guide to formal logic questions generalisation statements

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Guide to Formal Logic Questions –

Generalisation Statements

Contents Introduction - Premise Types .................................................................................. 1

Generalisation Statements...................................................................................... 1

Basic logical equivalents ......................................................................................... 2

Additional generalisation principles: ........................................................................ 7

Introduction - Premise Types Formal logic tests are comprised of questions in which an informative paragraph is given. These paragraphs should be considered as true. Multiple-choice answer responses are then presented, from which only one should be chosen. Your answer will be based either on the only response choice that can be validly inferred from the given text or the only response choice that cannot be validly inferred from the text (depending on the specific instructions presented in each question).

The informative paragraph is comprised of a few logical statements, which should be considered as premises. Each response choice will relate to at least one premise, and you should decide if it is logically equivalent or can be inferred from that premise.

In this series of guides, you will become familiar with all the possible premise types for these questions. You will also learn how to conclude if the statements presented in the response choices actually can or cannot be concluded from the given information.

There are two types of premises used to convey important information in logical reasoning questions:

1. Generalisation statements – stating that something always happens with something

else (A always leads to B).

2. Existence statements – stating that something sometimes happens with something

else (A sometimes leads to B).

In this guide, you will learn about generalisation statements.

Generalisation Statements The basic building block of logical reasoning is the ability to draw conclusions or inferences from generalising statements.

There are two basic forms of generalisation, which have a similar meaning:

1) 'All' statements: All A are B (all police officers on duty wear uniforms)

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2) 'If-then' conditional statements: If A then B (If a police officer is on duty then

he is wearing a uniform)

There are other ways of formulating these generalising statements (A is always B; whenever A then B, etc.). However, the two aforementioned statements are the most common and will be used interchangeably. A good rule of thumb is that as long as the sentence can be understood in a way that means that every time A happens (a police officer is on duty) then B follows (he is wearing uniform), these different formulations will have an identical meaning in the eyes of formal logic.

Important: Note that generalising statements (A leading to B) are comprised of two clauses: the condition clause (A) which always leads to the outcome clause (B). As long as the original directionality between condition and outcome stays the same, the sentence can be rephrased but stay logically equivalent to the original. An example of rephrasing whilst keeping the original meaning would be: They are wearing uniforms, if they are police officers on duty (B A). Although the clauses' positions were exchanged, the meaning stayed the same, as the clause roles (condition and outcome) were not changed. A still leads to B!

However, when directionality of condition and outcome are reversed, the new statement is not equivalent and cannot be inferred from the original: e.g. If they are wearing uniforms, then they are police officers on duty (B A). This is a wrong inference from the original statement, as there could be other uniformed people who are not police officers (like firefighters, office managers, etc.)! The whole meaning was changed as the cause and outcome clauses were transposed. Now B leads to A, which is an incorrect assumption.

The next step is understanding what can and cannot be inferred from a generalisation type statement…

Basic logical equivalents Most formal logic questions are based on inferring information from generalisation premises. Below, you will learn a system that will help in determining whether a statement is logically equivalent to the original premise (see in the next part, the 'NOT triangle'). But first, you will learn to intuitively identify what is logically equivalent to a generalisation.

Key operations - in generalisations A generalisation statement can be manipulated in certain ways whilst still maintaining its original meaning. It is useful to keep these possible manipulations in mind when comparing two statements to determine if they are logically equivalent (if one can be inferred from the other). Some key differences are crucial and should be actively sought out.

In generalisation type statements (if A then B), there are three important operations:

1. Transpose – the exchanging of the cause outcome clauses (if B then A).

2. Negative – the negating of one or both clauses (if not A then not B). Note that

double negatives resolve into positives.

3. Adding/removing the 'only' keyword – the use of which changes the logical meaning

of a generalisation (only if A then B).

The 'NOT triangle' These three operations are easily remembered by their initials–NOT (Negative, Only, Transpose), forming the 'NOT triangle':

Negative



Only Transpose

'NOT triangle' rule: The 'NOT triangle' accepts two or nothing! This means that using any two of the three possible operations does not change the meaning of the statement! However, using only one of the operations changes the meaning of the statement and therefore is not logically equivalent.

