Syllogisms Made Easy

baffo's writeup does a very good job of explaining the intricacies of formal syllogisms, but I'd like to tackle the same topic in a way that is perhaps a bit more accessible to those not versed in formal logic. In my opinion, it is tragic that the study of logic is so steeped in symbols and math-like structure that those who need it the most (i.e. those that are not logically or mathematically inclined) are those least likely to be attracted to it. It is my feeling that most people could benefit from just a basic understand of some the common logical arguments and fallacies.

A good starting point for an understanding of logic is the syllogism, as it is both easy to comprehend and used often (in various forms) in everyday arguments. With this in mind, what follows is my attempt to explain the syllogism to somebody who has no previous concept of formal logic:

A syllogism is a very common argument (in the context of logic, the word argument means "line of reasoning" rather than "point of contention" as it often does in everyday speech). The classic syllogism has three parts: two premises and a conclusion drawn from the premises. A syllogism is a way to compare or contrast groups or categories of things.

The Universal Syllogism

The easiest example of a syllogism to follow is that of the universal syllogism. This is a type of argument that compares the three categories of things. An example:

All poodles are dogs.
All dogs are animals.
Therefore, all poodles are animals.

The example is clear-cut; most syllogisms that you'll come across are not. Consider the following hypothetical passage:

"We all know that geeks are much smarter than the rest of the population. And common sense tells us that smart people make more money. This is why geeks make so much money."

Here we have a syllogism, though it may not be obvious at first. Break it down into its components, though, and you get:

All geeks are smarter than the rest of the population.
Smart people make more money (than average).
Therefore, geeks make more money than the average person."

The two examples of syllogisms we've looked at follow a specific form. They are examples of a universal syllogism. What this means is that the two premises and the conclusion make claims about all the members of a group.

Something worth noting is that we can switch the order of our two premises without changing the meaning of the syllogism. In other words:

All dogs are animals.
All poodles are dogs.
Therefore, all poodles are animals.

Is just as deductively valid as our first example. The term deductively valid (which we'll shorten to just "valid") means that, if we accept the premises as true, the conclusion must also be true. If all dogs are animals and all poodles are dogs, then all poodles must be animals. We'll learn how to evaluate syllogisms (determine if they're valid or not) in a little bit.

One thing you cannot change in a universal syllogism is the order of the claims within the premises and the conclusion. "All animals are dogs" is a very different statement than "All dogs are animals." The first category of a claim is called the subject and the second category is called the predicate. In the claim "All dogs are animals," the subject category is "dogs" and the predicate category is "animals." It is important to remember these two terms and to not mix them up while evaluating syllogisms.

The Universal-to-Particular Syllogism

Let's take a look at another type of syllogism: the universal-to-particular syllogism. This type of syllogism has as its first premise a universal claim, then makes a claim about a limited number of the members of a group. The conclusion then draws some relation between the two premises. These are a bit more complicated than the universal syllogism but are probably more common. An example:

All dogs are animals.
Some dogs are poodles.
Therefore, some animals are poodles.

This is also a valid syllogism. Like the universal syllogism, it doesn't matter which order the premises occur in (the "Some dogs are poodles" premise could come before the "All dogs are animals" premise without changing the validity). There is a weird catch when dealing with particular claims, though. Consider our previous example, now slightly altered:

All dogs are animals.
Some poodles are dogs.
Therefore, some animals are poodles.

Did you spot the difference? We've flipped the order of the claims in the particular premise. While this may seem weird at first, we can switch the subject and predicate categories in a particular claim without changing its meaning. "Some snakes are cold-blooded animals" is the logical equivalent of "Some cold-blooded animals are snakes." This is obviously not true of a universal claim (remember "All snakes are animals" is not the same as "All animals are snakes")!

