1050:
1015:
2232:
108:("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example,
395:
357:
321:
283:
241:
2221:
990:
in definitions of new terms. However, this usage of "if and only if" is relatively uncommon and overlooks the linguistic fact that the "if" of a definition is interpreted as meaning "if and only if". The majority of textbooks, research papers and articles (including
English Knowledge (XXG) articles)
926:
a statement of the form "P iff Q" by proving either "if P, then Q" and "if Q, then P", or "if P, then Q" and "if not-P, then not-Q". Proving these pairs of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An
1049:
1597:
Theorems which have the form "P if and only Q" are much prized in mathematics. They give what are called "necessary and sufficient" conditions, and give completely equivalent and hopefully interesting new ways to say exactly the same
991:
follow the linguistic convention of interpreting "if" as "if and only if" whenever a mathematical definition is involved (as in "a topological space is compact if every open cover has a finite subcover"). Moreover, in the case of a
1090:, either proper or improper, of Q. "P if Q", "if Q then P", and Q→P all mean that Q is a proper or improper subset of P. "P if and only if Q" and "Q if and only if P" both mean that the sets P and Q are identical to each other.
1014:
958:
It is somewhat unclear how "iff" was meant to be pronounced. In current practice, the single 'word' "iff" is almost always read as the four words "if and only if". However, in the preface of
646:
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275:
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221:
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1326:(the express mention of one thing excludes all others). Moreover, it underpins the application of logic programming to the representation of legal texts and legal reasoning.
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1950:
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838:) make a distinction between these, in which the first, ↔, is used as a symbol in logic formulas, while ⇔ is used in reasoning about those logic formulas (e.g., in
868:
505:
484:
1275:
Compared with the standard semantics for FOL, the database semantics has a more efficient implementation. Instead of reasoning with sentences of the form:
1192:
together with a "database (or logic programming) semantics". They give the example of the
English sentence "Richard has two brothers, Geoffrey and John".
1173:
1679:
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demands something less I use Halmos' 'iff'". The authors of one discrete mathematics textbook suggest: "Should you need to pronounce iff, really
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1836:
1761:
1738:
1418:
1323:
999:
half of the definition is interpreted as a sentence in the metalanguage stating that the sentences in the definition of a predicate are the
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146:
96:
between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is
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1465:
1709:
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1503:
1473:
1526:
1384:
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show logical relationships among events, properties, and so forth. "P only if Q", "if P then Q", and "P→Q" all mean that P is a
1558:
1936:
1907:
1584:
931:"(P and Q) or (not-P and not-Q)", which itself can be inferred directly from either of its disjuncts—that is, because "iff" is
962:, Kelley suggests that it should be read differently: "In some cases where mathematical content requires 'if and only if' and
1891:
1786:
179:. Some authors regard "iff" as unsuitable in formal writing; others consider it a "borderline case" and tolerate its use. In
1649:
2255:
180:
2201:
2196:
1437:
955:, who wrote "I invented 'iff,' for 'if and only if'—but I could never believe I was really its first inventor."
2185:
1672:"Jan Łukasiewicz > Łukasiewicz's Parenthesis-Free or Polish Notation (Stanford Encyclopedia of Philosophy)"
1960:
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31:
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1989:
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1114:
835:
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364:
1923:
1918:
1890:
Kowalski, R., Dávila, J., Sartor, G. and Calejo, M., 2023. Logical
English for law and education.
2236:
2129:
2035:
2031:
1237:(FOL) with the standard semantics, the same English sentence would need to be represented, using
874:
827:
93:
843:
1828:
1821:
1610:
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2006:
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so that people hear the difference from 'if'", implying that "iff" could be pronounced as .
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http://www.doc.ic.ac.uk/~rak/papers/Logical%20English%20for%20Law%20and%20Education%20.pdf
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826:, and sometimes "iff". These are usually treated as equivalent. However, some texts of
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is used outside the field of logic as well. Wherever logic is applied, especially in
1083:
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89:
1911:
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2001:
1997:
1226:
as expressing in the metalanguage that the sentences in the database represent the
1181:
923:
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In writing, phrases commonly used as alternatives to P "if and only if" Q include:
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knowledge that should be considered when drawing conclusions from the database.
1641:
1102:
979:
952:
928:
432:
73:
1924:
Southern
California Philosophy for philosophy graduate students: "Just in Case"
935:, "P iff Q" follows if P and Q have been shown to be both true, or both false.
