If and only if symbol word

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In logic and related fields such as mathematics and philosophy, «if and only if» (shortened as «iff«) is a biconditional logical connective between statements, where either both statements are true or both are false.

The connective is biconditional (a statement of material equivalence),^[1] and can be likened to the standard material conditional («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, P if and only if Q means that P is true whenever Q is true, and the only case in which P is true is if Q is also true, whereas in the case of P if Q, there could be other scenarios where P is true and Q is false.

In writing, phrases commonly used as alternatives to P «if and only if» Q include: Q is necessary and sufficient for P, for P it is necessary and sufficient that Q, P is equivalent (or materially equivalent) to Q (compare with material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q.^[2] Some authors regard «iff» as unsuitable in formal writing;^[3] others consider it a «borderline case» and tolerate its use.^[4]

In logical formulae, logical symbols, such as and ,^[5] are used instead of these phrases; see § Notation below.

Definition[edit]

The truth table of P Q is as follows:^[6][7]

P	Q	P Q	P Q	P  Q
T	T	T	T	T
T	F	F	T	F
F	T	T	F	F
F	F	T	T	T

It is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.^[8]

Usage[edit]

Notation[edit]

The corresponding logical symbols are «↔», ««,^[5] and «≡»,^[9] and sometimes «iff». These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) 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 metalogic). In Łukasiewicz’s Polish notation, it is the prefix symbol ‘E’.^[10]

Another term for the logical connective, i.e., the symbol in logic formulas, is exclusive nor.

In TeX, «if and only if» is shown as a long double arrow: via command iff.^[11]

Proofs[edit]

In most logical systems, one proves 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 pair of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the disjunction «(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 truth-functional, «P iff Q» follows if P and Q have been shown to be both true, or both false.

Origin of iff and pronunciation[edit]

Usage of the abbreviation «iff» first appeared in print in John L. Kelley’s 1955 book General Topology.^[12]
Its invention is often credited to Paul Halmos, who wrote «I invented ‘iff,’ for ‘if and only if’—but I could never believe I was really its first inventor.»^[13]

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 General Topology, Kelley suggests that it should be read differently: «In some cases where mathematical content requires ‘if and only if’ and euphony demands something less I use Halmos’ ‘iff'». The authors of one discrete mathematics textbook suggest:^[14] «Should you need to pronounce iff, really hang on to the ‘ff’ so that people hear the difference from ‘if'», implying that «iff» could be pronounced as [ɪfː].

Usage in definitions[edit]

Technically, definitions are «if and only if» statements; some texts — such as Kelley’s General Topology — follow the strict demands of logic, and use «if and only if» or iff in definitions of new terms.^[15]
However, this logically correct usage of «if and only if» 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 Wikipedia articles) follow the linguistic convention to interpret «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»).^[16]

Distinction from «if» and «only if»[edit]

• «Madison will eat the fruit if it is an apple.» (equivalent to «Only if Madison will eat the fruit, can it be an apple» or «Madison will eat the fruit ← the fruit is an apple»)

  This states that Madison will eat fruits that are apples. It does not, however, exclude the possibility that Madison might also eat bananas or other types of fruit. All that is known for certain is that she will eat any and all apples that she happens upon. That the fruit is an apple is a sufficient condition for Madison to eat the fruit.

• «Madison will eat the fruit only if it is an apple.» (equivalent to «If Madison will eat the fruit, then it is an apple» or «Madison will eat the fruit → the fruit is an apple»)

  This states that the only fruit Madison will eat is an apple. It does not, however, exclude the possibility that Madison will refuse an apple if it is made available, in contrast with (1), which requires Madison to eat any available apple. In this case, that a given fruit is an apple is a necessary condition for Madison to be eating it. It is not a sufficient condition since Madison might not eat all the apples she is given.

• «Madison will eat the fruit if and only if it is an apple.» (equivalent to «Madison will eat the fruit ↔ the fruit is an apple»)

  This statement makes it clear that Madison will eat all and only those fruits that are apples. She will not leave any apple uneaten, and she will not eat any other type of fruit. That a given fruit is an apple is both a necessary and a sufficient condition for Madison to eat the fruit.

Sufficiency is the converse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negation operator, i.e. «not»). This means that the relationship between P and Q, established by P→Q, can be expressed in the following, all equivalent, ways:

P is sufficient for Q

Q is necessary for P

¬Q is sufficient for ¬P

¬P is necessary for ¬Q

As an example, take the first example above, which states P→Q, where P is «the fruit in question is an apple» and Q is «Madison will eat the fruit in question». The following are four equivalent ways of expressing this very relationship:

If the fruit in question is an apple, then Madison will eat it.

Only if Madison will eat the fruit in question, is it an apple.

If Madison will not eat the fruit in question, then it is not an apple.

Only if the fruit in question is not an apple, will Madison not eat it.

Here, the second example can be restated in the form of if…then as «If Madison will eat the fruit in question, then it is an apple»; taking this in conjunction with the first example, we find that the third example can be stated as «If the fruit in question is an apple, then Madison will eat it; and if Madison will eat the fruit, then it is an apple».

In terms of Euler diagrams[edit]

• 

  A is a proper subset of B. A number is in A only if it is in B; a number is in B if it is in A.

• 

  C is a subset but not a proper subset of B. A number is in B if and only if it is in C, and a number is in C if and only if it is in B.

Euler diagrams 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 subset, 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.

More general usage[edit]

Iff is used outside the field of logic as well. Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).

The elements of X are all and only the elements of Y means: «For any z in the domain of discourse, z is in X if and only if z is in Y.»

See also[edit]

• Equivalence relation
• Logical biconditional
• Logical equality
• Logical equivalence
• Polysyllogism

References[edit]

External links[edit]

• «Tables of truth for if and only if». Archived from the original on 5 May 2000.
• Language Log: «Just in Case»
• Southern California Philosophy for philosophy graduate students: «Just in Case»

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In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.

In that it is biconditional, the connective can be likened to the standard material conditional («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. It is controversial whether the connective thus defined is properly rendered by the English «if and only if», with its pre-existing meaning. Of course, there is nothing to stop us stipulating that we may read this connective as «only if and if», although this may lead to confusion.

In writing, phrases commonly used, with debatable propriety, as alternatives to P «if and only if» Q include Q is necessary and sufficient for P, P is equivalent (or materially equivalent) to Q (compare material implication), P precisely if Q, P precisely (or exactly) when Q, P exactly in case Q, and P just in case Q. Many authors regard «iff» as unsuitable in formal writing; others use it freely.^{[citation needed]}

In logic formulae, logical symbols are used instead of these phrases; see the discussion of notation.

Definition

The truth table of p ↔ q is as follows:^[1]

Note that it is equivalent to that produced by the XNOR gate, and opposite to that produced by the XOR gate.

Usage

Notation

The corresponding logical symbols are «↔», «⇔» and «≡», and sometimes «iff». These are usually treated as equivalent. However, some texts of mathematical logic (particularly those on first-order logic, rather than propositional logic) 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 metalogic). In Łukasiewicz‘s notation, it is the prefix symbol ‘E’.

Another term for this logical connective is exclusive nor.

Proofs

In most logical systems, one proves a statement of the form «P iff Q» by proving «if P, then Q» and «if Q, then P». Proving this pair of statements sometimes leads to a more natural proof, since there are not obvious conditions in which one would infer a biconditional directly. An alternative is to prove the disjunction «(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 truth-functional, «P iff Q» follows if P and Q have both been shown true, or both false.

Origin of iff

Usage of the abbreviation «iff» first appeared in print in John L. Kelley‘s 1955 book General Topology.^[2]
Its invention is often credited to Paul Halmos, who wrote «I invented ‘iff,’ for ‘if and only if’—but I could never believe I was really its first inventor.»^[3]

Distinction from «if» and «only if»

1. «If the fruit is an apple, then Madison will eat it.» or «Madison will eat the fruit if it is an apple.» (equivalent to «Only if Madison will eat the fruit, is it an apple;» or «Madison will eat the fruit ← fruit is an apple»)

   This states only that Madison will eat fruits that are apples. It does not, however, preclude the possibility that Madison might also have occasion to eat bananas. Maybe she will, maybe she will not—the sentence does not tell us. All we know for certain is that she will eat any and all apples that she happens upon. That the fruit is an apple is a sufficient condition for Madison to eat the fruit.

