Validating Postfix (I need to validate prefix)

bool validate(const string& postfix)
{
    /*
    A postfix expression is valid if and only if:
        1) The first two elements are operands(values), and
        2) The last element is an operator, and
        3) For every n values there are n-1 operator(s), and
    */

    // The NOT character messes things up a lot for whatever reason
    // The NOT character can be the second element
    // And there can be an infinite number of not elements while only one character

    char first_element = postfix[0];
    char second_element = postfix[1];
    char last_element = postfix[postfix.size() - 1];


    if (priority(first_element) != 0)
        return false;

    if (priority(second_element == 2) || priority(second_element) == 3)
        return false;

    int num_letters = 0;
    int non_not_operands = 0;

    for (int i = 0; i < postfix.size(); i++)
    {
        if (priority(postfix[i]) == 0 && postfix[i] != ' ')
            num_letters++;
        else if (priority(postfix[i]) == 2 || priority(postfix[i] == 3))
            non_not_operands++;
    }

    return (num_letters == (non_not_operands + 1));
}