HOME

TheInfoList



OR:

In
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
and
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the logical biconditional, sometimes known as the material biconditional, is the
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
(\leftrightarrow) used to conjoin two statements and to form the statement "
if and only if 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 bic ...
", where is known as the '' antecedent'', and the ''
consequent A consequent is the second half of a hypothetical proposition. In the standard form of such a proposition, it is the part that follows "then". In an implication, if ''P'' implies ''Q'', then ''P'' is called the antecedent and ''Q'' is called ...
''. This is often abbreviated as " iff ". Other ways of denoting this operator may be seen occasionally, as a double-headed arrow (↔ or ⇔ may be represented in Unicode in various ways), a prefixed E "E''pq''" (in Łukasiewicz notation or Bocheński notation), an equality sign (=), an equivalence sign (≡), or ''EQV''. It is logically equivalent to both (P \rightarrow Q) \land (Q \rightarrow P) and (P \land Q) \lor (\neg P \land \neg Q) , and the
XNOR The XNOR gate (sometimes XORN'T, ENOR, EXNOR or NXOR and pronounced as Exclusive NOR. Alternatively XAND, pronounced Exclusive AND) is a digital logic gate whose function is the logical complement of the Exclusive OR (XOR) gate. It is equivale ...
(exclusive nor) boolean operator, which means "both or neither". Semantically, the only case where a logical biconditional is different from a
material conditional The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol \rightarrow is interpreted as material implication, a formula P \rightarrow Q is true unless P is true and Q i ...
is the case where the hypothesis is false but the conclusion is true. In this case, the result is true for the conditional, but false for the biconditional. In the conceptual interpretation, means "All 's are 's and all 's are 's". In other words, the sets and coincide: they are identical. However, this does not mean that and need to have the same meaning (e.g., could be "equiangular trilateral" and could be "equilateral triangle"). When phrased as a sentence, the antecedent is the ''subject'' and the consequent is the ''predicate'' of a
universal affirmative In logic, a categorical proposition, or categorical statement, is a proposition that asserts or denies that all or some of the members of one category (the ''subject term'') are included in another (the ''predicate term''). The study of arguments ...
proposition (e.g., in the phrase "all men are mortal", "men" is the subject and "mortal" is the predicate). In the propositional interpretation, P \leftrightarrow Q means that implies and implies ; in other words, the propositions are logically equivalent, in the sense that both are either jointly true or jointly false. Again, this does not mean that they need to have the same meaning, as could be "the triangle ABC has two equal sides" and could be "the triangle ABC has two equal angles". In general, the antecedent is the ''premise'', or the ''cause'', and the consequent is the ''consequence''. When an implication is translated by a ''hypothetical'' (or ''conditional'') judgment, the antecedent is called the ''hypothesis'' (or the ''condition'') and the consequent is called the ''thesis''. A common way of demonstrating a biconditional of the form P \leftrightarrow Q is to demonstrate that P \rightarrow Q and Q \rightarrow P separately (due to its equivalence to the conjunction of the two converse
conditional Conditional (if then) may refer to: *Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B has occurred *Conditional proof, in logic: a proof that asserts a co ...
s). Yet another way of demonstrating the same biconditional is by demonstrating that P \rightarrow Q and \neg P \rightarrow \neg Q. When both members of the biconditional are propositions, it can be separated into two conditionals, of which one is called a ''theorem'' and the other its ''reciprocal''. Thus whenever a theorem and its reciprocal are true, we have a biconditional. A simple theorem gives rise to an implication, whose antecedent is the ''hypothesis'' and whose consequent is the ''thesis'' of the theorem. It is often said that the hypothesis is the ''
sufficient condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
'' of the thesis, and that the thesis is the ''
necessary condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth o ...
'' of the hypothesis. That is, it is sufficient that the hypothesis be true for the thesis to be true, while it is necessary that the thesis be true if the hypothesis were true. When a theorem and its reciprocal are true, its hypothesis is said to be the necessary and sufficient condition of the thesis. That is, the hypothesis is both the cause and the consequence of the thesis at the same time.


Definition

Logical equality (also known as biconditional) is an
operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...
on two logical values, typically the values of two
proposition In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the no ...
s, that produces a value of ''true'' if and only if both operands are false or both operands are true.


Truth table

The following is a truth table for P \leftrightarrow Q (also written as P \equiv Q, , or P EQ Q): When more than two statements are involved, combining them with \leftrightarrow might be ambiguous. For example, the statement :x_1 \leftrightarrow x_2 \leftrightarrow x_3 \leftrightarrow \cdots \leftrightarrow x_n may be interpreted as :(((x_1 \leftrightarrow x_2) \leftrightarrow x_3) \leftrightarrow \cdots) \leftrightarrow x_n, or may be interpreted as saying that all are ''jointly true or jointly false'': :(x_1 \land \cdots \land x_n) \lor (\neg x_1 \land \cdots \land \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: The left Venn diagram below, and the lines ''(AB    )'' in these matrices represent the same operation.


Venn diagrams

Red areas stand for true (as in for ''
and or AND may refer to: Logic, grammar, and computing * Conjunction (grammar), connecting two words, phrases, or clauses * Logical conjunction in mathematical logic, notated as "∧", "⋅", "&", or simple juxtaposition * Bitwise AND, a boolea ...
'').


Properties

Commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
: Yes
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement ...
: Yes
Distributivity In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
: 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) Non
linearity Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
: 0 (the function is linear)


Rules of inference

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


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 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 bic ...
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

Biconditional elimination allows one to infer a
conditional Conditional (if then) may refer to: *Causal conditional, if X then Y, where X is a cause of Y *Conditional probability, the probability of an event A given that another event B has occurred *Conditional proof, in logic: a proof that asserts a co ...
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 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 bic ...
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


Colloquial usage

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.


See also

*
If and only if 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 bic ...
*
Logical equivalence In logic and mathematics, statements p and q are said to be logically equivalent if they have the same truth value in every model. The logical equivalence of p and q is sometimes expressed as p \equiv q, p :: q, \textsfpq, or p \iff q, depending o ...
* Logical equality * XNOR gate *
Biconditional elimination Biconditional elimination is the name of two valid rules of inference of propositional logic. It allows for one to infer a conditional from a biconditional. If P \leftrightarrow Q is true, then one may infer that P \to Q is true, and also that ...
*
Biconditional introduction In propositional logic, biconditional introductionCopi and Cohen is a valid rule of inference. It allows for one to infer a biconditional from two conditional statements. The rule makes it possible to introduce a biconditional statement into ...


References


External links

* {{DEFAULTSORT:Logical Biconditional Biconditional Equivalence (mathematics)