing the city's exhibition hall and art gallery, the Secession Building . Gustav Klimt painted a portrait of Wittgenstein's sister Margaret for her wedding, and Johannes Brahms and Gustav Mahler gave regular concerts in
23478-522: 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
23660-407: The north of the city. The landscape is mostly flat, though there is also some evidence of moraine and drumlins . Although there are some forest parks within the city limits, the area surrounding Leipzig is relatively unforested. During the 20th century, there were several open-pit mines in the region, many of which have been converted to lakes. Also see: Neuseenland Leipzig is also situated at
23842-454: The original Slavic name of the city has been reconstructed as *Lipьsko , which is also reflected in similar forms in neighbouring modern Slavic languages (Sorbian/Polish Lipsk , Czech Lipsko ). This has, however, been questioned by more recent onomastic research based on the very oldest forms like Libzi (c. 1015). Due to the etymology mentioned above, Lindenstadt or Stadt der Linden (City of Linden Trees) are common poetic epithets for
24024-409: 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
24206-444: 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
24388-429: The philosophy of mathematics, he abandoned epistemological idealism for Gottlob Frege 's conceptual realism . In later years, Wittgenstein was highly dismissive of Schopenhauer, describing him as an ultimately "shallow" thinker: Schopenhauer is quite a crude mind ... Where real depth starts, his comes to an end. Wittgenstein's relationship with Christianity and with religion in general, for which he always professed
24570-463: The point where Nazi officials were strong enough to evict the Jews from their own homes. They also had the power to force many of the Jews living in the city to sell their houses. Many people who sold their homes emigrated elsewhere, outside of Leipzig. Others moved to Judenhäuser, which were smaller houses that acted as ghettos, housing large groups of people. The Jews of Leipzig were greatly affected by
24752-485: 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,
24934-413: 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
25116-470: 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
25298-452: The press to support local and state militias, patriotic wartime mobilization, humanitarian relief and postwar commemorative practices and rituals. When it was made a terminus of the first German long-distance railway to Dresden (the capital of Saxony) in 1839, Leipzig became a hub of Central European railway traffic, with Leipzig Hauptbahnhof the largest terminal station by area in Europe. The railway station has two grand entrance halls,
25480-485: 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
25662-406: 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
25844-417: 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
26026-692: The publishing industry in Germany, resulting, in the 19th and 20th centuries, with the Reichsgericht (Imperial Court of Justice) and the German National Library being located here. During the Thirty Years' War , two battles took place in Breitenfeld , about 8 km (5 mi) outside Leipzig city walls. The first Battle of Breitenfeld took place in 1631 and the second in 1642. Both battles resulted in victories for
26208-645: The remaining 18 Jews still in Leipzig were deported from Leipzig to Auschwitz. According to records of the two waves of deportations to Auschwitz there were no survivors. According to records of the Theresienstadt deportation, only 53 Jews survived. During the German invasion of Poland at the start of World War II , in September 1939, the Gestapo carried out arrests of prominent local Poles , and seized
26390-436: 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
26572-492: 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
26754-418: 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
26936-604: 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
27118-518: 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
27300-404: 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
27482-415: 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
27664-870: 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
27846-474: 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
28028-533: The southernmost part of the North German Plain (the Leipzig Bay ), at the confluence of the White Elster and its tributaries Pleiße and Parthe , that form an extensive inland delta in the city known as Leipziger Gewässerknoten [ de ] , along which Leipzig Riverside Forest , Europe's largest intra-city riparian forest has developed. Leipzig is at the centre of Neuseenland ( new lake district ), consisting of several artificial lakes created from former lignite open-pit mines . Leipzig has been
28210-418: 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
28392-437: 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
28574-550: The strongest form of support: if their premises are true then their conclusion must also be true. This 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
28756-438: 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
28938-700: 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
29120-417: 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
29302-423: 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
29484-409: 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
29666-399: 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"
29848-906: The transport infrastructure. Leipzig is home to one of the oldest universities in Europe ( Leipzig University ). It is the main seat of the German National Library (the second is Frankfurt ), the seat of the German Music Archive , as well as of the German Federal Administrative Court . Leipzig Zoo is one of the most modern zoos in Europe and as of 2018 ranks first in Germany and second in Europe. Leipzig's late-19th-century Gründerzeit architecture consists of around 12,500 buildings. The city's central railway terminus Leipzig Hauptbahnhof is, at 83,460 square metres (898,400 sq ft), Europe's largest railway station measured by floor area. Since Leipzig City Tunnel came into operation in 2013, it has formed
30030-479: 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
30212-432: The university in the autumn of 1908. At the time, contemporary propeller designs were not advanced enough to actually put Wittgenstein's ideas into practice, and it would be years before a blade design that could support Wittgenstein's innovative design was created. Wittgenstein's design required air and gas to be forced along the propeller arms to combustion chambers on the end of each blade, where they were then compressed by
