**ARGUMENTS**by Dr. Debra J. Campbell, Ph.D.

(copyright March, 2017)

**DEFINITION**: An argument is a group of statements intended to prove some point. One of those statements is the statement you are trying to prove, and that is called the

**The other statement (or statements) provides the evidence or support for the conclusion and those statements are called the**

*conclusion.***Thus, an argument is made up of a single premise (or multiple premises) and a conclusion. It should be noted that each premise is a single statement and the conclusion is a single statement. Here are a couple of examples of arguments presented in what is called the “standard form” for an argument. [P1 and P2 are the premises and C indicates the conclusion.]**

*premise(s).*P1: All politicians desire power.

P2: No true statesmen desire power.

C: Therefore, no politicians are true statesmen.

P1: For all of recorded history, the sun has appeared to rise in the east.

C: So, the sun will rise in the east tomorrow.

Before looking at more examples of arguments, it should be noted that arguments fall into two categories:

**INDUCTIVE**and

**DEDUCTIVE**. Let’s look at inductive arguments first.

**INDUCTIVE ARGUMENTS**

DEFINITION: Inductive arguments are based on reasoning from past observation and experience to conclusions about the future. Because these arguments are based on previous experience and because the future can always change, this type of argument can lead only to

**conclusions.**

*probable*EXAMPLES OF INDUCTIVE ARGUMENTS:

**Prediction**– Five ASU students went to the bookstore to sell their books back. It’s quite likely there will be a big party tonight!!

**Inductive Generalization**– Maria dated three men who all complained that women spend too much money on beauty spa treatments. So, all men think that women spend too much money going to beauty spas.

**Argument by Analogy**– Barry took a logic course. He rarely attended class, rarely studied, but went to lots of parties! He received a “D” in the course. Mary is also taking a logic course, and she rarely attends class, never studies, but parties just as hard as Barry! So, Mary will probably receive a “D” in the course as well.

**Argument from Authority**– Stephanie Myers has written several best-selling books about vampires. In her latest book, she wrote that if you are pregnant with a half-vampire baby, you will need to drink a lot of blood. So, that’s probably true.

**Causal Inference**– Elaine’s car slowly sputtered to a stop as she was driving down the street. After pulling over and unleashing a string of curse words, she looked at the gas gauge and the pointer was below the “E” for empty. So, she probably ran out of gas.

**Arguments based on Signs**– The sign out in front of Four Peaks Brewery warns anyone trying to use a fake identification that all fake identifications will be confiscated. After seeing the sign, the three underage ASU students decided it would be best to go someplace else.

**DEDUCTIVE ARGUMENTS**

DEFINITION: A deductive argument is based on reasoning that will (when formulated correctly) lead to a necessary conclusion. The “necessity” of a deductive argument stems from a logical relationship such as one based on math or definition or rules of formal logic, as in a “syllogism.” Syllogistic reasoning was first recognized by the philosopher, Aristotle, who demonstrated that people often state arguments in a format of exactly two premises that lead to some necessary conclusion. Furthermore, if the premises of a deductive argument are true, then the conclusion that is necessitated by those premises will also have to be true. Here is one of the most famous examples of a deductive argument. This deductive argument is in the form of a “categorical” syllogism:

P1: All men are mortal.

P2: Socrates is a man.

C: Therefore, Socrates is mortal.

EXAMPLES OF DEDUCTIVE ARGUMENTS:

**Arguments based on Mathematics**– George Clooney is 56. His wife, Amal, is 39. So, George is 17 years older than his wife.

**Arguments based on Definition**– Zac Efron is a famous Hollywood bachelor. Therefore, Zac Efron is unmarried.

**Categorical syllogism**– According to the comedian, Tim Allen, “All men are pigs.” Tim Allen is a man. So, Tim Allen is a pig. [Tim Allen must not have taken a logic class.] J

**Hypothetical syllogism**– If Bella marries Edward, then Jacob will be mad. Bella married Edward. Jacob is mad.

**Disjunctive syllogism**– Either South Park or the PBS Newshour is a controversial TV show; the PBS Newshour is not controversial. Therefore, South Park is controversial.

**EXERCISES**

**(Induction vs. Deduction)**

Are the following arguments inductive or deductive? (Answers given at the bottom.)

1. In the history of the United States, the president of the United States has always been a man. Therefore, the next president will most likely be a man. ___________________________________________

2. Since A is greater than B, and B is greater than C, it necessarily follows that A is greater than C._______________________________

3. Without anyone touching them, the newspapers piled on the table suddenly fell to the floor. This happening proves that evil spirits do indeed exist. ___________________________________________

4. Either Taiwan or China is a member of the U.N. Taiwan is not recognized as a member. Therefore, China is a member. ___________________________

5. All men are lawyers. John is a man. Therefore, John is a lawyer. ____________________________________________.

Answers: (1) Inductive, (2) Deductive, (3) Inductive, (4) Deductive, (5) Deductive

**INDUCTIVE ARGUMENTS**

*STRENGTH AND COGENCY*Definition of

**: An inductive argument is**

*Strength***when the premises (that is, the reasons or evidence) support the conclusion in a highly probable way. So, if the premises are true, then the conclusion is very probably true as well. If, for any reason, the premises do not provide much support for the conclusion, then the argument is considered**

*strong*

*weak.*Example of a

**Strong**argument – All the previous presidents of the United States have been men. So, the next president of the United States will most likely be a man.

