A Guide to Syllogism: Definition, Types, Rules, Examples, & More

Syllogism is a logical argument in which you apply deductive reasoning to draw a valid conclusion from two premises assumed to be true. As the foundation of logic and critical thinking, syllogism can help you develop your reasoning, debating, and persuasion skills.

To gain a better understanding of this concept, we consulted our experts, who answered commonly asked questions about syllogism, such as what syllogism means, how it is formed, and how it can be effectively used.

In this article, we will share what they told us.

Key Takeaways

• Syllogism is a branch of logic that allows you to develop a conclusion from two propositions using deductive reasoning.
• There are three types of syllogism: categorical, conditional, and disjunctive syllogism.
• Syllogistic fallacies are mistakes in logic that can lead to the formation of wrong conclusions. Usually, they occur when you don’t accurately follow the rules of syllogism.
• Syllogism is important for developing sound arguments, arriving at truthful conclusions, and persuading your readers or listeners.

What Is Syllogism?

Syllogism is a logical argument that uses deductive reasoning to develop a conclusion from two propositions assumed or asserted to be true. Considering that it’s a type of deductive reasoning, syllogism starts from more general premises and arrives at a specific conclusion.

Typically, syllogism is written out using a three-line form such as the following:

All P is Q.

R is P.

Therefore, R is Q.

In this format, the first two sentences are true premises, while the last sentence is the valid conclusion that follows. The first premise (all P is Q) is usually considered the major premise, while the second one (R is P) is the minor premise, as it is more specific than the first general statement.

When expressed in natural language, syllogism looks like this:

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

If we analyze the premises and conclusion more closely, we can recognize three essential elements of syllogism: major, minor, and middle terms.

In this case, the major term is mortal, as it appears in the major premise and the conclusion. Socrates is the minor term present in the conclusion as well as the minor premise. Finally, man (or men) is the middle term that connects the two premises and allows you to draw specific conclusions.

Types of Syllogism

The commonly known types of syllogism include categorical, conditional, and disjunctive syllogism. Aside from these, branches of logic recognize legal syllogism, quasi-syllogism, polysyllogism, and other kinds, but the three mentioned above are the most important ones.

Below, we will examine the three basic types of syllogism in more detail.

#1. Categorical Syllogism

Categorical syllogism is the most basic type that often comes to mind first when we speak of syllogisms. In fact, the example we’ve given above is categorical syllogism, with two premises followed by a logical conclusion.

Still, there’s more to categorical syllogism than that. In the above example (all P is Q), we have a universal proposition as opposed to a particular proposition, which claims that some P is Q. There are two more examples of universal and particular propositions: no P is Q, and some P is not Q.

The following table will clarify the propositions of categorical syllogism further:

Universal proposition

All P is Q.

Example: All men are mortal.

No P is Q.

Example: No cats are birds.

Particular proposition

Some P is Q.

Example: Some cats are pets.

Some P is not Q.

Example: Some pets are not mammals.

#2. Conditional Syllogism

Conditional syllogism, otherwise known as hypothetical syllogism, contains one or two conditional statements from which you can draw a valid conclusion.

Here is a typical formula for conditional syllogism:

If P, then Q.

If Q, then R.

Therefore, if P, then R.

This is known as pure hypothetical syllogism, where both premises are conditional statements. In this case, the consequent (Q) of the first premise must match the antecedent of the second premise for the conclusion to be valid.

The other type of conditional syllogism is mixed hypothetical syllogism, with the following formula:

If P, then Q.

P.

Therefore, Q.

In this case, the first premise is a conditional, while the second affirms or denies either the antecedent or the consequent of the conditional. From that, we can deduce a valid conclusion.

#3. Disjunctive Syllogism

In disjunctive syllogism, the first premise contains a disjunctive statement, while the second one denies one of the elements. As a result, we can affirm that the other element is true.

Here is the formula:

P or Q.

Not P.

Therefore, Q.

When using disjunctive syllogism, it doesn’t matter if we deny the first or second element—whichever is not denied will be true. So, the formula above is still valid if Q is denied and P is affirmed.

However, it’s important to note that disjunctive syllogism doesn’t work in the following form:

P or Q.

P.

Therefore, not Q.

Although this may seem accurate, it’s actually known as the formal fallacy of affirming a disjunct since, in this case, both P and Q could be true. As a result, it’s impossible to draw a valid conclusion from these two premises.

The only time you can deny the other element by affirming the first one is when you’re using exclusive disjunction with the formula:

Either (only) P or (only) Q.

P.

Therefore, not Q.

In this case, only one element can be true, so it is possible to draw a conclusion.

Examples of Syllogism

If syllogism still seems somewhat abstract to you, we will try to illustrate it using examples in natural language. Let’s take a look at how each of the three types of syllogism mentioned above could be applied in real life.

Example 1: Categorical Syllogism

A typical example of a categorical syllogism is the one we’ve already given above:

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

However, this is only one kind of categorical syllogism—universal affirmative. There are three other important types: universal negative, particular affirmative, and particular negative.

