Is there a function that counts a cell if it contains a value rather than
equals that value?

For example, I'd like to count the number of cells that contain the word
"dog".

First cell is (dog, cat, bird)
Second cell is (dog)
Third cell is (cat, bird)
Forth cell is (dog, cat)

The formula would then return a value of 3, because the word "dog" shows up
in three cells.

Is this possible? If so, does anyone have any suggestions? Thanks in
advance.