Changes

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

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

| <math>p ~\text{implies}~ q.\!</math>

|

<math>\begin{array}{l}

| <math>\text{If}~ p ~\text{then}~ q.\!</math>

p ~\text{implies}~ q.

\\[6pt]

\text{if}~ p ~\text{then}~ q.

\end{array}</math>

|}

|}

Here <math>p\!</math> and <math>q\!</math> are propositional variables that stand for any propositions in a given language.  In a statement of the form <math>\text{if}~ p ~\text{then}~ q,\!</math>, the first term, <math>p\!</math>, is called the ''antecedent'' and the second term, <math>q\!</math>, is called the ''consequent'', while the statement as a whole is called either the ''conditional'' or the ''consequence''.  Assuming that the conditional statement is true, then the truth of the antecedent is a ''sufficient condition'' for the truth of the consequent, while the truth of the consequent is a ''necessary condition'' for the truth of the antecedent.

Here <math>p\!</math> and <math>q\!</math> are propositional variables that stand for any propositions in a given language.  In a statement of the form <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime},</math>, the first term, <math>p\!</math>, is called the ''antecedent'' and the second term, <math>q\!</math>, is called the ''consequent'', while the statement as a whole is called either the ''conditional'' or the ''consequence''.  Assuming that the conditional statement is true, then the truth of the antecedent is a ''sufficient condition'' for the truth of the consequent, while the truth of the consequent is a ''necessary condition'' for the truth of the antecedent.

'''Note.'''  Many writers draw a technical distinction between the form <math>\langle p ~\text{implies}~ q \rangle</math> and the form <math>\langle \text{if}~ p ~\text{then}~ q \rangle.</math>  In this usage, writing <math>\langle p ~\text{implies}~ q \rangle</math> asserts the existence of a certain relation between the logical value of <math>p\!</math> and the logical value of <math>q,\!</math> whereas writing <math>\langle \text{if}~ p ~\text{then}~ q \rangle</math> merely forms a compound statement whose logical value is a function of the logical values of <math>p\!</math> and <math>q\!</math>.  This will be discussed in detail below.

'''Note.'''  Many writers draw a technical distinction between the form <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}</math> and the form <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}.</math>  In this usage, writing <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime}</math> asserts the existence of a certain relation between the logical value of <math>p\!</math> and the logical value of <math>q,\!</math> whereas writing <math>{}^{\backprime\backprime} \text{if}~ p ~\text{then}~ q {}^{\prime\prime}</math> merely forms a compound statement whose logical value is a function of the logical values of <math>p\!</math> and <math>q\!</math>.  This will be discussed in detail below.

==Definition==

==Definition==

The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of <math>\mathrm{false}\!</math> just in case the first operand is true and the second operand is false.

The concept of logical implication is associated with an operation on two logical values, typically the values of two propositions, that produces a value of <math>\operatorname{false}</math> just in case the first operand is true and the second operand is false.

In the interpretation where <math>0 = \operatorname{false}\!</math> and <math>1 = \operatorname{true}\!</math>, the truth table associated with the statement <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime},\!</math> symbolized as <math>{}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime},\!</math> is as follows:

In the interpretation where <math>0 = \operatorname{false}</math> and <math>1 = \operatorname{true}</math>, the truth table associated with the statement <math>{}^{\backprime\backprime} p ~\text{implies}~ q {}^{\prime\prime},</math> symbolized as <math>{}^{\backprime\backprime} p \Rightarrow q {}^{\prime\prime},</math> is as follows:

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