logical biconditional

Notations to represent equivalence used in history include:

• $=$ in George Boole in 1847.{{cite book |last1=Boole |first1=G. |title=The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning |date=1847 |publisher=Macmillan, Barclay, & Macmillan/George Bell |location=Cambridge/London |page=17 |url=https://archive.org/details/mathematicalanal00booluoft}} Although Boole used $=$ mainly on classes, he also considered the case that $x,y$ are propositions in $x=y$, and at the time $=$ is equivalence.
• $\backslash equiv$ in Frege in 1879;{{cite book |last1=Frege |first1=G. |title=Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens |date=1879 |publisher=Verlag von Louis Nebert |location=Halle a/S. |page=15 |language=German}}
• $\backslash sim$ in Bernays in 1918;{{cite book |last1=Bernays |first1=P. |title=Beiträge zur axiomatischen Behandlung des Logik-Kalküls |date=1918 |publisher=Universität Göttingen |location=Göttingen |page=3}}
• $\backslash rightleftarrows$ in Hilbert in 1927 (while he used $\backslash sim$ as the main symbol in the article);{{cite journal |last1=Hilbert |first1=D. |title=Die Grundlagen der Mathematik |journal=Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität |orig-date=1927 |date=1928 |volume=6 |pages=65–85 |doi=10.1007/BF02940602 |language=German}}
• $\backslash leftrightarrow$ in Hilbert and Ackermann in 1928{{cite book |last1=Hilbert |first1=D. |last2=Ackermann |first2=W. |title=Grundzügen der theoretischen Logik |edition=1 |date=1928 |publisher=Verlag von Julius Springer |location=Berlin |page=4 |language=German}} (they also introduced $\backslash rightleftarrows,\backslash sim$ while they use $\backslash sim$ as the main symbol in the whole book; $\backslash leftrightarrow$ is adopted by many followers such as Becker in 1933{{cite book |last1=Becker |first1=A. |title=Die Aristotelische Theorie der Möglichkeitsschlösse: Eine logisch-philologische Untersuchung der Kapitel 13-22 von Aristoteles' Analytica priora I |date=1933 |publisher=Junker und Dünnhaupt Verlag |location=Berlin |page=4 |language=German}});
• $E$ (prefix) in Łukasiewicz in 1929{{cite book |last1=Łukasiewicz |first1=J. |editor1-last=Słupecki |editor1-first=J. |title=Elementy logiki matematycznej |orig-date=1929|date=1958 |publisher=Państwowe Wydawnictwo Naukowe |location=Warszawa |edition=2 |language=Polish}} and $Q$ (prefix) in Łukasiewicz in 1951;{{cite book |last1=Łukasiewicz |first1=J. |editor1-last=Słupecki |editor1-first=J. |title=Aristotle's Syllogistic from the Standpoint of Modern Formal Logic |orig-date=1951|date=1957 |publisher=Oxford University Press |location=Glasgow, New York, Toronto, Melbourne, Wellington, Bombay, Calcutta, Madras, Karachi, Lahore, Dacca, Cape Town, Salisbury, Nairobi, Ibadan, Accra, Kuala Lumpur and Hong Kong |edition=2 |language=Polish}}
• $\backslash supset\backslash subset$ in Heyting in 1930;{{cite journal |last1=Heyting |first1=A. |title=Die formalen Regeln der intuitionistischen Logik |journal=Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-mathematische Klasse |date=1930 |pages=42–56 |language=German}}
• $\backslash Leftrightarrow$ in Bourbaki in 1954;{{cite book |last1=Bourbaki |first1=N. |title=Théorie des ensembles |date=1954 |publisher=Hermann & Cie, Éditeurs |location=Paris |page=32 |language=French}}
• $\backslash subset\backslash supset$ in Chazal in 1996;{{cite book |last1=Chazal |first1=G. |title=Eléments de logique formelle |date=1996 |publisher=Hermes Science Publications |location=Paris}}

and so on. Somebody else also use $\backslash operatorname\{EQ\}$ or $\backslash operatorname\{EQV\}$ occasionally.{{citation needed|date=September 2023}}{{vague|date=September 2023}}{{clarify|date=September 2023}}

Logical equality (also known as biconditional) is an operation on two logical values, typically the values of two propositions, that produces a value of true if and only if both operands are false or both operands are true.

