The Principia Mathematica is a three-volume work on the foundations of mathematics written by Alfred North Whitehead and Bertrand Russell and published in 1910, 1912, 1913. In 1925–27, it appeared in a second edition with an important Introduction to the Second Edition, an Appendix A that replaced ✸9 and all-new Appendix B and Appendix C. PM is not to be confused with Russell's 1903 The Principles of Mathematics. PM was conceived as a sequel volume to Russell's 1903 Principles, but as PM states, this became an unworkable suggestion for practical and philosophical reasons: "The present work was intended by us to be comprised in a second volume of Principles of Mathematics... But as we advanced, it became evident that the subject is a much larger one than we had supposed. PM, according to its introduction, had three aims: to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of primitive notions and axioms, inference rules; this third aim motivated the adoption of the theory of types in PM.
The theory of types adopts grammatical restrictions on formulas that rules out the unrestricted comprehension of classes and functions. The effect of this is that formulas such as would allow the comprehension of objects like the Russell set turn out to be ill-formed: they violate the grammatical restrictions of the system of PM. There is no doubt that PM is of great importance in the history of mathematics and philosophy: as Irvine has noted, it sparked interest in symbolic logic and advanced the subject by popularizing it. Indeed, PM was in part brought about by an interest in Logicism, the view on which all mathematical truths are logical truths, it was in part thanks to the advances made in PM that, despite its defects, numerous advances in meta-logic were made, including Gödel's incompleteness theorems. For all that, PM is not used today: the foremost reason for this is its reputation for typographical complexity. Somewhat infamously, several hundred pages of PM precede the proof of the validity of the proposition 1+1=2.
Contemporary mathematicians tend to use a modernized form of the system of Zermelo–Fraenkel set theory. Nonetheless, the scholarly and philosophical interest in PM is great and ongoing: for example, the Modern Library placed it 23rd in a list of the top 100 English-language nonfiction books of the twentieth century; the Principia covered only set theory, cardinal numbers, ordinal numbers, real numbers. Deeper theorems from real analysis were not included, but by the end of the third volume it was clear to experts that a large amount of known mathematics could in principle be developed in the adopted formalism, it was clear how lengthy such a development would be. A fourth volume on the foundations of geometry had been planned, but the authors admitted to intellectual exhaustion upon completion of the third; as noted in the criticism of the theory by Kurt Gödel, unlike a formalist theory, the "logicistic" theory of PM has no "precise statement of the syntax of the formalism". Another observation is that immediately in the theory, interpretations are presented in terms of truth-values for the behaviour of the symbols "⊢", "~", "V".
Truth-values: PM embeds the notions of "truth" and "falsity" in the notion "primitive proposition". A raw formalist theory would not provide the meaning of the symbols that form a "primitive proposition"—the symbols themselves could be arbitrary and unfamiliar; the theory would specify only. By assignment of "values", a model would specify an interpretation of what the formulas are saying, thus in the formal Kleene symbol set below, the "interpretation" of what the symbols mean, by implication how they end up being used, is given in parentheses, e.g. "¬". But this is not a pure Formalist theory; the following formalist theory is offered as contrast to the logicistic theory of PM. A contemporary formal system would be constructed as follows: Symbols used: This set is the starting set, other symbols can appear but only by definition from these beginning symbols. A starting set might be the following set derived from Kleene 1952: logical symbols: "→", "&", "V", "¬", "∀", "∃". Symbol strings: The theory will build "strings" of these symbols by concatenation.
