PHIL102 Study Guide

2a. Construct valid and sound arguments using the standard form of an argument

• What is a valid argument?
• What is a sound argument?
• What is standard form for an argument?

The statements that constitute an argument are either true or false. When we argue, we typically believe the statements we assert are true. More specifically, we typically believe that the evidence or justification we provide for a claim is true, and that this truth makes the claim at issue true, as well. In other words, we typically believe our premises are good reasons to accept our conclusion. The statement that is supported is the conclusion, while the statement(s) that support(s) the conclusion are the premise(s).

There are plenty of occasions, however, when we don't know whether or not our premises are true. Moreover, there are premises some people may believe are true, while others do not – these premises themselves require justification. So, how do we evaluate such an argument?

One evaluation standard is validity. A valid argument is one whose premises guarantee the conclusion in virtue of the argument's structure. An invalid argument's premises do not guarantee the conclusion. In other words, when an argument is valid, it is impossible for the conclusion to be false, if the premises are true. When we assess an argument for validity, we assume the premises are true – even if we know they are false!

Consider these two examples:

The moon is made of blue or green cheese.
It's not the case that the moon is made of blue cheese.
The moon is made of green cheese.

The mathematical formula for the area of a triangle is either A = ¼ Base x Height or A = ½ Base x Height.
It's not the case that the mathematical formula for the area of a triangle is either A = ¼ Base x Height.
The mathematical formula for the area of a triangle is A = ½ Base x Height.

Notice that the form is the same for each argument (it is called disjunctive syllogism):

A or B
Not A
B

The content is irrelevant to the correctness of the form. So, whatever sentences we plug into the form, we have a valid argument. Another way to put this is that we assume the premises are true, even if we know, as in the case of the moon argument, that at least one is actually false. On this assumption, we cannot 'make' the conclusion false while the premises are true; doing so results in a contradiction. Think about it this way: If we have A or B, and it turns out we don't have A, then it's impossible not to have B – we must have B on pain of contradicting our premises.

Validity is a crucial concept in logic, but it is not the gold standard, as it were. A sound argument is even better. That is because a sound argument is valid, and its premises are true. So, the difference between the moon and math arguments is that the first is valid, but not sound (the first premise is false), while the second is sound (it's not only valid, but both its premises are true).

Finally, the presentation of each argument example is in standard form. Organizing your argument by writing out the premises and conclusion in list form makes the structure clearer. You can present it as seen above, or something like this:

P1: A or B
P2: Not A
C: B

Remember, just because an argument's premises and conclusion are (actually) true does not mean the argument is valid. A valid argument makes it impossible for the conclusion to be false. Remember also that the premises are just those statements we take to be true for the purpose of 'seeing' whether or not they yield the conclusion.

To review, see:

2b. Determine if a counterexample exists for a given argument

• How does the counterexample method work?

A valid argument's premises cannot yield a false conclusion. If we believe the argument is invalid, we can use the counterexample method to test it. The method consists in preserving the form of the reasoning, while replacing the content. More specifically, we replace the given conclusion with one we know is false, while replacing the original premises with statements we know to be true. If the argument resists the counterexample, it's very likely the argument is valid. The crucial point, however, is that, if successful, the counterexample method shows the argument is invalid.

Here are some examples, which are "imposters" of the valid arguments (see Validity, Soundness, and Valid Patterns):