Example 2: Conditional Syllogism

As we’ve seen before, a conditional syllogism can be either pure or mixed. If it’s pure, this is what it looks like in natural language:

If I don’t finish the task on time, I’ll have to do it tomorrow.

If I have to do it tomorrow, I won’t be able to rest.

Therefore, if I don’t finish the task on time, I won’t be able to rest.

On the other hand, mixed conditional syllogism may be phrased in the following way:

If a sparrow is a bird, it flies.

A sparrow is a bird.

Therefore, it flies.

Example 3: Disjunctive Syllogism

Finally, here is an example of disjunctive syllogism in its most basic form:

The sky is red, or it is blue.

It is not red.

Therefore, it is blue.

This can also be illustrated using exclusive disjunction:

The sky is either only red or it is only blue.

It is blue.

Therefore, it is not red.

Syllogism Rules

The rules of syllogism are important to follow, as they ensure that your deduction is sound and free of any logical fallacies. Below, we explore the six essential syllogistic rules.

#1. Rule 1

There must be exactly three terms in categorical syllogism, all of which must be used in the same context in all three statements.

For instance, if the term man denotes humans in one premise and an individual in the other, that could lead to confusion and a logical fallacy of four terms.

#2. Rule 2

Two affirmative premises can’t lead to a negative conclusion. Basically, if your conclusion is negative, at least one of the premises must also be negative.

#3. Rule 3

The middle term, which establishes a connection between the two premises, must be distributed at least once within the premises. Otherwise, the argument commits a logical fallacy called the undistributed middle.

#4. Rule 4

If a term appears in the conclusion, it must appear in at least one of the premises. In case it doesn’t, you commit the fallacy of illicit major or illicit minor.

#5. Rule 5

Categorical syllogism can’t have two negative premises, as no connection can be established and no conclusion drawn. This is usually called the fallacy of exclusive premises.

#6. Rule 6

If you have two universal premises, you can’t possibly draw a particular conclusion. Otherwise, your argument commits the so-called existential fallacy.

Syllogistic Fallacies Examples

When you don’t apply the rules of syllogism carefully enough, you can commit logical fallacies that lead to inaccurate conclusions.

As you’ve seen, there are many possible syllogistic fallacies, but we will explore the two most frequent ones below and show you what they look like in natural language.

The Fallacy of the Undistributed Middle

The fallacy of the undistributed middle occurs when the middle term isn’t properly distributed, and neither of the premises accounts for all members of the middle term. When that happens, the major and minor terms aren’t correctly linked, and a logical conclusion can’t be formed.

Here is an example:

All people are primates.

All gorillas are primates.

Therefore, all people are gorillas.

Of course, this is not the case because primates encompass much more than people or gorillas and can’t be used as a middle term between these two groups.

The Fallacy of Affirmative Conclusion from Negative Premises

As stated in Rule #5, categorical syllogism doesn’t work if there are two negative premises. In such a case, we may get something like this:

No people under the age of 66 are senior citizens.

No senior citizens are children.

Therefore, people under the age of 66 are children.

Of course, this is partly true—children are people under 66. However, not all people under this age are children, which is a false conclusion drawn from two negative premises.

Syllogism History

In its earliest form, syllogism was developed by Aristotle in his 350 BC book Prior Analytics. For the most part, Aristotle’s syllogism dealt with categorical propositions, and in that domain, it was considered largely complete.

So, when medieval logicians rediscovered Aristotle’s works, there wasn’t much to change or add. As a result, Aristotle’s syllogism entered the broader sphere of logic essentially unchanged, thanks to the French 14th-century philosopher John Buridan.

The first modifications to the Aristotelian syllogism were made in the 19th century to incorporate conditional and disjunctive premises. However, aside from that, philosophers still considered Aristotle’s syllogism complete and thought almost nothing needed to be changed or added.

However, that changed with the development of sentential and predicate logic, which largely replaced Aristotle’s syllogism in modern philosophy.

The Importance of Syllogisms in Writing and Speeches

Using syllogism correctly, you can deduce the truth, form valid arguments, and persuade people to consider your point of view. What’s more, the ability to use syllogistic reasoning indicates that your intelligence is highly developed, as it is an important aspect of deduction.

Here is how syllogism can help you more specifically:

• Deducing the truth. Since syllogism is considered pure, language-independent logic, it can help you come to accurate conclusions regardless of the context as long as you are able to apply its rules correctly.
• Forming valid arguments. Syllogisms can help whether you’re trying to come up with arguments in a debate, for your essays, or for a speech. If you follow its rules carefully, you’ll develop sound conclusions that are extremely difficult to refute.
• Persuasion. An excellent way to increase your credibility and support your claims is to use syllogism, as they often make the soundest arguments and, as such, tend to be the most convincing.

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Final Thoughts

Even though syllogism has existed for over 2000 years, it’s still relevant as a basis for logic, argumentation, and deductive reasoning. And considering that logic as a field rarely changes, it will likely stay relevant for many centuries to come.

Learning how to form and apply syllogism is an excellent way to improve your critical thinking and deduction, which can make you a significantly more persuasive writer and speaker. In addition, syllogisms can help you develop your intelligence and thrive in any setting that requires logical thinking.