406:, until the top entries of the stack contain the number of operands that fits to the top most operator (immediately beneath). This group of tokens at the stacktop (the last stacked operator and the according number of operands) is replaced by the result of executing the operator on these/this operand(s). Then the processing of the input continues in this manner. The rightmost operand in a valid prefix expression thus empties the stack, except for the result of evaluating the whole expression. When starting at the right, the pushing of tokens is performed similarly, just the evaluation is triggered by an operator, finding the appropriate number of operands that fits its arity already at the stacktop. Now the leftmost token of a valid prefix expression must be an operator, fitting to the number of operands in the stack, which again yields the result. As can be seen from the description, a
2497:
33:
401:
each, and all necessary operands are assumed to be explicitly given. A valid prefix expression always starts with an operator and ends with an operand. Evaluation can either proceed from left to right, or in the opposite direction. Starting at the left, the input string, consisting of tokens denoting
339:
of all involved operators (here the "−" denotes the binary operation of subtraction, not the unary function of sign-change), any well-formed prefix representation is unambiguous, and brackets within the prefix expression are unnecessary. As such, the above expression can be further simplified to
2003:
Die ältesten Texte in den 'Selected Works', in denen Łukasiewicz polnische
Notation verwendet, datieren relativ spät, sind aber Präsentationen vorangehender Arbeiten, die 'in the course of the years 1920–1930' (S. 131) stattgefunden haben, also auch keine genauere Zeitangabe
323:(infix). In more complex expressions, the operators still precede their operands, but the operands may themselves be expressions including again operators and their operands. For instance, the expression that would be written in conventional infix notation as
375:
When dealing with non-commutative operations, like division or subtraction, it is necessary to coordinate the sequential arrangement of the operands with the definition of how the operator takes its arguments, i.e., from left to right. For example,
247:, already had the idea of eliminating parentheses in logic formulas. In one of his papers Łukasiewicz stated that his notation is the most compact and the first linearly written parentheses-free notation, but not the first one as
1465:
The number of return values of an expression equals the difference between the number of operands in an expression and the total arity of the operators minus the total number of return values of the operators.
2043:
Worthy of remark is the parenthesis-free notation of Jan Łukasiewicz. In this the letters N, A, C, E, K are used in the roles of negation, disjunction, implication, equivalence, conjunction respectively.
392:
Prefix/postfix notation is especially popular for its innate ability to express the intended order of operations without the need for parentheses and other precedence rules, as are usually employed with
2221:
1439:
programming language, much like Lisp also uses Polish notation through the mathop library. The Ambi programming language uses Polish notation for arithmetic operations and program construction.
350:
notation, the innermost expressions are evaluated first, but in Polish notation this "innermost-ness" can be conveyed by the sequence of operators and operands rather than by bracketing.
825:
1018:
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689:
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784:
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648:
580:
1253:
488:
2386:
1752:
1332:. For classical propositional logic, it is a compatible extension of the notation of Łukasiewicz. But the notations are incompatible in the sense that Bocheński uses
1277:
1192:
512:
2425:
1109:
1053:
1410:
1390:
1370:
1350:
1130:
1074:
2363:(Tagungsband zum Kolloquium 14. November 2014 in Jena). GI Series: Lecture Notes in Informatics (LNI) – Thematics (in German). Vol. T-7. Bonn, Germany:
292:
to avoid brackets and that he had employed his notation in his logical papers since 1929. He then goes on to cite, as an example, a 1930 paper he wrote with
2212:
84:
2401:
2289:
223:
I came upon the idea of a parenthesis-free notation in 1924. I used that notation for the first time in my article Łukasiewicz (1), p. 610, footnote.
2470:
2501:
2016:
1887:"Reviewed work(s): Remarks on Nicod's Axiom and on "Generalizing Deduction" by Jan Łukasiewicz, Jerzy Słupecki, Państwowe Wydawnictwo Naukowe"
2086:
Martínez Nava, Xóchitl (2011-06-01), "Mhy bib I fail logic? Dyslexia in the teaching of logic", in
Blackburn, Patrick; van Ditmarsch, Hans;
1462:
syntax also allows functions to be called using prefix notation, while still supporting the unary postfix syntax common in other languages.
2321:
2527:
2418:
Friedrich L. Bauer's and Klaus
Samelson's works in the 1950s on the introduction of the terms cellar principle and cellar automaton
2413:
Friedrich L. Bauers und Klaus
Samelsons Arbeiten in den 1950er-Jahren zur Einführung der Begriffe Kellerprinzip und Kellerautomat
1999:
Logische
Notationen und deren Verarbeitung auf elektronischen Rechenanlagen aus theoretischer, praktischer und historischer Sicht
77:
2372:
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1840:
1728:
2113:
2411:
1447:
397:. Instead, the notation uniquely indicates which operator to evaluate first. The operators are assumed to have a fixed
70:
2466:
2264:
1714:
207:) and related programming languages define their entire syntax in prefix notation (and others use postfix notation).
