How to determine the figure of a syllogism and understand if the argument is correctly constructed?

Every day we face the need to defend our point of view, analyze others' arguments, and make well-considered decisions. At the core of rigorous, evidence-based thinking lies formal logic, and its classic tool is the simple categorical syllogism—a deductive inference consisting of two premises and a conclusion.

However, simply stating a conclusion is not enough. To understand whether the conclusion necessarily follows from the premises or if we are facing a logical trick, one must know how to analyze the structure of the inference. A key step in this analysis is determining the figure of the syllogism—the scheme that shows exactly how the so-called middle term is distributed in the statements.

This practical guide will help you quickly master the algorithm for analyzing syllogisms from scratch. You will learn to correctly identify the major and minor terms, distinguish between the major and minor premises, visualize logical connections using Euler circles, and instantly spot typical errors in reasoning.

Anatomy of an Inference: Identifying Terms S, P, and M

Before moving on to graphical schemes and figure rules, you must learn to dissect the inference like an experienced anatomist. Any simple categorical syllogism consists of three elements, which in logic are called terms. Their mutual arrangement determines the structure and correctness of the entire conclusion.

These three "pillars" are the subject (S), the predicate (P), and the middle term (M). An error in finding these elements will inevitably lead to an incorrect determination of the figure and false conclusions. Let's examine how to find these elements in the text without error and how to correctly separate the premises into major and minor.

How to Find the Extreme Terms (Subject S and Predicate P) in the Conclusion

Analysis of any syllogism must always begin with its end—the conclusion. This is where the key to deciphering the entire structure of the inference is hidden. In the text, the conclusion is easily recognizable by marker words such as: "therefore," "thus," "consequently," "it follows that."

As soon as you have isolated the conclusion statement, find the two extreme terms within it:

1. Subject of the conclusion ($S$) — this is what the conclusion is about (the minor term). Grammatically, it most often corresponds to the subject of the sentence.
2. Predicate of the conclusion ($P$) — this is what is asserted or denied about the subject (the major term). It is usually expressed by the predicate or the nominal part of the predicate.

Example:
In the conclusion *"Therefore, some lawyers ($S$) are advocates ($P$)"**:

• $S$ (subject) — "lawyers";
• $P$ (predicate) — "advocates".

By identifying $S$ and $P$, you lay a solid foundation for further analysis of the premises.

How to Calculate the Middle Term (M) and Differentiate Major and Minor Premises

After you have identified the extreme terms ($S$ and $P$) in the conclusion, finding the middle term ($M$, terminus medius) will be easy.

Step 1. Calculate the middle term ($M$)
Look closely at both premises. Find the concept that is duplicated in both but is completely absent in the conclusion itself. This is the middle term ($M$). It acts as the link between the subject and the predicate, allowing the transition to the conclusion, after which it "disappears."

Step 2. Differentiate the premises
In a simple categorical syllogism, the premises are strictly differentiated by the extreme terms they contain:

• Major premise — this is the statement that includes the major term ($P$, the predicate of the conclusion). In standard syllogistic notation, it is usually placed first.
• Minor premise — this is the statement that contains the minor term ($S$, the subject of the conclusion).

Example:

1. All metals ($M$) are conductive ($P$). (Major premise)
2. Copper ($S$) is a metal ($M$). (Minor premise)
3. Copper ($S$) is conductive ($P$). (Conclusion)

Now that the structure of the inference is fully analyzed, you can move on to the graphical determination of the syllogism's figure.

Step-by-Step Algorithm for Determining the Syllogism's Figure

After learning how to correctly find the middle term (M) and separate the premises into major and minor, the map of the inference opens up before us. In formal logic, the relative arrangement of terms in the premises is not random—it is subject to a strict geometry, which is called the figure of the syllogism. Determining the figure is a key step that allows you to understand the rules by which the thought unfolds and whether it contains hidden defects.

To ensure this process is not difficult during an exam or when analyzing real text, we will reduce it to a simple and clear algorithm. Next, we will visualize the structure of the premises using classic graphical schemes and break down a step-by-step instruction that will help you instantly recognize any of the four figures in specific practical examples.

Four Graphical Schemes for the Arrangement of the Middle Term

The figure of a simple categorical syllogism is its structural variation, determined exclusively by the position of the middle term ($M$) in the premises. Since the middle term links the extreme terms ($S$ and $P$) and is present in both premises, there are exactly four possible ways for it to be spatially distributed.

Four graphical schemes are used in logic for quick memorization and analysis. In these, the premises are conditionally depicted as horizontal lines, and the position of term $M$ is connected by a line segment:

• Figure 1: The middle term ($M$) occupies the subject position in the major premise and the predicate position in the minor premise. The graphical line is directed from top to bottom and left to right (forming a diagonal line \).

  $M$ — $P$
  $S$ — $M$

• Figure 2: The middle term ($M$) acts as the predicate in both premises. The connecting line forms a vertical line on the right side (|).

