A syllogism is a logical argument made up of three parts: a major premise, a minor premise and the conclusion deriving from the preceding ones. Thus we arrive at statements, referring to particular situations, which are generally true; by doing so, irrefutable and convincing arguments are obtained both in rhetoric and in literature. Syllogisms are a fundamental component for the formal study of logic and are often included in aptitude tests to verify candidates' logical reasoning skills.
Steps
Part 1 of 3: Becoming Familiar with the Definitions of Syllogisms
Step 1. Recognize how a syllogism forms an argument
To understand this you need to be familiar with the terms most used in discussions of logic. Simplifying as much as possible, a syllogism is the simplest sequence of logical premises leading to a conclusion; the premises are sentences used as proof in an argument, while the conclusion is the result of the logical elaboration based on the link between the premises.
Consider the conclusion of a syllogism as the "thesis" of an argument; in other words, the conclusion is the one that emerges from the premises
Step 2. Determine the three parts of the syllogism
Remember that it is made up of a major premise, a minor premise, and a conclusion. To give an example: "all human beings are mortal" may represent the major premise, since it indicates a fact universally accepted as true; "David Foster Wallace is a human" is the lesser premise.
- Note that the minor premise is more specific and closely related to the main one.
- If both propositions quoted above are considered to be true, the logical conclusion of the reasoning should be "David Foster Wallace is mortal".
Step 3. Find the major and minor term
Both must have a common term with the conclusion; what is present in both the major premise and the conclusion is called the "major term" and forms the nominal predicate of the conclusion (in other words, it indicates an attribute of the subject of the conclusion); the factor shared by the minor premise and the conclusion is called the "minor term" and will be the subject of the latter.
- Consider this example: "All birds are animals; parrots are birds. So, parrots are animals."
- In this case "animals" is the major term, since it is present in both the major premise and the conclusion.
- "Parrots" is the minor, being within the minor premise as well as the subject of the conclusion.
- Note that there is also another categorical term shared by the two premises, in this case "birds"; this is called the "middle term" and is of fundamental importance in determining the syllogism, as will be indicated in a later passage.
Step 4. Search for categorical terms
If you are preparing for a test of logic, or if you simply want to learn to understand syllogisms better, remember that most of the ones you will encounter will cover some categories; this means that they will be based on reasoning like this: "If _ are / are not [belonging to one category], then _ are / are not [members of the same / other category]".
Another way of schematizing the logical sequence of a syllogism concerning some categories is the following: "Some / all / none _ are / are not _"
Step 5. Understand the distribution of terms in a syllogism
Each of the three propositions of a syllogism can be presented in four different ways, based on how it "distributes" (or does not) the categorical terms present. Consider one of these terms as "distributed" if it refers to each element of the class it refers to; for example, in the premise "all human beings are mortal", the subject "human beings" is distributed because the proposition concerns all members of the category (in this case, they are referred to as "mortal"). Analyze how the four types differ in the way of distributing (or not distributing) the categorical terms:
- In the sentence "All Xs are Y" the subject (X) is distributed.
- In "No X is Y" both the subject (X) and the predicate (Y) are distributed.
- In the proposition "Some Xs are Y", subject and predicate are not distributed.
- In "Some Xs Are Not Y" only the predicate (Y) is distributed.
Step 6. Identify an entymeme
The entymemes (whose name derives from the Greek) are simply "compressed" syllogisms; they can also be described as one-sentence arguments, which can help you recognize the reasons why these are great logical tricks.
- In specific terms, an entymeme does not have the major premise and combines the minor with the conclusion.
- For example, consider this syllogism: "All dogs are canids; Lola is a dog. Lola is therefore a canid." The entymeme that summarizes the same logical sequence is instead: "Lola is a canid because she is a dog".
- Another example of an entymeme would be: "David Foster Wallace is mortal because he is a human being".
Part 2 of 3: Identifying an Invalid Syllogism
Step 1. Distinguish between "validity" and "truth"
Although a syllogism may be logically valid, it does not always mean that the conclusion it leads to is actually true: logical validity derives from a choice of premises such that the possible conclusion is unique; nevertheless, if the premises themselves are not valid, the conclusion could be totally false.
- If you want an example, think about the following syllogism: "All dogs can fly; Fido is a dog. Fido therefore knows how to fly." Logical validity is assured, but the conclusion is clearly unfounded, since the major premise is false.
- What is evaluated when verifying the validity of the syllogism is the logical reasoning underlying the argument.
Step 2. Check for any linguistic tricks that may indicate lack of logical validity
Look at the typology of the premises and the conclusion (affirmative or negative) when you are trying to determine the validity of the syllogism. Note that if both premises are negative, then the conclusion must be negative as well; if both premises are affirmative, so must the conclusion be; Finally, he recalls that at least one of the two premises must be affirmative, since no logical conclusion can be deduced from two negative premises. If any of these three rules are not followed, you can conclude that the syllogism is invalid.