2. «Only if the fruit is an apple, will Madison eat it.» or «Madison will eat the fruit only if it is an apple.» (equivalent to «If Madison will eat the fruit, then it is an apple» or «Madison will eat the fruit → fruit is an apple»)

   This states that the only fruit Madison will eat is an apple. It does not, however, preclude the possibility that Madison will refuse an apple if it is made available, in contrast with (1), which requires Madison to eat any available apple. In this case, that a given fruit is an apple is a necessary condition for Madison to be eating it. It is not a sufficient condition since Madison might not eat any and all apples she is given.

3. «If and only if the fruit is an apple will Madison eat it» or «Madison will eat the fruit if and only if it is an apple» or «Madison will eat the fruit ↔ fruit is an apple.»

   This, however, makes it quite clear that Madison will eat all and only those fruits that are apples. She will not leave any such fruit uneaten, and she will not eat any other type of fruit. That a given fruit is an apple is both a necessary and a sufficient condition for Madison to eat the fruit.

Sufficiency is the inverse of necessity. That is to say, given P→Q (i.e. if P then Q), P would be a sufficient condition for Q, and Q would be a necessary condition for P. Also, given P→Q, it is true that ¬Q→¬P (where ¬ is the negation operator, i.e. «not»). This means that the relationship between P and Q, established by P→Q, can be expressed in the following, all equivalent, ways:

P is sufficient for Q

Q is necessary for P

¬Q is sufficient for ¬P

¬P is necessary for ¬Q

As an example, take (1), above, which states P→Q, where P is «the fruit in question is an apple» and Q is «Madison will eat the fruit in question». The following are four equivalent ways of expressing this very relationship:

If the fruit in question is an apple, then Madison will eat it.

Only if Madison will eat the fruit in question, is it an apple.

If Madison will not eat the fruit in question, then it is not an apple.

Only if the fruit in question is not an apple, will Madison not eat it.

So we see that (2), above, can be restated in the form of if…then as «If Madison will eat the fruit in question, then it is an apple»; taking this in conjunction with (1), we find that (3) can be stated as «If the fruit in question is an apple, then Madison will eat it; AND if Madison will eat the fruit, then it is an apple».

Advanced considerations

Philosophical interpretation

A sentence that is composed of two other sentences joined by «iff» is called a biconditional. «Iff» joins two sentences to form a new sentence. It should not be confused with logical equivalence which is a description of a relation between two sentences. The biconditional «A iff B» uses the sentences A and B, describing a relation between the states of affairs which A and B describe. By contrast «A is logically equivalent to B» mentions both sentences: it describes a logical relation between those two sentences, and not a factual relation between whatever matters they describe. See use–mention distinction for more on the difference between using a sentence and mentioning it.

The distinction is a very confusing one, and has led many a philosopher astray. Certainly it is the case that when A is logically equivalent to B, «A iff B» is true. But the converse does not hold. Reconsidering the sentence:

If and only if the fruit is an apple will Madison eat it.

There is clearly no logical equivalence between the two halves of this particular biconditional. For more on the distinction, see W. V. Quine’s Mathematical Logic, Section 5.

One way of looking at «A if and only if B» is that it means «A if B» (B implies A) and «A only when B» (not B implies not A). «Not B implies not A» means A implies B, so then we get two way implication.

Definitions

In philosophy and logic, «iff» is used to indicate definitions, since definitions are supposed to be universally quantified biconditionals. In mathematics and elsewhere, however, the word «if» is normally used in definitions, rather than «iff». This is due to the observation that «if» in the English language has a definitional meaning, separate from its meaning as a propositional connective. This separate meaning can be explained by noting that a definition (for instance: A group is «abelian» if it satisfies the commutative law; or: A grape is a «raisin» if it is well dried) is not an equivalence to be proved, but a rule for interpreting the term defined.

