An argument containing two premises and a conclusion.  

Categorical Syllogism.— A syllogism containing only categorical propositions. E.g.: 

All men are mortal.   

Socrates is a man.   

Therefore, Socrates is mortal.   

Hypothetical Syllogism (also called Compound Syllogism).— A syllogism whose major premise is a hypothetical proposition of some kind.  Although some classical logicians have held that the hypothetical syllogism is reducible to categorical form, others have argued that hypothetical syllogisms represent a sui generis manner of articulating the relationship between propositions. 

Conditional Syllogism.— A hypothetical syllogism whose first premise is standardly a conditional proposition. E.g.: 

If the weather is poor, then it is unpleasant. 

But the weather is poor. 

Therefore, it is unpleasant. 

Conjunctive Syllogism.— A hypothetical syllogism whose first premise is standardly a conjunctive proposition. N.B.: There is only one valid form of a conjunctive syllogism, and it begins with a negative conjunctive proposition. E.g.,  

No train comes from Milan and arrives on time.  

This train is coming from Milan. 

Therefore, it will not arrive on time. 

Disjunctive Syllogism.— A hypothetical syllogism whose first premise is standardly a disjunctive proposition. E.g.: 

The economy is strong or it is weak. 

But the economy is strong. 

Therefore, it is not weak. 

See also the entries for demonstration and inference.