So basically, I'm trying to design a logic circuit which does the following:

There's a big gate that gets opened by stepping on a pressure plate. There are 4 plates total, 2 on each side of the big gate. The gate is opened when any (or both) plates are stepped on. Find the simplified boolean function to make this work.

So my input/output look somewhat like this. (Assume S1,S2 is inside and S3,S4 is outside)

S1 S2 S3 S4 | F
0 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 1
0 1 0 0 1
0 1 0 1 1
0 1 1 0 1
0 1 1 1 1
1 0 0 0 1
1 0 0 1 1
1 0 1 0 1
1 0 1 1 1
1 1 0 0 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 1


Then, there's a problem. There's few cases where some inputs are an impossibility. For instance, you cannot step on the outside plate and inside plate at the same time or step on more than 2 plates at the same time so this eliminates many possible inputs (Such as 1,1,1,1 or 1,0,0,1). How do I eliminate this condition to simplify my equation as much as possible when drawing my K-map?

 

May be I didnt understood your question or may be this is what you are asking for....
To eliminate we use "Don't care cell in Karnaugh Map".

S1 S2 S3 S4 | F
0 0 0 0 0
0 0 0 1 1
0 0 1 0 1
0 0 1 1 1
0 1 0 0 1
0 1 0 1 1
0 1 1 0 1
0 1 1 1 1
1 0 0 0 1
1 0 0 1 X
1 0 1 0 1
1 0 1 1 1
1 1 0 0 1
1 1 0 1 1
1 1 1 0 1
1 1 1 1 X

Good Luck

 

debjit used "don' care" terms where you indicated. If you want to add more of them, do so.
Remember that when you go to form the Karnaugh map afterwards, you can substituted X with either 1 or 0, depending on what gives you a simpler expression.