17:
2156:
1894:
In Prolog: The Next 50 Years (pp. 287-299). Cham: Springer Nature
Switzerland.
1335:
992:
77:
1877:
1985:
1222:
the knowledge relevant for problem solving in a given domain. It interprets
839:
746:
1480:
While it can be a real time-saver, we don't recommend it in formal writing.
1413:(Second ed.). Upper Saddle River, NJ: Pearson Education. p. 197.
1214:
The database semantics interprets the database (or program) as containing
2180:
2109:
2095:
1928:
1495:
Engineering
Writing by Design: Creating Formal Documents of Lasting Value
1196:
1105:
discussions, it has the same meaning as above: it is an abbreviation for
750:
1434:
2014:
963:
2138:
1184:
note (page 282), in effect, that it is often more natural to express
1087:
394:
356:
320:
282:
240:
2105:
1433:
Weisstein, Eric W. "Iff." From MathWorld--A Wolfram Web
Resource.
69:
1932:
982:
are "if and only if" statements; some texts — such as Kelley's
885:
1461:
Reading, Writing, and
Proving: A Closer Look at Mathematics
1322:
The database semantics is analogous to the legal principle
943:
Usage of the abbreviation "iff" first appeared in print in
1245:
interpreted in the object language, in some such form as:
1781:(3rd ed.). Boca Raton, Fla.: CRC Press. p. 60.
986:— follow this convention, and use "if and only if" or
1254:
1203:, this could be represented simply by two sentences:
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X(Brother(Richard, X) iff X = Geoffrey or X = John).
888:, "if and only if" is shown as a long double arrow:
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1753:Handbook of writing for the mathematical sciences
1492:Rothwell, Edward J.; Cloud, Michael J. (2014),
157:P is equivalent (or materially equivalent) to Q
53:
44:"⟺" and "⇔" redirect here. For other uses, see
1944:
1777:Maurer, Stephen B.; Ralston, Anthony (2005).
1409:Copi, I. M.; Cohen, C.; Flage, D. E. (2006).
8:
1003:determining the extension of the predicate.
37:"↔" redirects here. Not to be confused with
153:for P it is necessary and sufficient that Q
1969:
1951:
1937:
1929:
1856:Artificial Intelligence: A Modern Approach
1174:Artificial Intelligence: A Modern Approach
899:
895:
463:
30:"Iff" redirects here. For other uses, see
1827:, American Mathematical Society, p.
1809:iff it is finite or countably infinite."
1253:
910:via command \iff or \Longleftrightarrow.
893:
877:, i.e., the symbol in logic formulas, is
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811:
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745:It is equivalent to that produced by the
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223:, are used instead of these phrases; see
208:
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1109:, indicating that one statement is both
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1058:is a subset but not a proper subset of
1010:
766:The corresponding logical symbols are "
749:, and opposite to that produced by the
622:
595:
568:
541:
508:
132:, there could be other scenarios where
1611:"XOR/XNOR/Odd Parity/Even Parity Gate"
104:), and can be likened to the standard
1435:http://mathworld.wolfram.com/Iff.html
1324:expressio unius est exclusio alterius
1113:for the other. This is an example of
951:. Its invention is often credited to
128:is also true, whereas in the case of
7:
1908:"Tables of truth for if and only if"
1712:from the original on 22 October 2019
1520:
1518:
1510:It is common in mathematical writing
120:is true, and the only case in which
1652:from the original on 3 October 2020
1756:(2nd ed.). SIAM. p. 24.
1682:from the original on 9 August 2019
1466:Undergraduate Texts in Mathematics
1255:
641:{\displaystyle P\leftrightarrow Q}
533:{\displaystyle \neg P\land \neg Q}
524:
515:
418:{\displaystyle P\leftrightarrow Q}
270:{\displaystyle \neg P\land \neg Q}
261:
252:
25:
1621:from the original on 7 April 2022
1583:, UHM Department of Mathematics,
1527:"Conditionals and Biconditionals"
2230:
2219:
1823:A Primer of Mathematical Writing
1779:Discrete Algorithmic Mathematics
1361:If and only if in logic programs
1168:When "if" means "if and only if"
1048:
1013:
799:{\displaystyle \Leftrightarrow }
779:{\displaystyle \leftrightarrow }
451:{\displaystyle \leftrightarrow }
393:
355:
319:
281:
239:
216:{\displaystyle \Leftrightarrow }
196:{\displaystyle \leftrightarrow }
1587:from the original on 5 May 2000
1285:it uses sentences of the form:
939:Origin of iff and pronunciation
169:P precisely (or exactly) when Q
1125:in statements of definition).
