Alfred Tarski  

Born  Alfred Teitelbaum January 14, 1901 
Died  October 26, 1983  (aged 82)
Nationality  Polish, American 
Education  University of Warsaw (Ph.D., 1924) 
Known for 

Scientific career  
Fields  Mathematics, logic, formal language 
Institutions 

Thesis  O wyrazie pierwotnym logistyki (On the Primitive Term of Logistic) (1924) 
Doctoral advisor  Stanisław Leśniewski 
Doctoral students  
Other notable students  Evert Willem Beth 
Alfred Tarski (/ˈtɑːrski/, born Alfred Teitelbaum;^{[1]}^{[2]}^{[3]} January 14, 1901 – October 26, 1983) was a PolishAmerican^{[4]} logician and mathematician.^{[5]} A prolific author best known for his work on model theory, metamathematics, and algebraic logic, he also contributed to abstract algebra, topology, geometry, measure theory, mathematical logic, set theory, and analytic philosophy.
Educated in Poland at the University of Warsaw, and a member of the Lwów–Warsaw school of logic and the Warsaw school of mathematics, he immigrated to the United States in 1939 where he became a naturalized citizen in 1945. Tarski taught and carried out research in mathematics at the University of California, Berkeley, from 1942 until his death in 1983.^{[6]}
His biographers Anita Burdman Feferman and Solomon Feferman state that, "Along with his contemporary, Kurt Gödel, he changed the face of logic in the twentieth century, especially through his work on the concept of truth and the theory of models."^{[7]}
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The Banach–Tarski Paradox

ALFRED TARSKI

The Simplest Math Problem No One Can Solve  Collatz Conjecture

Who Was Alfred Tarski? (Theories of Truth)

The Banach Tarski paradox  is it nonsense?  Sociology and Pure Mathematics  N J Wildberger
Transcription
Hey, Vsauce. Michael here. There's a famous way to seemingly create chocolate out of nothing. Maybe you've seen it before. This chocolate bar is 4 squares by 8 squares, but if you cut it like this and then like this and finally like this you can rearrange the pieces like so and wind up with the same 4 by 8 bar but with a leftover piece, apparently created out of thin air. There's a popular animation of this illusion as well. I call it an illusion because it's just that. Fake. In reality, the final bar is a bit smaller. It contains this much less chocolate. Each square along the cut is shorter than it was in the original, but the cut makes it difficult to notice right away. The animation is extra misleading, because it tries to cover up its deception. The lost height of each square is surreptitiously added in while the piece moves to make it hard to notice. I mean, come on, obviously you cannot cut up a chocolate bar and rearrange the pieces into more than you started with. Or can you? One of the strangest theorems in modern mathematics is the BanachTarski paradox. It proves that there is, in fact, a way to take an object and separate it into 5 different pieces. And then, with those five pieces, simply rearrange them. No stretching required into two exact copies of the original item. Same density, same size, same everything. Seriously. To dive into the mind blow that it is and the way it fundamentally questions math and ourselves, we have to start by asking a few questions. First, what is infinity? A number? I mean, it's nowhere on the number line, but we often say things like there's an infinite "number" of blahblahblah. And as far as we know, infinity could be real. The universe may be infinite in size and flat, extending out for ever and ever without end, beyond even the part we can observe or ever hope to observe. That's exactly what infinity is. Not a number per se, but rather a size. The size of something that doesn't end. Infinity is not the biggest number, instead, it is how many numbers there are. But there are different sizes of infinity. The smallest type of infinity is countable infinity. The number of hours in forever. It's also the number of whole numbers that there are, natural number, the numbers we use when counting things, like 1, 2, 3, 4, 5, 6 and so on. Sets like these are unending, but they are countable. Countable means that you can count them from one element to any other in a finite amount of time, even if that finite amount of time is longer than you will live or the universe will exist for, it's still finite. Uncountable infinity, on the other hand, is literally bigger. Too big to even count. The number of real numbers that there are, not just whole numbers, but all numbers is uncountably infinite. You literally cannot count even from 0 to 1 in a finite amount of time by naming every real number in between. I mean, where do you even start? Zero, okay. But what comes next? 0.000000... Eventually, we would imagine a 1 going somewhere at the end, but there is no end. We could always add another 0. Uncountability makes this set so much larger than the set of all whole numbers that even between 0 and 1, there are more numbers than there are whole numbers on the entire endless number line. Georg Cantor's famous diagonal argument helps illustrate this. Imagine listing every number between zero and one. Since they are uncountable and can't be listed in order, let's imagine randomly generating them forever with no repeats. Each number regenerate can be paired with a whole number. If there's a one to one correspondence between the two, that is if we can match one whole number to each real number on our list, that would mean that countable and uncountable sets are the same size. But we can't do that, even though this list goes on for ever. Forever isn't enough. Watch this. If we go diagonally down our endless list of real numbers and take the first decimal of the first number and the second of the second number, the third of the third and so on and add one to each, subtracting one if it happens to be a nine, we can generate a new real number that is obviously between 0 and 1, but since we've defined it to be different from every number on our endless list and at least one place it's clearly not contained in the list. In other words, we've used up every single whole number, the entire infinity of them and yet we can still come up with more real numbers. Here's something else that is true but counterintuitive. There are the same number of even numbers as there are even and odd numbers. At first, that sounds ridiculous. Clearly, there are only half as many even numbers as all whole numbers, but that intuition is wrong. The set of all whole numbers is denser but every even number can be matched with a whole number. You will never run out of members either set, so this one to one correspondence shows that both sets are the same size. In other words, infinity divided by two is still infinity. Infinity plus one is also infinity. A good illustration of this is Hilbert's paradox up the Grand Hotel. Imagine a hotel with a countably infinite number of rooms. But now, imagine that there is a person booked into every single room. Seemingly, it's fully booked, right? No. Infinite sets go against common sense. You see, if a new guest shows up and wants a room, all the hotel has to do is move the guest in room number 1 to room number 2. And a guest in room 2 to room 3 and 3 to 4 and 4 to 5 and so on. Because the number of rooms is never ending we cannot run out of rooms. Infinity 1 is also infinity again. If one guest leaves the hotel, we can shift every guest the other way. Guest 2 goes to room 1, 3 to 2, 4 to 3 and so on, because we have an infinite amount of guests. That is a never ending supply of them. No room will be left empty. As it turns out, you can subtract any finite number from infinity and still be left with infinity. It doesn't care. It's unending. BanachTarski hasn't left our sights yet. All of this is related. We are now ready to move on to shapes. Hilbert's hotel can be applied to a circle. Points around the circumference can be thought of as guests. If we remove one point from the circle that point is gone, right? Infinity tells us it doesn't matter. The circumference of a circle is irrational. It's the radius times 2Pi. So, if we mark off points beginning from the whole, every radius length along the circumference going clockwise we will never land on the same