Typical pair vs. overcard
 

Villain is very loose/neutral. I feel like I should see a showdown heads-up with this hand.
On the flop, I'm raising for information and to buy a free card. I know this will induce a bet on the river, but I'm willing to do that.
Post-flop cost this way: 2 BB
Post-flop cost if I bet the turn and fold: 2 BB
Post-flop cost if I bet the turn and call down: 3-4 BB.
Is my logic wrong?
Thanks.

Party Poker 2/4 Hold'em (10 handed) converter

Preflop: Hero is CO with J[img]/images/graemlins/diamond.gif[/img], J[img]/images/graemlins/club.gif[/img].
<font color="#666666">6 folds</font>, <font color="#CC3333">Hero raises</font>, <font color="#666666">1 fold</font>, <font color="#CC3333">SB 3-bets</font>, <font color="#666666">1 fold</font>, <font color="#CC3333">Hero caps</font>, SB calls.

Flop: (9 SB) 7[img]/images/graemlins/spade.gif[/img], A[img]/images/graemlins/spade.gif[/img], 5[img]/images/graemlins/heart.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">SB bets</font>, <font color="#CC3333">Hero raises</font>, SB calls.

Turn: (6.50 BB) 6[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(2 players)</font>
SB checks, Hero checks.

River: (6.50 BB) A[img]/images/graemlins/club.gif[/img] <font color="#0000FF">(2 players)</font>
<font color="#CC3333">SB bets</font>, Hero calls.

Final Pot: 8.50 BB

Re: Typical pair vs. overcard
 

I'd call here - your check on the turn means he's going to bet no matter what he had...but his check on the turn leads me to be believe he's not confident in his hand. I'd put him on either a small PP, maybe and A-rag, or a busted flush draw.

Re: Typical pair vs. overcard
 

After you cap the flop it will be highly unusual for villian to CR the turn without something that is way ahead of your JJ, so I would bet the turn and bet the river. If he CR the turn you can safely fold this.