896:
793:
773:
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605:
587:{\displaystyle P\rightarrow Q}
578:
445:
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344:{\displaystyle P\rightarrow Q}
335:
210:
190:
1:
614:{\displaystyle P\leftarrow Q}
380:{\displaystyle P\leftarrow Q}
224:
59:Logical symbols representing
1919:Language Log: "Just in Case"
927:alternative is to prove the
1750:Nicholas J. Higham (1998).
1207:Brother(Richard, Geoffrey).
1117:(although, as noted above,
183:, logical symbols, such as
72:and related fields such as
2282:
1819:Krantz, Steven G. (1996),
1007:In terms of Euler diagrams
850:, it is the prefix symbol
43:
36:
29:
2266:Necessity and sufficiency
2216:
1967:
1854:; Norvig, Peter (2020) .
1498:, CRC Press, p. 98,
1280:conclusion iff conditions
88:") is paraphrased by the
2261:Mathematical terminology
1805:, p. 25: "A set is
1468:, Springer, p. 52,
1440:13 November 2018 at the
1290:conclusion if conditions
1261:{\displaystyle \forall }
1121:is more often used than
1111:necessary and sufficient
1074:if and only if it is in
1066:if and only if it is in
560:{\displaystyle P\land Q}
306:{\displaystyle P\land Q}
147:necessary and sufficient
1561:18 October 2016 at the
1210:Brother(Richard, John).
830:(particularly those on
819:{\displaystyle \equiv }
84:" (often shortened as "
2237:Mathematics portal
1706:Art of Problem Solving
1389:sites.millersville.edu
1262:
1023:is a proper subset of
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2226:Philosophy portal
1646:mathworld.wolfram.com
1385:"Logical Connectives"
1346:Logical biconditional
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1070:, and a number is in
905:
903:{\displaystyle \iff }
873:Another term for the
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39:Bidirectional traffic
1341:Equivalence relation
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161:material implication
106:material conditional
102:material equivalence
32:IFF (disambiguation)
2256:Logical connectives
1801:For instance, from
1640:Weisstein, Eric W.
1411:Essentials of Logic
1356:Logical equivalence
1146:domain of discourse
1115:mathematical jargon
968:hang on to the 'ff'
836:propositional logic
173:P exactly in case Q
1852:Russell, Stuart J.
1676:plato.stanford.edu
1567:Wolfram|Alpha
1537:on 24 October 2020
1258:
1094:More general usage
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875:logical connective
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110:P if and only if Q
94:logical connective
27:Logical connective
2243:
2242:
2211:
2210:
1869:978-0-13-461099-3
1838:978-0-8218-0635-7
1763:978-0-89871-420-3
1739:978-0-387-90125-1
1731:General Topology,
1420:978-0-13-238034-8
1235:first-order logic
1062:. A number is in
1035:; a number is in
1031:only if it is in
1027:. A number is in
863:{\displaystyle E}
832:first-order logic
743:
742:
500:{\displaystyle Q}
479:{\displaystyle P}
116:is true whenever
16:(Redirected from
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1910:. Archived from
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1862:. p. 1136.
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1803:General Topology
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1351:Logical equality
1271:Geoffrey ≠ John.
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1265:
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1140:means: "For any
1136:the elements of
1128:The elements of
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984:General Topology
978:Conventionally,
960:General Topology
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100:(a statement of
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1563:Wayback Machine
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1531:web.mnstate.edu
1525:Peil, Timothy.
1524:
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1454:Daepp, Ulrich;
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1156:if and only if
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1927:
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1921:
1916:
1914:on 5 May 2000.
1902:
1901:External links
1899:
1897:
1896:
1883:
1868:
1858:(4 ed.).
