I don't think they suggest capital punishment is the reason for more crime. They are pointing out that is does not act as a deterrent though (and there are studies showing that).
There is actually also studies that show that in the US for example, states with the death penalty actually have higher homicide rates. (However I would assume there is a correlation between the states that have the death penalty and the states with more lax gun laws/higher gun ownership etc).
I dont believe the death penalty is a "deterrent" its more (or should be) a quick solution to remove those most evil from society who cannot be rehabilitated and freeing up more space in prisons for violent/sexual offenders.
The current issue in our society is we cant lock dangerous people up that long as the system is too clogged up - so you have a situation often where violent offenders are released prematurely back into society due to the sentencing guidelines which are influenced to an extent by prison capacity.
Studies show a good portion of these people go onto reoffend at some stage which means more needless victims in our society and studies show that statistically more victims = more offenders (the bullied often become the bully, the abuse victims often become the abuser) so you have a society where we have a mental health crisis, kids stabbing other kids to death daily and the solution apparently is to keep less people in prison for shorter durations !
I know people will say capital punishment its cruel but I disagree - death by firing squad is much more "humane" for those who cannot be rehabilitated than locking them up for 40+ years to rot in a lonely cell until they die (whilst being in constant fear of attacks/rapes from other inmates).