  $P$ — $M$
  $S$ — $M$

• Figure 3: The middle term ($M$) is the subject in both premises. The connecting line forms a vertical line on the left side (|).

  $M$ — $P$
  $M$ — $S$

• Figure 4: The middle term ($M$) occupies the predicate position in the major premise and the subject position in the minor premise. The graphical line is directed from top to bottom and right to left (forming a diagonal line /).

  $P$ — $M$
  $M$ — $S$

These geometric patterns allow for the instant visualization of the inference's structure and its preparation for checking logical correctness.

Step-by-Step Instruction for Identifying the Figure in a Practical Example

Let's examine the step-by-step analysis of a classic syllogism:

1. All metals conduct electricity.
2. Copper is a metal.
3. Copper conducts electricity.

Step 1. Analyze the conclusion.
First, we find the conclusion: "Copper conducts electricity." The subject of the conclusion (S) is "copper," and the predicate (P) is "conducts electricity."

Step 2. Determine the premises.
The major premise contains the predicate (P): "All metals conduct electricity." The minor premise contains the subject (S): "Copper is a metal."

Step 3. Finding the Middle Term (M).
The concept present in both premises but absent in the conclusion is the "metal." We denote it as M.

Step 4. Determining the Figure.
We write down the scheme of term arrangement:

• In the major premise: M — P (the middle term is in the subject position).
• In the minor premise: S — M (the middle term is in the predicate position).

Visually connecting M, we get a diagonal line running from left to right and top to bottom. This is the classic first figure of the syllogism.

Analysis of the Four Figures and Their Special Rules

Determining the graphical scheme of a syllogism is only half the battle. To guarantee the truth of the conclusion when the premises are true, it is necessary to check whether the special rules prescribed for each specific figure are observed. In formal logic, these rules act as strict filters: even if the inference appears convincing externally, violating even one local requirement instantly renders it logically unsound.

Each of the four figures has its own unique cognitive significance and strict requirements regarding the quality and quantity of the premises. Let's examine in detail which laws govern the structure of each figure and how these rules help instantly rule out false deductive inferences.

Features and Rules of the First and Second Figures

The first and second figures of the simple categorical syllogism are of fundamental importance in logic, as the most obvious deductive inferences and refutation procedures are built upon them.

First Figure (M — P, S — M):
In this figure, the middle term ($M$) is the subject in the major premise and the predicate in the minor premise. It is subject to two strict rules:

1. The major premise must be a universal proposition ($A$ or $E$).
2. The minor premise must be an affirmative proposition ($A$ or $I$).

Valid moods: AAA, EAE, AII, EIO.

Second Figure (P — M, S — M):
Here, the middle term ($M$) occupies the predicate position in both premises. Its rules require:

1. The major premise must be universal ($A$ or $E$).
2. One of the premises (and the conclusion itself) must be negative ($E$ or $O$).

Valid moods: EAE, AEE, EIO, AOO.

Violating any of these rules automatically makes the inference logically incorrect, even if the premises seem true.

Features and Rules of the Third and Fourth Figures

Moving on to the analysis of the third and fourth figures of the simple categorical syllogism, it is important to note that they are subject to their own specific laws, different from the first two figures.

Third Figure (the middle term M occupies the subject position in both premises):

• Minor Premise Rule: The minor premise must necessarily be an affirmative proposition (A or I).
• Conclusion Rule: The conclusion in this figure must always be a particular proposition (I or O).

Example: All bamboos (M) are grains (P). All bamboos (M) are plants (S). Some plants (S) are grains (P).

Fourth Figure (the middle term M is the predicate in the major premise and the subject in the minor premise):
This figure is considered the least natural for our thinking, but it is strictly formalized:

• If the major premise is affirmative (A or I), then the minor premise must be universal (A or E).
• If the minor premise is affirmative (A or I), then the conclusion must be particular (I or O).
• If one of the premises is negative (E or O), then the major premise must be universal (A or E).

Adhering to these rules guarantees the formal correctness of the inference for these structures.

Methods for Checking the Logical Validity of an Inference

After determining the figure of the syllogism and comparing it with the special rules, we take an important step, but this is not always sufficient for a final verdict. To completely rule out error in reasoning, it is necessary to proceed to a direct check of the logical correctness of the inference structure. In formal logic, two main methods are traditionally applied for this: analytical and graphical.

The first method is based on comparing the syllogism's structure with a table of valid moods that have been tested over centuries. The second method is visual, using Euler circles to model the relationships between the volumes of concepts. Combining these approaches guarantees absolute accuracy in analyzing any simple categorical syllogism.

Analysis of Valid Moods for Each Figure

Each figure of the simple categorical syllogism has its own valid moods—varieties of inferences distinguished by the quantitative and qualitative characteristics of the premises and conclusion ($A, I, E, O$). Out of 256 theoretically possible combinations, only 24 are logically valid (6 for each figure, including weakened moods).

