Jan Łukasiewicz was a Polish logician and philosopher born in Lemberg, a city in the Galician kingdom of Austria-Hungary. His work centred on philosophical logic, mathematical logic, history of logic, he thought innovatively about traditional propositional logic, the principle of non-contradiction and the law of excluded middle. Modern work on Aristotle's logic builds on the tradition started in 1951 with the establishment by Łukasiewicz of a revolutionary paradigm; the Łukasiewicz approach was reinvigorated in the early 1970s in a series of papers by John Corcoran and Timothy Smiley—which inform modern translations of Prior Analytics by Robin Smith in 1989 and Gisela Striker in 2009. Łukasiewicz is regarded as one of the most important historians of logic. He grew up in Lwów and was the only child of Paweł Łukasiewicz, a captain in the Austrian army, Leopoldina, née Holtzer, the daughter of a civil servant, his family was Roman Catholic. He finished his gymnasium studies in philology and in 1897 went on to Lwów University, where he studied philosophy and mathematics.
He was a pupil of philosopher Kazimierz Twardowski. In 1902 he received a Doctor of Philosophy degree under the patronage of Emperor Franz Joseph I of Austria, who gave him a special doctoral ring with diamonds, he spent three years as a private teacher, in 1905 he received a scholarship to complete his philosophy studies at the University of Berlin and the University of Louvain in Belgium. Łukasiewicz continued studying for his habilitation qualification and in 1906 submitted his thesis to the University of Lwów. In 1906 he was appointed a lecturer at the University of Lwów where he was appointed Extraordinary Professor by Emperor Franz Joseph I, he taught there until the First World War. In 1915 he was invited to lecture as a full professor at the University of Warsaw which had re-opened after being closed down by the Tsarist government in the 19th century. In 1919 Łukasiewicz left the university to serve as Polish Minister of Religious Denominations and Public Education in the Paderewski government until 1920.
Łukasiewicz led the development of a Polish curriculum replacing the Russian and Austrian curricula used in partitioned Poland. The Łukasiewicz curriculum emphasized the early acquisition of mathematical concepts. In 1928 he married Regina Barwińska, he remained a professor at the University of Warsaw from 1920 until 1939 when the family house was destroyed by German bombs and the university was closed under German occupation. He had been a rector of the university twice. In this period Łukasiewicz and Stanisław Leśniewski founded the Lwów–Warsaw school of logic, made internationally famous by Alfred Tarski, Leśniewski's student. At the beginning of World War II he worked at the Warsaw Underground University as part of the secret system of education in Poland during World War II, he and his wife wanted to move to Switzerland but couldn't get permission from the German authorities. Instead, in the summer of 1944, they left Poland with the help of Heinrich Scholz and spent the last few months of the war in Münster, Germany hoping to somehow go on further to Switzerland.
Following the war, he worked at University College Dublin until his death. Jan Łukasiewicz's papers are held by the University of Manchester Library. A number of axiomatizations of classical propositional logic are due to Łukasiewicz. A elegant axiomatization features a mere three axioms and is still invoked to the present day, he was a pioneer investigator of multi-valued logics. He wrote on the philosophy of science, his approach to the making of scientific theories was similar to the thinking of Karl Popper. Łukasiewicz invented the Polish notation for the logical connectives around 1920. There is a quotation from his paper, Remarks on Nicod's Axiom and on "Generalizing Deduction", page 180. I used that notation for the first time in my article Łukasiewicz, p. 610, footnote." The reference cited by Łukasiewicz above is a lithographed report in Polish. The referring paper by Łukasiewicz Remarks on Nicod's Axiom and on "Generalizing Deduction" published in Polish in 1931, was reviewed by H. A. Pogorzelski in the Journal of Symbolic Logic in 1965.
In Łukasiewicz 1951 book, Aristotle's Syllogistic from the Standpoint of Modern Formal Logic, he mentions that the principle of his notation was to write the functors before the arguments to avoid brackets and that he had employed his notation in his logical papers since 1929. He goes on to cite, as an example, a 1930 paper he wrote with Alfred Tarski on the sentential calculus; this notation is the root of the idea of the recursive stack, a last-in, first-out computer memory store proposed by several researchers including Turing and Hamblin, first implemented in 1957. In 1960, Łukasiewicz notation concepts and stacks were used as the basis of the Burroughs B5000 computer designed by Robert S. Barton and his team at Burroughs Corporation in California; the concepts led to the design of the English Electric multi-programmed KDF9 computer system of 1963, which had two such hardware register stacks. A similar concept underlies the reverse Polish notation of the Friden EC-130 calculator and its successors, many Hewlett Packard calculators, the Forth programming language, the PostScript page description langu
An Euler diagram is a diagrammatic means of representing sets and their relationships. They involve overlapping shapes, may be scaled, such that the area of the shape is proportional to the number of elements it contains, they are useful for explaining complex hierarchies and overlapping definitions. They are confused with Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships; the first use of "Eulerian circles" is attributed to Swiss mathematician Leonhard Euler. In the United States, both Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement of the 1960s. Since they have been adopted by other curriculum fields such as reading as well as organizations and businesses. Euler diagrams consist of simple closed shapes in a two dimensional plane that each depict a set or category. How or if these shapes overlap demonstrates the relationships between the sets.