1860:
2532:
2407:
1595:
407:
1918:
1651:
2093:
Tools for
Teaching Logic: Third International Congress, TICTTL 2011, Salamanca, Spain, 1–4 June 2011, Proceedings
1811:. Amsterdam and London/Warszawa: North-Holland Publishing Company/Polish Scientific Publishers. pp. 179–196.
346:
The processing of the product is deferred until its two operands are available (i.e., 5 minus 6, and 7). As with
19:
This article is about a prefix notation in mathematics and computer sciences. For the similarly named logic, see
804:
2517:
1900:
1665:
2424:(in German). Jena, Germany: Institut für Informatik, Christian-Albrechts-Universität zu Kiel. pp. 19–29.
2364:
1942:(1924). "Über die Bausteine der mathematischen Logik" [On the building blocks of mathematical logic].
1891:
1550:
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1528:
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2064:
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1544:
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200:
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1321:
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403:
2452:
1783:
968:
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668:
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926:
736:
600:
433:'s notation in modern logic. Some letters in the Polish notation table stand for particular words in
1150:
532:
369:
changes the meaning and the result of the expression. This version is written in Polish notation as
1983:
1882:
1747:
891:
227:
The reference cited by Łukasiewicz, i.e., Łukasiewicz (1), is apparently a lithographed report in
1700:
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831:
763:
695:
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418:
188:
148:
129:
41:
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History of informatics in German-speaking countries - Programming languages and compiler design
2310:
2220:. Collection Synthese (in French). Vol. 2. Bussum, Pays-Bas, Netherlands: F. G. Kroonder.
1742:
1733:
297:
289:
266:
2091:
1427:, where the parentheses are required since the operators in the language are themselves data (
1944:
1807:(1970). "Comments on Nicod's Axiom and on 'Generalizing Deduction'". In Borkowski, L. (ed.).
1788:
Księga pamiątkowa
Polskiego Towarzystwa Filozoficznego We Lwowie, 12. II. 1904–1912. II. 1929
1669:
1372:(for nonimplication and converse nonimplication) in propositional logic and Łukasiewicz uses
274:
1259:
1174:
494:
1501:
1428:
1094:
1038:
525:
192:
185:
1939:
244:
20:
8:
2120:
Polish or prefix notation has come to disuse given the difficulty that using it implies.
1971:
1741:(5). Mathematics Department, Santa Monica College, Santa Monica, California, USA: 26–29.
1329:
661:
354:
417:
The above sketched stack manipulation works—with mirrored input—also for expressions in
384:, with 7 left to 6, has the meaning of 7 − 6 (read as "subtract from 7 the operand 6").
2010:
1961:
1904:
1706:
1587:
1533:
1395:
1375:
1355:
1335:
1325:
1315:
1115:
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262:
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1914:
1913:(NB. The original 1931 paper "Uwagi o aksjomacie Nicoda i 'dedukcji uogólniającej" by
1856:
1804:
1779:
1692:
1657:
1643:
430:
380:, with 10 to the left of 5, has the meaning of 10 ÷ 5 (read as "divide 10 by 5"), or
353:
In the conventional infix notation, parentheses are required to override the standard
216:
167:
155:
their operands. It does not need any parentheses as long as each operator has a fixed
2378:
2368:
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1965:
1836:
1710:
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232:
1926:
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Geschichte der deutschsprachigen
Informatik - Programmiersprachen und Übersetzerbau
2101:
1953:
1601:
304:
270:
240:
2087:
1790:(in Polish). Lwów: Wydawnictwo Polskie Towarzystwo Filozoficzne. pp. 366–383.
2448:
2317:
2105:
2097:
1823:
1577:
1555:
1506:
434:
285:
252:
228:
2166:
Comptes Rendus des Séances de la Société des
Sciences et des Lettres de Varsovie
2069:
Comptes Rendus des Séances de la Société des
Sciences et des Lettres de Varsovie
1832:
1478:
1474:
729:
394:
315:
The expression for adding the numbers 1 and 2 is written in Polish notation as
303:
While no longer used much in logic, Polish notation has since found a place in
140:
50:
1605:
1318:
ranged over propositional values in Łukasiewicz's work on many-valued logics.
