Only if

According to the grammar of ‘only if’ ‘p only if q’ is equivalent to ‘if p then q’.  I think we can see this from a syntactical point of view and from a semantical point of view.  I’ll start with the semantic claim first.

Semantic
The following seem to be semantically equivalent:
a. p only if q
b. p occurs only if q occurs
c. p is true only if q is true
d. if q fails to happen, then p fails to happen.

What seems to be the common element in b-d is q appears to function as a necessary condition for the occurrence of p.  Perhaps d. captures the necessary condition idea the clearest, but it seems to be there in b. and c. as well.  If that is right, then in a. q is a necessary condition for p.  But ‘if p, then q’ states that q is a necessary condition for p. Furthermore, ‘if p, then q’ is logically equivalent to ‘if ~q, then ~p.’  But ‘if ~q, then ~p’ is d. above.  Hence, ‘p only if q’ is equivalent to ‘if p, then q.’

Syntactic
Syntactically we know that 
1. p if, and only if q 

is equivalent to

2. if p, then q, and if q, then p

1. can be written as follows: 

1*. p if q and p only if q.

p if q 
is equivalent to 
if q, then p.

So that leaves the ‘if p, then q’ part of 2 above and the ‘p only if q’ in 1* above.  So,

If p, then q

            Is equivalent to

p only if q.



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