Formation rules: The theory specifies the rules of syntax as a recursive definition that starts with "0" and specifies how to build acceptable strings or "well-formed formulas". This includes a rule for "substitution" of strings for the symbols called "variables". Transformation rule: The axioms that specify the behaviours of the symbols and
Aristotle was a philosopher during the Classical period in Ancient Greece, the founder of the Lyceum and the Peripatetic school of philosophy and Aristotelian tradition. Along with his teacher Plato, he is considered the "Father of Western Philosophy", his writings cover many subjects – including physics, zoology, logic, aesthetics, theatre, rhetoric, linguistics, economics and government. Aristotle provided a complex synthesis of the various philosophies existing prior to him, it was above all from his teachings that the West inherited its intellectual lexicon, as well as problems and methods of inquiry; as a result, his philosophy has exerted a unique influence on every form of knowledge in the West and it continues to be a subject of contemporary philosophical discussion. Little is known about his life. Aristotle was born in the city of Stagira in Northern Greece, his father, died when Aristotle was a child, he was brought up by a guardian. At seventeen or eighteen years of age, he joined Plato's Academy in Athens and remained there until the age of thirty-seven.
Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon, tutored Alexander the Great beginning in 343 BC. He established a library in the Lyceum which helped him to produce many of his hundreds of books on papyrus scrolls. Though Aristotle wrote many elegant treatises and dialogues for publication, only around a third of his original output has survived, none of it intended for publication; the fact that Aristotle was a pupil of Plato contributed to his former views of Platonism, following Plato's death, Aristotle developed an increased interest in natural sciences and adopted the position of immanent realism. Aristotle's views on physical science profoundly shaped medieval scholarship, their influence extended from Late Antiquity and the Early Middle Ages into the Renaissance, were not replaced systematically until the Enlightenment and theories such as classical mechanics. Some of Aristotle's zoological observations found in his biology, such as on the hectocotyl arm of the octopus, were disbelieved until the 19th century.
His works contain the earliest known formal study of logic, studied by medieval scholars such as Peter Abelard and John Buridan. Aristotle's influence on logic continued well into the 19th century He influenced Islamic thought during the Middle Ages, as well as Christian theology the Neoplatonism of the Early Church and the scholastic tradition of the Catholic Church. Aristotle was revered among medieval Muslim scholars as "The First Teacher" and among medieval Christians like Thomas Aquinas as "The Philosopher", his ethics, though always influential, gained renewed interest with the modern advent of virtue ethics, such as in the thinking of Alasdair MacIntyre and Philippa Foot. In general, the details of Aristotle's life are not well-established; the biographies written in ancient times are speculative and historians only agree on a few salient points. Aristotle, whose name means "the best purpose" in Ancient Greek, was born in 384 BC in Stagira, about 55 km east of modern-day Thessaloniki.
His father Nicomachus was the personal physician to King Amyntas of Macedon. Both of Aristotle's parents died when he was about thirteen, Proxenus of Atarneus became his guardian. Although little information about Aristotle's childhood has survived, he spent some time within the Macedonian palace, making his first connections with the Macedonian monarchy. At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Plato's Academy, he remained there for nearly twenty years before leaving Athens in 348/47 BC. The traditional story about his departure records that he was disappointed with the Academy's direction after control passed to Plato's nephew Speusippus, although it is possible that he feared the anti-Macedonian sentiments in Athens at that time and left before Plato died. Aristotle accompanied Xenocrates to the court of his friend Hermias of Atarneus in Asia Minor. After the death of Hermias, Aristotle travelled with his pupil Theophrastus to the island of Lesbos, where together they researched the botany and zoology of the island and its sheltered lagoon.
While in Lesbos, Aristotle married Hermias's adoptive daughter or niece. She bore him a daughter, whom they named Pythias. In 343 BC, Aristotle was invited by Philip II of Macedon to become the tutor to his son Alexander. Aristotle was appointed as the head of the royal academy of Macedon. During Aristotle's time in the Macedonian court, he gave lessons not only to Alexander, but to two other future kings: Ptolemy and Cassander. Aristotle encouraged Alexander toward eastern conquest and Aristotle's own attitude towards Persia was unabashedly ethnocentric. In one famous example, he counsels Alexander to be "a leader to the Greeks and a despot to the barbarians, to look after the former as after friends and relatives, to deal with the latter as with beasts or plants". By 335 BC, Aristotle had returned to Athens. Aristotle conducted courses at the school for the next twelve years. While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stagira, who bore him a son whom he named after his father, Nicomachus.