For ease of memorization, special mnemonic names have been used since the Middle Ages, where the vowels indicate the type of proposition in the syllogism's structure (major premise, minor premise, and conclusion):

• Figure I (yields all types of conclusions): Barbara (AAA), Celarent (EAE), Darii (AII), Ferio (EIO).
• Figure II (conclusion is always negative): Cesare (EAE), Camestres (AEE), Festino (EIO), Baroco (AOO).
• Figure III (conclusion is always particular): Darapti (AAI), Disamis (IAI), Datisi (AII), Felapton (EAO), Bocardo (OAO), Ferison (EIO).
• Figure IV (does not yield universal conclusions): Bramantip (AAI), Camenes (AEE), Dimaris (IAI), Fesapo (EAO), Fresison (EIO).

If the structure of your syllogism matches one of these moods, the inference is deductively sound.

Graphical Verification Method Using Euler Circles

The graphical method using Euler circles is the most reliable way to visualize a simple categorical syllogism and instantly check its validity without memorizing rules and moods.

The basis of the method is the depiction of concept volumes as circles. Since a syllogism involves three terms—the subject of the conclusion (S), the predicate of the conclusion (P), and the middle term (M)—we work with three circles in the scheme, whose relative positions are strictly dictated by the premises.

Step-by-step verification algorithm:

1. Illustrate the major premise. Draw circles reflecting the relationship between the middle term (M) and the predicate (P).
2. Overlay the minor premise onto the scheme. Add the subject circle (S), strictly observing its relationship to the middle term (M).
3. Evaluate the resulting scheme. Check whether the relationship between S and P stated in the conclusion follows from it. If the circles can be drawn such that the premises remain true, but the conclusion becomes false, the syllogism is constructed incorrectly.

This visual tool eliminates subjective errors and helps quickly verify the logical conclusion in an exam or discussion.

Even with graphical verification methods such as Euler circles, it is easy to make a mistake in practice if control over the structure of concepts is weakened. Formal logic demands impeccable accuracy: the slightest shift in the meaning of a word or the ignoring of specific rules for a particular figure instantly destroys the logical necessity of the conclusion. As a result, we arrive at a false or unreliable conclusion based on outwardly convincing premises.

To protect one's reasoning from destructive distortions, it is necessary to study typical logical fallacies in detail. Most often, they are related either to incorrect manipulation of the terms within the syllogism itself or to a subtle violation of those very special rules of the figures that we discussed earlier. Let's examine these traps in more detail to learn how to instantly recognize them in everyday discussion and academic problems.

The Effect of Term Quaternization and How to Recognize It

The effect of term quaternization (quaternio terminorum) is one of the most insidious errors in a simple categorical syllogism. According to the rules of logic, a syllogism must strictly contain three terms: the subject ($S$), the predicate ($P$), and the middle term ($M$). The error occurs when the same word is used in the premises with different meanings. As a result, the middle term is split, and we end up with four terms instead of three, which makes the conclusion false or logically unconnected.

Consider a classic example:

1. Movement is eternal (here "movement" is a philosophical category).
2. Going to university is movement (here "movement" is a physical action).
3. Therefore, going to university is eternal.

How to recognize term quaternization in practice?

• Check the unambiguity of concepts: Ensure that the middle term ($M$) in the major and minor premises has an absolutely identical meaning.
• Beware of homonyms and metaphors: Words with double meanings (e.g., "world," "key," "bow") are the main sources of this error.
• Analyze the context: If a term is taken in a collective sense (as a class of objects) in one premise, and in a distributive sense (as a specific element) in another, you have term quaternization.

Logical Traps When Violating Special Figure Rules

Violating the special rules of specific figures is another common source of logical errors. Each figure imposes strict requirements on the quality and quantity of the premises; failing to comply destroys the rigor of the conclusion.

• Ignoring the rules of the first figure. Here, the minor premise must be affirmative, and the major premise must be universal. If you take a negative minor premise: "All students (M) take exams (P). This person (S) is not a student (M)", a logically valid conclusion cannot be drawn. The attempt to conclude that "This person does not take exams" is erroneous because the predicate of the conclusion ($P$) turns out to be distributed, even though it was not distributed in the premise.
• The trap of affirmative premises in the second figure. The second figure requires that one of the premises be negative. Violating this rule leads to the undistributed middle term. For example: "All true scientists (P) love truth (M). Petrov (S) loves truth (M)". The conclusion "Petrov is a true scientist" is logically unsound, because dilettantes can also love truth.

Knowledge of these rules saves one from false conclusions that appear convincing on the surface.

Conclusion

The ability to determine the figure of a syllogism and check its correctness is not just an academic skill for passing exams, but a reliable foundation for critical thinking. Systematic analysis of arguments step-by-step—from finding the extreme and middle terms ($S$, $P$, $M$) to graphical analysis and checking against the special rules of the figures—allows one to instantly recognize manipulations and logical errors in everyday discussion, legal practice, or scientific research.

Regular practice quickly brings this algorithm to automation. Use the studied schemes, Euler circles, and mood rules as a filter for incoming information so that your own arguments always remain impeccable, provable, and logically consistent.