There are only 3 possible relationships between any 2 sets. This is referred to as containment, overlap or neither or in mathematics, it may be referred to as subset and disjoint; each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, the exterior, which represents all elements that are not members of the set. Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets. A curve, contained within the interior zone of another represents a subset of it. Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all 2n logically possible zones of overlap between its n curves, representing all combinations of inclusion/exclusion of its constituent sets. Regions not part of the set are indicated by coloring them black, in contrast to Euler diagrams, where membership in the set is indicated by overlap as well as color; when the number of sets grows beyond 3 a Venn diagram becomes visually complex compared to the corresponding Euler diagram.
The difference between Euler and Venn diagrams can be seen in the following example. Take the three sets: A = B = C = The Euler and the Venn diagrams of those sets are: In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a universe of discourse. In the examples below, the Euler diagram depicts that the sets Animal and Mineral are disjoint since the corresponding curves are disjoint, that the set Four Legs is a subset of the set of Animals; the Venn diagram, which uses the same categories of Animal and Four Legs, does not encapsulate these relationships. Traditionally the emptiness of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent emptiness either by the absence of a region. A set of well-formedness conditions are imposed. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the adjacent diagram, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations.
However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs; as shown in the illustration to the right, Sir William Hamilton in his posthumously published Lectures on Metaphysics and Logic erroneously asserts that the original use of circles to "sensualize... the abstractions of Logic" was not Leonhard Paul Euler but rather Christian Weise in his Nucleus Logicae Weisianae that appeared in 1712 posthumously, the latter book was written by Johann Christian Lange rather than Weise. He references Euler's Letters to a German Princess In Hamilton's illustration the four categorical propositions that can occur in a syllogism as symbolized by the drawings A, E, I and O are: A: The Universal Affirmative, Example: "All metals are elements". E: The Universal Negative, Example: "No metals are compound substances".
I: The Particular Affirmative, Example: "Some metals are brittle". O: The Particular Negative, Example: "Some metals are not brittle". In his 1881 Symbolic Logic Chapter V "Diagrammatic Representation", John Venn comments on the remarkable prevalence of the Euler diagram: "...of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose:-somewhat at random, as they happened to be most accessible:-it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian Scheme." But he contended, "the inapplicability of this scheme for the purposes of a general Logic" (pag
In transportation infrastructure, a bidirectional traffic system divides travelers into two streams of traffic that flow in opposite directions. In the design and construction of tunnels, bidirectional traffic can markedly affect ventilation considerations. Microscopic traffic flow models have been proposed for bidirectional automobile and railway traffic. Bidirectional traffic can be observed in ant trails and this has been researched for insight into human traffic models. In a macroscopic theory proposed by Laval, the interaction between fast and slow vehicles conforms to the Newell kinematic wave model of moving bottlenecks. In air traffic control traffic is separated by elevation, with east bound flights at odd thousand feet elevations and west bound flights at thousand feet elevations. Above 28,000 ft only odd flight levels are used, with FL 290, 330, 370, etc. for eastbound flights and FL 310, 350, 390, etc. for westbound flights. Entry to and exit from airports is always one-way traffic, as runways are chosen to allow aircraft to take off and land into the wind, to reduce ground speed.
In no wind cases, a preferred calm wind runway and direction is chosen and used by all flights, to avoid collisions. In uncontrolled airports, airport information can be obtained from anyone at the airport. Traffic follows a specific traffic pattern, with designated entry and exits. Radio announcements are made, whether anyone is listening or not, to allow any other traffic to be aware of other traffic in the area. In the earliest days of railways in the United Kingdom, most lines were built double tracked because of the difficulty of coordinating operations in pre-telegraphy times. Most modern roads carry bidirectional traffic, although one-way traffic is common in dense urban centres. Bidirectional traffic flow is believed to influence the rate of traffic collisions. In an analysis of head-on collisions, rear-end collisions, lane-changing collisions based on the Simon-Gutowitz bidirectional traffic model, it was concluded that "the risk of collisions is important when the density of cars in one lane is small and that of the other lane is high enough", that "heavy vehicles cause an important reduction of traffic flow on the home lane and provoke an increase of the risk of car accident".