2511:
2382:
2060:
2030:
1424:
797:
293:
258:
248:
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Polish notation, usually in postfix form, is the chosen notation of certain
410:
with no capability of arbitrary stack inspection suffices to implement this
1521:
1459:
2246:
2261:"Google Code Archive - Long-term storage for Google Code Project Hosting"
1829:
1228:
1143:
593:
184:
When Polish notation is used as a syntax for mathematical expressions by
160:
2352:
Keller, Stack und automatisches Gedächtnis – eine Struktur mit Potenzial
2349:
Fothe, Michael; Wilke, Thomas, eds. (2015) . Written at Jena, Germany.
1957:
1908:
1886:
1470:
1451:
1786:[Comments on Nicod's Axiom and on 'Generalizing Deduction'].
196:
1511:
463:
164:
2260:
1697:
Aristotle's Syllogistic from the Standpoint of Modern Formal Logic
284:, he mentions that the principle of his notation was to write the
282:
Aristotle's Syllogistic from the Standpoint of Modern Formal Logic
411:
136:
1324:
introduced a system of Polish notation that names all 16 binary
2496:
2357:
Cellar, stack and automatic memory - a structure with potential
1922:
1087:
1974:, ed. (1967). "On the building blocks of mathematical logic".
1031:
398:
336:
265:
as worthy of remark in notational systems even contrasted to
156:
32:
2096:, Lecture Notes in Artificial Intelligence, vol. 6680,
2001:(Diploma thesis) (in German). Vienna, Austria. p. 88:
1440:
2465:(in German). Karlsruhe, Germany: Fakultät für Informatik,
1436:
1576:
Jorke, Günter; Lampe, Bernhard; Wengel, Norbert (1989).
2249:. Dordrecht, Netherlands: D. Reidel Publishing Company.
2164:[Investigations into the Sentential Calculus].
2067:[Investigations into the Sentential Calculus].
1477:. At a lower level, postfix operators are used by some
1784:"Uwagi o aksjomacie Nicoda i 'dedukcji uogólniającej'"
402:
operators or operands, is pushed token for token on a
16:
Mathematics notation with operators preceding operands
1398:
1378:
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739:
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671:
630:
603:
562:
535:
497:
473:
357:, since, referring to the above example, moving them
231:. The referring paper by Łukasiewicz was reviewed by
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846:
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751:
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683:
642:
615:
574:
547:
506:
482:
2150:
2055:
1586:] (in German) (1 ed.). Berlin, Germany:
2509:
2176:
2125:
2049:
1849:
1579:Arithmetische Algorithmen der Mikrorechentechnik
1575:
1990:
1875:
1861:"O znaczeniu i potrzebach logiki matematycznej"
1794:
1767:
1676:
1619:
2201:
1976:A Source Book in Mathematical Logic, 1879–1931
1970:
261:mentions this notation in his classic book on
219:in 1931 states how the notation was invented:
2085:
2079:
1420:Prefix notation has seen wide application in
273:'s logical notational exposition and work in
78:
1825:Data structures and other objects using Java
1304:
1219:
1134:
1078:
1022:
952:
882:
788:
720:
652:
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516:
424:
2185:
2156:
2134:
2023:
1938:
1881:
1855:
1803:
1778:
1720:
1691:
1656:
1642:
1443:filter syntax uses Polish prefix notation.
181:) to also include reverse Polish notation.
2441:
2348:
2278:
2015:: CS1 maint: location missing publisher (
1996:
1932:
1650:(in Polish) (1 ed.). Warsaw, Poland:
820:{\displaystyle \phi \leftrightarrow \psi }
85:
71:
2406:
2303:
2237:
2207:
2162:"Untersuchungen über den Aussagenkalküls"
2065:"Untersuchungen über den Aussagenkalküls"
1746:
1569:
1006:
978:
936:
908:
159:. The description "Polish" refers to the
1815:
170:, who invented Polish notation in 1924.
1925:, Poland in 1961 in a volume edited by
1726:
1584:Arithmetic algorithms in microcomputers
387:
177:is sometimes taken (as the opposite of
2510:
2029:
128:, is a mathematical notation in which
329:can be written in Polish notation as
2447:
2410:(2015) . Written at Kiel, Germany.
2253:
1821:
1448:stack-oriented programming languages
204:
2367:(GI) / Köllen Druck + Verlag GmbH.
2189:(1953). "A System of Modal Logic".
1608:. MPN 5539165. License 201.370/4/89
243:, editor in 1924 of the article of
13:
2342:
2311:"HP calculators - HP 35s RPN Mode"
2138:(1939). "Der Äquivalenzenkalkül".