point twice, ever. We can count off each point we mark with a whole number. So this set is neverending, but countable, just like guests and rooms in Hilbert's hotel. And like those guests, even though one has checked out, we can just shift the rest. Move them counterclockwise and every room will be filled Point 1 moves to fill in the hole, point 2 fills in the place where point 1 used to be, 3 fills in 2 and so on. Since we have a unending supply of numbered points, no hole will be left unfilled. The missing point is forgotten. We apparently never needed it to be complete. There's one last needo consequence of infinity we should discuss before tackling BanachTarski. Ian Stewart famously proposed a brilliant dictionary. One that he called the Hyperwebster. The Hyperwebster lists every single possible word of any length formed from the 26 letters in the English alphabet. It begins with "a," followed by "aa," then "aaa," then "aaaa." And after an infinite number of those, "ab," then "aba," then "abaa", "abaaa," and so on until "z, "za," "zaa," et cetera, et cetera, until the final entry in infinite sequence of "z"s. Such a dictionary would contain every single word. Every single thought, definition, description, truth, lie, name, story. What happened to Amelia Earhart would be in that dictionary, as well as every single thing that didn't happened to Amelia Earhart. Everything that could be said using our alphabet. Obviously, it would be huge, but the company publishing it might realize that they could take a shortcut. If they put all the words that begin with a in a volume titled "A," they wouldn't have to print the initial "a." Readers would know to just add the "a," because it's the "a" volume. By removing the initial "a," the publisher is left with every "a" word sans the first "a," which has surprisingly become every possible word. Just one of the 26 volumes has been decomposed into the entire thing. It is now that we're ready to investigate this video's titular paradox. What if we turned an object, a 3D thing into a Hyperwebster? Could we decompose pieces of it into the whole thing? Yes. The first thing we need to do is give every single point on the surface of the sphere one name and one name only. A good way to do this is to name them after how they can be reached by a given starting point. If we move this starting point across the surface of the sphere in steps that are just the right length, no matter how many times or in what direction we rotate, so long as we never backtrack, it will never wind up in the same place twice. We only need to rotate in four directions to achieve this paradox. Up, down, left and right around two perpendicular axes. We are going to need every single possible sequence that can be made of any finite length out of just these four rotations. That means we will need lef, right, up and down as well as left left, left up, left down, but of course not left right, because, well, that's backtracking. Going left and then right means you're the same as you were before you did anything, so no left rights, no right lefts and no up downs and no down ups. Also notice that I'm writing the rotations in order right to left, so the final rotation is the leftmost letter. That will be important later on. Anyway. A list of all possible sequences of allowed rotations that are finite in lenght is, well, huge. Countably infinite, in fact. But if we apply each one of them to a starting point in green here and then name the point we land on after the sequence that brought us there, we can name a countably infinite set of points on the surface. Let's look at how, say, these four strings on our list would work. Right up left. Okay, rotating the starting point this way takes us here. Let's colour code the point based on the final rotation in its string, in this case it's left and for that we will use purple. Next up down down. That sequence takes us here. We name the point DD and color it blue, since we ended with a down rotation. RDR, that will be this point's name, takes us here. And for a final right rotation, let's use red. Finally, for a sequence that end with up, let's colour code the point orange. Now, if we imagine completing this process for every single sequence, we will have a countably infinite number of points named and colorcoded. That's great, but not enough. There are an uncountably infinite number of points on a sphere's surface. But no worries, we can just pick a point we missed. Any point and color it green, making it a new starting point and then run every sequence from here. After doing this to an uncountably infinite number of starting point we will have indeed named and colored every single point on the surface just once. With the exception of poles. Every sequence has two poles of rotation. Locations on the sphere that come back to exactly where they started. For any sequence of right or left rotations, the polls are the north and south poles. The problem with poles like these is that more than one sequence can lead us to them. They can be named more than once and be colored in more than one color. For example, if you follow some other sequence to the north or south pole, any subsequent rights or lefts will be equally valid names. In order to deal with this we're going to just count them out of the normal scheme and color them all yellow. Every sequence has two, so there are a countably infinite amount of them. Now, with every point on the sphere given just one name and just one of six colors, we are ready to take the entire sphere apart. Every point on the surface corresponds to a unique line of points below it all the way to the center point. And we will be dragging every point's line along with it. The lone center point we will set aside. Okay, first we cut out and extract all the yellow poles, the green starting points, the orange up points, the blue down points and the red and purple left and right points. That's the entire sphere. With just these pieces you could build the whole thing. But take a look at the left piece. It is defined by being a piece composed of every point, accessed via a sequence ending with a left rotation. If we rotate this piece right, that's the same as adding an "R" to every point's name. But left and then right is a backtrack, they cancel each other out. And look what happens when you reduce them away. The set becomes the same as a set of all points with names that end with L, but also U, D and every point reached with no rotation. That's the full set of starting points. We have turned less than a quarter of the sphere into nearly threequarters just by rotating it. We added nothing. It's like the Hyperwebster. If we had the right piece and the poles of rotation and the center point, well, we've got the entire sphere again, but with stuff left over. To make a second copy, let's rotate the up piece down. The down ups cancel because, well, it's the same as going nowhere and we're left with a set of all starting points, the entire up piece, the right piece and the left piece, but there's a problem here. We don't need this extra set of starting points. We still haven't used the original ones. No worries, let's just start over. We can just move everything from the up piece that turns into a starting point when rotated down. That means every point whose final rotation is up. Let's put them in the piece. Of course, after rotating points named UU will just turn into points named U, and that would give us a copy here and here. So, as it turns out, we need to move all points with any name that is just a string of Us. We will put them in the down piece and rotate the up piece down, which makes it congruent to the up right and left pieces, add in the down piece along with some up and the starting point piece and, well, we're almost done. The poles of rotation and center are missing from this copy, but no worries. There's a countably infinite number of holes, where the poles of rotations used to be, which means there is some pole around which we can rotate this sphere such that every pole hole orbits around without hitting another. Well, this is just a bunch of circles with one point missing. We fill them each like we did earlier. And we do the same for the centerpoint. Imagine