 Name of Invalid Form Invalid Form Counterexample Affirming the Consequent If A then BB        A If I have 10 pennies, I have 10 cents.I have 10 cents.I have 10 pennies.   The conclusion could be false, e.g., I could have a dime. Denying the Antecedent If A then BNot A Not B If I have 10 pennies, I have 10 cents.It's not the case that I have 10 pennies.It's not the case that I have 10 cents.   The conclusion could be false, e.g., I could have a dime. No Name in Common Use If A then BIf A then CIf B then C If I have 10 pennies, I have 10 cents.If I have 10 pennies, I have a dime.If I have 10 cents, I have a dime.   The conclusion could be false, e.g., I could have two nickels. No Name in Common Use A or BA        Not B I have 10 pennies or I have a dime.I have 10 pennies.I don't have a dime.   This one is tricky, because we typically think of "either…or" claims as exclusive, i.e., we think of the "either…or" as one or the other, but not both. In logic, however, the or is inclusive. Think of it this way: If I have 10 pennies, it follows I have 10 pennies or a dime. Since we don't know whether or not I do have a dime, it's possible the statement, "I have a dime", is true. Hence, the disjunctive statement is true when I have at least one of the disjuncts (and I could have both)! No Name in Common Use If A then B, and, if C then DA or CNot either B or D If I have 10 pennies, I have 10 cents, and if I have four quarters, I have a dollar.I have 10 cents or a dollar.I have 10 pennies or four quarters.   The conclusion could be false, e.g., I could have two nickels and a dollar bill. No Name in Common Use Some A are BSome B are CSome A are C Some apples are green fruits.Some green fruits are avocados.Some apples are avocados. No Name in Common Use All A are BAll A are CAll B are C All dogs are four-legged creatures.All dogs are canines.All four-legged creatures are canines.   The conclusion is false, e.g., horses are four-legged, but not canines.

To review, see Validity, Soundness, and Valid Patterns

2c. Illustrate valid argument patterns such as modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, constructive dilemma, and reductio ad absurdum

• What is modus ponens?
• What is modus tollens?
• What is hypothetical syllogism?
• What is disjunctive syllogism?
• What is constructive dilemma?
• What is a reductio ad absurdum?

In formal logic, the actual truth value of a sentence is not at issue. That said, like other systems of deductive argumentation, natural deduction aims at certainty by way of a valid argument form. So, when we consider any valid inference in this system, the possible truth values for a given connective drive the assembling and disassembling of a statement.

There are many more valid arguments than those presented in this unit. Modus ponens (MP) is a Latin phrase translated as "mode of affirmation", whereas modus tollens (MT) is a Latin phrase translated as "mode of denial". Hypothetical syllogism (HS), disjunctive syllogism (DS), and constructive dilemma (CD) are arguably the most common – you'll see them in almost any logic text.

The reductio ad absurdum (RAA) is any argument that proceeds by arguing to a logical absurdity, which is a contradiction. So, for example, modus tollens is a version of the reductio, as is destructive dilemma, and the negation introduction or indirect proof in symbolic logic.

Recall the discussion of necessary conditions and sufficient conditions from Unit 1. These are helpful to understanding how MP, MT, and CD work. We can also re-formulate the disjunctive claim in DS as a conditional claim, which is equivalent to the original. In so doing, we can see that the concepts of sufficiency and necessity apply here, as well.

Let's look at each focusing on the notion of assembling and dismantling a statement at its connectives or "joint". We will use that focus to see how a statement's truth-functionality plays out in a given inference. As you look through each argument form, think about how the truth definitions guarantee each inference – and remember the definition of a valid argument: If the premises are true, the conclusion can't be false.

Modus ponens, also known as "affirming the antecedent", yields the consequent of a conditional claim when the antecedent is affirmed. In other words, the conditional claim asserts that the antecedent (what comes after the "if" clause) guarantees the consequent (what comes after the "then" clause). Notice, however, that the conditional claim does not assert that the antecedent obtains. After all, it is a hypothetical. This is where the affirmation of the antecedent comes in – it is the premise in the reasoning that, combined with the conditional claim in the other premise, yields the consequent.

Form of Modus Ponens:

If A then B
A
B

Example:

If the animal is a dog, then it's a canine.
The animal is a dog.
The animal is a canine.

Modus tollens denies the consequent of a conditional statement, thereby concluding (i.e., on the basis of the conditional statement and denial), that the antecedent is false. Think of it as a way to show that, when the necessary condition is false, that of which it is a condition is also false. Another way to think about it is in terms of what happens when we reason something like this: "Without fuel, a fire cannot start. So, since there is no fuel to burn, we cannot start a fire".

Form of Modus Tollens:

If A then B
Not B
Not A

Example:

If the animal is a dog, then it's a canine.
The animal is not a canine.
The animal is not a dog.