According to the Suda, he had an erômenos, Palaephatus of Abydus. This period in Athens, between 335 and 323 BC, is when Aristotle is believed to have composed many of his works, he wrote many dialogues. Those works that have survived are in treatise form and were not
Graham Priest is Distinguished Professor of Philosophy at the CUNY Graduate Center, as well as a regular visitor at the University of Melbourne where he was Boyce Gibson Professor of Philosophy and at the University of St Andrews. Priest was educated at the St John's College and the London School of Economics, his thesis advisor was John Lane Bell. He holds a DLitt from the University of Melbourne, he is known for his defence of dialetheism, his in-depth analyses of the logical paradoxes, his many writings related to paraconsistent and other non-classical logics. Priest, a long-time resident of Australia, now residing in New York City, is the author of numerous books, has published articles in nearly every major philosophical and logical journal, he was a frequent collaborator with the late Richard Sylvan, a fellow proponent of dialetheism and paraconsistent logic. Priest has published on metaphilosophy. In addition to his work in philosophy and logic, Priest practiced Karate-do, he is International Karate-do Shobukai.
Presently, he practices Taichi. Graham Priest personal website -- free pdfs of papers available for download An in-depth autobiographical interview with Graham Priest Priest archive on the CUNY Philosophy Commons Video of Graham Priest & Maureen Eckert on Deviant Logic Rationally Speaking, Podcast of Graham Priest on Paradoxes and Paraconsistent Logic Arche Foundations of Logical Consequence Workshop 2009, "Is the Ternary R Depraved?" The Monthly: Graham priest on Gottlob Frege The Philosopher's Zone, 10 July 2010: "It's All about Me: A Forum on the Philosophy of the Self" The New York Times, The Stone Blog, 28 November 2010: "Paradoxical Truth" Graham Priest Photograph Wikimedia Sept. 2010 Philosophy TV, 10 January 2011: Discussion of Deviant Logic, teaching logic and logic with Maureen Eckert Two-Part Interview on Florida Student Philosophy Blog: Part 1 General Questions and Part 2 Technical Questions
Capitalism is an economic system based on the private ownership of the means of production and their operation for profit. Characteristics central to capitalism include private property, capital accumulation, wage labor, voluntary exchange, a price system, competitive markets. In a capitalist market economy, decision-making and investment are determined by every owner of wealth, property or production ability in financial and capital markets, whereas prices and the distribution of goods and services are determined by competition in goods and services markets. Economists, political economists and historians have adopted different perspectives in their analyses of capitalism and have recognized various forms of it in practice; these include welfare capitalism and state capitalism. Different forms of capitalism feature varying degrees of free markets, public ownership, obstacles to free competition and state-sanctioned social policies; the degree of competition in markets, the role of intervention and regulation, the scope of state ownership vary across different models of capitalism.
The extent to which different markets are free as well as the rules defining private property are matters of politics and policy. Most existing capitalist economies are mixed economies, which combine elements of free markets with state intervention and in some cases economic planning. Market economies have existed under many forms of government and in many different times and cultures. Modern capitalist societies—marked by a universalization of money-based social relations, a large and system-wide class of workers who must work for wages, a capitalist class which owns the means of production—developed in Western Europe in a process that led to the Industrial Revolution. Capitalist systems with varying degrees of direct government intervention have since become dominant in the Western world and continue to spread. Over time, capitalist countries have experienced consistent economic growth and an increase in the standard of living. Critics of capitalism argue that it establishes power in the hands of a minority capitalist class that exists through the exploitation of the majority working class and their labor.