a circle that contains it inside the sphere and just fill in from infinity and look what we've done. We have taken one sphere and turned it into two identical spheres without adding anything. One plus one equals 1. That took a while to go through, but the implications are huge. And mathematicians, scientists and philosophers are still debating them. Could such a process happen in the real world? I mean, it can happen mathematically and math allows us to abstractly predict and describe a lot of things in the real world with amazing accuracy, but does the BanachTarski paradox take it too far? Is it a place where math and physics separate? We still don't know. History is full of examples of mathematical concepts developed in the abstract that we did not think would ever apply to the real world for years, decades, centuries, until eventually science caught up and realized they were totally applicable and useful. The BanachTarski paradox could actually happen in our realworld, the only catch of course is that the five pieces you cut your object into aren't simple shapes. They must be infinitely complex and detailed. That's not possible to do in the real world, where measurements can only get so small and there's only a finite amount of time to do anything, but math says it's theoretically valid and some scientists think it may be physically valid too. There have been a number of papers published suggesting a link between by BanachTarski and the way tiny tiny subatomic particles can collide at high energies and turn into more particles than we began with. We are finite creatures. Our lives are small and can only scientifically consider a small part of reality. What's common for us is just a sliver of what's available. We can only see so much of the electromagnetic spectrum. We can only delve so deep into extensions of space. Common sense applies to that which we can access. But common sense is just that. Common. If total sense is what we want, we should be prepared to accept that we shouldn't call infinity weird or strange. The results we've arrived at by accepting it are valid, true within the system we use to understand, measure, predict and order the universe. Perhaps the system still needs perfecting, but at the end of day, history continues to show us that the universe isn't strange. We are. And as always, thanks for watching. Finally, as always, the description is full of links to learn more. There are also a number of books linked down there that really helped me wrap my mind kinda around BanachTarski. First of all, Leonard Wapner's "The Pea and the Sun." This book is fantastic and it's full of lot of the preliminaries needed to understand the proof that comes later. He also talks a lot about the ramifications of what BanachTarski and their theorem might mean for mathematics. Also, if you wanna talk about math and whether it's discovered or invented, whether it really truly will map onto the universe, Yanofsky's "The Outer Limits of Reason" is great. This is the favorite book of mine that I've read this entire year. Another good one is E. Brian Davies' "Why Beliefs Matter." This is actually Corn's favorite book, as you might be able to see there. It's delicious and full of lots of great information about the limits of what we can know and what science is and what mathematics is. If you love infinity and math, I cannot more highly recommend Matt Parker's "Things to Make and Do in the Fourth Dimension." He's hilarious and this book is very very great at explaining some pretty awesome things. So keep reading, and if you're looking for something to watch, I hope you've already watched Kevin Lieber's film on Field Day. I already did a documentary about Whittier, Alaska over there. Kevin's got a great short film about putting things out on the Internet and having people react to them. There's a rumor that Jake Roper might be doing something on Field Day soon. So check out mine, check out Kevin's and subscribe to Field Day for upcoming Jake Roper action, yeah? He's actually in this room right now, say hi, Jake. [Jake:] Hi. Thanks for filming this, by the way. Guys, I really appreciate who you all are. And as always, thanks for watching.
Life
Early life and education
Alfred Tarski was born Alfred Teitelbaum (Polish spelling: "Tajtelbaum"), to parents who were Polish Jews in comfortable circumstances. He first manifested his mathematical abilities while in secondary school, at Warsaw's Szkoła Mazowiecka.^{[8]} Nevertheless, he entered the University of Warsaw in 1918 intending to study biology.^{[9]}
After Poland regained independence in 1918, Warsaw University came under the leadership of Jan Łukasiewicz, Stanisław Leśniewski and Wacław Sierpiński and quickly became a worldleading research institution in logic, foundational mathematics, and the philosophy of mathematics. Leśniewski recognized Tarski's potential as a mathematician and encouraged him to abandon biology.^{[9]} Henceforth Tarski attended courses taught by Łukasiewicz, Sierpiński, Stefan Mazurkiewicz and Tadeusz Kotarbiński, and in 1924 became the only person ever to complete a doctorate under Leśniewski's supervision. His thesis was entitled O wyrazie pierwotnym logistyki (On the Primitive Term of Logistic; published 1923). Tarski and Leśniewski soon grew cool to each other, mainly due to the latter's increasing antisemitism.^{[7]} However, in later life, Tarski reserved his warmest praise for Kotarbiński, which was reciprocated.
In 1923, Alfred Teitelbaum and his brother Wacław changed their surname to "Tarski". The Tarski brothers also converted to Roman Catholicism, Poland's dominant religion. Alfred did so even though he was an avowed atheist.^{[10]}^{[11]}
Career
After becoming the youngest person ever to complete a doctorate at Warsaw University, Tarski taught logic at the Polish Pedagogical Institute, mathematics and logic at the university, and served as Łukasiewicz's assistant. Because these positions were poorly paid, Tarski also taught mathematics at a Warsaw secondary school;^{[12]} before World War II, it was not uncommon for European intellectuals of research caliber to teach high school. Hence between 1923 and his departure for the United States in 1939, Tarski not only wrote several textbooks and many papers, a number of them groundbreaking, but also did so while supporting himself primarily by teaching highschool mathematics. In 1929 Tarski married fellow teacher Maria Witkowska, a Pole of Catholic background. She had worked as a courier for the army in the Polish–Soviet War. They had two children; a son Jan Tarski, who became a physicist, and a daughter Ina, who married the mathematician Andrzej Ehrenfeucht.^{[13]}
Tarski applied for a chair of philosophy at Lwów University, but on Bertrand Russell's recommendation it was awarded to Leon Chwistek.^{[14]} In 1930, Tarski visited the University of Vienna, lectured to Karl Menger's colloquium, and met Kurt Gödel. Thanks to a fellowship, he was able to return to Vienna during the first half of 1935 to work with Menger's research group. From Vienna he traveled to Paris to present his ideas on truth at the first meeting of the Unity of Science movement, an outgrowth of the Vienna Circle. Tarski's academic career in Poland was strongly and repeatedly impacted by his heritage. For example, in 1937, Tarski applied for a chair at Poznań University but the chair was abolished to avoid assigning it to Tarski (who was undisputedly the strongest applicant) because he was a Jew.^{[15]} Tarski's ties to the Unity of Science movement likely saved his life, because they resulted in his being invited to address the Unity of Science Congress held in September 1939 at Harvard University. Thus he left Poland in August 1939, on the last ship to sail from Poland for the United States before the German and Soviet invasion of Poland and the outbreak of World War II. Tarski left reluctantly, because Leśniewski had died a few months before, creating a vacancy which Tarski hoped to fill. Oblivious to the Nazi threat, he left his wife and children in Warsaw. He did not see them again until 1946. During the war, nearly all his Jewish extended family were murdered at the hands of the German occupying authorities.