Bidirectional traffic is the most common form of flow observed in trails, some larger pedestrian concourses exhibit multidirectional traffic. "Structural design issues". Transportation research record. National Research Council, Transportation Research Board. 2000. ISBN 978-0-309-06744-7. Two-lane rural highways with bidirectional traffic Michael S. Bernick. Transit villages in the 21st century. McGraw-Hill. ISBN 978-0-07-005475-2. Satellite subcenters would function as countermagnets to central Stockholm, leading to efficient bidirectional traffic flows. "Annual meeting". Compendium of technical papers. Institute of Transportation Engineers. 53. 1983. Madras conducted some field studies to evolve a relationship between speed and volume of traffic on single and two lane bidirectional traffic roads. Bickel, John. Tunnel engineering handbook. University of California: Van Nostrand Reinhold Company. P. 499. ISBN 978-0-442-28127-4; this results in bidirectional traffic in a single tunnel. Highway Research Board.
"Proceedings of a workshop held May 17–19, 1971". Systems building for bridges. P. 46. ISBN 978-0-309-02063-3; the hazards are far more numerous and the need for better headlighting much greater on these millions of miles of streets and highways carrying bidirectional traffic because so few of these miles have mediocre fixed highway lighting. National Research Council. Pavement management systems. National Academy Press. P. 24. ISBN 978-0-309-05468-3. Transportation Research Board. "Report 65: "Evaluation of Bus Bulbs"". Transit Cooperative Research Program: 20. Retrieved 2009-09-09. Conversely, at 44 sq ft, passing slower pedestrian traffic is easier, crossing bidirectional traffic is nearly unhindered, traveling through the zone is less affected by other walking or standing pedestrians. Stone, H. David. Vital rails: the Charleston & Savannah Railroad and the Civil War in coastal South Carolina. University of South Carolina Press. ISBN 978-1-57003-716-0. Single tracking and the relative lack of sidings made bidirectional traffic difficult — a difficulty that became more obvious during the conflict with the North.
Behrens, John. "A staff technical report". Recommendation for the Chicago area freight system for 1995. Chicago Area Transportation Study; the average yard handle 350 vans of bidirectional traffic each day. National Research Council. "Transportation system management and travel demand management". Transportation Research Record. 1404: 39. ISBN 978-0-309-05550-5; this is true for physically challenged users and the elderly, who have to weave through bidirectional traffic. Borndörfer, Ralf. "Optimal Fares for Public Transport". In Haasis, Hans-Dietrich. Operations Research Proceedings 2005. 2005. Bremen: Springer Berlin Heidelberg. Doi:10.1007/3-540-32539-5. It consists of a network containing 23 nodes and a corresponding upper-triangular origin-destination matrix with 210 nonzero entries that account for a symmetric bidirectional traffic. Burger, H.. "Options for tunnelling". Developments in Geotechnical Engineering: 35. ISBN 978-0-444-89935-4. A tunnel with an inner diameter of 9.75 meters allowing a bidi
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Domain of discourse
In the formal sciences, the domain of discourse called the universe of discourse, universal set, or universe, is the set of entities over which certain variables of interest in some formal treatment may range. The domain of discourse is identified in the preliminaries, so that there is no need in the further treatment to specify each time the range of the relevant variables. Many logicians distinguish, sometimes only tacitly, between the domain of a science and the universe of discourse of a formalization of the science. Giuseppe Peano formalized number theory taking its domain to be the positive integers and the universe of discourse to include all numbers, not just integers. For example, in an interpretation of first-order logic, the domain of discourse is the set of individuals over which the quantifiers range. A proposition such as ∀ x is ambiguous. In one interpretation, the domain of discourse could be the set of real numbers. If the domain of discourse is the set of real numbers, the proposition is false, with x = √2 as counterexample.
The term universe of discourse refers to the collection of objects being discussed in a specific discourse. In model-theoretical semantics, a universe of discourse is the set of entities that a model is based on; the concept universe of discourse is attributed to Augustus De Morgan but the name was used for the first time by George Boole on page 42 of his Laws of Thought. Boole's definition is quoted below; the concept discovered independently by Boole in 1847, played a crucial role in his philosophy of logic in his principle of wholistic reference. In every discourse, whether of the mind conversing with its own thoughts, or of the individual in his intercourse with others, there is an assumed or expressed limit within which the subjects of its operation are confined; the most unfettered discourse is that in which the words we use are understood in the widest possible application, for them the limits of discourse are co-extensive with those of the universe itself. But more we confine ourselves to a less spacious field.
