It is “written on every heart” as in it is part of our nature. So, yes reason alone can know the natural law but it may be unclear to many people.
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1956 The natural law, present in the heart of each man and established by reason, is universal in its precepts and its authority extends to all men. It expresses the dignity of the person and determines the basis for his fundamental rights and duties:
For there is a true law: right reason. It is in conformity with nature, is diffused among all men, and is immutable and eternal; its orders summon to duty; its prohibitions turn away from offense . . . . To replace it with a contrary law is a sacrilege; failure to apply even one of its provisions is forbidden; no one can abrogate it entirely.9
1957 Application of the natural law varies greatly; it can demand reflection that takes account of various conditions of life according to places, times, and circumstances. Nevertheless, in the diversity of cultures, the natural law remains as a rule that binds men among themselves and imposes on them, beyond the inevitable differences, common principles.
1958 The natural law is *immutable *and permanent throughout the variations of history;10 it subsists under the flux of ideas and customs and supports their progress. The rules that express it remain substantially valid. Even when it is rejected in its very principles, it cannot be destroyed or removed from the heart of man. It always rises again in the life of individuals and societies
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