Once in the United States, Tarski held a number of temporary teaching and research positions: Harvard University (1939), City College of New York (1940), and thanks to a Guggenheim Fellowship, the Institute for Advanced Study in Princeton (1942), where he again met Gödel. In 1942, Tarski joined the Mathematics Department at the University of California, Berkeley, where he spent the rest of his career. Tarski became an American citizen in 1945.^{[16]} Although emeritus from 1968, he taught until 1973 and supervised Ph.D. candidates until his death.^{[17]} At Berkeley, Tarski acquired a reputation as an astounding and demanding teacher, a fact noted by many observers:
His seminars at Berkeley quickly became famous in the world of mathematical logic. His students, many of whom became distinguished mathematicians, noted the awesome energy with which he would coax and cajole their best work out of them, always demanding the highest standards of clarity and precision.^{[18]}
Tarski was extroverted, quickwitted, strongwilled, energetic, and sharptongued. He preferred his research to be collaborative — sometimes working all night with a colleague — and was very fastidious about priority.^{[19]}
A charismatic leader and teacher, known for his brilliantly precise yet suspenseful expository style, Tarski had intimidatingly high standards for students, but at the same time he could be very encouraging, and particularly so to women — in contrast to the general trend. Some students were frightened away, but a circle of disciples remained, many of whom became worldrenowned leaders in the field.^{[20]}
Tarski supervised twentyfour Ph.D. dissertations including (in chronological order) those of Andrzej Mostowski, Bjarni Jónsson, Julia Robinson, Robert Vaught, Solomon Feferman, Richard Montague, James Donald Monk, Haim Gaifman, Donald Pigozzi, and Roger Maddux, as well as Chen Chung Chang and Jerome Keisler, authors of Model Theory (1973),^{[21]} a classic text in the field.^{[22]}^{[23]} He also strongly influenced the dissertations of Alfred Lindenbaum, Dana Scott, and Steven Givant. Five of Tarski's students were women, a remarkable fact given that men represented an overwhelming majority of graduate students at the time.^{[23]} However, he had extramarital affairs with at least two of these students. After he showed another of his female student's^{[who?]} work to a male colleague^{[who?]}, the colleague published it himself, leading her to leave the graduate study and later move to a different university and a different advisor.^{[24]}
Tarski lectured at University College, London (1950, 1966), the Institut Henri Poincaré in Paris (1955), the Miller Institute for Basic Research in Science in Berkeley (1958–60), the University of California at Los Angeles (1967), and the Pontifical Catholic University of Chile (1974–75). Among many distinctions garnered over the course of his career, Tarski was elected to the United States National Academy of Sciences, the British Academy and the Royal Netherlands Academy of Arts and Sciences in 1958,^{[25]} received honorary degrees from the Pontifical Catholic University of Chile in 1975, from Marseilles' Paul Cézanne University in 1977 and from the University of Calgary, as well as the Berkeley Citation in 1981. Tarski presided over the Association for Symbolic Logic, 1944–46, and the International Union for the History and Philosophy of Science, 1956–57. He was also an honorary editor of Algebra Universalis.^{[26]}
Work in mathematics
Tarski's mathematical interests were exceptionally broad. His collected papers run to about 2,500 pages, most of them on mathematics, not logic. For a concise survey of Tarski's mathematical and logical accomplishments by his former student Solomon Feferman, see "Interludes I–VI" in Feferman and Feferman.^{[27]}
Tarski's first paper, published when he was 19 years old, was on set theory, a subject to which he returned throughout his life.^{[28]} In 1924, he and Stefan Banach proved that, if one accepts the Axiom of Choice, a ball can be cut into a finite number of pieces, and then reassembled into a ball of larger size, or alternatively it can be reassembled into two balls whose sizes each equal that of the original one. This result is now called the Banach–Tarski paradox.^{[29]}
In A decision method for elementary algebra and geometry, Tarski showed, by the method of quantifier elimination, that the firstorder theory of the real numbers under addition and multiplication is decidable. (While this result appeared only in 1948, it dates back to 1930 and was mentioned in Tarski (1931).) This is a very curious result, because Alonzo Church proved in 1936 that Peano arithmetic (the theory of natural numbers) is not decidable. Peano arithmetic is also incomplete by Gödel's incompleteness theorem. In his 1953 Undecidable theories, Tarski et al. showed that many mathematical systems, including lattice theory, abstract projective geometry, and closure algebras, are all undecidable. The theory of Abelian groups is decidable, but that of nonAbelian groups is not.
In the 1920s and 30s, Tarski often taught high school geometry.^{[where?]} Using some ideas of Mario Pieri, in 1926 Tarski devised an original axiomatization for plane Euclidean geometry, one considerably more concise than Hilbert's.^{[30]} Tarski's axioms form a firstorder theory devoid of set theory, whose individuals are points, and having only two primitive relations. In 1930, he proved this theory decidable because it can be mapped into another theory he had already proved decidable, namely his firstorder theory of the real numbers.
In 1929 he showed that much of Euclidean solid geometry could be recast as a secondorder theory whose individuals are spheres (a primitive notion), a single primitive binary relation "is contained in", and two axioms that, among other things, imply that containment partially orders the spheres. Relaxing the requirement that all individuals be spheres yields a formalization of mereology far easier to exposit than Lesniewski's variant. Near the end of his life, Tarski wrote a very long letter, published as Tarski and Givant (1999), summarizing his work on geometry.^{[31]}
Cardinal Algebras studied algebras whose models include the arithmetic of cardinal numbers. Ordinal Algebras sets out an algebra for the additive theory of order types. Cardinal, but not ordinal, addition commutes.
In 1941, Tarski published an important paper on binary relations, which began the work on relation algebra and its metamathematics that occupied Tarski and his students for much of the balance of his life. While that exploration (and the closely related work of Roger Lyndon) uncovered some important limitations of relation algebra, Tarski also showed (Tarski and Givant 1987) that relation algebra can express most axiomatic set theory and Peano arithmetic. For an introduction to relation algebra, see Maddux (2006). In the late 1940s, Tarski and his students devised cylindric algebras, which are to firstorder logic what the twoelement Boolean algebra is to classical sentential logic. This work culminated in the two monographs by Tarski, Henkin, and Monk (1971, 1985).^{[32]}
Work in logic
Tarski's student, Robert Lawson Vaught, has ranked Tarski as one of the four greatest logicians of all time — along with Aristotle, Gottlob Frege, and Kurt Gödel.^{[7]}^{[33]}^{[34]} However, Tarski often expressed great admiration for Charles Sanders Peirce, particularly for his pioneering work in the logic of relations.
