Preheat oven to 350 degrees F (175 degrees C). Lightly grease cookie sheets. Stir together the flour, baking soda, cream of tartar, and salt; set aside.
In a large bowl, cream together the butter and sugar until smooth. Beat in eggs, vegetable oil, and vanilla. Gradually stir in flour mixture until smooth. Stir in nuts. Drop dough by rounded teaspoonfuls onto the prepared cookie sheets. Flatten cookies with a fork.
Bake in the preheated oven for 10 to 12 minutes, or until the edges begin to brown. Cool on baking sheets for a minute before transferring to wire racks to cool completely.