Example 1:

The statement that all A are B (all police officers on duty wear uniforms) is equivalent to the following three statements:

1. Transpose + negative: All not B are not A

(All non-uniformed personnel are not police officers on duty) 2. Transpose + Only: Only if B then A

(Only if a person is wearing uniform is he a police officer on duty) 3. Negative + Only: Only if not A then not B

(Only if a person is not a police officer on duty is he not wearing a uniform)

Example 2:

The statement that all A are not B (all FBI agents have no criminal records) is equivalent to the following three statements:

1. Transpose + negative: If B then not A

(If you have a criminal record then you are not an FBI agent). *Note that when negating the original clause–not B (no criminal record), which was already negative–the double negative not not B cancels out into a positive.

2. Transpose + Only: Only if not B then A

(Only if a person doesn’t have a criminal record is he an FBI agent) 3. Negative + Only: Only if not A then B

(Only if a person is not an FBI agent does he have a criminal record) *Again, the negation of an already negative clause–not B–cancels out into a positive.



Acquiring an intuitive understanding of logical equivalents: Through the use of Venn diagrams, logical reasoning principles can be understood intuitively.

Let's consider the initial statement (all A are B) using Venn diagrams:

The small A circle is an event that could happen in the world, whilst the B circle is a second possible event that includes A (A is totally engulfed in B) but could also include other events besides A. The Ω symbol means all other possible events in the world.

Transposing: all A are B is not logically equivalent to all B are A Because event A is completely enclosed in B, if event A has happened in the world (a person is a police officer on duty), then B has also happened as well (a person is wearing a uniform).

The transposing of the initial statement (all B are A–all people wearing uniforms are police officers) is not logically equivalent, because the initial statement regards a generalisation of all of event A. Whilst we have knowledge about all police officers on duty (the whole A circle–all A are B), we lack complete knowledge of event B (i.e. about all people who wear uniforms).

Example 3:

The statement that only if A then B (only if a firearm was fired would it have left cartridge discharge residue) is equivalent to the following three statements:

1. Transpose + negative: only if not B then not A

(Only if there is no cartridge discharge residue a firearm has not fired) 2. Transpose + Only: if B then A

(If cartridge discharge residue was left a firearm had fired) *Note that when the keyword 'only' is already present in the original statement, the 'NOT triangle' operation is to remove the word 'only' instead of adding it.

3. Negative + Only: if not A then not B

(If a firearm was not fired no cartridge discharge residue could have been left) *Again, the keyword 'only' is removed because it was present in the original statement.

A

B Ω



As can be seen, if event B has happened (a person is wearing a uniform), event A may either have happened (he is a police officer on duty) or not have happened. Some other event C could have happened instead of A (he is actually a firefighter on duty).

Negating: all A are B is not logically equivalent to all not A are not B As mentioned, we have complete knowledge of what event A entails (what happens if a person is a police officer on duty). However, we can't assume any knowledge of what happens if A has not occurred (if a person is not a police officer on duty)!

As can be seen in the diagram, there are a lot of options if event A does not happen! Thus, if A did not happen (a person is not a police officer on duty), then B could still possibly have happened (the person is not a police officer and is uniformed). However, it could also not have happened (the person is not a police officer and is not uniformed).

Adding the 'only' keyword: all A are B is not logically equivalent to only A are B (or only if A then B) The use of the keyword 'only' can be counterintuitive at times, making the 'NOT triangle' system valuable for avoiding confusion. Basically, as mentioned before, we have full knowledge of A (all officers on duty). However, we cannot assume from this that B is exclusive to A (not all uniformed personnel are police officers on duty). Thus, it cannot be inferred that only A are B. (It is not correct to assume that only police officers on duty wear uniforms.)

A

B Ω

A

B Ω

B

A Ω



As can be seen from the diagram, the generalisation that only A are B is equivalent to stating that all B are A (using the 'NOT triangle' rule: transposing and removing the 'only' keyword). Neither of these statements are logically equivalent to the initial statement that all A are B.

Using the 'NOT triangle' rule to find logical equivalents: Next, let's examine the 'NOT triangle' rule, according to which using two 'NOT' operations doesn't change the meaning of a generalisation.

