9. What deductive arguments are there?

How can you analyse arguments?

There are two techniques for creating and evaluating deductive arguments. This chapter is mainly about categorical logic. This is logic based on the relationships of inclusion and exclusion between categories in categorical claims. Categorical logic is useful in clarifying and analysing deductive arguments. When we understand how this works, we can be more critical and precise with regard to propositions and arguments and avoid ambiguity.

Categorical claims

A categorical claim says something about categories of objects. A standard-form categorical claim is a claim that arises when names or descriptions are added to categories. Here are four types of:

• The A-claim: "All ... are ...". Example: "All Protestants are Christians."
• The E-claim: "None ... are ...". Example: "No atheists are Christians."
• The I claim: "Some ... are ...". Example: "Some Christians are Arabic."
• The O-claim: "Some ... are not ...". Example: "Some Christians are not Catholic."

By "some" we mean "at least one."

Terms

The words that appear on the dotted lines above are called terms. The word that appears on the first dotted line in a claim is called the "subject term". The word that appears on the second dot line is called the "predicate term."

The words that serve as "subject term" and "predicate term" in a sentence are collectively also called classes. The above claims can also be processed and displayed in Venn diagrams. Such a Venn diagram is a graphical representation of all possible hypothetical logical relationships between a finite set of statements. Visually, this is a circle for each category, overlapping the moment they have a community. Thanks to the overlap between some statements, you can draw conclusions from the statements; proportions are visible.

The claim "some dogs bite" would therefore be represented by two overlapping circles - one circle for "dogs" and one circle for "bite". The overlap is therefore "dogs that bite". Because this claim concerns all dogs, but for some dogs you put a cross in the overlapping piece to indicate that at least one dog is biting.

The A and I claims are called affirmative claims because they include part of another class. The E and O claims are called negative claims because they exclude a part of one class from another.

The conversion of claims

It is important to be able to convert a claim into a standard-form categorical claim that means the same. We say that two claims are the same ("equivalent claims") when they are both the same in exactly every situation. This conversion must be done precisely so that the meaning of the claim is not changed. For some claims that is easy. The claim "Every rose is a flower" can easily be transformed into an A claim, namely: "All roses are flowers." However, sometimes it is more difficult to transform a claim into one of the four standard-form categorical claims. It is therefore important, when having a discussion or debate or paper, to first determine the terms that appear in a claim.

What is the square of opposition?

We say that two categorical claims correspond when they have the same subject term and the same predicate term. So the claim "All Protestants are Christians" corresponds to "Some Protestants are Christians." In both claims, "Protestants" is the subject term, while "Christians" is the predicate term. The claim "Some Christians are not Protestant" does not correspond to the above two claims, because the places of the subject term and the predicate term are interchanged in this claim. Logical relationships between A, E, I, and O claims can be explained in a figure: the square of opposition ("square of opposition" see page 263).

• A-claims and E-claims are called "contrary claims" because they can both be false, but both cannot be true.
• I-claims and O-claims are called "subcontrary claims" because they can both be true, but not both can be false.
• The A and O claims and the E and I claims are collectively called "contradictory claims" because they can never all be (equally) true and must therefore be contradictory.

With the help of the square of opposition we can often read the truth values ​​of the claims. There are a number of limitations to this:

• If the A and / or E claim is / are true, or if the I and / or O claim is / are false, we can derive the truth values ​​from the remaining claims.
• When the A and / or E claim is (are) true, or when the I and / or O claim is (are) true, we can only determine the truth value of the "contradictory claim".

Which categorical actions can be undertaken?

1. Conversion

Converting a standard-form categorical claim can be done by reversing the position of the subject term and the predicate term. Only the E and I claims contain the same information as their conversions. Therefore, the conclusion is: Only E and I claims, but not the A and O claims, are equal to their conversations. Schematically; P = Q, Q = P

Examples are:

• E-claim: "No Chinese are Africans" and "No Africans are Chinese." These claims are therefore the same.
• I claim: "Some capitals are major cities" and "Some major cities are capitals".

2. Turn around - ("obversion")

In addition to conversion, obversion is a second categorical implementation. Before this concept is explained, two other concepts must first be understood:

• Universe of discourse: the claims we make are context-bound. When a teacher enters the classroom and says that everyone has passed, students know that it is not about everyone in the whole world, but about people in the classroom itself. The context of the claim therefore basically determines the population of the claim.
• Complementary class: for every category within a universe or discourse there is a complementary category, for example "students" and "non-students". These are called complementary terms. Often this can be done simply enough by pasting "no" before, although in some cases there are specific words for it.

Finding the reverse ('obverse') of a claim can be done in two ways: (1) turn an affirmative claim into a negative claim or vice versa, so make an A claim an E claim or turn an O claim an I claim and (2) replace the predicate term with the complementary term.

Schematically; ~ P = Q, P = ~ Q

Example:

• E-claim: "No fish are mammals" changes to A-claim: "All fish are not mammals".

All categorical claims, whether they belong in the A, E, I or O category, are the same as their opposite form.