Supporters argue that it provides better products and innovation through competition, disperses wealth to all productive people, promotes pluralism and decentralization of power, creates strong economic growth, yields productivity and prosperity that benefit society. The term "capitalist", meaning an owner of capital, appears earlier than the term "capitalism" and it dates back to the mid-17th century. "Capitalism" is derived from capital, which evolved from capitale, a late Latin word based on caput, meaning "head"—also the origin of "chattel" and "cattle" in the sense of movable property. Capitale emerged in the 12th to 13th centuries in the sense of referring to funds, stock of merchandise, sum of money or money carrying interest. By 1283, it was used in the sense of the capital assets of a trading firm and it was interchanged with a number of other words—wealth, funds, assets, property and so on; the Hollandische Mercurius uses "capitalists" in 1654 to refer to owners of capital. In French, Étienne Clavier referred to capitalistes in 1788, six years before its first recorded English usage by Arthur Young in his work Travels in France.
In his Principles of Political Economy and Taxation, David Ricardo referred to "the capitalist" many times. Samuel Taylor Coleridge, an English poet, used "capitalist" in his work Table Talk. Pierre-Joseph Proudhon used the term "capitalist" in his first work, What is Property?, to refer to the owners of capital. Benjamin Disraeli used the term "capitalist" in his 1845 work Sybil; the initial usage of the term "capitalism" in its modern sense has been attributed to Louis Blanc in 1850 and Pierre-Joseph Proudhon in 1861. Karl Marx and Friedrich Engels referred to the "capitalistic system" and to the "capitalist mode of production" in Capital; the use of the word "capitalism" in reference to an economic system appears twice in Volume I of Capital, p. 124 and in Theories of Surplus Value, tome II, p. 493. Marx did not extensively use the form capitalism, but instead those of capitalist and capitalist mode of production, which appear more than 2,600 times in the trilogy The Capital. According to the Oxford English Dictionary, the term "capitalism" first appeared in English in 1854 in the novel The Newcomes by novelist William Makepeace Thackeray, where he meant "having ownership of capital".
According to the OED, Carl Adolph Douai, a German American socialist and abolitionist, used the phrase "private capitalism" in 1863. Capitalism in its modern form can be traced to the emergence of agrarian capitalism and mercantilism in the early Renaissance, in city states like Florence. Capital has existed incipiently on a small scale for centuries in the form of merchant and lending activities and as small-scale industry with some wage labour. Simple commodity exchange and simple commodity production, which are the initial basis for the growth of capital from trade, have a long history. Classical Islam promulgated capitalist economic policies such as free banking, their use of Indo-Arabic
Emil Leon Post
Emil Leon Post was an American mathematician and logician. He is best known for his work in the field that became known as computability theory. Post was born in Augustów, Suwałki Governorate, Russian Empire into a Polish-Jewish family that immigrated to New York City in May 1904, his parents were Pearl Post. Post at the age of twelve lost his left arm in a car accident; this loss was a significant obstacle to being a professional astronomer. He decided to pursue mathematics, rather than astronomy. Post attended the Townsend Harris High School and continued on to graduate from City College of New York in 1917 with a B. S. in Mathematics. After completing his Ph. D. in mathematics at Columbia University, supervised by Cassius Jackson Keyser, he did a post-doctorate at Princeton University in the 1920–1921 academic year. Post became a high school mathematics teacher in New York City. Post married Gertrude Singer in 1929, with whom he had a daughter, Phyllis Goodman. Post spent at most three hours a day on research on the advice of his doctor in order to avoid manic attacks, which he had been experiencing since his year at Princeton.
In 1936, he was appointed to the mathematics department at the City College of New York. He died in 1954 of a heart attack following electroshock treatment for depression. In his doctoral thesis shortened and published as "Introduction to a General Theory of Elementary Propositions", Post proved, among other things, that the propositional calculus of Principia Mathematica was complete: all tautologies are theorems, given the Principia axioms and the rules of substitution and modus ponens. Post devised truth tables independently of Wittgenstein and C. S. Peirce and put them to good mathematical use. Jean van Heijenoort's well-known source book on mathematical logic reprinted Post's classic article setting out these results. While at Princeton, Post came close to discovering the incompleteness of Principia Mathematica, which Kurt Gödel proved in 1931. Post failed to publish his ideas as he believed he needed a'complete analysis' for them to be accepted. In 1936, Post developed, independently of Alan Turing, a mathematical model of computation, equivalent to the Turing machine model.
