Can I do a correlation between a likert scale and a dichotomous scale with the same list of items I am trying to do a correlation between a 4 level likert scale with 11 items and a dichotomous present-not present scale of the same 11 items.  Can I do this?
 A: I am assuming that each item has four response options and for one variable you are coding the response options 1,2,3,4, and for the other variable you are coding the response options 0,0,1,1.
You certainly could create total scores using these two versions of the variables and correlate the two total scores. Under typical conditions they would correlate highly. 
The main question is why would you want to correlate these two versions of the variables? 
A legitimate reasons might be that you want to know whether the choice of coding makes much of a difference; if the two codings correlate above .99 or something then for most purposes it wont matter which you use. If this was your aim, you'd probably also like to see how well the two versions of summation predict relevant criterion.
Here's a little R simulation that you can play with. It shows that under the conditions of the simulation as the number of items increases, so does the correlation between the two versions of the test.
set.seed(1234)

cor_x2x4 <- function(items=11) {
    n <- 1000
    latent <- rnorm(n, 0, 1)
    x <-  sapply(seq(items),
           function(X) rnorm(n, latent, 1))
    x <- data.frame(x)
    x4 <- sapply(x, function(X) cut(X, 
                                   c(-Inf, -1, 0, 1, Inf), 
                                   labels=FALSE))

    x2 <- sapply(data.frame(x4), function(X) as.numeric(X > 2))
    totals <- data.frame(x2=apply(x2, 1, sum), 
               x4= apply(x4, 1, sum))

    cor(totals)[1,2]
}

nitems <- c(1,2,3,4,5,10,15,20, 50, 100, 1000) 
r <- sapply(nitems, function(X) cor_x2x4(X) )

cbind(nitems, r)

The results look like this:
      nitems         r
 [1,]      1 0.8926544
 [2,]      2 0.9197795
 [3,]      3 0.9320194
 [4,]      4 0.9409412
 [5,]      5 0.9532897
 [6,]     10 0.9716208
 [7,]     15 0.9783644
 [8,]     20 0.9834715
 [9,]     50 0.9916207
[10,]    100 0.9941332
[11,]   1000 0.9970657