Hypothetical syllogism is another three-statement argument form, but here the statements are all hypothetical, which means they are supposed, conditional, or theoretical. In this type of argument, there is a transitive relation between the two terms in each of the statements by virtue of a term common in each.

Form of Hypothetical Syllogism:

If A then B
If B then C
If A then C

Example:

If the animal is a dog, it is a canine.
If the animal is a canine, it is a mammal.
If the animal is a dog, it is a mammal.

You already know a bit about the argument form known as disjunctive syllogism. A disjunction is an "either-or" statement, and a syllogism is a three-statement argument. So, a disjunctive syllogism is a three-statement argument that involves a disjunction. One statement is disjunctive – it's the "or", or "either…or" part of the argument. Another statement denies one side of the disjunction, and the conclusion, based on the two premises, is that the other side – the other disjunct – must be true. Another way to think of this argument form is in terms of elimination: When there are two options, and one is eliminated, the other is what is left.

Form of Disjunctive Syllogism:

Either A or B
Not A
B

Example:

The animal is a cat or a dog.
The animal is not a cat.
The animal is a dog.

Constructive dilemma is a combination of elements from modus ponens and disjunctive syllogism. This argument form is also a bit complicated by multiple elements, so describing it is probably more confusing than simply showing the form and giving examples:

Form of Constructive Dilemma

If A then B, and, if C then D
Either A or C
Either B or D

Example:

If the animal is a dog, it's a canine, and if the animal is a cat, it's a feline.
The animal is a dog or a cat.
The animal is a canine or a feline.

Notice that constructive dilemma takes two conditional statements and affirms that the antecedent of one or the other is the case, which means that the consequent of one or the other is the case.

To review, see Validity, Soundness, and Valid Patterns

2d. Identify hidden and implicit assumptions in an argument

• What is a hidden or implicit assumption?
• What is a method for determining the hidden or implicit assumption in an argument?

Understanding and uncovering hidden or implicit assumptions can be difficult, as so much of our daily communication occurs in a sort of "shorthand". Consider this conversation between Adam and his next-door neighbor, Rosario, that illustrates the sorts of inferences we make, but don't always think about:

Rosario:    Do you want to go ride bikes tomorrow?
Adam:      No, I'm going to the Lemmings basketball game with a bunch of the guys.
Rosario:    Oh, that sounds like fun! I love basketball and always root for the Lemmings.
Adam:      What? No way, girls don't like basketball. Besides, basketball is a guy sport; the WNBA is totally irrelevant in our culture because nobody is interested in watching girls play basketball.
Rosario:    Adam, I thought you were smarter than that! Just because the WNBA isn't doing well doesn't mean girls can't like basketball. What a silly thing to say. Besides, you've seen me playing ball in my front driveway with Robert and some of the other kids.
Adam:      Yeah, but I thought you were just trying to impress Robert because you like him.
Rosario:    Who says I like him?

Well, you're always all giggly and acting flighty whenever you're around him. That means you have a crush on him.

Rosario:    Oh, boy. I have no idea what you mean by "acting flighty", but I always giggle.
Adam:      Not around me, you don't. Anyway, that's just what girls do when they like a boy. They act silly.
Rosario:    I think you're a really nice person and all, Adam, but I don't giggle around you because, well, you're not very funny. Robert, on the other hand, is hysterical. He's always telling jokes and doing impressions of people.
Adam:      I am funny! I can't believe you think I'm not. Is it because I play computer games? You think I'm a nerd, and nerds can't be funny?
Rosario:    Um, no. I just think you're a very serious person. It's not a criticism.
Rosario:    Well, you look serious all the time. You don't smile very much.
Rosario:     And you always go around with your nose in books – books with titles I can't even pronounce. You don't say things that are funny or laugh when someone does.
Adam:      I like science, so what? What's wrong with quantum physics? And anyway, I don't spend my time thinking about funny stuff, but that doesn't mean I can't be funny.
Rosario:     Well, okay. Maybe I jumped to conclusions.
Adam:      Yeah, I guess I did, too. Want to come to the game tomorrow then?
Rosario:     I'd love to.

