## If_Then Variations

### Subjects to be Learned

• different ways of saying if_then: only if, necessary, sufficient

### Contents

If-then statements appear in various forms in practice. The following list presents some of the variations. These are all logically equivalent, that is as far as true or false of statement is concerned there is no difference between them. Thus if one is true then all the others are also true, and if one is false all the others are false.

• If p , then q.
• p implies q.
• If p,   q.
• p only if q.
• p is sufficient for q.
• q if p.
• q whenever p.
• q is necessary for p.
• It is necessary for p that q.
For instance, instead of saying "If she smiles then she is happy", we can say "If she smiles, she is happy", "She is happy whenever she smiles", "She smiles only if she is happy" etc. without changing their truth values.

"Only if" can be translated as "then". For example, "She smiles only if she is happy" is equivalent to "If she smiles, then she is happy".
Note that "She smiles only if she is happy" means "If she is not happy, she does not smile", which is the contrapositive of "If she smiles, she is happy".
You can also look at it this way: "She smiles only if she is happy" means "She smiles only when she is happy". So any time you see her smile you know she is happy. Hence "If she smiles, then she is happy". Thus they are logically equivalent.

Also "If she smiles, she is happy" is equivalent to "It is necessary for her to smile that she is happy". For "If she smiles, she is happy" means "If she smiles, she is always happy". That is, she never fails to be happy when she smiles. "Being happy" is inevitable consequence/necessity of "smile". Thus if "being happy" is missing, then "smile" can not be there either. "Being happy" is necessary "for her to smile" or equivalently "It is necessary for her to smile that she is happy".

### Test Your Understanding of If_Then Variations

Which of the following pairs of propositions are logically equivalent ?

Click Yes if equivalent, else No , then Submit. There are two sets of questions.

Next -- From English to Proposition

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