2090:; Soler-Toscano, Fernando (eds.),
2035:Introduction to Mathematical Logic
1921:(National Scientific Publishers),
1415:
1098:
1042:
1013:{\displaystyle \varSigma p\,\phi }
972:
902:
474:
429:The table below shows the core of
14:
2549:
2489:
2467:Karlsruhe Institute of Technology
1446:Postfix notation is used in many
139:, in contrast to the more common
2528:Science and technology in Poland
2495:
2191:The Journal of Computing Systems
985:{\displaystyle \exists p\,\phi }
915:{\displaystyle \forall p\,\phi }
684:{\displaystyle \phi \land \psi }
143:, in which operators are placed
31:
2476:from the original on 2022-05-19
2431:from the original on 2022-11-14
2392:from the original on 2020-04-12
2327:from the original on 2022-01-21
2292:from the original on 2022-10-14
2267:from the original on 2017-09-28
2227:from the original on 2023-08-03
1755:from the original on 2022-07-01
1295:{\displaystyle \varGamma \phi }
1210:{\displaystyle \varDelta \phi }
943:{\displaystyle \varPi p\,\phi }
752:{\displaystyle \phi \mid \psi }
616:{\displaystyle \phi \lor \psi }
199:for the same. Because of this,
2243:A Precis of Mathematical Logic
2214:Précis de logique mathématique
2211:(1949). Written at Fribourg.
2037:. Princeton, New Jersey, USA:
1997:Gottschall, Christian (2005).
1901:Association for Symbolic Logic
1662:Elements of Mathematical Logic
1431:). Lisp functions may also be
1163:{\displaystyle \Diamond \phi }
811:
548:{\displaystyle \phi \to \psi }
539:
310:
251:proposed his parentheses-free
1:
2502:Polish notation (mathematics)
2440:(11 pages) (NB. Published in
1919:Państwowe Wydawnictwo Naukowe
1892:The Journal of Symbolic Logic
1727:Kennedy, John (August 1982).
1652:Państwowe Wydawnictwo Naukowe
1648:Elementy logiki matematycznej
1562:
1551:Head-directionality parameter
2106:10.1007/978-3-642-21350-2_19
1666:Wojtasiewicz, Olgierd Adrian
1529:Polish School of Mathematics
280:In Łukasiewicz's 1951 book,
215:A quotation from a paper by
191:, it is readily parsed into
7:
2365:Gesellschaft für Informatik
1517:Lisp (programming language)
1488:
873:{\displaystyle Q\phi \psi }
847:{\displaystyle E\phi \psi }
779:{\displaystyle D\phi \psi }
711:{\displaystyle K\phi \psi }
643:{\displaystyle A\phi \psi }
575:{\displaystyle C\phi \psi }
195:and can, in fact, define a
10:
2554:
2039:Princeton University Press
1248:{\displaystyle \Box \phi }
483:{\displaystyle \neg \phi }
255:notation in 1879 already.
210:
151:(RPN), in which operators
18:
1883:Pogorzelski, Henry Andrew
1024:kwantyfikator szczegółowy
425:Polish notation for logic
319:(prefix), rather than as
237:Journal of Symbolic Logic
197:one-to-one representation
1984:Harvard University Press
1980:Bauer-Mengelberg, Stefan
2533:Operators (programming)
1701:Oxford University Press
1496:Reverse Polish notation
1483:Burroughs large systems
419:reverse Polish notation
149:reverse Polish notation
2239:Bocheński, Józef Maria
2209:Bocheński, Józef Maria
1822:Main, Michael (2006).
1734:PPC Calculator Journal
1406:
1386:
1366:
1346:
1305:
1296:
1273:
1272:{\displaystyle L\phi }
1249:
1220:
1211:
1188:
1187:{\displaystyle M\phi }
1164:
1135:
1126:
1105:
1079:
1070:
1049:
1023:
1014:
986:
962:Existential quantifier
953:
944:
916:
883:
874:
848:
821:
789:
780:
753:
721:
712:
685:
653:
644:
617:
585:
576:
549:
517:
508:
507:{\displaystyle N\phi }
484:
225:
122:Polish prefix notation
106:normal Polish notation
2518:Mathematical notation
1945:Mathematische Annalen
1670:The MacMillan Company
1429:first-class functions
1407:
1387:
1367:
1347:
1297:
1274:
1250:
1212:
1189:
1165:
1127:
1106:
1104:{\displaystyle \bot }
1071:
1050:
1048:{\displaystyle \top }
1015:
987:
945:
917:
875:
849:
822:
781:
754:
713:
686:
645:
618:
577:
550:
509:
485:
275:Principia Mathematica
221:
193:abstract syntax trees
147:operands, as well as
2504:at Wikimedia Commons
2286:"LDAP Filter Syntax"
2100:, pp. 162–169,
1972:van Heijenoort, Jean
1917:was re-published at
1502:Function application
1396:
1376:
1356:
1336:
1283:
1260:
1236:
1198:
1175:
1151:
1116:
1095:
1060:
1039:
997:
969:
954:kwantyfikator ogólny
927:
899:
892:Universal quantifier
858:
832:
805:
764:
737:
696:
669:
628:
601:
560:
533:
526:Material conditional
495:
471:
388:Evaluation algorithm
233:Henry A. Pogorzelski
186:programming language
114:Łukasiewicz notation
2538:Logical expressions
1986:. pp. 355–366.
