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:

:* <math>p\!</math> implies <math>q\!</math>.

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

| <math>\text{If}~ p ~\text{then}~ q.\!</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 "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 "if <math>p\!</math> then <math>q\!</math>", the first term, <math>p\!</math>, is called the ''[[antecedent (logic)|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\!</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.