Both Rosario and Adam infer things about each other based on outward appearances and stereotypes. A stereotype is a classification of a person or groups of people, usually negative, based on too few samples. Both made inferences that were off-base – or at least not very well supported by the evidence. Let's look a little more closely at the inferences Rosario and Adam made about each other.

Adam infers that Rosario cannot like basketball. This inference is made based on his belief that girls don't like basketball. Since Rosario is a girl, she won't like basketball, either. It's not clear just why it is that Adam holds this belief, but he does state that basketball is a "guy sport", so perhaps he infers that girls do not like basketball because, basketball is a guy sport, and since girls are not guys, they will not like it.

Adam does not provide explicit support for his claim that basketball is a guy sport, but he does assert that the WNBA is "totally irrelevant". You can see that, upon closer inspection, Adam is making an inference based on rather flimsy evidence, and he is rather vague about his reasons for thinking Rosario can't like basketball. You have probably noticed that Adam has made a few inferences in order to get to his conclusion that girls don't like basketball, and there are some "missing" premises.

When evaluating arguments, it is important to determine what, if anything, is missing from an argument. The second text in this series will include a focus on a specific type of argument in which a premise or conclusion is missing, and we will see how the missing piece can affect the argument as a whole. For now, remember that you should account for all the components of the argument, even the missing, or implicit, premises and conclusions. Recall from Unit 1 that an implicit claim is implied rather than stated, and is typically taken to be understood by a listener or reader without being expressed by the speaker or writer. Here's one way to construct an argument for Adam's conclusion, including the missing premises:

Premise 1: Basketball is a guy's sport.
Premise 2: Anything that is a guy's sport is not liked by girls.

Notice that Adam's first premise is essentially unsupported. Many arguments' premises are in need of some sort of support, which means they are not accepted facts. For example, "Leprechauns exist", is a statement. As such, it is either true or false, but not everyone agrees about how to prove its truth-value, or even if proof is possible. To assert that the statement is true, some would claim, is to assert an improvable belief. Adam's assertion that basketball is a guy sport is not as difficult to prove true or false as is the statement about God's existence, and it would only help Adam's inference that girls don't like basketball if he were to support it. His reference to the failure of the WNBA seems to suggest some sort of support, so let's see how it would look constructed as an argument, including the missing premises:

Premise 1: Nobody is interested in watching girls play basketball.
Premise 2: The WNBA is a women's basketball league.
Premise 3: Whatever isn't interesting will fail.
Conclusion: The WNBA is failing.

The conclusion then becomes the first premise of another argument that supports Adam's claim that basketball is a guy's sport:

Premise 1: The WNBA is failing.
Premise 2: Girls aren't interested in things that fail.
Conclusion: Girls aren't interested in the WNBA.

Wait a minute! What does girls' interest in the WNBA have to do with an interest in basketball and rooting for the Lemmings? If you answered, "Nothing", you're right. Adam is making claims that just aren't related to one another in such a way that it is reasonable for him to conclude that girls don't like basketball. He seems to be confusing the failure of the WNBA with a lack of interest on the part of girls in the sport of basketball (regardless of who's playing). Moreover, he is assuming that the failure of the WNBA is evidence of basketball being of interest only to guys. It would be more reasonable to conclude, instead, that women playing basketball is just not interesting to guys. In any event, there is a bit of confusion on Adam's part in terms of what he is asserting and its relevance to the ultimate claim that he makes.

Adam also makes an erroneous inference that Rosario has a crush on her neighbor, Robert, based on the fact that she giggles and "acts flighty" when he's around. Adam presupposes that girls who giggle and act flighty (whatever that may mean) around boys are girls who have crushes on those boys.

No sooner does Rosario set Adam straight, she makes an inference that is not particularly good. She claims that Adam does not have a sense of humor, and her evidence of this is that his facial expression is serious, ("Well, you look serious all the time"), he does not smile much, and he always has his nose in books with titles that are hard to pronounce. The unstated inference is that books with difficult to pronounce titles are serious, which is then supposed to support her claim that Adam's facial expression is serious. That, we just saw, is meant to serve as evidence for the claim that Adam does not have a sense of humor. The poor quality of the reasoning is highlighted when what is unstated is made explicit.