1545:Verb–object–subject
1539:Verb–subject–object
1330:propositional logic
298:sentential calculus
2140:Collectanea Logica
1958:10.1007/BF01448013
1940:Schönfinkel, Moses
1885:(September 1965).
1707:Garland Publishing
1588:VEB Verlag Technik
1534:Hungarian notation
1402:
1382:
1362:
1342:
1292:
1269:
1245:
1207:
1184:
1160:
1122:
1101:
1066:
1045:
1010:
982:
940:
912:
870:
844:
817:
776:
749:
708:
681:
640:
613:
572:
545:
504:
480:
263:mathematical logic
157:number of operands
45:("Reverse Polish")
2523:Polish inventions
2500:Media related to
2442:Fothe & Wilke
2374:978-3-88579-426-4
2172:(Cl. III): 51–77.
2075:(Cl. III): 30–50.
1842:978-0-321-37525-4
1729:"RPN Perspective"
1668:. New York, USA:
1405:{\displaystyle M}
1385:{\displaystyle L}
1365:{\displaystyle M}
1345:{\displaystyle L}
1312:
1311:
1125:{\displaystyle O}
1080:prawda, prawdziwy
1069:{\displaystyle V}
363:or removing them
335:Assuming a given
245:Moses Schönfinkel
104:), also known as
95:
94:
21:Łukasiewicz logic
2545:
2499:
2484:
2482:
2481:
2475:
2464:
2439:
2437:
2436:
2430:
2423:
2400:
2398:
2397:
2391:
2362:
2336:
2335:
2333:
2332:
2326:
2315:
2307:
2301:
2300:
2298:
2297:
2282:
2276:
2275:
2273:
2272:
2257:
2251:
2250:
2245:. Translated by
2235:
2233:
2232:
2226:
2219:
2205:
2199:
2198:
2187:Łukasiewicz, Jan
2183:
2174:
2173:
2158:Łukasiewicz, Jan
2154:
2148:
2147:
2136:Łukasiewicz, Jan
2132:
2123:
2122:
2115:978-3-64221349-6
2083:
2077:
2076:
2057:Łukasiewicz, Jan
2053:
2047:
2046:
2027:
2021:
2020:
2014:
2006:
1994:
1988:
1987:
1978:. Translated by
1968:
1952:(3–4): 305–316.
1936:
1930:
1912:
1879:
1873:
1872:
1857:Łukasiewicz, Jan
1853:
1847:
1846:
1819:
1813:
1812:
1805:Łukasiewicz, Jan
1801:
1792:
1791:
1780:Łukasiewicz, Jan
1776:
1765:
1763:
1761:
1760:
1750:
1724:
1718:
1704:
1693:Łukasiewicz, Jan
1689:
1674:
1673:
1664:. Translated by
1658:Łukasiewicz, Jan
1654:
1644:Łukasiewicz, Jan
1640:
1617:
1616:
1614:
1613:
1606:978-3-34100515-6
1573:
1412:in modal logic.
1411:
1409:
1408:
1403:
1391:
1389:
1388:
1383:
1371:
1369:
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1363:
1351:
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989:
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983:
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947:
946:
941:
921:
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913:
886:
879:
877:
876:
871:
853:
851:
850:
845:
826:
824:
823:
818:
792:
785:
783:
782:
777:
758:
756:
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750:
724:
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715:
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690:
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656:
649:
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641:
622:
620:
619:
614:
588:
581:
579:
578:
573:
554:
552:
551:
546:
520:
513:
511:
510:
505:
489:
487:
486:
481:
440:
439:
383:
379:
355:precedence rules
322:
318:
305:computer science
271:Bertrand Russell
267:Alfred Whitehead
241:Heinrich Behmann
87:
80:
73:
60:
46:
42:Postfix notation
35:
28:
27:
2553:
2552:
2548:
2547:
2546:
2544:
2543:
2542:
2508:
2507:
2492:
2479:
2477:
2473:
2462:
2434:
2432:
2428:
2421:
2408:Langmaack, Hans
2395:
2393:
2389:
2375:
2360:
2345:
2343:Further reading
2340:
2339:
2330:
2328:
2324:
2318:Hewlett-Packard
2313:
2309:
2308:
2304:
2295:
2293:
2284:
2283:
2279:
2270:
2268:
2259:
2258:
2254:
2230:
2228:
2224:
2217:
2206:
2202:
2184:
2177:
2155:
2151:
2133:
2126:
2116:
2098:Springer Nature
2084:
2080:
2054:
2050:
2028:
2024:
2008:
2007:
1995:
1991:
1937:
1933:
1915:Jan Łukasiewicz
1880:
1876:
1854:
1850:
1843:
1835:. p. 334.