Intending this as the first of a series of models of equivalent power but increasing complexity, he titled his paper Formulation 1. This model is sometimes called "Post's machine" or a Post–Turing machine, but is not to be confused with Post's tag machines or other special kinds of Post canonical system, a computational model using string rewriting and developed by Post in the 1920s but first published in 1943. Post's rewrite technique is now ubiquitous in programming language specification and design, so with Church's lambda-calculus is a salient influence of classical modern logic on practical computing. Post devised a method of'auxiliary symbols' by which he could canonically represent any Post-generative language, indeed any computable function or set at all. Correspondence systems were introduced by Post in 1946 to give simple examples of undecidability, he showed that the Post Correspondence Problem of satisfying their constraints is, in general, undecidable. With 2 string pairs, PCP was shown to be decidable in 1981.
It is known to be undecidable. The undecidability of his Post correspondence problem turned out to be what was needed to obtain undecidability results in the theory of formal languages. In an influential address to the American Mathematical Society in 1944, he raised the question of the existence of an uncomputable recursively enumerable set whose Turing degree is less than that of the halting problem; this question, which became known as Post's problem, stimulated much research. It was solved in the affirmative in the 1950s by the introduction of the powerful priority method in recursion theory. Post made a fundamental and still-influential contribution to the theory of polyadic, or n-ary, groups in a long paper published in 1940, his major theorem showed that a polyadic group is the iterated multiplication of elements of a normal subgroup of a group, such that the quotient group is cyclic of order n − 1. He demonstrated that a polyadic group operation on a set can be expressed in terms of a group operation on the same set.
The paper contains many other important results. Post, Emil Leon. "Introduction to a General Theory of Elementary Propositions". American Journal of Mathematics. 43: 163–185. Doi:10.2307/2370324. Post, Emil Leon. "Finite Combinatory Processes – Formulation 1". Journal of Symbolic Logic. 1: 103–105. Doi:10.2307/2269031. Post, Emil Leon. "Polyadic groups". Transactions of the American Mathematical Society. 48: 208–350. Doi:10.2307/1990085. Post, Emil Leon. "Formal Reductions of the General Combinatorial Decision Problem". American Journal of Mathematics. 65: 197–215. Doi:10.2307/2371809. Post, Emil Leon. "Recursively enumerable sets of positive integers and their decision problems". Bulletin of the American Mathematical Society. 50: 284–316. Doi:10.1090/s0002-9904-1944-08111-1. Introduces the important concept of many-one reduction. Arithmetical hierarchy Functional completeness List of multiple discoveries List of pioneers in computer science Post's inversion formula Post's lattice Post's theorem Stillwell, John, "Emil Post and His Anticipation of Gödel and Turing", Mathematics Magazine, 77: 3–14, doi:10.2307/3219226, JSTOR 3219226 Urquhart, Alasdair.
"Emil Post". In Gabbay, Dov M.. Logic from Russell to Church. Handbook of the History of
Plato was an Athenian philosopher during the Classical period in Ancient Greece, founder of the Platonist school of thought, the Academy, the first institution of higher learning in the Western world. He is considered the pivotal figure in the history of Ancient Greek and Western philosophy, along with his teacher and his most famous student, Aristotle. Plato has often been cited as one of the founders of Western religion and spirituality; the so-called Neoplatonism of philosophers like Plotinus and Porphyry influenced Saint Augustine and thus Christianity. Alfred North Whitehead once noted: "the safest general characterization of the European philosophical tradition is that it consists of a series of footnotes to Plato."Plato was the innovator of the written dialogue and dialectic forms in philosophy. Plato appears to have been the founder of Western political philosophy, his most famous contribution bears his name, the doctrine of the Forms known by pure reason to provide a realist solution to the problem of universals.
