pairwise.t.test doesn't set an alpha value, it gives adjusted p-values.
pairwise.t.test returns the adjusted p-values themselves, so it's up to you to decide on your own "alpha" (confidence level cutoff), if you're going to use the so-called Neyman-Pearson approach to dichotomize results into "reject null hypothesis" vs. "fail to reject null hypothesis". Adapting the example from
airquality$Month <- factor(airquality$Month, labels = month.abb[5:9])
Pairwise comparisons using t tests with pooled SD
data: Ozone and Month
May Jun Jul Aug
Jun 1.00000 - - -
Jul 0.00029 0.10225 - -
Aug 0.00019 0.08312 1.00000 -
Sep 1.00000 1.00000 0.00697 0.00485
P value adjustment method: bonferroni
So you need to look at the values in the table (which are the Bonferroni-adjusted p-values) and decide on the basis of your own alpha. For example, for the comparison of August and June (adjusted p-value = 0.08312), you could decide to reject the null hypothesis if your alpha=0.1, or fail to reject it if your alpha=0.05 ...
Unsolicited PS: unless you absolutely must, there's no reason to use Bonferroni instead of the default Holm correction.
?p.adjust even says:
There seems no reason to use the
unmodified Bonferroni correction because it is dominated by Holm's
method, which is also valid under arbitrary assumptions.