Sometimes, in discoursing of men we imply that it is of men only under certain circumstances and conditions that we speak, as of civilized men, or of men in the vigour of life, or of men under some other condition or relation. Now, whatever may be the extent of the field within which all the objects of our discourse are found, that field may properly be termed the universe of discourse. Furthermore, this universe of discourse is in the strictest sense the ultimate subject of the discourse. Domain of a function Domain theory Interpretation Term algebra Universe
Necessity and sufficiency
In logic and sufficiency are terms used to describe a conditional or implicational relationship between statements. For example, in the conditional statement "If P Q", we say that "Q is necessary for P" because P cannot be true unless Q is true. We say that "P is sufficient for Q" because P being true always implies that Q is true, but P not being true does not always imply that Q is not true; the assertion that a statement is a "necessary and sufficient" condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either true or false. In ordinary English, "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. Being a male sibling is a necessary and sufficient condition for being a brother. In the conditional statement, "if S N", the expression represented by S is called the antecedent and the expression represented by N is called the consequent; this conditional statement may be written in many equivalent ways, for instance, "N if S", "S only if N", "S implies N", "N is implied by S", S → N, S ⇒ N, or "N whenever S".
In the above situation, we say that N is a necessary condition for S. In common language this is saying that if the conditional statement is a true statement the consequent N must be true if S is to be true. Phrased differently, the antecedent S cannot be true without N being true. For example, in order for someone to be called Socrates, it is necessary for that someone to be Named. In the above situation, we can say S is a sufficient condition for N. Again, consider the third column of the truth table below. If the conditional statement is true if S is true, N must be true. In common terms, "S guarantees N". Continuing the example, knowing that someone is called Socrates is sufficient to know that someone has a Name. A necessary and sufficient condition requires that both of the implications S ⇒ N ⇒ S hold. From the first of these we see that S is a sufficient condition for N, from the second that S is a necessary condition for N; this is expressed as "S is necessary and sufficient for N ", "S if and only if N ", or S ⇔ N.
The assertion that Q is necessary for P is colloquially equivalent to "P cannot be true unless Q is true" or "if Q is false P is false". By contraposition, this is the same thing as "whenever P is true, so is Q"; the logical relation between P and Q is expressed as "if P Q" and denoted "P ⇒ Q". It may be expressed as any of "P only if Q", "Q, if P", "Q whenever P", "Q when P". One finds, in mathematical prose for instance, several necessary conditions that, taken together, constitute a sufficient condition, as shown in Example 5. Example 1 For it to be true that "John is a bachelor", it is necessary that it be true that he is unmarried, adult, since to state "John is a bachelor" implies John has each of those three additional predicates. Example 2 For the whole numbers greater than two, being odd is necessary to being prime, since two is the only whole number, both and prime. Example 3 Consider thunder, the sound caused by lightning. We say. Whenever there's lightning, there's thunder; the thunder does not cause the lightning, but because lightning always comes with thunder, we say that thunder is necessary for lightning.
Example 4 Being at least 30 years old is necessary for serving in the U. S. Senate. If you are under 30 years old it is impossible for you to be a senator; that is, if you are a senator, it follows that you are at least 30 years old. Example 5 In algebra, for some set S together with an operation ⋆ to form a group, it is necessary that ⋆ be associative, it is necessary that S include a special element e such that for every x in S it is the case that e ⋆ x and x ⋆ e both equal x. It is necessary that for every x in S there exist a corresponding element x″ such that both x ⋆ x″ and x″ ⋆ x equal the special element e. None of these three necessary conditions by itself is sufficient, but the conjunction of the three is. If P is sufficient for Q knowing P to be true is adequate grounds to conclude that Q is true; the logical relation is, as before, expressed as "if P Q" or "P ⇒ Q". This can be expressed as "P only if Q", "P implies Q" or several other variants, it may be the case that several sufficient conditions, when taken together, constitute a single necessary condition, as illustrated in example 5.
Example 1 "John is a king" implies. So knowing that it is true that John is a king is sufficient to know that he is a male. Example 2 A number's being divisible by 4 is sufficient for its being but being divisible by 2 is both sufficient and necessary. Example 3 An occurrence of thunder is a sufficient condition for the occurrence of lightning in the sense
Pamela Gorkin is an American mathematician specializing in complex analysis and operator theory. She is a professor of mathematics at Bucknell University. Gorkin earned bachelor's and master's degrees in statistics from Michigan State University in 1976, she shifted to pure mathematics for her doctoral studies, completing her Ph. D. at Michigan State in 1982, the same year she joined the Bucknell Faculty. Her dissertation, Decompositions of the Maximal Ideal Space of L, was supervised by Sheldon Axler. At Bucknell, she was Presidential Professor from 2001 to 2004. With Ulrich Daepp, Gorkin is the author of the undergraduate textbook Reading and Proving: A Closer Look at Mathematics. With Daepp, Andrew Shaffer, Karl Voss, she is the author of Finding Ellipses: What Blaschke Products, Poncelet’s Theorem, the Numerical Range Know about Each Other. Gorkin is the 2018 AWM/MAA Falconer Lecturer, her lecture was on "Finding Ellipses", the topic of one of her books. She is the recipient of Bucknell's Lindback Award for Distinguished Teaching, the Crawford Distinguished Teaching Award of the Mathematical Association of America