Thursday, May 28, 2015

Statistical Syllogism

A statistical syllogism is an inductive argument in which a statistical generalization is applied to a particular case. For example:

Most surgeons carry malpractice insurance.
Dr. Jones is a surgeon.
Therefore, Dr. Jones likely carries malpractice insurance.

This sort of argument can be written in the general form:

P1. Most A's are B
P2. x is an A
C. Therefore, probably x is a B

When the proportions are known the form can be written as:

P1. Z percent of A's are B
P2. x is an A
C. Therefore, it is probable to the .Z degree that x is B

In the general forms presented above, A is called the reference class,  B the attribute class and x is the individual object.

We often use informal versions of the statistical syllogism in everyday reasoning. For instance, if you read in the New York times that the President is visiting China and you believe it to be true, on what basis do you justify this belief? Most people understand that you can't believe everything you read in a newspaper but recognize that certain kinds of reports published in certain newspapers tend to be true. This is one of those kind of reports so it is likely true.

Strength/Weakness of a Statistical Syllogism
There are two primary standards which determine the strength of a statistical syllogism. First is the strength condition which is, the closer to 100% the reference class is to the attribute class the greater the confidence in the truth of the conclusion. Conversely, the closer to 0% the weaker the argument.

Second is the available evidence condition (also called the rule of total evidence) which requires using all available evidence in constructing or assessing such arguments. With statistical syllogisms this essentially means questioning if there is additional relevant information available concerning the individual object (x) that has not been included in the premises? Another way of saying this is that the individual object must be included in the reference class most specifically relevant to the conclusion. Failure to use all available evidence is commonly referred to as the Fallacy of Incomeplete Evidence.

For example:

P1. Sixty percent of students at the University believe in God.
P2. Fred is a student at the University.
C.  It is sixty percent probable Fred believes in God.

But if we also know that Fred is a history major and that only forty percent of history majors believe in God then it would not be appropriate to use the reference class in the example since it excludes this relevant information.

Due care must be taken when judging individuals using statistical syllogisms as their misuse can contribute to stereotyping and prejudice.

A Practical Study of Argument

Critical Thinking: An Introduction to Basic Skills

Critical Reasoning and Philosophy: A Concise Guide to Reading, Evaluating and Writing Philosophical Works

Wednesday, May 20, 2015

Analogical Arguments

An analogical argument is the use of a comparison between two or more things which have some similarity and from this basis inferring that they share some other property. The central topic which we want to draw a conclusion about is often referred to as the primary subject and the thing(s) to which the primary subject is compared to is called the analogue. The things the analogues and primary subject have in common are referred to as shared attributes. The attribute which the analogues possess that is being inferred to the primary subject are called the target attribute.

As described by Govier "An argument based on analogy begins by using one case (usually agreed on and relatively easy to understand) to illuminate or clarify another (usually less clear). It then seeks to justify a conclusion about the second case on the basis of considerations about the first. The grounds for drawing the conclusion are the relevant similarities between the cases, which show a commonality of structure."

The general form of an analogical argument is:

P1. A has properties p, q, r
P2. B has properties p, q, r
P3. A has property s
C. Therefore B probably has property s


Analogue = A
Primary Subject = B
Shared Attributes = p, q, r
Target Attribute = s

For example:

P1. John's brother and parents smoked two packs of cigarettes a day and ate fatty foods.
P2. John smoked two packs of cigarettes a day and ate fatty foods.
P3. John's brother and parents all died prematurely of heart attacks.
C. Therefore, John will probably die prematurely of a heart attack.

Here is another example in non-standard language:

Tom goes to Las Vegas for his first time. He goes into huge casino with lots of slot machines, gambling tables, bars and an all you can eat buffet. He goes into a second huge casino that also has lots of slot machines, gambling tables and bars. He becomes hungry and remembers the first casino had an all you can eat buffet and concludes that this casino probably as one as well.

Evaluating Analogical Arguments
The strength or weakness of an analogical argument depends upon a number of considerations:

Similarity - Verify that the properties proposed as being shared among the comparison group (shared attributes) do indeed exist. As analogical arguments are rarely actually presented in the form above, it may even be necessary to first list just how it is assumed the comparison groups are similar. Here is a simple example. "John is like Mike. Mike is smart. Therefore John must be smart". In this example none of the assumed similarities between John and Mike have been presented. Before the argument can carry any weight these similarities must be listed and verified.

Relevance - The more relevant the shared attributes are to the target attribute, the stronger the argument. Here is an example of an analogical argument which lacks relevance. "Book A and Book B both have a hardbound cover, pages, words on the pages and numbers at the bottom of the pages. Book A is a boring story. Therefore we can assume that Book B has a boring story." Though I have given a number of similar properties between Book A and Book B, none of these properties are relevant and thus do nothing to increase the probability that Book B is boring.

Number - The more shared attributes the primary subject and analogues share in common with each other, the stronger the argument. This is based on the notion that the more two things are alike, the more likely they also share the property stated in the conclusion. As stated above, relevance plays a key role in determining how much weight these similarities are given.

Disanalogy - Relevant disanalogies or dissimilarities must also be considered when determining the strength or weakness of an analogy. For example if I say, "I have known three people who have had surgery at this hospital with the same surgeon and they have all turned out successfully. Therefore Jane's surgery will also be a success." But what if the three success stories all had minor surgery and Jane is scheduled for a high risk procedure? This of course would be a very relevant disanalogy.

