Changes

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

Close approximations to the concept of logical implication are expressed in ordinary language by means of linguistic forms like the following:

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

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

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

'''Note.'''  Many writers draw a technical distinction between the form "<math>p\!</math> implies <math>q\!</math>" and the form "if <math>p\!</math> then <math>q\!</math>". In this usage, writing "<math>p\!</math> implies <math>q\!</math>" asserts the existence of a certain relation between the logical value of <math>p\!</math> and the logical value of <math>q\!</math> while writing "if <math>p\!</math> then <math>q\!</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>p ~\text{implies}~ q\!</math> and the form <math>\text{if}~ p ~\text{then}~ q.\!</math>  In this usage, writing <math>p ~\text{implies}~ q\!</math> asserts the existence of a certain relation between the logical value of <math>p\!</math> and the logical value of <math>q\!</math> while writing <math>\text{if}~ p ~\text{then}~ q\!</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==