1820:
1816:
1802:
1795:
1777:
1768:
1758:
1756:
1725:
1721:
1690:
1677:
1641:
1620:
1611:
1609:
1598:
1574:
1570:
1565:
1560:
1507:Lambda calculus
1491:
1475:Hewlett-Packard
1473:, notably from
1418:
1416:Implementations
1397:
1394:
1393:
1377:
1374:
1373:
1357:
1354:
1353:
1337:
1334:
1333:
1284:
1281:
1280:
1261:
1258:
1257:
1237:
1234:
1233:
1199:
1196:
1195:
1176:
1173:
1172:
1152:
1149:
1148:
1136:fałsz, fałszywy
1117:
1114:
1113:
1096:
1093:
1092:
1061:
1058:
1057:
1040:
1037:
1036:
998:
995:
994:
970:
967:
966:
928:
925:
924:
900:
897:
896:
859:
856:
855:
833:
830:
829:
806:
803:
802:
765:
762:
761:
738:
735:
734:
730:Non-conjunction
697:
694:
693:
670:
667:
666:
629:
626:
625:
602:
599:
598:
561:
558:
557:
534:
531:
530:
496:
493:
492:
472:
469:
468:
457:
452:
447:
431:Jan Łukasiewicz
427:
408:push-down store
390:
381:
377:
373:
367:
361:
344:
333:
327:
320:
316:
313:
253:Begriffsschrift
217:Jan Łukasiewicz
213:
175:Polish notation
168:Jan Łukasiewicz
126:prefix notation
118:Warsaw notation
98:Polish notation
91:
62:
58:
57:
56:Prefix notation
48:
44:
43:
24:
17:
12:
11:
5:
2551:
2541:
2540:
2535:
2530:
2525:
2520:
2506:
2505:
2491:
2490:External links
2488:
2487:
2486:
2451:(2017-08-07).
2445:
2404:
2373:
2344:
2341:
2338:
2337:
2302:
2277:
2252:
2236:Translated as
2200:
2175:
2149:
2124:
2114:
2088:Manzano, Maria
2078:
2061:Tarski, Alfred
2048:
2041:. p. 38.
2031:Church, Alonzo
2022:
1989:
1931:
1927:Jerzy Słupecki
1874:
1848:
1841:
1833:Addison-Wesley
1828:(3 ed.).
1814:
1809:Selected Works
1793:
1766:
1748:10.1.1.90.6448
1719:
1705:(Reprinted by
1699:(2 ed.).
1675:
1618:
1596:
1567:
1566:
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1561:
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1548:
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1526:
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1524:
1514:
1509:
1504:
1499:
1492:
1490:
1487:
1479:stack machines
1417:
1414:
1401:
1381:
1361:
1341:
1314:Note that the
1310:
1309:
1302:
1291:
1288:
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827:
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395:infix notation
389:
386:
371:
365:
359:
342:
331:
325:
312:
309:
212:
209:
179:infix notation
141:infix notation
93:
92:
90:
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82:
75:
67:
64:
63:
54:
51:Infix notation
40:
37:
36:
15:
9:
6:
4:
3:
2:
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2455:
2450:
2449:Goos, Gerhard
2446:
2443:
2427:
2419:
2415:
2414:
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2405:
2402:
2388:
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2323:
2319:
2312:
2306:
2291:
2287:
2281:
2266:
2262:
2256:
2248:
2247:Bird, Otto A.
2244:
2240:
2223:
2216:
2215:
2210:
2204:
2197:(1): 111–149.
2196:
2192:
2188:
2182:
2180:
2171:
2168:(in German).
2167:
2163:
2159:
2153:
2145:
2142:(in German).
2141:
2137:
2131:
2129:
2121:
2117:
2111:
2107:
2103:
2099:
2095:
2094:
2089:
2082:
2074:
2071:(in German).