Because of this, syllogisms with particular claims get tricky to recognize and evaluate. We will discuss how to evaluate them in a moment. It is worth mentioning that there is no way to construct a valid syllogism that only has particular claims and the reason for this will become obvious when we learn how to evaluate them. Before we get to that, let's recap:

  • A syllogism consists of three parts: two premises (the first of which is traditionally called the "major premise" while the second is the "minor premise") and a conclusion.
  • A universal claim is a claim about all members of a group or category and usually starts with the word "All," though it can just as easily start with a word like "Every."
  • A particular claim is a claim about a particular set of members within a larger group or category and usually starts with a the word "Some" but can also start with words like "Most" or "Many."
  • A universal syllogism is a syllogism in which both premises make universal claims. A universal-to-particular syllogism is one which has a universal and a particular claim as premises.
  • Changing the order of premises in either type of syllogism does not affect the syllogism at all.
  • Changing the order of the subject and predicate categories in a universal claim does affect the statement. This is not true of particular claims.

Evaluating Universal Syllogisms

We understand the basic structure of two common syllogisms (there are more complex forms which will be dealt with later). How do we determine if a syllogism is valid or invalid?

The universal syllogism is easy to evaluate. All valid universal syllogisms follow the following form:

All A is B.
All B is C.
Therefore, All A is C.

The only tricky thing to remember is that we can switch the order of the two premises ("All B is C, All A is B, Therefore all A is C") without changing the validity of the syllogism. For the syllogism to be valid, just make sure that the predicate category in one of the premises is the subject category in the other and that the conclusion draws a relationship between the other two categories.

A common counterfeit universal syllogism follows this form:

All A is B.
All A is C.
Therefore, all B is C.

An example of this sort of syllogism:

All dogs are animals.
All dogs are furry.
Therefore, all animals are furry.

It's easy to see, using that example, how silly this reasoning is. Saying "All dogs are animals" does not imply that "All animals are dogs" and thus we cannot say that because all dogs are furry that all animals must be. For all we know, dogs could be the only furry animals. That dogs are a subset of a larger group (animals) does not imply anything about other members of that group.

This counterfeit can be harder to spot than you might think, especially if they're not clearly set up as syllogisms. Look at the following example:

"Obviously, all gun-owners are nutjobs who want to overthrow the government! Consider that everyone who's a Republican owns a gun. It becomes clear when you realize that Republicans secretly want to overthrow the government."

The reasoning is slightly more muddled, and the conclusion is stated first, but the argument is a syllogism:

All Republicans own guns.
All Republicans want to overthrow the government.
Therefore, all gun-owners want to overthrow the government.

Again, we see that just because the group "Republican" intersects with the group "gun-owner" it doesn't mean that all gun-owners are Republican. This can be very tricky to watch out for, especially on issues that you have strong personal feelings about (and as such are less likely to think critically about).

The easiest way to determine validity for a universal syllogism when you identify one in print or speech is to simplify the argument, write it clearly as a syllogism and ensure it follows the proper form. So long as the syllogism follows the correct form, it must be valid (remember that just because it is valid does not mean that the conclusion is true).

Evaluating Universal-to-Particular Syllogisms

The universal-to-particular syllogism can be a bit harder to evaluate than the universal syllogism. This is because the form is less clear. Generally speaking, the valid form of the syllogism looks like this:

All A are B.
Some A are C.
Therefore, some B is C.

The above is a perfectly valid argument. What's confusing is that the following is also valid:

All A are B.
Some C are A.
Therefore, some B is C.

Remember that we can switch the order of the subject and predicate categories in a particular claim (that's the second part of the universal-to-particular syllogism) without adjusting its meaning! This throws a monkey-wrench into our evaluation process: this adds a new layer of complexity that means you would have to memorize a bunch of forms to check for validity.

Fortunately, there's a way to simply evaluate these syllogisms without memorizing forms. Here's the key: for a universal-to-particular syllogism to be valid, the subject category of the universal claim must appear in the particular claim. IT CAN BE EITHER THE SUBJECT OR THE PREDICATE CATEGORY, SO LONG AS IT APPEARS.

This seems kind of weird, at first, but it makes sense once you try constructing a few universal-to-particular syllogisms and swap the subject and predicate in the particular claim. As an example:

All E2 users are smart.
Some E2 users are good-looking.
Therefore, some smart people are good-looking.

The above example (valid, though I'd question its soundness) is equivalent to saying the following:

All E2 users are smart.
Some good-looking people are E2 users.
Therefore, some smart people are good-looking.

