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The usage of the terms '''''logical implication''''' and '''''material conditional''''' varies from field to field and even across different contexts of discussion.  One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.

The usage of the terms '''''logical implication''''' and '''''material conditional''''' varies from field to field and even across different contexts of discussion.  One way to minimize the potential confusion is to begin with a focus on the various types of formal objects that are being discussed, of which there are only a few, taking up the variations in language as a secondary matter.

The main formal object under discussion is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' just in case the first operand is true and the second operand is false.  By way of a temporary name, the logical operation in question may be written as Cond (''p'', ''q''), where ''p'' and ''q'' are logical values.  The [[truth table]] associated with this operation is as follows:

The main formal object under discussion is a [[logical operation]] on two [[logical value]]s, typically the values of two [[proposition]]s, that produces a value of ''false'' just in case the first operand is true and the second operand is false.  By way of a temporary name, the logical operation in question may be written as Cond (''p'', ''q''), where ''p'' and ''q'' are logical values.  The [[truth table]] associated with this operation is as follows:

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"

{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:60%"

|+ '''Conditional Operation : B² → B'''

|+ '''Conditional Operation : B × B → B'''

|- style="background:paleturquoise"

|- style="background:paleturquoise"

! style="width:15%" | p

! style="width:15%" | p

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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–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 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.

 

: <math>\begin{matrix}

: <math>\begin{matrix}

p \rightarrow q & \quad & \quad & p \Rightarrow q \\

p \rightarrow q             & \quad & \quad & p \Rightarrow q \\

\mbox{if}\ p \ \mbox{then}\ q & \quad & \quad & p \ \mbox{implies}\ q

\mbox{if}\ p\ \mbox{then}\ q & \quad & \quad & p\ \mbox{implies}\ q \\

\end{matrix}</math>

\end{matrix}</math>

Let <math>\mathbb{B} = \{\mathbf{F},\ \mathbf{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} = \{\mathbf{F},\ \mathbf{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: