# The Differences Between Statistical & Logical Arguments

Instructor: Paul Bohan-Broderick

Paul has been teaching many subjects in many different ways since he received his PhD in 2001.

Logical arguments and statistical arguments are both important, rigorous ways in which premises can support conclusions; however, there are several important differences, which we'll discuss in this lesson.

## Arguments in Real Life

A logical argument might argue from the knowledge that all people die to the knowledge that some particular person (say Socrates) will die. A statistical argument might argue that since the mammals in a sample give birth to live young (that is, don't lay eggs), then all mammals give birth to live young. Although this is a standard way to talk about these sorts of arguments, not all logical arguments move from general rule to particular observation. Many statistical arguments do not invoke a general rule in either their premises or their conclusions.

## Logical Arguments

### Premises and Conclusions

Logical arguments are also called deductive arguments. The argument about human mortality and the mortality of Socrates is probably the most famous deductive argument in history.

The conclusion of a deductive argument is implied by its premises with no possibility of error. There is no way for the conclusion to be false when the premises are true. Here's another way of saying this: if the premises of a deductive argument are true, then the conclusion must be true.

Please note that this definition does not claim that the premises of a logical argument are true! Validity is a relationship between the premises and the conclusion of a logical argument. Any argument in which the premises and the conclusion have this relationship is called valid. A logical argument that is valid and has true premises is called sound.

### Validity

Validity is a very high level of proof; in fact, it is the highest level of proof. How could an argument ever provide such a strong guarantee?

An old-fashioned way to answer this question is to claim that the conclusion can be found in the premises. A more modern approach would be to observe that all information in the conclusion is in the premises. A logical argument only uses information expressed by its premises. There is no missing information and no information is assumed.

The logical form of an argument expresses the relationship among its terms or elements. Deductive arguments are possible because they have a valid form. Any argument having a valid form will be valid. The truth of the premises will guarantee the truth of the conclusion, no matter what the argument is actually about.

### Example Argument: Modus Ponens

Consider the following (valid) argument with two premises and a conclusion:

• Premise 1: If the light is on, then the machine is working.
• Premise 2: The light is on.
• Conclusion: The machine is working.

To discover the form of this argument, remove any phrases that refer to the world outside the argument and replace them with letters. In these cases, the referential phrases are:

• ''the light is on'' (call this P) and
• ''the machine is working'' (call this Q)

The resulting form is:

• Premise 1: If P then Q
• Premise 2: P
• Conclusion: Q

This form is called modus ponens and it is valid. Try substituting P and Q with two phrases, any two phrases. Because of the relationship between the terms, it is impossible to substitute phrases for P and Q and produce an argument with true premises and a false conclusion. This can be proven more rigorously, but for now it should be enough to just try substituting phrases into the forms.

## Statistical Arguments

Statistical arguments belong to the class of inductive arguments. Inductive arguments do not prove that their conclusions cannot be false; however, they do present evidence for the conclusion.

### Strong Arguments

Successful statistical arguments are called strong arguments. A strong argument does not prove that its conclusions are true. Instead statistical arguments present evidence that the conclusion is true based on observations. Knowing that the premises of a statistical argument should make the conclusion more acceptable.

For example, suppose researchers are interested in discovering what types of television shows are most popular. As it would be very difficult to poll everyone who watches television, a researcher will ask a much smaller group of television watchers, called a sample. Suppose the poll shows that murder mysteries are the most popular type of show. Does this mean that murder mysteries are in fact the most popular type of show among all televisions viewers?

Maybe.

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