3. Exchange ("contraposition")

A third categorical implementation is called contraposition. To find the contraposition of a categorical claim, (1) the subject term must be placed in the place of the predicate term, while the predicate term is placed in the place of the subject term. In addition (2) both terms must be replaced by complementary terms. Schematically; P = Q, ~ P ​​= ~ Q

Example;

• A-claim: "All Arabs are Muslims." When counterposition is applied, it becomes "All non-Muslims are not Arabs."
• O-claim: "Some citizens are not voters". When contraposition is applied, it becomes "Some non-voters are not non-citizens".

Only A and O claims are the same as their counter position.

What are categorical syllogisms?

A monergism is a deductive argument that consists of two premises. A categorical syllogism is a syllogism that consists of standard-form categorical claims, where three terms of each claim must occur exactly twice in two of the claims.

An example:

1. All Americans are consumers.
2. Some consumers are not Democrats.
3. That is why some Americans are not Democrats.

All terms ("Americans", "consumers" and "Democrats") appear exactly twice in two different claims.

The terms of a syllogism get the following label:

• Major term (P): the term that appears as a predicate term in the conclusion of a syllogism. In the example this is "democrats"
• Minor term (S): the term that appears as the subject term in the conclusion of a syllogism. In the example this is "Americans.
• Middle term (M): the term that occurs in both premises, but not in the conclusion. In the example this is "consumers"

When S and P are connected by means of M, then an argument is valid. An argument is called valid if it is not possible for the premises to be true, while the conclusion is false. A Venn diagram can be used to find out the relationship between S, P and M, so that it can be seen whether an argument is valid (for explanation and examples, see "The Venn diagram method of testing for validity" on pages 274 and 275).

A Venn diagram consists of three circles: on the left is the minor term, on the right is the major term and below that is the middle term. When one of the premises is an I or O premise, there may be confusion about where the "X" should be placed. A decision can sometimes be made using the following rules:

• If one premise is an A or E premise, and the other is an I or O premise, the A or E premise must first be placed in the diagram. It is then immediately clear where the "X" should end up in the diagram.
• An "X" that can be placed in two "areas" is placed on the line that separates the two areas.
• If both premises are an A or E claim, and the conclusion is an I or O claim, placing the premises in the diagrams cannot produce a conclusion. This is because the A and E claims yield coloured areas, and I and O claims require an "X" that can be read from the diagram. This is solved as follows: when a circle has one non-coloured area, the "X" must be placed there.

Categorical syllogisms can also be hidden in unspoken premises. It is then important to name the unspoken premises and to write out the categorical syllogisms step by step.

How can validity be tested?

In addition to drawing up a Venn diagram, there is an easier method to test the validity. This method is based on three simple rules (see below). These rules are based on two concepts: (1) affirmative and negative categorical claims and (2) the concept of distribution. Distribution occurs when a claim says something about each member of a category. There is no distribution if a claim does not say something about every member of a category.

A syllogism is valid if the following three rules are met:

• The number of negative claims in the premises must be the same as the number of negative claims in the conclusion.
• At least one claim must be the distribution of the "middle term" (M).
• A term that is distributed in the conclusion of a syllogism must also be distributed in the premises.

Example: (1) "All students are people," (2) "Some people are not employees." Conclusion: (3) "Some students are not employees". The term "people" is the M and is not distributed in both premises. The first premise is an A-claim and is not distributed in terms of predicate term and the second premise (an O-claim) is not distributed in terms of subject term. This syllogism therefore does not meet the criteria of rule two. This means that this argument is not valid.

Practice questions

1. What are categorical claims? Which four main types can you distinguish? Give an example of each.
2. What is a syllogism? What are the most important concepts that occur here? Why are they important to science?
3. Which four types of claims exist?
4. Which model can be used to describe these claims?
5. What does the "square of opposition" mean?
6. Name three categorical techniques that can be used to transform a claim.
7. What are categorical syllogisms?

Answers

1. Categorical statements are statements that say something about the group (category) of certain things. Categorical statements are statements about a specific category. The four main types are General Laws Affirmative (A), General Laws Denial (E), Observations Affirmative (I) and Observations Denial (O). Examples:

• A: All metals are conductors.
• E: No plastics are conductors.
• I: Some metals are conductors.
• O: Some metals are not conductors.

1. A syllogism is a deductive argument that is derived from two premises. The most important terms are:

• Major term: the term that serves as the predication term for the conclusion of syllogism, this is indicated by the letter P.
• Minor term: the term that serves as the subject term of the conclusion of the syllogism is indicated by the letter S.
• The middle term: the term that occurs in both premises but not in the conclusion is indicated by the letter M.

Syllogisms are important for science because with syllogisms you can draw a conclusion that is true from two arguments that are true. So you can check whether an argument is valid. An example:

• All Dutch people are consumers
• Some consumers are not VVD people
• Some Dutch people are not VVD people (conclusion)
• No VVD members = P
• Dutch = S
• Consumers = M

3. There are four types of claims: A- ('all ... are ...') ,, I- ('some ... are ...'), E- ('no ... are ...'), and O- ('some ... are not ... claims).
4. These claims can be described by means of Venn diagrams.
5. The square of opposition shows the relationships between different types of claims.
6. Three categorical techniques can be used to transform claims are: conversion, obversion and contraposition.
7. Categorical syllogisms are standardized deductive arguments.