- Furthermore, at least one premise of a valid syllogism must have a universal formula; if both premises are particular, no logically valid conclusion can be obtained. For example, "some cats are black" and "some black things are tables" are particular propositions, so it cannot follow a conclusion such as "some cats are tables".
- Very often you will realize the invalidity of a syllogism that does not respect these rules without even thinking about it, since it will immediately sound illogical.
Step 3. Think carefully about conditional syllogisms
These are hypothetical arguments and their conclusions are not always valid, since they depend on the possibility of a not universally true premise coming true. Conditional syllogisms include reasoning similar to "If _, then _". These arguments are invalid if they include other factors that may contribute to the conclusion.
- For example: "If you continue to eat a lot of sweets every day, you risk getting diabetes. Stefano doesn't eat sweets every day. Therefore, Stefano doesn't risk diabetes."
- This syllogism is not valid for various reasons: among these, Stefano could eat a considerable amount of sweets on various days of the week (but not daily), which would still make him at risk of diabetes; alternatively, he could eat one cake a day and similarly risk getting sick.
Step 4. Beware of syllogistic fallacies
A syllogism can imply a wrong conclusion if it starts from wrong premises. Discuss this example: "Jesus walked on water; the feathered basilisk can walk on water. The feathered basilisk is Jesus." The conclusion is obviously false, since the median term (in this case the ability to walk on the surface of the water) is not distributed in the conclusion.
- To take another example: "All dogs love to eat" and "John likes to eat" do not necessarily imply "John is a dog". This error is called the "fallacy of the undistributed medium", because the term that connects the two sentences is never completely distributed.
- Another mistake to pay close attention to is the "fallacy of illicit treatment of the major term", present in this reasoning: "All cats are animals; no dog is a cat. No dog is an animal". In this case the syllogism is invalid because the major term "animals" is not distributed in the major premise: not all animals are cats, but the conclusion is based on this insinuation.
- The same goes for the illicit treatment of the minor term, as in: "All cats are mammals; all cats are animals. All animals are therefore mammals." The invalidity lies, similarly to before, in the fact that not all animals are cats, but the conclusion is based on this erroneous idea.
Part 3 of 3: Determine the Mode and Figure of a Categorical Syllogism
Step 1. Recognize the various types of propositions
If both premises of a syllogism are accepted as valid, then the conclusion may also be valid; the logical validity, however, also depends on the "mode" and the "figure" of the syllogism, which descend from the propositions used. In categorical syllogisms, four different forms are used to compose the premises and the conclusion.
- The propositions of form "A" are affirmative universals, that is, "all [category or characteristic term] are [a different category or characteristic]"; for example, "all cats are felines".
- The "E" propositions are just the opposite, that is, negative universals. For example, "no [category or characteristic] is [different category or quality]", as in "no dog is a feline".
- The "I" forms are the affirmative particulars, in which some elements of the first group have a certain characteristic or belong to another group: for example, "some cats are black".
- The "O" forms are the negative particulars, in which it is stated that some elements do not have a particular characteristic or belonging: "some cats are not black".
Step 2. Identify the "mode" of the syllogism by analyzing the propositions
By verifying to which of the four forms each proposition belongs, the syllogism can be reduced to a succession of three letters, in order to easily check if it is a valid form for the figure to which it belongs (the various figures will be described in the next step). For now concentrate on the possibility of "labeling" each sentence of a syllogism (both the premises and the conclusion) according to the type of proposition that is used, thus managing to identify the way of reasoning.
- To give an example, this is a categorical syllogism of the AAA mode: "All Xs are Y; all Ys are Z. Therefore, all Xs are Z".
- The mode refers only to the forms of propositions that are used in a "common" syllogism (major premise - minor premise - conclusion) and can also be the same for two reasonings belonging to different figures.
Step 3. Recognize the "figure" of the syllogism
This can be identified on the basis of the role of the medium term, or if this is a subject or predicate in the premises. Remember that the subject is the "protagonist" of the sentence, while the predicate is a quality or a characteristic (or a belonging group) that is attributed to the subject of the sentence.
- In a syllogism of the first figure, the middle term is subject in the major premise and predicated in the minor one: "All birds are animals; all parrots are birds. All parrots are animals."
- In the second figure, the middle term is predicated in both the major and minor premises: "No fox is a bird; all parrots are birds. No parrot is a fox."
- In the syllogisms of the third figure the middle term is subject in both premises: "All birds are animals; all birds are mortal. Some mortals are animals."
- In the case of the fourth figure, the middle term is predicated in the major and subject premise of the minor: "No bird is a cow; all cows are animals. Some animals are not birds."
Step 4. Identify valid syllogistic modes
Although there are 256 possible forms of syllogism (since there are 4 possible forms for each proposition and 4 different figures of syllogism) only 19 ways are logically valid.
- For the syllogisms of the first figure, these are AAA, EAE, AII, and EIO.
- For the second figure, only EAE, AEE, EIO and AOO are valid.
- In the case of the third figure, only the AAI, IAI, AII, EAO, OAO and EIO modes must be considered.
- For the syllogisms of the fourth figure the modes AAI, AEE, IAI, EAO and EIO are valid.
