I am also confused about your question here. The reason we model the log-odds in logistic regression is to map the dependent variable onto the real line. Why do you want $\alpha + \beta^tX$ to be positive?
If you really want to restrict the $\alpha + \beta^tX$ to the positive real line, one possible solution is to use $-log(p)$ as your response, though I am not sure why you would want do this and how you would do interpretation. But since $p\in(0, 1)$, then $log(p)\in(-\infty, 0)$, thus$-log(p)\in(0, \infty)$ as you wish.
Note that even your independent variables are always positive, $\alpha + \beta^tX$ can still be negative, since the estimates of $a$ and $b$ could be negative, unless you really want to constrain them to be positive.