Example of a

**Weak**argument – Madonna was considered one of the “hottest” female vocalists of the 1980’s. Therefore, she is still one of the hottest female vocalists today.

Definition of

**: An inductive argument is considered**

*Cogency***when it is both**

*cogent***and has**

*strong***. If an inductive argument is strong but has false premises, then it is automatically uncogent.**

*true premises*Example of a

**Strong and Cogent**argument – All of the previous presidents of the United States have been men. So, the next president of the United States will most likely be a man. (

**Strong inductive reasoning and true premise, so cogent**)

Example of a

**Strong but Uncogent**

**argument because it has a false premise**– All of the previous presidents of the United States have been women. So, the next president of the United States will most likely be a woman. (

**Strong inductive reasoning, but false premise, so uncogent**)

**There is no example of a Weak and Cogent**argument because, by definition, none can exist. A Cogent argument must first be Strong, and then it must also have true premises. If an inductive argument is weak, then it may have true or false premises, but it is still uncogent because it is weak:

Example of a

**Weak (Uncogent) argument with a true premise**– Since no woman has ever been the Republican candidate for the Presidency of the United States that is long overdue! So, the next Republican candidate will most likely be a woman!! (

**Weak inductive reasoning, but true premise, still uncogent**)

Example of a

**Weak (Uncogent) argument with a false premise**– Elvis Presley is still alive somewhere, so if he wanted to make a comeback his music would still be wildly popular. (

**Weak inductive reasoning and false premise, so uncogent**)

**EXERCISES**

**(Induction: Strength and Cogency)**

Please note whether the following inductive arguments are weak or strong. (Answers given below.)

1. I went to Houston last Thursday and I saw at least three people wearing cowboy boots. Therefore, all Texans must wear cowboy boots. ____________________

2. The rainfall in Seattle has been over 15 inches per year for the last 30 years. Therefore, next year the rainfall in Seattle will probably be over 15 inches. __________________________________________________

Answers: (1) Weak, (2) Strong

Please note whether the following inductive arguments are cogent or uncogent. Remember that a

**argument exhibits**

*cogent***inductive reasoning and the premise(s) must be**

*strong***. (Answers given below.)**

*true*- In the past, all the previous presidents of the United States have also held the position of Commander in Chief of our armed forces. So, the next president of the United States will also hold the position of Commander in Chief. __________________________________________________________________

- There are a significant number of beautiful mansions in Beverly Hills, California. So, everyone who lives in California must be very rich.___________________________

Answers: (1) Cogent, (2) Uncogent

**DEDUCTIVE ARGUMENTS**

*VALIDITY AND SOUNDNESS*Definition of

**: A deductive argument is**

*Validity***when there is a certain conclusion necessitated by the premises based on math, or definition, or a rule-based logical relationship (as in a syllogism). Consequently, if the premises of a valid deductive argument are true, then the conclusion that follows necessarily (logically) must be true as well. If the premises of a deductive argument are true and the conclusion turns out to be false, then either at least one of the premises must be false, or the math, definition, or logical relationship is not correct and therefore the argument is invalid.**

*valid*Example of a

**Valid**argument – Liam Hemsworth is a bachelor. Therefore, he is not married.

**(Valid because it follows necessarily from the definition)**

Example of an

**Invalid**argument – Jennifer Aniston is 43. Angelina Jolie is 36. Therefore, Jennifer Aniston is 10 years older than Angelina Jolie.

**(Invalid because the math is done incorrectly)**

Definition of

**– A deductive argument is deemed**

*Soundness***when it is both**

*sound***and has**

*valid***. So, if the reasoning of a deductive argument makes it invalid, then it cannot be sound whether the premises are true or false. If a deductive argument is valid, but has even one false premise, then it is unsound as well.**

*true premises*Example of a

**Valid and Sound**argument – Oprah Winfrey is a billionaire. She is also a famous television star. So, there is at least one famous television star that is a billionaire. (

**Valid reasoning and true premises**)

Example of a

**Valid but Unsound**argument – All women are bad drivers. Lindsay Lohan is a woman. Therefore, Lindsay Lohan is a bad driver. (

**Valid reasoning, but a false premise, so unsound, even though the conclusion might be true.)**

Examples of

**Invalid and Unsound**arguments – Some millionaires pay more than 30% of their income in taxes. Mitt Romney is a millionaire. Therefore, Mitt Romney pays more than 30% of his income in taxes. (

**Invalid reasoning, so even though the premises are true, it is still Unsound)**

Some millionaires pay more than 30% of their income in taxes. The average student at Mesa Community College is a millionaire. Therefore, all Mesa Community College students should pay more than 30% of their income in taxes. (

**Invalid reasoning and false premises, so Unsound)**

**There is no example of an Invalid and Sound argument**because, by definition, such an argument cannot exist. If an argument is Invalid, meaning that the reasoning process (or logical relationship of the premises to the conclusion) does not guarantee that the truth of the premises will necessitate the truth of the conclusion, then the argument cannot be sound. The definition of a sound argument is one that is VALID and has true premises.

