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Some logicians draw a firm distinction between the conditional connective (the syntactic sign "<math>\rightarrow\!</math>"), and the implication relation (the formal object denoted by the sign "<math>\Rightarrow\!</math>"). These logicians use the phrase ''if&ndash;then'' for the conditional connective and the term ''implies'' for the implication relation.  Some explain the difference by saying that the conditional is the ''contemplated'' relation while the implication is the ''asserted'' relation.  In most fields of mathematics, it is treated as a variation in the usage of the single sign "<math>\Rightarrow\!</math>", not requiring two separate signs.  Not all of those who use the sign "<math>\rightarrow\!</math>" for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called ''[[syncategorematic sign]]'', that is, a sign with a purely syntactic function.  For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign "<math>\rightarrow\!</math>" to denote the [[boolean function]] that is associated with the [[truth table]] of the material conditional.  These considerations result in the following scheme of notation.

Some logicians draw a firm distinction between the conditional connective, the symbol <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime},</math> and the implication relation, the object denoted by the symbol <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}.</math>)  These logicians use the phrase ''if&ndash;then'' for the conditional connective and the term ''implies'' for the implication relation.  Some explain the difference by saying that the conditional is the ''contemplated'' relation while the implication is the ''asserted'' relation.  In most fields of mathematics, it is treated as a variation in the usage of the single sign <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime},</math> not requiring two separate signs.  Not all of those who use the sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> for the conditional connective regard it as a sign that denotes any kind of object, but treat it as a so-called ''syncategorematic sign'', that is, a sign with a purely syntactic function.  For the sake of clarity and simplicity in the present introduction, it is convenient to use the two-sign notation, but allow the sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> to denote the [[boolean function]] that is associated with the [[truth table]] of the material conditional.  These considerations result in the following scheme of notation.

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Let <math>\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}</math> be the ''[[boolean domain]]'' of two logical values.  The truth table shows the ordered triples of a [[triadic relation]] <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined as follows:

Let <math>\mathbb{B} = \{ \operatorname{F}, \operatorname{T} \}</math> be the ''[[boolean domain]]'' consisting of two logical values.  The truth table shows the ordered triples of a [[triadic relation]] <math>L \subseteq \mathbb{B} \times \mathbb{B} \times \mathbb{B}\!</math> that is defined as follows:

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The relationship between <math>\operatorname{Cond}</math> and <math>L\!</math> exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.

The relationship between <math>\operatorname{Cond}</math> and <math>L\!</math> exemplifies the standard association that exists between any binary operation and its corresponding triadic relation.

The conditional sign "<math>\rightarrow\!</math>" denotes the same formal object as the function name "<math>\operatorname{Cond}\mbox{ }\!</math>", the only difference being that the first is written infix while the second is written prefix.  Thus we have the following equation:

The conditional sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> denotes the same formal object as the function name <math>{}^{\backprime\backprime} \operatorname{Cond} {}^{\prime\prime},</math> the only difference being that the first is written infix while the second is written prefix.  Thus we have the following equation:

{| align="center" cellspacing="10" width="90%"

{| align="center" cellspacing="10" width="90%"

The same object is achieved in the following way.  Begin with the binary operation:

The same object is achieved in the following way.  Begin with the binary operation:

: <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}\,.\!</math>

| <math>\operatorname{Cond} : \mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>

Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{T}</math>, notated as follows:

Form the binary relation that is called the ''fiber'' of <math>\operatorname{Cond}</math> at <math>\operatorname{T},</math> notated as follows:

: <math>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math>

| <math>\operatorname{Cond}^{-1}(\operatorname{T}) \subseteq \mathbb{B} \times \mathbb{B}.</math>

This object is defined as follows:

This object is defined as follows:

: <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond} (p, q) = \operatorname{T} \}.</math>

| <math>\operatorname{Cond}^{-1}(\operatorname{T}) = \{ (p, q) \in \mathbb{B} \times \mathbb{B} : \operatorname{Cond}(p, q) = \operatorname{T} \}.</math>

The implication sign "<math>\Rightarrow\!</math>" denotes the same formal object as the relation names "<math>L_{..T}\mbox{ }\!</math>" and "<math>\operatorname{Cond}^{-1}(T)\mbox{ }\!</math>", the only differences being purely syntactic.  Thus we have the following logical equivalence:

The implication sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> denotes the same formal object as the relation names <math>{}^{\backprime\backprime} L_{\underline{~} \, \underline{~} \, \operatorname{T}} {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \operatorname{Cond}^{-1}(T) {}^{\prime\prime},</math> the only differences being purely syntactic.  Thus we have the following logical equivalence:

: <math>(p \Rightarrow q) \iff (p,\ q) \in L_{..T} \iff (p,\ q) \in \operatorname{Cond}^{-1}(T)\,.\!</math>

| <math>(p \Rightarrow q) \iff (p, q) \in L_{\underline{~} \, \underline{~} \, \operatorname{T}} \iff (p, q) \in \operatorname{Cond}^{-1}(T).</math>

This completes the derivation of the mathematical objects that are denoted by the signs "<math>\rightarrow\!</math>" and "<math>\Rightarrow\!</math>" in this discussion.  It needs to be remembered, though, that not all writers observe this distinction in every context.  Especially in mathematics, where the single arrow sign "<math>\rightarrow\!</math>" is reserved for function notation, it is common to see the double arrow sign "<math>\Rightarrow\!</math>" being used for both concepts.

This completes the derivation of the mathematical objects that are denoted by the signs <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> and <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}</math> in this discussion.  It needs to be remembered, though, that not all writers observe this distinction in every context.  Especially in mathematics, where the single arrow sign <math>{}^{\backprime\backprime} \rightarrow {}^{\prime\prime}</math> is reserved for function notation, it is common to see the double arrow sign <math>{}^{\backprime\backprime} \Rightarrow {}^{\prime\prime}</math> being used for both concepts.

==References==

==References==