Building a Syllogism

A syllogism needs:

A major premise (M) and a minor premise (m).

Two terms and a middle term that is used in both premises.

A conclusion (“Therefore, etc.”) that uses both premises.

You can also use  ∴  for “Therefore.”

Shake together and you have a valid syllogism!

E.g.,

(M) Chadwick is a human being.
(m) Mr. Curly the Guinea Pig is smarter than all human beings.
Therefore, Mr. Curly the Guinea Pig is smarter than Chadwick.

This is a valid syllogism. It presents two premises and a conclusion, three terms (Mr. Curly, human beings, Chadwick), and a middle term (human being/s) which does not show up in the conclusion.

In the conclusion, the verb “to be” asserts a relation though the six uses of "to be" is another topic! 

OF course, the next question is whether it is sound. We may agree that Chadwick is a human being, but we likely would not agree that Mr. Curly is smarter than all human beings.   

And it may be questionable whether Chadwick is a human being.

Some rules:

The middle term must be distributed at least once (that is “all” or “no” should be understood).

From two negative premises no conclusion can be made.

From two particular premises (“some”) no conclusion can be made.

If one premise is negative then the conclusion must be negative.

If one premise is particular then the conclusion must be particular.

From two affirmative premises one cannot make a negative conclusion.