(a) The following children, when issued certificates of exemption by the county superintendent of education, where they reside in territory under the control and supervision of the county board of education, or the city superintendent of schools, where they reside in territory under the control and supervision of a city board of education, shall not be required to attend school, or to be instructed by a private tutor:
(1) Children whose physical or mental condition is such as to prevent or render inadvisable attendance at school or application to study. Before issuing such certificate of exemption, the superintendent shall require a certificate from the county health officer in counties which have a health unit, and from a regularly licensed, practicing physician in counties which do not have a health unit, that such a child is physically or mentally incapacitated for school work;
(2) Children 16 years of age and upward or children who have completed the course of study of the public schools of the state through high school as now constituted;
(3) Where because of the distance children reside from school and the lack of public transportation such children would be compelled to walk over two miles to attend a public school;
(4) Where the children are legally and regularly employed under the provisions of the law relating to child labor and hold permits to work granted under the terms of said child labor law.
(b) Nothing in this section shall be construed so as to deny any right to any child granted under the provisions of Sections 16-39-1 through 16-39-12.