Here are some more brief examples of poor inference-making, the weakness of which is uncovered when we include the relevant implicit assumption:

1. I have trouble with spelling. I must not be smart. [Implicit assumption: Smart people don't have trouble with spelling].
2. This Doberman is not a good watchdog. He obviously doesn't care about his human family. [Implicit assumption: Only good watchdogs care about their human families].
3. My parents went to Europe and brought home the best chocolate from Switzerland. The Swiss must have the best ingredients in the world. [Implicit assumption: The best chocolate in the world is made out of the best ingredients in the world].
4. The house is quiet. Everyone must be asleep. [Implicit assumption: If everyone is asleep, the house is quiet].
5. This computer's no good. It doesn't do what I want. [Implicit assumption: The computer is good, if it does what I want].

Having trouble with spelling does not imply a lack of intelligence. Lots of people who are plenty smart find spelling difficult. There are a number of reasons that could account for trouble with spelling, and these reasons are probably much more likely to account for the spelling difficulty than is the inference that I am not smart. It is possible, for example, that I am dyslexic. Dyslexia is a neurological disorder in which the brain processes something different from what the eye sees. Quite often a person with dyslexia will transpose letters in a word, or transpose words in a sentence. A bad speller might also not have much experience spelling, perhaps because he doesn't read much or doesn't read broadly enough to come across lots of different words.

Similarly with the spelling inference, the Doberman inference fails because there are other, possibly better reasons, to account for its being a poor watchdog. The Swiss chocolate inference fails, in part, for the same reason as the Doberman and spelling examples. In addition, there is an assumption that what is true of the whole, in this case the goodness of the chocolate cake, is also true of its parts. The inference is, then, that the ingredients of the chocolate cake are as good as the cake itself. A similar mistaken inference could be made if I was to claim that every individual European country makes good chocolate cake just because the cake came from one of those European countries.

The fourth and fifth examples also fail as good inferences because other inferences could be made that would work just as well, if not better than, the ones actually made. From these examples it should be clear that one can make any inference one wants from the facts at hand, but this does not mean they're any good. The examples show that good inferences rely on a strong connection between the facts, or evidence, and the conclusions inferred from them.

To review, see Hidden Assumptions, Inductive Reasoning, and Good Arguments

2e. Explain the differences between deductive reasoning and inductive reasoning

•     What is deductive reasoning?
•     What is inductive reasoning?

The study of logic in a course like this divides arguments into two types: deductive and inductive. Though inductive arguments are far more common in our everyday use than deductive arguments, it is fair to say that we generally want our arguments to have the certainty of deduction.

We know that an argument consists of two or more statements, one of which is supported by the other(s). The statement that is supported is the conclusion, while the statement(s) that support(s) the conclusion are the premise(s). An argument is characterized by an inferential flow: The conclusion follows from the premises; the premises imply the conclusion. Another way to say this is that the conclusion is drawn from, or implied by, the premises.

An argument is evaluated in terms of the strength of the connection between the premise(s) and conclusion. The strongest connection between the premises and conclusion is entailment. When the conclusion is a necessary consequence of the premises, the conclusion is entailed by the premises. This means that you cannot have true premises and a false conclusion at the same time. The conclusion is said to be true on the basis of accepting that the premises are true. So, if the premises are true, it would be contradictory for the conclusion to be false. All such arguments' power comes from the structure of the premise-conclusion relation.

As statements, premises and conclusions have truth-values. The truth-value of a statement is its truth or falsity. So, statements are about what is the case; they are not about beliefs, opinions, questions, or commands. It is true that beliefs can be asserted as statements, but their truth-value is more difficult to determine. When you say, "He's handsome" or "She's beautiful", the statement is true or false depending on whether or not he is handsome or she is beautiful; and determining that is not quite so easy as determining the truth-value of something like, "He is in the room".