The following is a truth table for $A\; \backslash leftrightarrow\; B$:

{{2-ary truth table|1|0|0|1|$A\; \backslash leftrightarrow\; B$}}

{{-}}

When more than two statements are involved, combining them with $\backslash leftrightarrow$ might be ambiguous. For example, the statement

:$x\_1\; \backslash leftrightarrow\; x\_2\; \backslash leftrightarrow\; x\_3\; \backslash leftrightarrow\; \backslash cdots\; \backslash leftrightarrow\; x\_n$

may be interpreted as

:$(((x\_1\; \backslash leftrightarrow\; x\_2)\; \backslash leftrightarrow\; x\_3)\; \backslash leftrightarrow\; \backslash cdots)\; \backslash leftrightarrow\; x\_n$,

or may be interpreted as saying that all {{math|x_i}} are jointly true or jointly false:

:$(x\_1\; \backslash land\; \backslash cdots\; \backslash land\; x\_n)\; \backslash lor\; (\backslash neg\; x\_1\; \backslash land\; \backslash cdots\; \backslash land\; \backslash neg\; x\_n)$

As it turns out, these two statements are only the same when zero or two arguments are involved. In fact, the following truth tables only show the same bit pattern in the line with no argument and in the lines with two arguments:

File:Variadic logical XAND.svg

File:Variadic logical all or nothing.svg

{{-}}

The left Venn diagram below, and the lines (AB    ) in these matrices represent the same operation.

Red areas stand for true (as in 40px for and).

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Commutativity: Yes

Associativity: Yes

Distributivity: Biconditional doesn't distribute over any binary function (not even itself), but logical disjunction distributes over biconditional.

Idempotency: No

Monotonicity: No

Truth-preserving: Yes

When all inputs are true, the output is true.

Falsehood-preserving: No

When all inputs are false, the output is not false.

Walsh spectrum: (2,0,0,2)

Nonlinearity: 0 (the function is linear)

{{Main|Rules of inference}}

Like all connectives in first-order logic, the biconditional has rules of inference that govern its use in formal proofs.

{{Main|Biconditional introduction}}

Biconditional introduction allows one to infer that if B follows from A and A follows from B, then A if and only if B.

For example, from the statements "if I'm breathing, then I'm alive" and "if I'm alive, then I'm breathing", it can be inferred that "I'm breathing if and only if I'm alive" or equivalently, "I'm alive if and only if I'm breathing." Or more schematically:

B → A   

A → B   

∴ A ↔ B

B → A   

A → B   

∴ B ↔ A

Biconditional elimination allows one to infer a conditional from a biconditional: if A ↔ B is true, then one may infer either A → B, or B → A.

For example, if it is true that I'm breathing if and only if I'm alive, then it's true that if I'm breathing, then I'm alive; likewise, it's true that if I'm alive, then I'm breathing. Or more schematically:

A ↔ B  

∴ A → B

A ↔ B  

∴ B → A

One unambiguous way of stating a biconditional in plain English is to adopt the form "b if a and a if b"—if the standard form "a if and only if b" is not used. Slightly more formally, one could also say that "b implies a and a implies b", or "a is necessary and sufficient for b". The plain English "if'" may sometimes be used as a biconditional (especially in the context of a mathematical definitionIn fact, such is the style adopted by Wikipedia's manual of style in mathematics.). In which case, one must take into consideration the surrounding context when interpreting these words.

For example, the statement "I'll buy you a new wallet if you need one" may be interpreted as a biconditional, since the speaker doesn't intend a valid outcome to be buying the wallet whether or not the wallet is needed (as in a conditional). However, "it is cloudy if it is raining" is generally not meant as a biconditional, since it can still be cloudy even if it is not raining.

{{Portal|Philosophy|Psychology}}

• If and only if
• Logical equivalence
• Logical equality
• XNOR gate
• Biconditional elimination
• Biconditional introduction

{{reflist}}

• {{Commonscatinline}}

{{Logical connectives}}

{{Mathematical logic}}

{{PlanetMath attribution|id=484|title=Biconditional}}

{{DEFAULTSORT:Logical Biconditional}}

Biconditional

Category:Equivalence (mathematics)