Tarski produced axioms for logical consequence and worked on deductive systems, the algebra of logic, and the theory of definability. His semantic methods, which culminated in the model theory he and a number of his Berkeley students developed in the 1950s and 60s, radically transformed Hilbert's prooftheoretic metamathematics. Around 1930, Tarski developed an abstract theory of logical deductions that models some properties of logical calculi. Mathematically, what he described is just a finitary closure operator on a set (the set of sentences). In abstract algebraic logic, finitary closure operators are still studied under the name consequence operator, which was coined by Tarski. The set S represents a set of sentences, a subset T of S a theory, and cl(T) is the set of all sentences that follow from the theory. This abstract approach was applied to fuzzy logic (see Gerla 2000).
In [Tarski's] view, metamathematics became similar to any mathematical discipline. Not only can its concepts and results be mathematized, but they actually can be integrated into mathematics. ... Tarski destroyed the borderline between metamathematics and mathematics. He objected to restricting the role of metamathematics to the foundations of mathematics.^{[35]}
Tarski's 1936 article "On the concept of logical consequence" argued that the conclusion of an argument will follow logically from its premises if and only if every model of the premises is a model of the conclusion.^{[36]} In 1937, he published a paper presenting clearly his views on the nature and purpose of the deductive method, and the role of logic in scientific studies.^{[28]} His high school and undergraduate teaching on logic and axiomatics culminated in a classic short text, published first in Polish, then in German translation, and finally in a 1941 English translation as Introduction to Logic and to the Methodology of Deductive Sciences.^{[37]}
Tarski's 1969 "Truth and proof" considered both Gödel's incompleteness theorems and Tarski's undefinability theorem, and mulled over their consequences for the axiomatic method in mathematics.
Truth in formalized languages
In 1933, Tarski published a very long paper in Polish, titled "Pojęcie prawdy w językach nauk dedukcyjnych",^{[38]} "Setting out a mathematical definition of truth for formal languages." The 1935 German translation was titled "Der Wahrheitsbegriff in den formalisierten Sprachen", "The concept of truth in formalized languages", sometimes shortened to "Wahrheitsbegriff". An English translation appeared in the 1956 first edition of the volume Logic, Semantics, Metamathematics. This collection of papers from 1923 to 1938 is an event in 20thcentury analytic philosophy, a contribution to symbolic logic, semantics, and the philosophy of language. For a brief discussion of its content, see Convention T (and also Tschema).
Some recent^{[when?]} philosophical debate examines the extent to which Tarski's theory of truth for formalized languages can be seen as a correspondence theory of truth. The debate centers on how to read Tarski's condition of material adequacy for a true definition. That condition requires that the truth theory have the following as theorems for all sentences p of the language for which truth is being defined:
 "p" is true if and only if p.
(where p is the proposition expressed by "p")
The debate amounts to whether to read sentences of this form, such as
"Snow is white" is true if and only if snow is white
as expressing merely a deflationary theory of truth or as embodying truth as a more substantial property (see Kirkham 1992). It is important to realize that Tarski's theory of truth is for formalized languages, so examples in natural language are not illustrations^{[why?]} of the use of Tarski's theory of truth.^{[citation needed]}
Logical consequence
In 1936, Tarski published Polish and German versions of a lecture, “On the Concept of Following Logically",^{[39]} he had given the preceding year at the International Congress of Scientific Philosophy in Paris. A new English translation of this paper, Tarski (2002), highlights the many differences between the German and Polish versions of the paper and corrects a number of mistranslations in Tarski (1983).^{[39]}
This publication^{[which?]} set out the modern modeltheoretic definition of (semantic) logical consequence, or at least the basis for it. Whether Tarski's notion was entirely the modern one turns on whether he intended to admit models with varying domains (and in particular, models with domains of different cardinalities).^{[citation needed]} This question is a matter of some debate in the current^{[when?]} philosophical literature. John Etchemendy stimulated much of the recent discussion about Tarski's treatment of varying domains.^{[40]}
Tarski ends by pointing out that his definition of logical consequence depends upon a division of terms into the logical and the extralogical and he expresses some skepticism that any such objective division will be forthcoming. "What are Logical Notions?" can thus be viewed as continuing "On the Concept of Logical Consequence".^{[citation needed]}
Logical notions
Another theory of Tarski's attracting attention in the recent^{[when?]} philosophical literature is that outlined in his "What are Logical Notions?" (Tarski 1986). This is the published version of a talk that he gave originally in 1966 in London and later in 1973 in Buffalo; it was edited without his direct involvement by John Corcoran. It became the most cited paper in the journal History and Philosophy of Logic.^{[41]}
In the talk, Tarski proposed demarcation of logical operations (which he calls "notions") from nonlogical. The suggested criteria were derived from the Erlangen program of the 19thcentury German mathematician Felix Klein. Mautner (in 1946), and possibly^{[clarification needed]} an article by the Portuguese mathematician José Sebastião e Silva, anticipated Tarski in applying the Erlangen Program to logic.^{[citation needed]}
That program^{[which?]} classified the various types of geometry (Euclidean geometry, affine geometry, topology, etc.) by the type of oneone transformation of space onto itself that left the objects of that geometrical theory invariant. (A onetoone transformation is a functional map of the space onto itself so that every point of the space is associated with or mapped to one other point of the space. So, "rotate 30 degrees" and "magnify by a factor of 2" are intuitive descriptions of simple uniform oneone transformations.) Continuous transformations give rise to the objects of topology, similarity transformations to those of Euclidean geometry, and so on.^{[citation needed]}
As the range of permissible transformations becomes broader, the range of objects one is able to distinguish as preserved by the application of the transformations becomes narrower. Similarity transformations are fairly narrow (they preserve the relative distance between points) and thus allow us to distinguish relatively many things (e.g., equilateral triangles from nonequilateral triangles). Continuous transformations (which can intuitively be thought of as transformations which allow nonuniform stretching, compression, bending, and twisting, but no ripping or glueing) allow us to distinguish a polygon from an annulus (ring with a hole in the centre), but do not allow us to distinguish two polygons from each other.^{[citation needed]}
Tarski's proposal^{[which?]} was to demarcate the logical notions by considering all possible onetoone transformations (automorphisms) of a domain onto itself. By domain is meant the universe of discourse of a model for the semantic theory of logic. If one identifies the truth value True with the domain set and the truthvalue False with the empty set, then the following operations are counted as logical under the proposal:
 Truthfunctions: All truthfunctions are admitted by the proposal. This includes, but is not limited to, all nary truthfunctions for finite n. (It also admits of truthfunctions with any infinite number of places.)
