Aristotelian vs. Modern Syllogism
Consider the following argument:
All dogs are animals.
Therefore, some animals are dogs.
On the surface, it seems pretty reasonable. All dogs are animals, and some animals are dogs. Both are true statements. Even so, let's consider the next argument, with exactly the same form:
All unicorns are animals.
Therefore, some animals are unicorns.
Intuitively, we know that this is not true, but how have we arrived at a true conclusion and a false one with the same form? Obviously, the form cannot be valid, right? Actually, the answer is both yes and no. In Aristotelian logic, each term in a syllogism is assumed to be true. In effect, we are assuming a second premise, namely: Some Unicorns are in existence.
In modern logic, we do not make such a presumption. This avoids a rather embarrassing quandary. Suppose I want to run a contest for my employees. I will say, "Each employee who has sold 1000 widgets will get an extra day of paid vacation." In a strictly Aristotelian sense, I can't make this statement, since at this point, none of my employees has sold 1000 widgets, and perhaps none will. It is not known whether I am referring to any real beings, so technically, I can't make any logical conclusions about them.
This may seem entirely nitpicky, and in a way, it probably is. Even so, it is a mistake that people make from time to time. Using just the modern approach to syllogism, we can say that both of the original arguments are invalid because they lack the correct form. We are missing a premise. In such a simple example, it's easy to catch, but next time you meet someone who could sell ice to an Eskimo, you might be surprised to observe how often such simple mistakes in critical thinking can be exploited by clever salespeople.