30394-413: The war ended in 1945. On 22 May 1930, Carl Friedrich Goerdeler was elected mayor of Leipzig. He later became an opponent of the Nazi regime . He resigned in 1937 when, in his absence, his Nazi deputy ordered the destruction of the city's statue of Felix Mendelssohn . On Kristallnacht in 1938, the 1855 Moorish Revival Leipzig synagogue , one of the city's most architecturally significant buildings,
30576-479: Was repeatedly struck by Allied bombing raids , beginning in 1943 and lasting until 1945. The first raid occurred on the morning of 4 December 1943, when 442 bombers of the Royal Air Force (RAF) dropped a total amount of almost 1,400 tons of explosives and incendiaries on the city, destroying large parts of the city centre. This bombing was the largest up to that time. Due to the close proximity of many of
30758-477: Was 14 years old. Subsequently, for three years, he attended a school. After the deaths of Hans and Rudi, Karl relented and allowed Paul and Ludwig to be sent to school. Waugh writes that it was too late for Wittgenstein to pass his exams for the more academic Gymnasium in Wiener Neustadt; having had no formal schooling, he failed his entrance exam and only barely managed after extra tutoring to pass
30940-568: Was a Bohemian Jew, and her mother was an Austrian- Slovene Catholic – she was Wittgenstein's only non-Jewish grandparent. Poldi was an aunt of the Nobel Prize laureate Friedrich Hayek on his maternal side. Wittgenstein was born at 8:30 PM on 26 April 1889 in the "Villa Wittgenstein" at what is today Neuwaldegger Straße 38 in the suburban parish Neuwaldegg [ de ] next to Vienna. Karl and Poldi had nine children in all – four girls: Hermine, Margaret (Gretl), Helene, and
31122-616: Was baptized as an infant by a Catholic priest and received formal instruction in Catholic doctrine as a child, as was common at the time. In an interview, his sister Gretl Stonborough-Wittgenstein says that their grandfather's "strong, severe, partly ascetic Christianity " was a strong influence on all the Wittgenstein children. While he was at the Realschule , he decided he lacked religious faith and began reading Arthur Schopenhauer per Gretl's recommendation. He nevertheless believed in
31304-587: Was deliberately destroyed. Goerdeler was later executed by the Nazis on 2 February 1945. Several thousand forced labourers were stationed in Leipzig during the Second World War. Beginning in 1933, many Jewish citizens of Leipzig were members of the Gemeinde , a large Jewish religious community spread throughout Germany, Austria and Switzerland. In October 1935, the Gemeinde helped found the Lehrhaus (English:
31486-614: Was destroyed. Following the war, the Communist Party of Germany provided aid for the reconstruction of the city. In 1933, a census recorded that over 11,000 Jews were living in Leipzig. In the 1939 census, the number had fallen to roughly 4,500, and by January 1942 only 2,000 remained. In that month, these 2,000 Jews began to be deported. On 13 July 1942, 170 Jews were deported from Leipzig to Auschwitz concentration camp . On 19 September 1942, 440 Jews were deported from Leipzig to Theresienstadt concentration camp . On 18 June 1943,
31668-405: 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
31850-402: Was extremely polluting, making Leipzig an even less attractive city to live in. Between 1950 and the end of the German Democratic Republic, the population of Leipzig fell from 600,000 to 500,000. In October 1989, after prayers for peace at St. Nicholas Church , established in 1983 as part of the peace movement, the Monday demonstrations started as the most prominent mass protest against
32032-677: Was hailed as a musical prodigy. At the age of four, writes Alexander Waugh , Hans could identify the Doppler effect in a passing siren as a quarter-tone drop in pitch, and at five started crying "Wrong! Wrong!" when two brass bands in a carnival played the same tune in different keys . But he died in mysterious circumstances in May 1902, when he ran away to the US and disappeared from a boat in Chesapeake Bay , most likely having committed suicide. Two years later, aged 22 and studying chemistry at
32214-491: Was merely thinking that a certain doctrine was true. From this time on, Wittgenstein viewed religious faith as a way of living and opposed rational argumentation or proofs for God. With age, a deepening personal spirituality led to several elucidations and clarifications, as he untangled language problems in religion—attacking, for example, the temptation to think of God's existence as a matter of scientific evidence. In 1947, finding it more difficult to work, he wrote: I have had
32396-680: Was reported at the time that he had sought advice from the Scientific-Humanitarian Committee , an organization that was campaigning against Paragraph 175 of the German Criminal Code, which prohibited homosexual sex. Ludwig himself was a closeted homosexual, who separated sexual intercourse from love, despising all forms of the former. His father forbade the family from ever mentioning his name again. The second eldest brother, Kurt, an officer and company director, shot himself on 27 October 1918 just before
32578-430: Was strongly marked by religious thought and feeling. I am inclined to think that he was more deeply religious than are many people who correctly regard themselves as religious believers. Toward the end, Wittgenstein wrote: Bach wrote on the title page of his Orgelbüchlein , 'To the glory of the most high God, and that my neighbour may be benefited thereby.' That is what I would have liked to say about my work. While
32760-585: Was the largest battle in Europe before the First World War and the coalition victory ended Napoleon 's presence in Germany and would ultimately lead to his first exile on Elba . The Monument to the Battle of the Nations celebrating the centenary of this event was completed in 1913. In addition to stimulating German nationalism, the war had a major impact in mobilizing a civic spirit in numerous volunteer activities. Many volunteer militias and civic associations were formed, and collaborated with churches and
32942-460: Was the way they were wrong that was interesting. In a letter dated 23 August 1931, Wittgenstein wrote the following to G. E. Moore : Dear Moore, Thanks for your letter. I can quite imagine that you don't admire Weininger very much, what with that beastly translation and the fact that W. must feel very foreign to you. It is true that he is fantastic but he is great and fantastic. It isn't necessary or rather not possible to agree with him but
33124-710: Was unusually adept at whistling lengthy and detailed musical passages. He also learnt to play the clarinet in his 30s. A fragment of music (three bars), composed by Wittgenstein, was discovered in one of his 1931 notebooks, by Michael Nedo , director of the Wittgenstein Institute in Cambridge. Ray Monk writes that Karl's aim was to turn his sons into captains of industry; they were not sent to school lest they acquire bad habits but were educated at home to prepare them for work in Karl's industrial empire. Three of
#128871