 Individuals: No individuals, provided the domain has at least two members.
 Predicates:
 the oneplace total and null predicates, the former having all members of the domain in its extension and the latter having no members of the domain in its extension
 twoplace total and null predicates, the former having the set of all ordered pairs of domain members as its extension and the latter with the empty set as extension
 the twoplace identity predicate, with the set of all orderpairs <a,a> in its extension, where a is a member of the domain
 the twoplace diversity predicate, with the set of all order pairs <a,b> where a and b are distinct members of the domain
 nary predicates in general: all predicates definable from the identity predicate together with conjunction, disjunction and negation (up to any ordinality, finite or infinite)
 Quantifiers: Tarski explicitly discusses only monadic quantifiers and points out that all such numerical quantifiers are admitted under his proposal. These include the standard universal and existential quantifiers as well as numerical quantifiers such as "Exactly four", "Finitely many", "Uncountably many", and "Between four and 9 million", for example. While Tarski does not enter into the issue, it is also clear that polyadic quantifiers are admitted under the proposal. These are quantifiers like, given two predicates Fx and Gy, "More(x, y)", which says "More things have F than have G."
 SetTheoretic relations: Relations such as inclusion, intersection and union applied to subsets of the domain are logical in the present sense.
 Set membership: Tarski ended his lecture with a discussion of whether the set membership relation counted as logical in his sense. (Given the reduction of (most of) mathematics to set theory, this was, in effect, the question of whether most or all of mathematics is a part of logic.) He pointed out that set membership is logical if set theory is developed along the lines of type theory, but is extralogical if set theory is set out axiomatically, as in the canonical Zermelo–Fraenkel set theory.
 Logical notions of higher order: While Tarski confined his discussion to operations of firstorder logic, there is nothing about his proposal that necessarily restricts it to firstorder logic. (Tarski likely restricted his attention to firstorder notions as the talk was given to a nontechnical audience.) So, higherorder quantifiers and predicates are admitted as well.^{[citation needed]}
In some ways the present proposal is the obverse of that of Lindenbaum and Tarski (1936), who proved that all the logical operations of Bertrand Russell's and Whitehead's Principia Mathematica are invariant under onetoone transformations of the domain onto itself. The present proposal is also employed in Tarski and Givant (1987).^{[42]}
Solomon Feferman and Vann McGee further discussed Tarski's proposal^{[which?]} in work published after his death. Feferman (1999) raises problems for the proposal and suggests a cure: replacing Tarski's preservation by automorphisms with preservation by arbitrary homomorphisms. In essence, this suggestion circumvents the difficulty Tarski's proposal has in dealing with a sameness of logical operation across distinct domains of a given cardinality and across domains of distinct cardinalities. Feferman's proposal results in a radical restriction of logical terms as compared to Tarski's original proposal. In particular, it ends up counting as logical only those operators of standard firstorder logic without identity.^{[citation needed]}
Vann McGee (1996) provides a precise account of what operations are logical in the sense of Tarski's proposal in terms of expressibility in a language that extends firstorder logic by allowing arbitrarily long conjunctions and disjunctions, and quantification over arbitrarily many variables. "Arbitrarily" includes a countable infinity.^{[43]}
Selected publications
 Anthologies and collections
 1986. The Collected Papers of Alfred Tarski, 4 vols. Givant, S. R., and McKenzie, R. N., eds. Birkhäuser.
 Givant Steven (1986). "Bibliography of Alfred Tarski". Journal of Symbolic Logic. 51 (4): 913–41. doi:10.2307/2273905. JSTOR 2273905. S2CID 44369365.
 1983 (1956). Logic, Semantics, Metamathematics: Papers from 1923 to 1938 by Alfred Tarski, Corcoran, J., ed. Hackett. 1st edition edited and translated by J. H. Woodger, Oxford Uni. Press.^{[44]} This collection contains translations from Polish of some of Tarski's most important papers of his early career, including The Concept of Truth in Formalized Languages and On the Concept of Logical Consequence discussed above.
 Original publications of Tarski
 1930 Une contribution à la théorie de la mesure. Fund Math 15 (1930), 42–50.
 1930. (with Jan Łukasiewicz). "Untersuchungen uber den Aussagenkalkul" ["Investigations into the Sentential Calculus"], Comptes Rendus des seances de la Societe des Sciences et des Lettres de Varsovie, Vol, 23 (1930) Cl. III, pp. 31–32 in Tarski (1983): 38–59.
 1931. "Sur les ensembles définissables de nombres réels I", Fundamenta Mathematicae 17: 210–239 in Tarski (1983): 110–142.
 1936. "Grundlegung der wissenschaftlichen Semantik", Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935, vol. III, Language et pseudoproblèmes, Paris, Hermann, 1936, pp. 1–8 in Tarski (1983): 401–408.
 1936. "Über den Begriff der logischen Folgerung", Actes du Congrès international de philosophie scientifique, Sorbonne, Paris 1935, vol. VII, Logique, Paris: Hermann, pp. 1–11 in Tarski (1983): 409–420.
 1936 (with Adolf Lindenbaum). "On the Limitations of Deductive Theories" in Tarski (1983): 384–92.
 1937. Einführung in die Mathematische Logik und in die Methodologie der Mathematik. Springer, Wien (Vienna).
 1994 (1941).^{[45]}^{[46]} Introduction to Logic and to the Methodology of Deductive Sciences. Dover.
 1941. "On the calculus of relations", Journal of Symbolic Logic 6: 73–89.
 1944. "The Semantical Concept of Truth and the Foundations of Semantics," Philosophy and Phenomenological Research 4: 341–75.
 1948. A decision method for elementary algebra and geometry. Santa Monica CA: RAND Corp.^{[47]}
 1949. Cardinal Algebras. Oxford Univ. Press.^{[48]}
 1953 (with Mostowski and Raphael Robinson). Undecidable theories. North Holland.^{[49]}
 1956. Ordinal algebras. NorthHolland.
 1965. "A simplified formalization of predicate logic with identity", Archiv für Mathematische Logik und Grundlagenforschung 7: 6179
 1969. "Truth and Proof", Scientific American 220: 63–77.