The two examples differ only in their minor premise, and all we've done is to switch the subject with the predicate. But they still both mean the same thing. If this is unclear, try building a few example syllogisms of your own (the more outrageous the better) and I suspect you will understand what's going on.

Here is a common counterfeit universal-to-particular syllogism:

All A is B.
Some B is C.
Therefore, Some A is C.

You should recognize immediately why this is invalid: the subject category in the universal premise is not present in the particular premise. An example may be helpful:

All E2 users are smart.
Some smart people are astronauts.
Therefore, some E2 users are astronauts.

While there may be an astronaut on E2 (you never know), the conclusion certainly doesn't follow from the premises. Again, I invite the reader to try and construct a valid universal-to-particular syllogism that doesn't have the subject category from the universal premise somewhere in the particular premise. It just doesn't work.

So evaluating these types of syllogisms is pretty simple: merely check to make sure the subject category of the universal claim is present in the particular claim and it should be clear whether or not the syllogism is valid. A final example, first how you might read the argument in the real world, then in a clear syllogism form:

"Every player in the NFL is rich! A lot of those rich people got that way by screwing over the poor. Obviously, some of those pro-football players have been ripping off poor people."

In the simple form:

All pro-football players are rich.
Some rich people steal from poor people.
Therefore, some pro-football players steal from poor people.

Odds, Ends & Exercises

This writeup is growing quite lengthy, so I will only briefly touch on a few other points:

There are quite a few other types of syllogisms you should be aware of. There is the hypothetical syllogism which is valid in this form:

If A, then B.
If B, then C.
Therefore, if A, then C.

You can add multiple premises, so long as they always follow each other ("If A, then B. If B, then C. If C, then D. If D, then E. Therefore, if A then E."). And of course you can switch the order of the premises (though that makes things quite messy).

Another very common syllogism is the disjunctive syllogism which works as follows:

Either A or B.
Not B.
Therefore, A.

The validity of this syllogism should be self-evident. Those still interested in logic would do well to read up on conditional statements, particularly the forms Modus Ponens and Modus Tollens. Evaluating simple conditionals is very similar to evaluating syllogisms in that both require only a basic understanding of their forms.

The hard part about logic is recognizing these forms when they appear in real life. It is uncommon that, upon hearing a bad argument, someone says, "Ah, but that's a counterfeit universal syllogism!" or, "That can't be right -- you just denied the antecedent!"

With that in mind, I leave the still-interested reader with a few exercises in recognizing and then evaluating syllogisms. I am confident that this writeup contains the tools needed to solve the exercises without too much difficulty. Answers are at the bottom of the writeup, but I strongly suggest you work them out for yourself -- this often reveals flaws in your reasoning that you wouldn't catch if you just looked at solutions.

Exercises: Determine what sort of syllogism each argument is. Write it out in a simplified form. Determine if each argument is valid or invalid (remember that an argument can be valid even if its conclusion is untrue). Some of them may have words, phrases or sentences that aren't directly related to the argument -- this isn't sneaky, it's realistic!

  1. "You say that sappy movies are always bad? I can prove that isn't true. For instance, Steven Spielberg always makes great movies! But some of his movies are sappy. So, some sappy movies are also great."
  2. "Since smoking is so bad for you, anybody who smokes must be unhealthy. Also, unhealthy people don't live as long as normal people. That means all smokers are going to die sooner than the average person!"
  3. " History has shown that wars are always expensive for the country that starts them. Not only that, but wars are bad. It's clear that anything expensive for a country is bad."
  4. "I hate most computers. They're all complicated! Since a lot of complicated things are a pain to use, most computers are a pain to use."


Most of my information for this WU is pretty basic logic (and was introduced to me in a beginning logic class at Palomar College). I did, however, get several of the terms from "Open Minds and Everyday Reasoning," a tremendous introductory logic text by Zachary Seech (who just so happened to be the professor for the class).

Answers for the exercises below (if you can't read them, try pasting them into a text editor such as Notepad):

  1. Valid universal-to-particular syllogism
  2. Valid universal syllogism
  3. Invalid counterfeit universal syllogism
  4. Invalid counterfeit universal-to-particular syllogism