Examples

Here are some examples of true statements that use «iff» — true biconditionals (the first is an example of a definition, so it should normally have been written with «if»):

• A person is a bachelor iff that person is a marriageable man who has never married.
• «Snow is white» in English is true iff «Schnee ist weiß» in German is true.
• For any p, q, and r: (p & q) & r iff p & (q & r). (Since this is written using variables and «&«, the statement would usually be written using «↔», or one of the other symbols used to write biconditionals, in place of «iff»).
• For any real numbers x and y, x=y+1 iff y=x−1.
• A subset containing n elements of an n-dimensional vector space is linearly independent iff it spans the vector space.
• The triangular number ⁿ⁽ⁿ⁺¹⁾/₂ is an even perfect number iff n = 2^p-1 is a Mersenne prime, with p being a prime number. As of Feb 2013, only 48 such even perfect numbers and Mersenne primes have been discovered.
• y(x) is a solution to the differential equation y=f(x,y) if and only if, the curve associated with y(x) is an integral curve of the direction field associated with y=f(x,y).

Analogs

Other words are also sometimes emphasized in the same way by repeating the last letter; for example orr for «Or and only Or» (the exclusive disjunction).

The statement «(A iff B)» is equivalent to the statement «(not A or B) and (not B or A),» and is also equivalent to the statement «(not A and not B) or (A and B)».

It is also equivalent to: not[(A or B) and (not A or not B)],

or more simply:

¬ [ ( ¬A ∨ ¬B ) ∧ ( A ∨ B ) ]

which converts into

[ ( ¬A ∧ ¬B) ∨ (A ∧ B) ]

and

[ ( ¬A ∨ B) ∧ (A ∨ ¬B) ]

which were given in verbal interpretations above.

More general usage

Iff is used outside the field of logic, wherever logic is applied, especially in mathematical discussions. It has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon. (However, as noted above, if, rather than iff, is more often used in statements of definition.)

The elements of X are all and only the elements of Y is used to mean: «for any z in the domain of discourse, z is in X if and only if z is in Y.»

See also

• Covariance
• Logical biconditional
• Logical equality
• Necessary and sufficient condition
• Polysyllogism

Notes

Part I: Propositional Logic

Выражение может быть именем закладки, строкой текста, числом, вложенным полем, возвращающим значение, или математической формулой.

Коды полей: IF

{ IF Выражение1 Оператор Выражение2 Текст1 Текст2 }

В зависимости от результата сравнения двух выражений вставляет тот или иной текст. В основном документе слияния поле IF можно использовать для анализа записей данных, таких как почтовые индексы или номеров банковских счетов. Например, можно разослать письма только клиентам, проживающим в определенном городе.

Инструкции

Выражение1, Выражение2
Значения, которые требуется сравнить. Выражение может быть именем закладки, строкой текста, числом, вложенным полем, возвращающим значение, или математической формулой. Если выражение содержит пробелы, оно должно быть заключено в кавычки.

Оператор 
Оператор сравнения. Пробелы перед оператором и после него являются обязательными.
Оператор    Описание
=    Равно
<>    Не равно
>    Больше
<    Меньше
>=    Больше или равно
<=    Меньше или равно

Текст1, Текст2
Текст, который вставляется в случае истинности (Текст1) или ложности (Текст2) логического выражения. Если Текст2 не указан, а логическое выражение ложно, не будет вставлено никакого текста. Каждая строка, содержащая несколько слов, должна быть заключена в кавычки.

Примечание.  Если используется оператор = или <>, то Выражение2 может содержать вопросительный знак (?), заменяющий любой отдельный знак, или звездочку (*), заменяющую любую строку символов. Такое выражение должно быть заключено в кавычки. Если Выражение2 содержит звездочку, то сумма длин фрагмента Выражения1, соответствующего этой звездочке, и остальных знаков значения Выражения2 не должна превышать 128 знаков.