The premises and conclusion of a deductive argument have a specific relationship. The conclusion is supposed to follow from the premises with certainty, or necessity, and the premises are said to be sufficient to infer the conclusion. In other words, the relationship between the premises and conclusion is one in which the premises are claimed to support the conclusion such that it is impossible for the premise(s) to be true while the conclusion is false. Before moving on to discuss the structure of deductive arguments, let us spend a bit more time looking at the relationship between the premises and conclusion. An example might help make clear just how that relationship works:

All Chihuahuas are dogs.
All dogs are canines.
So, all Chihuahuas are canines.

Besides thinking to yourself, 'Duh, this isn't proving to me anything I don't already know,' you might also think, 'Of course Chihuahuas have to be canines. After all, if it's true that Chihuahuas are dogs, and dogs are canines, Chihuahuas also have to be canines.' What you're expressing is the sufficient relationship between the premises and conclusion, and the necessary relationship between the conclusion and premises. Another way to assert the two ideas just expressed is to say that, assuming the premises are true, you are committed to asserting the truth of the conclusion – to do otherwise is to assert a contradiction, that is, that the premises can be true while the conclusion is false.

When we see an argument whose conclusion cannot be doubted while we accept the premises, we see an argument that succeeds by its structure. Here another example, which highlights how the meanings of certain relations, when structured in specific ways, reveal this idea:

Gerrardo is taller than Stanley.
Stanley is taller than Kumar.
Gerrardo is taller than Kumar.

Now consider the following argument, which uses the same relational predicate, "taller", but with a change of the argument's structure, the premises no longer guarantee the conclusion:

Gerrardo is taller than Stanley.
Gerrardo is taller than Kumar.
Stanley is taller than Kumar.

Here, we could easily falsify the conclusion, while accepting the truth of the premises. Suppose that Gerrardo is 5'10" tall, Stanley is 5'8" tall, and Kumar is 5'9" tall. At 5'10" tall, Gerrardo is taller than both Stanley and Kumar. So, the premises are true. At 5'8" tall, Stanley is not taller than Kumar, who is 5' 9" tall. So, the conclusion is false.

Perhaps our most common, everyday mode of reasoning is experiential. Consider the following examples:

• Rosa leaves for school at 8:00 a.m. Rosa is always on time. Rosa infers that she will always be on time for school, if she leaves at 7:00 a.m.
• The cost of the fabric was $1.00. The cost of labor to manufacture the fabric was$.50. The sales price of the resulting item was \$5.00. So, the manufacturer made a good profit.
• Every snowstorm in this area comes from the northeast. There is wind and flakes of snow coming from the northeast, so, a new snowstorm is coming from the northeast.
• Javier is showing a new car to his friend, Ahmed. So, Javier must have bought a new car.
• The living room walls are painted yellow, as are the kitchen and dining room walls. So, all the walls in the house are painted red.
• Every time you touch poison ivy, you get a rash. So, you're allergic to poison ivy.
• Every dog you know barks. So, every dog barks.
• Half of the students at this university receive student aid. Therefore, half of all university students receive student aid.
• All of the children in the class are brown-haired, therefore all children in this neighborhood are brown-haired.
• Michelle just moved here from Los Angeles. Michelle has blonde hair, therefore people from Los Angeles have blonde hair.
• All the players on my basketball team are tall, so all basketball players must be tall.

We call experiential reasoning inductive. The premises and conclusions of inductive arguments do not claim the same relationship as those of deductive arguments. Instead, the relationship is only probable. Probability expresses the likelihood that something is the case. This means that, instead of expressing a supposedly necessary relationship between premises and conclusion, an inductive argument asserts that, if the premises are true, the conclusion is likely to be true. Another way to express probability is in terms of chances. In an inductive argument, if the premises are true, the chances of the conclusion being true increase. Often, we use inductive reasoning to express our conviction, that is, to reason to a claim that expresses we are convinced something is the case.

Here are some more examples of inductive arguments:

Example 1:

She'll like this new book about Grizzlies.