2070:
2066:
2062:
2058:
2052:
2045:
2040:
2036:
2032:
2026:
2018:
2012:
2005:
2000:
1993:
1985:
1981:
1977:
1973:
1967:
1963:
1959:
1955:
1951:
1948:(in German).
1947:
1946:
1941:
1935:
1928:
1924:
1920:
1916:
1910:
1906:
1902:
1898:
1894:
1893:
1888:
1884:
1878:
1870:
1867:(in Polish).
1866:
1862:
1858:
1852:
1844:
1838:
1834:
1831:
1827:
1826:
1818:
1810:
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1775:
1773:
1771:
1754:
1749:
1744:
1740:
1736:
1735:
1730:
1723:
1716:
1715:0-8240-6924-2
1712:
1708:
1702:
1698:
1694:
1688:
1686:
1684:
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1680:
1671:
1667:
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1467:
1463:
1461:
1457:
1453:
1449:
1444:
1442:
1438:
1434:
1430:
1426:
1425:S-expressions
1423:
1413:
1399:
1379:
1359:
1339:
1331:
1328:of classical
1327:
1323:
1319:
1317:
1307:
1303:
1289:
1286:
1266:
1263:
1256:
1242:
1239:
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1218:
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1145:
1142:
1141:
1137:
1133:
1119:
1112:
1091:
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1077:
1063:
1056:
1035:
1033:
1030:
1029:
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1021:
1007:
1003:
1000:
993:
979:
975:
965:
963:
960:
959:
955:
951:
937:
933:
930:
923:
909:
905:
895:
893:
890:
889:
885:
881:
867:
864:
861:
841:
838:
835:
828:
814:
808:
801:
799:
798:Biconditional
796:
795:
791:
787:
773:
770:
767:
760:
746:
743:
740:
733:
731:
728:
727:
723:
719:
705:
702:
699:
692:
678:
675:
672:
665:
663:
660:
659:
655:
651:
637:
634:
631:
624:
610:
607:
604:
597:
595:
592:
591:
587:
583:
569:
566:
563:
556:
542:
536:
529:
527:
524:
523:
519:
515:
501:
498:
491:
477:
467:
465:
462:
461:
455:
450:
445:
442:
441:
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436:
432:
422:
420:
415:
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409:
405:
400:
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385:
370:
364:
358:
356:
351:
349:
341:
338:
330:
324:
308:
306:
301:
299:
295:
294:Alfred Tarski
291:
287:
283:
278:
276:
272:
268:
264:
260:
259:Alonzo Church
256:
254:
250:
249:Gottlob Frege
246:
242:
238:
234:
230:
224:
220:
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206:
202:
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190:
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146:
142:
138:
134:
131:
127:
123:
119:
115:
111:
107:
103:
99:
88:
83:
81:
76:
74:
69:
68:
66:
65:
61:
53:
52:
47:
39:
38:
34:
30:
29:
26:
22:
2478:. Retrieved
2458:
2453:
2433:. Retrieved
2417:
2412:
2394:. Retrieved
2356:
2351:
2329:. Retrieved
2305:
2294:. Retrieved
2280:
2269:. Retrieved
2255:
2242:
2229:. Retrieved
2213:
2203:
2194:
2190:
2169:
2165:
2152:
2143:
2139:
2119:
2092:
2081:
2072:
2068:
2051:
2042:
2034:
2025:
2002:
1998:
1992:
1975:
1949:
1943:
1934:
1896:
1890:
1877:
1868:
1865:Nauka Polska
1864:
1851:
1824:
1817:
1808:
1787:
1757:. Retrieved
1738:
1732:
1722:
1696:
1661:
1647:
1610:. Retrieved
1597:3-34100515-3
1583:
1578:
1571:
1556:WFF 'N PROOF
1522:S-expression
1481:such as the
1468:
1464:
1460:CoffeeScript
1445:
1419:
1320:
1313:
884:ekwiwalencja
446:Conventional
437:, as shown:
428:
416:
391:
374:
368:
362:
352:
347:
345:
334:
328:
314:
302:
281:
279:
257:
236:
226:
222:
214:
189:interpreters
183:
178:
174:
172:
152:
144:
132:
125:
121:
117:
113:
109:
105:
101:
97:
96:
55:
49:
25:
1903:: 376–377.
1830:Pearson PLC
1471:calculators
1326:connectives
1316:quantifiers
1306:konieczność
1144:Possibility
662:Conjunction
654:alternatywa
594:Disjunction
360:5 − (6 × 7)
332:× (− 5 6) 7
326:(5 − 6) × 7
311:Explanation
288:before the
161:nationality
2512:Categories
2485:(11 pages)
2480:2022-11-14
2435:2022-11-14
2403:(77 pages)
2396:2020-04-12
2331:2022-11-14
2296:2022-11-14
2271:2022-11-14
2231:2023-11-12
2146:: 145–169.