**EXERCISES**

**(Deduction: Validity and Soundness)**

Please note whether the following arguments are valid or invalid. (Answers given below.)

- Because triangle A is exactly the same as triangle B, and triangle A is an isosceles triangle, it follows that triangle B is an isosceles triangle. _______________________________
- Some women are mothers. Some mothers are over-protective. So, all women are over-protective. _______________________________

Please note whether the following arguments are sound or unsound. Remember that a

**argument exhibits**

*sound***deductive reasoning and has**

*valid***premises.**

*true*- All men are mortal. Barack Obama is a man. Therefore, Barack Obama is mortal. __________________________
- All men are lawyers. John is a man. Therefore, John is a lawyer. __________________________________

*A LESS FORMAL EXPLANATION OF HOW TO KEEP INDUCTIVE AND DEDUCTIVE STRAIGHT***(I wrote this as an informal answer to a friend)**

Whether an argument is inductive or deductive is not about how "likely" the conclusion is to be true . If you are talking about "the odds," then it is inductive reasoning. In fact, if you are using the word "probability" it is, by definition, "inductive." Deductive arguments, on the other hand, are arguments that can be presented such that, if the premises are true, the conclusion will follow NECESSARILY (not probably) based on the way the argument is constructed, that is, it is based on math, definition, or logic.

Here's one way to keep the distinction clear: if your reasoning is based on past observation or experience (which usually means you are making a prediction), then it is necessarily inductive, because you cannot predict the future with certainty. For an explanation of Hume's argument on "the problem of induction" see: plato.stanford.edu/entries/induction-problem/

Inductive reasoning is based on past experience and therefore is probable, not necessary. The only way you can make a "deductive" argument or "deduce" something is if you are basing it on logic (syllogistic, propositional, or predicate), math, or definition. The examples below are similar to the examples in my handout:

Example #1:

A bachelor is an unmarried man.

John is a bachelor.

Therefore, John is an unmarried man.

Based on the syllogistic presentation, this argument is classified as deductive. It is constructed in the format of a valid categorical syllogism.

Of course, whether any of those statements are true or false is a separate question. But IF you assume the two premises are true, then the truth of the conclusion follows necessarily (by the logical structure). There is no way to say that those premises are true and the conclusion is false without admitting that one of the premises must be false. That is the definition of deductive validity.

Example #2:

John is a bachelor.

Therefore, John is an unmarried man.

This argument is not a "syllogism" but it is "deductive" because it is an argument based on the definition of "bachelor." If the definition is correct, then the argument is valid. If the definition is wrong, such as, "this object has four sides, therefore it is a triangle," then that argument is still deductive (the reasoning is still based on an argument by definition), but now it is invalid because the definition is incorrect.

Example # 3:

There were 25 students on my roster.

I removed 5 students from the roster.

Therefore, there are 20 students remaining on my roster.

This is deductive, because it is an argument based on mathematics. Assuming all of those statements are true, the conclusion will be true necessarily based on the math. There is no way to avoid the truth of that conclusion without saying that either the math is wrong, or one of the statements is false. Again, that is what we mean by deductively "valid."

But, if I say something like, "last time I saw John he was unmarried, therefore I think he is a bachelor," that is actually inductive reasoning, because I am basing it on past experience and predicting something about the future. Even though this sounds a bit like an argument based on definition and therefore "deductive," you have to carefully consider the wording and this person is arguing that "the last time I saw John he was unmarried," and then draws the conclusion about his marital status today, so that is inductive reasoning. At best, that conclusion only follows probably, and not necessarily.

Make sense? The wording of the argument can make all the difference.

Here's another example:

Smoking is bad for you. So, if you keep smoking, you will probably get sick.

That's inductive. It's a prediction.

But if you put it in this format:

Smoking is bad for you.

You smoke.

Therefore, you are doing something that is bad for you.

That's a deductive categorical syllogism. If the premises are true, then the conclusion follows necessarily.

Here's another way to state the argument deductively:

If you continue to smoke, you are more likely to develop lung disease.

You continue to smoke.

Therefore, you are more likely to develop lung disease.

This is a valid hypothetical syllogism (based on if/then reasoning).

It's all in how you state your argument.

To summarize, if your reasoning is based on past observation and experience, then it's inductive. To be "deductive" it must be based on math, definition, or presented in a formal manner following the rules of categorical or propositional logic.