Example 2:

Yesterday Michael visited his aunt.
As she's done with previous visits, she gave him some candy.
Michael is going to his aunt's house again today.
His aunt will give him some candy.

Example 3:

The dishes in the dish rack are still wet.
Whoever was washing dishes just finished a little while ago.

Example 4:

Danielle and I were exploring in the hills yesterday.
We found a bunch of seashells embedded in the ground.
I bet the hills were under water thousands of years ago.

Notice that the probability of the conclusion being true is strengthened by how well the premises support the conclusion. In the first example, based on the similarities between Polar Bears and Grizzly Bears, it's inferred that Kayla's love of reading about Polar Bears means she's likely to enjoy a book on Grizzlies. The second example relies on predicting something about the future based on repeated past experiences. This is a very common way of thinking, and we use it every day to order our lives. Just imagine how hard it would be to get through your day if you didn't rely on past experience! The third example makes a connection between the wetness of the dishes and the timeline for their having been washed. The fourth example concludes something about the past based on present experience. It's basically the reverse of predicting something about the future based on the past.

Here are some examples of inductive arguments in which the chance of the conclusion following from the (true) premises is far less than in the previous examples:

Example 1:

Carmen fell yesterday when trying out her new bicycle.
I just know she's going to fall off every time she rides it.

Example 2:

Our new neighbors' youngest child, Chloe, is a little brat.
Our new neighbors must be so rude.

Example 3:

I hate broccoli, which is a cruciferous vegetable.
Brussels sprouts are also cruciferous vegetables.
I know I'll hate Brussels sprouts.

Example 4:

Samantha had a cold, but she took this special potion.
Her cold disappeared in a week.
The special potion cured Samantha's cold.

Carmen hasn't fallen off her new bike enough times to warrant the conclusion that she's going to fall off every time she rides it in the future. Though all predictions are uncertain, some are more reliable than others. This prediction, however, fails due to insufficient evidence. The second example is problematic because it applies to every member of the household something that is only known to be true of one of its members. By similar reasoning, you could try to claim that the new neighbors are all youngest children on the basis of the youngest child being the youngest – and that's utter nonsense!  (What about the broccoli example?) Finally, there is no good reason to conclude that the "special potion" Samantha took cured her cold. After all, colds typically run their course and are over within a week.

Though there are no inductive forms of arguments, there are types and patterns of inductive reasoning that are similar enough in their functions to categorize them as follows:

 Inductive Reasoning Pattern Definition of the Pattern Prediction This argument pattern involves a prediction based on past and/or present evidence. Analogy This argument pattern involves an inference about an object or event based on comparing it with another, similar object or event. Cause and Effect This argument pattern involves drawing a conclusion that one event is either the cause or effect of a related (temporally or proximally) event or object. Statistical Argument This argument pattern involves an inference about a target population based on a representative or random sampling. Authority This argument pattern involves drawing a conclusion based on an authority. Generalization This argument pattern involves inferring a general claim based on one or more individual objects or events. Argument to the Past This argument pattern involves an inference about the past based on present evidence.

We will see later that there are ways in which inductive arguments are twisted or so bad that they confuse people into believing something that's not true.

To review, see Hidden Assumptions, Inductive Reasoning, and Good Arguments

2f. Explain the pattern of inductive reasoning called an analogical argument

• What is an analogical argument?
• Why is an analogical argument a type of inductive argument?

One of the most common ways of reasoning in our daily lives is by way of a comparison between two or more objects or events. Suppose you are looking at produce. You want a ripe avocado. On the basis of previous experience with the features of a ripe avocado, you begin looking through the pile in front of you, taking out an avocado at random, gently squeezing it, and then putting it back. Eventually, you find one that "feels right". That's the one that goes in your basket.

Or suppose you need to purchase a car. On the basis of your previous experience with a particular make and model you owned, you conclude that buying another car of the same make and model will suit you very well.

Now suppose that you are an astronomer researching the possibility of life on other planets. You will make comparisons between the requisite features for life found here on Earth, and, say, the chemical composition of materials found on another planet.