1843:
1837:
1811:
1794:
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1723:
1702:"LaTeX:Symbol"
1693:
1663:
1632:
1615:www.cburch.com
1602:
1580:If and only if
1570:
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1484:
1474:
1456:Gorkin, Pamela
1445:
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1239:if and only if
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1186:if and only if
1169:
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1107:if and only if
1095:
1092:
1084:Euler diagrams
1081:
1080:
1054:
1047:
1045:
1019:
1012:
1008:
1005:
1001:only sentences
975:
972:
945:John L. Kelley
940:
937:
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567:
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550:
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529:
526:
523:
520:
517:
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496:
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475:
447:
414:
411:
408:
399:
392:
391:
390:
376:
373:
370:
361:
354:
353:
340:
337:
334:
325:
318:
317:
316:
302:
299:
296:
287:
280:
279:
266:
263:
260:
257:
254:
245:
238:
237:
236:
235:
234:
232:
229:
212:
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159:(compare with
124:is true is if
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2053:
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2039:
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2025:
2024:contradiction
2020:
2016:
2011:
2008:
2003:
1999:
1994:
1991:
1987:
1982:
1977:
1972:
1971:
1966:
1962:
1954:
1949:
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1942:
1940:
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1920:
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1366:Polysyllogism
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1151:
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98:biconditional
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90:biconditional
87:
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1886:
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1705:
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1675:
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1645:
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1635:
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1614:
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1579:
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1535:the original
1530:
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136:is true and
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2202:existential
1656:4 September
1541:4 September
1313:conclusions
1305:conclusions
980:definitions
953:Paul Halmos
929:disjunction
844:Łukasiewicz
433:truth table
112:means that
74:mathematics
2250:Categories
1788:1568811667
1716:22 October
1686:22 October
1625:22 October
1591:16 October
1372:References
1336:Definition
1317:conditions
1301:conditions
231:Definition
140:is false.
78:philosophy
2186:universal
2064:therefore
2052:therefore
2007:tautology
1878:359890490
1807:countable
1309:backwards
1256:∀
1172:In their
897:⟺
840:metalogic
814:≡
794:⇔
774:↔
747:XNOR gate
633:↔
606:←
579:→
552:∧
525:¬
522:∧
516:¬
446:↔
410:↔
372:←
336:→
298:∧
262:¬
259:∧
253:¬
211:⇔
191:↔
2139:superset
2050:entails,
2036:entails,
1733:reissue
1710:Archived
1680:Archived
1650:Archived
1619:Archived
1585:archived
1559:Archived
1458:(2011),
1438:Archived
1330:See also
1197:database
918:In most
762:Notation
751:XOR gate
2155:
2134:implies
2122:implies
2104:
2076:because
1984:
1959:Common
1243:only if
1241:, with
1224:only if
1178:Russell
1144:in the
997:only if
964:euphony
806:", and
227:below.
2149:
2098:
2038:proves
1978:
1876:
1866:
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1598:thing.
1502:
1472:
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1182:Norvig
1160:is in
1152:is in
1088:subset
995:, the
924:proves
922:, one
914:Proofs
842:). In
175:, and
130:P if Q
2019:false
1986:&
1452:E.g.
1311:from
1299:from
1195:In a
757:Usage
149:for P
145:Q is
70:logic
2173:nand
2002:true
1874:OCLC
1864:ISBN
1833:ISBN
1783:ISBN
1758:ISBN
1735:ISBN
1718:2019
1688:2019
1658:2020
1627:2019
1593:2016
1543:2020
1500:ISBN
1470:ISBN
1415:ISBN
1396:2023
1228:only
1220:only
1218:and
1180:and
1132:are
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431:The
203:and
92:, a
76:and
56:↔⇔≡⟺
2161:iff
2110:not
1990:and
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1307:or
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1295:to
1233:In
1216:all
1199:or
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886:TeX
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1981:or
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2197:∃
2181:∀
2169:|
2157:≡
2147:↔
2136:,
2130:⊃
2118:→
2106:~
2096:¬
2084:∨
2072:∵
2060:∴
2046:⊨
2032:⊢
2021:,
2015:⊥
2004:,
1998:⊤
1976:∧
1952:e
1945:t
1938:v
1880:.
1791:.
1766:.
1720:.
1690:.
1660:.
1629:.
1545:.
1423:.
1398:.
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1078:.
1076:B
1072:C
1068:C
1064:B
1060:B
1056:C
1043:.
1041:A
1037:B
1033:B
1029:A
1025:B
1021:A
858:E
739:T
736:T
733:T
730:T
727:F
724:T
721:T
716:F
713:T
710:F
707:F
704:F
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698:T
693:F
690:F
687:T
684:F
681:F
678:T
675:F
670:T
667:T
664:T
661:F
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655:F
652:F
636:Q
630:P
609:Q
603:P
582:Q
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555:Q
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528:Q
519:P
495:Q
474:P
459:Q
437:P
413:Q
407:P
375:Q
369:P
339:Q
333:P
301:Q
295:P
265:Q
256:P
138:Q
134:P
126:Q
122:P
118:Q
114:P
48:.
41:.
34:.
20:)
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