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{| align="center" cellspacing="10" width="90%"
 
{| align="center" cellspacing="10" width="90%"
| <math>p ~\text{implies}~ q.\!</math>
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|-
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<math>\begin{array}{l}
| <math>\text{If}~ p ~\text{then}~ q.\!</math>
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p ~\text{implies}~ q.
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\\[6pt]
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\text{if}~ p ~\text{then}~ q.
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\end{array}</math>
 
|}
 
|}
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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.
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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.
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'''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.
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'''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==
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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.
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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.
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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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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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