All possible combinations of two 'NOT' operations yield three logical equivalents. This means that every generalisation can be manipulated whilst maintaining its original meaning in the following ways:

1. Transposing + negating (transposed negative) – the most common one.

2. Transposing + adding/removing the 'only' keyword.

3. Negating + adding/removing the 'only' keyword.

1. Transposing + negating (transposed negative): all not B are not A

A generalising statement (all A are B) is logically equivalent to the transposed negative formulation of the same statement (all not B are not A). Note that double negatives are resolved into positives. E.g.: All police officers on duty wear uniforms (police officer on duty uniform) is logically equivalent to its transposed negative–if a person is not wearing a uniform, he is not a police officer on duty (no uniform not a police officer on duty).

*This logical equivalent is by far the most common one and will likely be used frequently on your logical reasoning test.

As can be seen, if event B has not happened, event A could not have happened as well. Therefore, stating that all not B are not A is a valid conclusion from the initial statement that all A are B.

2. Transposing + adding/removing the 'only' keyword: only B is not A A generalising statement (all A are B) is logically equivalent to its transposed formulation whilst adding the 'only' keyword (only if B then A). Note that if the keyword 'only' is already present in the initial statement, then the word 'only' is cancelled out and removed. E.g.: If a person is an officer on duty, then he is wearing a uniform (officer on duty uniformed) is logically equivalent to its transposed version whilst adding the word 'only'–only if a person is wearing uniform, can he be an officer on duty (only if uniformed then a police officer on duty).

A

B Ω



As can be seen in the diagram, the statement that only if B then A means that event B happening is a requirement for event A to happen. This is logically equivalent to the original statement that if A then B.

3. Negating + adding/removing the 'only' keyword: only if not A then not B A generalising statement (all A are B) is logically equivalent to its negative formulation (negating both clauses) if the 'only' keyword is also added (only if not A then not B). Note that if the keyword 'only' is already present in the initial statement, then the word 'only' is cancelled out and removed. E.g.: If a person is an officer on duty, then he is wearing a uniform (officer on duty uniformed) is logically equivalent to its negated version with the word 'only' added–only if a person is not an officer on duty, can he not be uniformed (only if not a police officer on duty then not uniformed).

* This is the rarest and most counterintuitive logical equivalent.

As can be seen in the diagram, the statement that only if not A then not B means that event A not happening is a requirement for event B not to happen. If A happens, then event B had to have happened, which is equivalent to the initial statement if A then B.

Additional generalisation principles:

Negative generalisations To avoid confusion, it is advised to write down generalisation statements in positive form, using the words 'all' or 'if-then', rather than the words 'no' or 'never'. In logical formulation, use the generalisations all A are B or if A then B, rather than their negative form equivalents–there is no A which is not B or A is never not B. E.g., you should write down the negatively formulised generalisation–there are no police officers on duty who do not wear uniforms–as a positive one–all police officers on duty wear uniforms. (Notice how the double negatives–no, do not wear uniforms–resolve into a positive). Note that the phrasing, 'there are no A which are', is a negative formulation signifying what never happens when event A does happen (for all police officers on duty, it is never the case that they do not wear uniforms). This is not the same as 'not A', which is the negation of event A (as in

A

B Ω

A

B Ω



the 'NOT triangle'), and signifies what happens when event A does not happen (if someone in not a police officer).

If and only if (IFF) If and only if A then B is a biconditional statement, combining the condition if A then B with the condition only if A then B. This combination means that the topic and subject of the statement are exclusive to each other and are thus interchangeable. In other words, A always happens with B and never without B, whilst B always happens with A and never without A! In logical formulation, if and only if A then B is logically equivalent to the following four statements: if A then B; if B then A; if not A then not B; if not B then not A.

Train of generalisations If A B and B C, then A C. This principle is valid when there is a shared clause between two generalisations, connecting them. E.g., from the statements if you are an officer on duty then you are wearing a uniform and if you are wearing a uniform then you look formal, it can be inferred that if you are an officer on duty then you look formal.

One outcome, two conditions: If an outcome has two conditions (if A then C and also if B then C), the logical equivalents apply to all conditions (if not C then not A and not B). For example, if a person is a police officer on duty, or if he is a firefighter on duty, then he is wearing a uniform. The transposed negative equivalent of this generalisation is (transposing and negating both clauses is logically equivalent according to the 'NOT triangle'): If a person is not uniformed, then he cannot be a police officer on duty or a firefighter on duty.

Good luck!


guide to formal logic questions generalisation statements

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