[education] The conscience clause was an important term in education in England throughout much of 19th century. In this context, it referred to permitting parents of schoolchildren to withdraw them from Church of England worship services or other school activities that violated the parents` religious principles. ...
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http://en.wikipedia.org/wiki/Conscience_clause_(education)
[medical] Conscience clauses are legal clauses attached to laws in some parts of the United States which permit pharmacists, physicians, and other providers of health care not to provide certain medical services for reasons of religion or conscience. In many cases, the clauses also permit health care providers to refuse to refer patients to...
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http://en.wikipedia.org/wiki/Conscience_clause_(medical)

A conscience clause is a clause in certain British Acts of Parliament which dispenses people from certain duties if they have religious objections or other conscientious scruples to their performance.
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http://www.probertencyclopaedia.com/browse/AC1.HTM

a clause or article in an act or law that exempts persons whose conscientious or religious scruples forbid their compliance.
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https://www.infoplease.com/dictionary/conscience-clause
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