 1971 (with Leon Henkin and Donald Monk). Cylindric Algebras: Part I. NorthHolland.
 1985 (with Leon Henkin and Donald Monk). Cylindric Algebras: Part II. NorthHolland.
 1986. "What are Logical Notions?", Corcoran, J., ed., History and Philosophy of Logic 7: 143–54.
 1987 (with Steven Givant). A Formalization of Set Theory Without Variables. Vol.41 of American Mathematical Society colloquium publications. Providence RI: American Mathematical Society. ISBN 9780821810415. Review
 1999 (with Steven Givant). "Tarski's system of geometry", Bulletin of Symbolic Logic 5: 175–214.
 2002. "On the Concept of Following Logically" (Magda Stroińska and David Hitchcock, trans.) History and Philosophy of Logic 23: 155–196.
See also
 History of philosophy in Poland
 Cylindric algebra
 Interpretability
 Weak interpretability
 List of things named after Alfred Tarski
 Timeline of Polish science and technology
References
 ^ Alfred Tarski, "Alfred Tarski", Encyclopædia Britannica.
 ^ School of Mathematics and Statistics, University of St Andrews, "Alfred Tarski", School of Mathematics and Statistics, University of St Andrews.
 ^ "Alfred Tarski". Oxford Reference.
 ^ GomezTorrente, Mario (March 27, 2014). "Alfred Tarski  Philosophy  Oxford Bibliographies". Oxford University Press. Retrieved October 24, 2017.
 ^ Alfred Tarski, "Alfred Tarski", Stanford Encyclopedia of Philosophy.
 ^ Feferman A.
 ^ ^{a} ^{b} ^{c} Feferman & Feferman, p.1
 ^ Feferman & Feferman, pp.1718
 ^ ^{a} ^{b} Feferman & Feferman, p.26
 ^ Feferman & Feferman, p.294
 ^ "Most of the Socialist Party members were also in favor of assimilation, and Tarski's political allegiance was socialist at the time. So, along with its being a practical move, becoming more Polish than Jewish was an ideological statement and was approved by many, though not all, of his colleagues. As to why Tarski, a professed atheist, converted, that just came with the territory and was part of the package: if you were going to be Polish then you had to say you were Catholic." Anita Burdman Feferman, Solomon Feferman, Alfred Tarski: Life and Logic (2004), page 39.
 ^ "The Newsletter of the Janusz Korczak Association of Canada" (PDF). September 2007. Number 5. Retrieved 8 February 2012.
 ^ Feferman & Feferman (2004), pp. 239–242.
 ^ Feferman & Feferman, p. 67
 ^ Feferman & Feferman, pp. 102103
 ^ Feferman & Feferman, Chap. 5, pp. 124149
 ^ Robert Vaught; John Addison; Benson Mates; Julia Robinson (1985). "Alfred Tarski, Mathematics: Berkeley". University of California (System) Academic Senate. Retrieved 20081226.
 ^ Obituary in Times, reproduced here
 ^ Gregory Moore, "Alfred Tarski" in Dictionary of Scientific Biography
 ^ Feferman
 ^ Chang, C.C., and Keisler, H.J., 1973. Model Theory. NorthHolland, Amsterdam. American Elsevier, New York.
 ^ Alfred Tarski at the Mathematics Genealogy Project
 ^ ^{a} ^{b} Feferman & Feferman, pp. 385386
 ^ Feferman & Feferman, pp. 177–178 and 197–201.
 ^ "Alfred Tarski (1902  1983)". Royal Netherlands Academy of Arts and Sciences. Retrieved 17 July 2015.
 ^ O'Connor, John J.; Robertson, Edmund F., "Alfred Tarski", MacTutor History of Mathematics Archive, University of St Andrews
 ^ Feferman & Feferman, pp. 4352, 6975, 109123, 189195, 277287, 334342
 ^ ^{a} ^{b} "Alfred Tarski". mathshistory.standrews.ac.uk. Retrieved 28 April 2023.
 ^ Katie Buchhorn (8 August 2012). "The BanachTarski Paradox". arXiv:2108.05714 [math.HO].
 ^ Adam Grabowski. "Tarski's Geometry and the Euclidean Plane in Mizar" (PDF). ceurws.org. Retrieved 28 April 2023.
 ^ Tarski, Alfred; Givant, Steven (1999). "Tarski's System of Geometry". The Bulletin of Symbolic Logic. 5 (2): 175–214. doi:10.2307/421089. JSTOR 421089. S2CID 18551419.
 ^ "Tarski's conventionT and inductive definition?". goodmancoaching.nl. 22 May 2022. Retrieved 28 April 2023.
 ^ Vaught, Robert L. (Dec 1986). "Alfred Tarski's Work in Model Theory". Journal of Symbolic Logic. 51 (4): 869–882. doi:10.2307/2273900. JSTOR 2273900. S2CID 27153078.
 ^ Restall, Greg (2002–2006). "Great Moments in Logic". Archived from the original on 6 December 2008. Retrieved 20090103.
 ^ Sinaceur, Hourya (2001). "Alfred Tarski: Semantic Shift, Heuristic Shift in Metamathematics". Synthese. 126 (1–2): 49–65. doi:10.1023/A:1005268531418. ISSN 00397857. S2CID 28783841.
 ^ GómezTorrente, Mario (1996). "Tarski on Logical Consequence". Notre Dame Journal of Formal Logic. 37. doi:10.1305/ndjfl/1040067321. S2CID 13217777.
 ^ "Introduction To Logic And To The Methodology Of Deductive Sciences". archive.org. Retrieved 28 April 2023.
 ^ Alfred Tarski, "POJĘCIE PRAWDY W JĘZYKACH NAUK DEDUKCYJNYCH", Towarszystwo Naukowe Warszawskie, Warszawa, 1933. (Text in Polish in the Digital Library WFISUWIFISPANPTF) Archived 20160304 at the Wayback Machine.
 ^ ^{a} ^{b} Tarski, Alfred (2002). "On the Concept of Following Logically". History and Philosophy of Logic. 23 (3): 155–196. doi:10.1080/0144534021000036683. S2CID 120956516.
 ^ Etchemendy, John (1999). The Concept of Logical Consequence. Stanford CA: CSLI Publications. ISBN 9781575861944.
 ^ "History and Philosophy of Logic".
