Yes, you can relax logic gates into continuous versions which makes the system differentiable. An AND gate can be constructed with the function x*y and NOT by 1-x (on inputs in the range [0,1]. From there you can construct a NAND gate, which is universal and can be used to construct all other gates. Sigmoid can be used to squash the inputs into [0,1] if necessary.
This paper lists out all 16 possible logic gates in Table 1 if you're interested in this sort of thing: https://arxiv.org/abs/2210.08277
This paper lists out all 16 possible logic gates in Table 1 if you're interested in this sort of thing: https://arxiv.org/abs/2210.08277