A proposition in an argument whose truth, when taken together with the truth of at least one other proposition, is supposed to entail the truth of the conclusion. 

Major Premise.— In a categorical syllogism, it contains the major term and minor term and is standardly the first premise. E.g.: 

(MAJOR PREMSE) All men are mortal. 

Socrates is a man.  

Therefore, Socrates is mortal.  

Minor Premise.— In a categorical syllogism, it contains the minor term and middle term and is standardly the second premise. E.g.: 

All men are mortal. 

(MINOR PREMSE) Socrates is a man.  

Therefore, Socrates is mortal.