I have another question regarding linear logic: I want to get to the proof E, using the premises in (1-4). Is this at all possible?

1: $A$

2: $C$

3: $(A\multimap B)\otimes(C\multimap D)$

4: $B\multimap (D\multimap E)$

I originally hoped that I could simply reverse the tensor in (3) (making $(A\multimap B)$ and $(C\multimap D)$ individual premises), so I could inject the premises from (1) and (2). If that had been possible, I could have used assumed premises for (4) in order to use the multiplication substitution rule (which essentially states that I could replace the assumed $B$ and $D$ with $(B\otimes D)$ to derive E. Since I was told that this impossible, however, I am looking for another way to get to E, using the premises described above.

I hope that this question is less vague than my last one. If it is not, I apologise. I cannot see how I can make my question any clearer with my current level of knowledge in logic (or mathematical formalism).


I'm neither a logican nor mathematician; I'm currently looking into glue semantics, which uses Linear Logic to derive the meaning of a sentence. So I am grateful for any and all assistance.