 ^ Németi, István (12 March 2014). "Alfred Tarski and Steven Givant. A formalization of set theory without variables. American Mathematical Society colloquium publications, vol. 41. American Mathematical Society, Providence1987, xxi + 318 pp". The Journal of Symbolic Logic. 55 (1): 350–352. doi:10.2307/2274990. JSTOR 2274990. Retrieved 28 April 2023.
 ^ McGee, Vann (1997). "Revision". Philosophical Issues. 8: 387–406. doi:10.2307/1523019. JSTOR 1523019. Retrieved 28 April 2023.
 ^ Halmos, Paul (1957). "Review: Logic, semantics, metamathematics. Papers from 1923 to 1938 by Alfred Tarski; translated by J. H. Woodger" (PDF). Bull. Amer. Math. Soc. 63 (2): 155–156. doi:10.1090/S000299041957101153.
 ^ Quine, W. V. (1938). "Review: Einführung in die mathematische Logik und in die Methodologie der Mathematik by Alfred Tarski. Vienna, Springer, 1937. x+166 pp" (PDF). Bull. Amer. Math. Soc. 44 (5): 317–318. doi:10.1090/s000299041938067316.
 ^ Curry, Haskell B. (1942). "Review: Introduction to Logic and to the Methodology of Deductive Sciences by Alfred Tarski" (PDF). Bull. Amer. Math. Soc. 48 (7): 507–510. doi:10.1090/s000299041942076981.
 ^ McNaughton, Robert (1953). "Review: A decision method for elementary algebra and geometry by A. Tarski" (PDF). Bull. Amer. Math. Soc. 59 (1): 91–93. doi:10.1090/s000299041953096641.
 ^ Birkhoff, Garrett (1950). "Review: Cardinal algebras by A. Tarski" (PDF). Bull. Amer. Math. Soc. 56 (2): 208–209. doi:10.1090/s00029904195009394x.
 ^ Gál, Ilse Novak (1954). "Review: Undecidable theories by Alfred Tarski in collaboration with A. Mostowsku and R. M. Robinson" (PDF). Bull. Amer. Math. Soc. 60 (6): 570–572. doi:10.1090/S000299041954098580.
Further reading
 Biographical references
 Feferman, Anita Burdman (1999). "Alfred Tarski". American National Biography. Vol. 21. Oxford University Press. pp. 330–332. ISBN 9780195128000.
 Feferman, Anita Burdman; Feferman, Solomon (2004). Alfred Tarski: Life and Logic. Cambridge University Press. ISBN 9780521802406. OCLC 54691904.
 FrostArnold, Greg (2013). Carnap, Tarski, and Quine at Harvard: Conversations on Logic, Mathematics, and Science. Chicago: Open Court. ISBN 9780812698374.
 Givant Steven (1991). "A portrait of Alfred Tarski". Mathematical Intelligencer. 13 (3): 16–32. doi:10.1007/bf03023831. S2CID 122867668.
 Patterson, Douglas. Alfred Tarski: Philosophy of Language and Logic (Palgrave Macmillan; 2012) 262 pages; biography focused on his work from the late1920s to the mid1930s, with particular attention to influences from his teachers Stanislaw Lesniewski and Tadeusz Kotarbinski.
 Logic literature
 The December 1986 issue of the Journal of Symbolic Logic surveys Tarski's work on model theory (Robert Vaught), algebra (Jonsson), undecidable theories (McNulty), algebraic logic (Donald Monk), and geometry (Szczerba). The March 1988 issue of the same journal surveys his work on axiomatic set theory (Azriel Levy), real closed fields (Lou Van Den Dries), decidable theory (Doner and Wilfrid Hodges), metamathematics (Blok and Pigozzi), truth and logical consequence (John Etchemendy), and general philosophy (Patrick Suppes).
 Blok, W. J.; Pigozzi, Don, "Alfred Tarski's Work on General Metamathematics", The Journal of Symbolic Logic, Vol. 53, No. 1 (Mar., 1988), pp. 36–50
 Chang, C.C., and Keisler, H.J., 1973. Model Theory. NorthHolland, Amsterdam. American Elsevier, New York.
 Corcoran, John, and Sagüillo, José Miguel, 2011. "The Absence of Multiple Universes of Discourse in the 1936 Tarski ConsequenceDefinition Paper", History and Philosophy of Logic 32: 359–80. [1]
 Corcoran, John, and Weber, Leonardo, 2015. "Tarski's convention T: condition beta", South American Journal of Logic. 1, 3–32.
 Etchemendy, John, 1999. The Concept of Logical Consequence. Stanford CA: CSLI Publications. ISBN 1575861941
 Feferman Solomon (1999). "Logic, Logics, and Logicism" (PDF). Notre Dame Journal of Formal Logic. 40: 31–54. doi:10.1305/ndjfl/1039096304.
 Gerla, G. (2000) Fuzzy Logic: Mathematical Tools for Approximate Reasoning. Kluwer Academic Publishers.
 GrattanGuinness, Ivor, 2000. The Search for Mathematical Roots 18701940. Princeton Uni. Press.
 Kirkham, Richard, 1992. Theories of Truth. MIT Press.
 Maddux, Roger D., 2006. Relation Algebras, vol. 150 in "Studies in Logic and the Foundations of Mathematics", Elsevier Science.
 Mautner F. I. (1946). "An Extension of Klein's Erlanger Program: Logic as InvariantTheory". American Journal of Mathematics. 68 (3): 345–84. doi:10.2307/2371821. JSTOR 2371821.
 McGee Van (1996). "Logical Operations". Journal of Philosophical Logic. 25 (6): 567–80. doi:10.1007/bf00265253. S2CID 32381037.
 Popper, Karl R., 1972, Rev. Ed. 1979, "Philosophical Comments on Tarski's Theory of Truth", with Addendum, Objective Knowledge, Oxford: 319–340.
 Sinaceur H (2001). "Alfred Tarski: Semantic shift, heuristic shift in metamathematics". Synthese. 126: 49–65. doi:10.1023/a:1005268531418. S2CID 28783841.
 Smith, James T., 2010. "Definitions and Nondefinability in Geometry", American Mathematical Monthly 117:475–89.
 Wolenski, Jan, 1989. Logic and Philosophy in the Lvov–Warsaw School. Reidel/Kluwer.
External links
 Stanford Encyclopedia of Philosophy:
 Tarski's Truth Definitions by Wilfred Hodges.
 Alfred Tarski by Mario GómezTorrente.
 Algebraic Propositional Logic by Ramon Jansana. Includes a fairly detailed discussion of Tarski's work on these topics.
 Tarski's Semantic Theory on the Internet Encyclopedia of Philosophy.