He is the namesake of Platonic love and the Platonic solids. His own most decisive philosophical influences are thought to have been along with Socrates, the pre-Socratics Pythagoras and Parmenides, although few of his predecessors' works remain extant and much of what we know about these figures today derives from Plato himself. Unlike the work of nearly all of his contemporaries, Plato's entire oeuvre is believed to have survived intact for over 2,400 years. Although their popularity has fluctuated over the years, the works of Plato have never been without readers since the time they were written. Due to a lack of surviving accounts, little is known about education. Plato belonged to an influential family. According to a disputed tradition, reported by doxographer Diogenes Laërtius, Plato's father Ariston traced his descent from the king of Athens and the king of Messenia, Melanthus. Plato's mother was Perictione, whose family boasted of a relationship with the famous Athenian lawmaker and lyric poet Solon, one of the seven sages, who repealed the laws of Draco.
Perictione was sister of Charmides and niece of Critias, both prominent figures of the Thirty Tyrants, known as the Thirty, the brief oligarchic regime, which followed on the collapse of Athens at the end of the Peloponnesian War. According to some accounts, Ariston tried to force his attentions on Perictione, but failed in his purpose; the exact time and place of Plato's birth are unknown. Based on ancient sources, most modern scholars believe that he was born in Athens or Aegina between 429 and 423 BC, not long after the start of the Peloponnesian War; the traditional date of Plato's birth during the 87th or 88th Olympiad, 428 or 427 BC, is based on a dubious interpretation of Diogenes Laërtius, who says, "When was gone, joined Cratylus the Heracleitean and Hermogenes, who philosophized in the manner of Parmenides. At twenty-eight, Hermodorus says, went to Euclides in Megara." However, as Debra Nails argues, the text does not state that Plato left for Megara after joining Cratylus and Hermogenes.
In his Seventh Letter, Plato notes that his coming of age coincided with the taking of power by the Thirty, remarking, "But a youth under the age of twenty made himself a laughingstock if he attempted to enter the political arena." Thus, Nails dates Plato's birth to 424/423. According to Neanthes, Plato was six years younger than Isocrates, therefore was born the same year the prominent Athenian statesman Pericles died. Jonathan Barnes regards 428 BC as the year of Plato's birth; the grammarian Apollodorus of Athens in his Chronicles argues that Plato was born in the 88th Olympiad. Both the Suda and Sir Thomas Browne claimed he was born during the 88th Olympiad. Another legend related that, when Plato was an infant, bees settled on his lips while he was sleeping: an augury of the sweetness of style in which he would discourse about philosophy. Besides Plato himself and Perictione had three other children; the brothers Adeimantus and Glaucon are mentioned in the Republic as sons of Ariston, brothers of Plato, though some have argued they were uncles.
In a scenario in the Memorabilia, Xenophon confused the issue by presenting a Glaucon much younger than Plato. Ariston appears to have died in Plato's childhood, although the precise dating of his death is difficult. Perictione married Pyrilampes, her mother's brother, who had served many times as an ambassador to the Persian court and was a friend of Pericles, the leader of the democratic faction in Athens. Pyrilampes had a son from a previous marriage, famous for his beauty. Perictione gave birth to Pyrilampes' second son, the half-brother of Plato, who appears in Parmenides. In contrast to his reticence about himself, Plato introduced his distinguished relatives into his dialogues, or referred to them with some precision. In addition to Adeimantus and Glaucon in the Republic, Charmides has a dialogue named after him; these and other references suggest a considerable amount of family pride and enable us to reconstruct Plato's family tree. According to Burnet, "the opening scene of the Ch
Square of opposition
The square of opposition is a diagram representing the relations between the four basic categorical propositions. The origin of the square can be traced back to Aristotle making the distinction between two oppositions: contradiction and contrariety, but Aristotle did not draw any diagram. This was done several centuries by Apuleius and Boethius. In traditional logic, a proposition is a spoken assertion, not the meaning of an assertion, as in modern philosophy of language and logic. A categorical proposition is a simple proposition containing two terms and predicate, in which the predicate is either asserted or denied of the subject; every categorical proposition can be reduced to one of four logical forms. These are: The so-called'A' proposition, the universal affirmative, whose form in Latin is'omne S est P' translated as'every S is a P'. The'E' proposition, the universal negative, Latin form'nullum S est P' translated as'no S are P'. The'I' proposition, the particular affirmative, Latin'quoddam S est P' translated as'some S are P'.