Critical Thinking Web: Analogical Arguments

Monday, May 18, 2015

Hasty Generalization

The hasty generalization is an informal fallacy in which an inductive generalization is made from a sample that is inadequate to support the generalization in the conclusion. As discussed in the post on inductive generalization, this may be because the sample is too small or biased.

Hasty generalizations often result from anecdotal arguments, which are short stories typically taken from the personal experience of the arguer. Generally, these anecdotal arguments describe only one or a few episodes which are then used to generalize about the population.

For example:

"Acupuncture works. My friend Tom tried it and he said it cured his back pain.".


"Smoking isn't harmful. My dad smoked a pack a day and lived until 95."

The Nizkor Project: Hasty Generalization
Fallacy Files: Hasty Generalization

Thursday, May 14, 2015

Inductive Generalization

An inductive generalization is an argument that moves from particular premises to a generalized claim. As defined by Trudy Govier "In inductive generalizations, the premises describe a number of observed objects or events as having some particular feature, and the conclusion asserts, on the basis of these observations, that all or most objects or events of the same type will have that feature."

P1 - Pavlovian conditioning caused dog Fido to salivate when a bell rings.
P2 - Pavlovian conditioning caused dog Rover to salivate when a bell rings. 
P3 - Pavlovian conditioning caused dog Spot to salivate when a bell rings.
P4 - (etc.)
C - Therefore, Pavlovian conditioning causes all dogs to salivate when a bell rings.

It seems intuitive that the strength of the example above largely relies upon how many particular instances Pavlovian conditioning resulted in a dog salivating. A thousand instances of a salivating dog would be a stronger argument than only ten instances. This leads us to the concept of sample.

"In inductive generalizations, features that have been observed for some cases are projected to others. Following established practice in statistics and in science, we call the observed cases the sample and the cases we are trying to generalize about the population." Statistical sampling methodologies are beyond the scope of this post but the basic idea is that the strength of an inductive generalization largely depends on sample size and how representative it is.  

In general, increased sample size is associated with a decrease in sampling error as it is more likely to represent the population (though there are diminishing returns). For more on why this is so, see the law of large numbers and central limit theorem.

A representative sample is one in which the selected segment closely parallels the whole population in terms of the characteristics that are under examination (for example, if one third of the population has relevant characteristic X, then one third of the sample should have characteristic X). We try to make samples representative by choosing them in such a way that the variety in the sample will reflect variety in the population.

Sampling methods include Random Sampling, Stratified Sampling, Systematic Sampling, Convenience Sampling, Quota Sampling and Purposive Sampling.

Monday, May 11, 2015

Induction: Inductively Strong & Inductively Cogent

Induction is the process of reasoning from one or more premises to reach a conclusion which is likely, though not certainly true. Hence, ainductive argument is one in which if the premises were true, then the conclusion is likely to be true.

"In the most general sense, Inductive reasoning is that in which we extrapolate from experience to what we have not experienced. The assumption behind inductive reasoning is that known cases can provide information about unknown cases."1 Govier goes on to describe inductive arguments as having the following characteristics:

1. The premises and the conclusion are all empirical propositions.
2. The conclusion is not deductively entailed by the premises.
3. The reasoning used to infer the conclusion from the premises is based on the underlying assumption that the regularities described in the premises will persist.
4. The inference is either that unexamined cases will resemble examined ones or that evidence makes an explanatory hypothesis probable.

Inductively Strong Arguments - An inductively strong (forceful) argument is one in which, if the premises were considered to be true, the conclusion is probably true. In other words, if we assume the premises are true the likelihood that the conclusion of an inductively strong argument is true is greater than 50%. If the probability that the conclusion is true is 50% or less, than the argument is inductively week.

Inductively Cogent Arguments - An inductively cogent argument is one which is inductively strong and all of its premises are actually true. 

To fit into our informal logic model, instead of requiring that the premises be true we would require that they be acceptable. This, of course, is due to the difficulty often encountered in establishing with certainty whether or not something is true. 

Types of inductive arguments include inductive generalization, statistical syllogism and analogical arguments.

1 A Practical Study of Argument

Critical Reasoning and Philosophy, A Concise Guide to Reading, Evaluating, and Writing Philosophical Works 

Probability and Induction

Thursday, May 7, 2015


An outline is a method of presenting the main and subordinate ideas of a document by organizing them hierarchically.

Alphanumeric outline
An alphanumeric outline uses Roman numerals, capitalized letters, Arabic numerals and lowercase letters as prefix headings. 

The Chicago Manual of Style (CMS) uses the following outline format:

I. (Roman numeral) 
     A. (Capital letter) 
          1. (Number) 
               a) (Lowercase letter followed by closing parenthesis) 
                    (1) (Number enclosed in parenthesis) 
                         (a) (Lowercase letter enclosed in parenthesis) 
                              i) (Roman numeral with lowercase letters followed by a closing parenthesis)

The Modern Language Association (MLA) uses essentially the same method except the first lowercase letter is followed by a period instead of a closing parenthesis.

Decimal outline
The decimal outline uses only numbers as prefix headings making it easier to see how every item relates within the hierarchy. It uses the following outline format:

Here is a sample decimal outline:

1.0 Choose Desired College
     1.1 Visit and evaluate college campuses 
     1.2 Visit and evaluate college websites 
          1.2.1 Look for interesting classes 
          1.2.2 Note important statistics