1895:(Review).
1871:: 604–620.
1764:(12 pages)
1759:2022-07-02
1612:2015-12-01
1563:References
1452:PostScript
790:dysjunkcja
722:koniunkcja
586:implikacja
372:− 5 × 6 7.
124:or simply
59:("Polish")
2383:1614-3213
2011:cite book
1966:118507515
1743:CiteSeerX
1709:in 1987,
1695:(1957) .
1322:Bocheński
1290:ϕ
1287:Γ
1267:ϕ
1243:ϕ
1240:◻
1229:Necessity
1221:możliwość
1205:ϕ
1202:Δ
1182:ϕ
1158:ϕ
1155:◊
1099:⊥
1043:⊤
1008:ϕ
1001:Σ
980:ϕ
973:∃
938:ϕ
931:Π
910:ϕ
903:∀
868:ψ
865:ϕ
842:ψ
839:ϕ
815:ψ
812:↔
809:ϕ
774:ψ
771:ϕ
747:ψ
744:∣
741:ϕ
706:ψ
703:ϕ
679:ψ
676:∧
673:ϕ
638:ψ
635:ϕ
611:ψ
608:∨
605:ϕ
570:ψ
567:ϕ
543:ψ
540:→
537:ϕ
502:ϕ
478:ϕ
475:¬
366:5 − 6 × 7
343:× − 5 6 7
290:arguments
239:in 1965.
205:see below
173:The term
130:operators
2471:Archived
2426:Archived
2387:Archived
2322:Archived
2290:Archived
2265:Archived
2241:(1959).
2222:Archived
2160:(1930).
2063:(1930).
2033:(1944).
1859:(1929).
1782:(1931).
1753:Archived
1660:(1963).
1646:(1929).
1512:Currying
1489:See also
1433:variadic
464:Negation
453:notation
448:notation
286:functors
165:logician
137:operands
2469:(KIT).
1909:2269644
518:negacja
443:Concept
412:parsing
296:on the
235:in the
211:History
145:between
133:precede
2461:]
2420:]
2381:
2371:
2359:]
2112:
2004:geben.
1964:
1923:Warsaw
1907:
1839:
1745:
1713:
1604:
1594:
1435:. The
1088:Falsum
456:Polish
451:Polish
435:Polish
378:÷ 10 5
229:Polish
153:follow
135:their
2474:(PDF)
2463:(PDF)
2457:[
2429:(PDF)
2422:(PDF)
2416:[
2390:(PDF)
2361:(PDF)
2355:[
2325:(PDF)
2314:(PDF)
2225:(PDF)
2218:(PDF)
1962:S2CID
1905:JSTOR
1899:(3).
1582:[
1547:(VOS)
1541:(VSO)
1498:(RPN)
1456:Forth
1450:like
1032:Verum
458:term
404:stack
399:arity
382:− 7 6
337:arity
321:1 + 2
317:+ 1 2
2379:ISSN
2369:ISBN
2110:ISBN
2017:link
1837:ISBN
1711:ISBN
1592:ISBN
1454:and
1441:LDAP
1422:Lisp
1392:and
1352:and
269:and
201:Lisp
2102:doi
1954:doi
1602:EAN
1437:Tcl
1279:or
1194:or
854:or
348:any
163:of
112:),
110:NPN
2514::
2444:.)
2385:.
2377:.
2320:.
2316:.
2288:.
2263:.
2193:.
2178:^
2170:23
2127:^
2118:,
2108:,
2073:23
2059:;
2013:}}
2009:{{
1982:.
1969:;
1960:.
1950:92
1929:.)
1897:30
1889:.
1869:10
1863:.
1796:^
1769:^
1751:.
1737:.
1731:.
1717:.)
1678:^
1655:;
1621:^
1600:.
1590:.
1485:.
1458:.
421:.
414:.
307:.
300:.
277:.
120:,
116:,
102:PN
2483:.
2438:.
2399:.
2334:.
2299:.
2274:.
2234:.
2195:3
2144:1
2104::
2019:)
1956::
1911:.
1845:.
1762:.
1739:9
1703:.
1672:.
1615:.
1400:M
1380:L
1360:M
1340:L
1264:L
1179:M
1120:O
1064:V
1004:p
976:p
934:p
906:p
862:Q
836:E
768:D
700:K
632:A
564:C
499:N
203:(
108:(
100:(
86:e
79:t
72:v
23:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