Примеры
В приведенном ниже примере указано, что если заказ клиента равен или больше 100 единиц продукции, то будет выведено слово «Спасибо», а если клиент заказывает менее 100 единиц продукции, то будет выведено уведомление «Минимальный заказ равен 100 единиц».
{IF заказ>=100 «Спасибо» «Минимальный заказ равен 100 единиц»}

Примеры полей IF

Поле IF выполняет одну из двух операций в зависимости от указанных условий. Например, утверждение «Если погода будет хорошей, мы прогуляемся по парку, а если нет — пойдем в кино» определяет условие, (хорошая погода), при котором произойдет определенное событие (прогулка по парку). Если это условие не соблюдено, происходит альтернативное событие (поход в кино).
Приведенное ниже поле IF представляет собой утверждение того же типа. Если текущая запись данных содержит значение «Саратов» в поле «Город», в документ, полученный в результате слияния с этой записью данных, будет включен текст, содержащийся в первых кавычках. В противном случае будет добавлен второй текст. Следующий пример использует поля IF и MERGEFIELD.
{ IF { MERGEFIELD Город } = «Саратов» «Пожалуйста, позвоните в местное отделение нашей фирмы.» «Пожалуйста, отправьте заполненный формуляр в приложенном конверте.» }
Чтобы вставить поле IF в документ, на панели инструментов Слияние нажмите кнопку Добавить поле Word, а затем выберите поле IF…THEN…ELSE.
Ниже приведены некоторые примеры способов изменения кодов поля IF.

Сравнение поля данных с другим полем или закладкой
В приведенном ниже примере выполняется сравнение значения поля данных «Ставка» со значением поля данных «Скидка». Если числа равны, печатается указанный текст.
{ IF { MERGEFIELD Ставка } = { MERGEFIELD Скидка } «Мы предложим вам еще большую скидку. » » » }
(Если число в поле данных «Ставка» не равно числу в поле данных «Скидка», текст не печатается.)

Вставка других полей Word в поле IF
Пример 1
В случае выполнения условия, указанного в приведенном ниже поле IF, в документ вставляется содержимое файла, указанного в поле INCLUDETEXT. При задании местоположения и имени включаемого документа следует использовать двойные обратные косые черты (), как показано в примере. Если условие не будет выполнено, будет вставлен элемент списка автотекста «Авторское право».
{ IF { MERGEFIELD Штат } = » Нью-Йорк » «{ INCLUDETEXT c:\worddocs\contract.doc }» «{ AUTOTEXT Авторское право }» }

Пример 2
Если поле «Компания» в записи данных содержит какую-либо информацию — т. е. не является пустым, — печатается текст из поля «АдресКомпании». В противном случае печатается текст из поля «ДомашнийАдрес».
{ IF { MERGEFIELD Компания } <> » » «{ MERGEFIELD АдресКомпании }» «{ MERGEFIELD ДомашнийАдрес }» }

Выполнение вычислений в поле IF
В приведенном ниже примере первое поле = (Формула) вычисляет текущее сальдо для каждого счета. Если сальдо положительно, печатается значение второго поля = (Формула). Если вычисленное сальдо равно 0 (нулю) или отрицательно, печатается альтернативный текст «Ваш счет пуст. Спасибо.».
{ IF { = { MERGEFIELD Сальдо } — {MERGEFIELD Платеж } } > 0 «Текущее сальдо { = { MERGEFIELD Сальдо } — { MERGEFIELD Платеж } # «#,##0.00р.; » }.» «Ваш счет пуст. Спасибо» }

Задание нескольких условий
Чтобы задать несколько условий в поле IF, в него следует включить поле COMPARE или другие поля IF.

Пример 1
В приведенном ниже примере при слиянии каждой записи данных поле COMPARE сравнивает поля данных «НомерКлиента» и «РангКлиента». Функция AND в поле = (Формула) возвращает значение «1» (истина), если значения обоих полей указывают на удовлетворительное состояние счета. В этом случае печатается текст из первых кавычек.
{ IF { = AND ( { COMPARE { MERGEFIELD НомерКлиента } >= 4 }, { COMPARE { MERGEFIELD РангКлиента } <= 3 } ) } = 1 » Удовлетворительно » » Неудовлетворительно «}

Пример 2
В приведенном ниже примере жителям Калифорнии и Вашингтона предоставляются особые скидки. Если в поле «Штат» указан другой штат, никакой текст не печатается, поскольку во втором поле IF не указан альтернативный текст.
{IF {MERGEFIELD Штат } = » Калифорния » » Для жителей Калифорнии предлагаются особые скидки на посещение Азии и Японии.» «{IF {MERGEFIELD Штат } = «Вашингтон» «Для жителей Вашингтона предлагаются особые скидки на посещение Азии и Японии.» » «} «}