The'O' proposition, the particular negative, Latin'quoddam S non est P' translated as'some S are not P'. In tabular form: *While the standard form "No S is P" is unambiguous, the form "All S is not P" is ambiguous and so is not a standard form: because it can be either an E or O proposition, it requires a context to determine the form. Aristotle states, that there are certain logical relationships between these four kinds of proposition, he says that to every affirmation there corresponds one negation, that every affirmation and its negation are'opposed' such that always one of them must be true, the other false. A pair of affirmative and negative statements he calls a'contradiction'. Examples of contradictories are'every man is white' and'not every man is white','no man is white' and'some man is white'.'Contrary' statements, are such that both cannot at the same time be true. Examples of these are the universal affirmative'every man is white', the universal negative'no man is white'; these cannot be true at the same time.
However, these are not contradictories. For example, it is false, yet it is false that no man is white, since there are some white men. Since every statement has a contradictory opposite, since a contradictory is true when its opposite is false, it follows that the opposites of contraries can both be true, but they cannot both be false. Since subcontraries are negations of universal statements, they were called'particular' statements by the medieval logicians. Another logical opposition implied by this, though not mentioned explicitly by Aristotle, is'alternation', consisting of'subalternation' and'superalternation'. Alternation is a relation between a particular statement and a universal statement of the same quality such that the particular is implied by the other; the particular is the subaltern of the universal, the particular's superaltern. For example, if'every man is white' is true, its contrary'no man is white' is false. Therefore, the contradictory'some man is white' is true; the universal'no man is white' implies the particular'not every man is white'.
In summary: Universal statements are contraries:'every man is just' and'no man is just' cannot be true together, although one may be true and the other false, both may be false. Particular statements are subcontraries.'Some man is just' and'some man is not just' cannot be false together. The particular statement of one quality is the subaltern of the universal statement of that same quality, the superaltern of the particular statement because in Aristotelian semantics'every A is B' implies'some A is B' and'no A is B' implies'some A is not B'. Note that modern formal interpretations of English sentences interpret'every A is B' as'for any x, x is A implies x is B', which does not imply'some x is A'; this is a matter of semantic interpretation and does not mean, as is sometimes claimed, that Aristotelian logic is'wrong'. The universal affirmative and the particular negative are contradictories. If some A is not B, not every A is B. Conversely, though this is not the case in modern semantics, it was thought that if every A is not B, some A is not B.
This interpretation has caused difficulties. While Aristotle's Greek does not represent the particular negative as'some A is not B', but as'not every A is B', someone in his commentary on the Peri hermaneias, renders the particular negative as'quoddam A non est B', literally'a certain A is not a B', in all medieval writing on logic it is customary to represent the particular proposition in this way; these relationships became the basis of a diagram originating with Boethius and used by medieval logicians to classify the logical relationships. The propositions are placed in the four corners of a square, the relations represented as lines drawn between them, whence the name'The Square of Opposition'. Subcontraries, which medieval logicians represented in the form'quoddam A est B' and'quoddam A non est B' cannot both be false, since their universal contradictory statements cannot both be true; this le