Finally – although many more examples are to be had – suppose you are a legal expert. You have before you a case of a self-driving car that has been involved in an accident. The car went off the road and ran into someone's yard. While no one was injured, the property owner's fence and garden was ruined. The question of liability needs to be settled, but suppose there is no precedent – there is no case law under which the current circumstance falls. The legal expert must think about commonalities. Is the autonomous vehicle more like a horse-drawn carriage than an ordinary, human-driven vehicle? Is the autonomous vehicle more like a personal computer, with hardware and software? Determining the relevant comparison can make an enormous difference to how to make and interpret the law.

Analogical reasoning generally follows this pattern:

• Two (though sometimes more) objects or events are compared.
• One of those objects (the target, or secondary analogue) is lesser known than the other (the primary or original analogue).
• On the basis of the comparison, a conclusion is drawn about the lesser known of the two objects.
• The number of relevant similarities and dissimilarities will determine whether or not the inference is strong.

Here are two ways to view the general structure of analogical reasoning:

P and Q are similar in respects a, b, c, etc.
f is true of P.
Therefore, f is also true of Q.

P has attributes a, b, c, and f.
Q has attributes a, b, c.
Therefore, f is also true of Q.

Analogical reasoning can go wrong. Consider, for example, the following story: A woman in her first pregnancy experiences feeling and seeing puffiness, "like I ate too much pizza the night before", and seriously low energy. Despite eating a healthy diet and being a marathon runner, her symptoms continued to worsen. When her blood pressure spiked into the 190s over 110, she went to the hospital, where doctors worried they might have to deliver her fetus at only 25 weeks of gestation.

She responded well to blood pressure medication, so an emergency c-section was postponed. After all, if her symptoms indicated a problem that removing the fetus would not solve, the premature baby's health would be at risk. The question remained, did the woman have preeclampsia? Part of the thinking was that, apart from the high blood pressure, the woman did not have other signs of preeclampsia, a disorder that is the leading cause of maternal death, worldwide. Untreated, high blood pressure negatively affects a pregnant woman's organs.

Because the woman and her doctors were initially unconcerned about her symptoms, and because even in a critical state, her symptoms were apparently not enough like preeclampsia to warrant the diagnosis. The woman did go on to have a C-section at just over 30 weeks. Since her experience, the woman's daughter has grown into a healthy toddler. When she became pregnant again, and began to experience the same symptoms as she had during her first pregnancy, she immediately got help.

This woman's story is part of a larger picture emerging in maternal health. Historically, much of the medical focus surrounding pregnancy, delivery, and post-partum has been on fetal and infant health. There are, for example, fetal health centers around the United States, where care plans are far more customized than would be the case with uncomplicated situations. There have not, however, been health centers devoted to maternal health. The Mothers Center at New York-Presbyterian/Columbia University Irving Medical Center is the first center to focus on at-risk pregnant women – and it's modeled on the hospital's own fetal health center.

We can see from the mistakes made in the case discussed above, that analogical reasoning does not always go well. We can also see that analogical reasoning is at work in the thinking of the medical professionals who inaugurated the Mothers Health Center at New York-Presbyterian/Columbia University Irving Medical Center.

To review, see the section on Analogical Arguments.

2g. Construct an argument map for an argument

• How can a modus ponens argument form be mapped?

We know the MP argument has the following form:

If A then B
A
B

Here is an ordinary language example: The dog is barking. So, someone's at the door. That's because, if the dog is barking, someone's at the door.

One we identify the conclusion (the conclusion indicator word, "So" alerts us), we can number our elements:

1) The dog is barking.
2) Someone's at the door.
3) If the dog is barking, someone's at the door.

The argument map looks like this:

To review, see Argument Mapping.

2h. Apply the criteria for evaluating the strength of an argument to any given argument

•     What is a strong argument?
•     How can analogical arguments be evaluated?

Just as deductive arguments are "good" or "bad", so are inductive arguments. But the criteria for evaluating an inductive argument are different from those for deductive arguments – with one exception. Like deductive arguments, inductive arguments are evaluated based on how well the premises support the conclusion. In the case of deductive arguments, there are validity and soundness. With inductive arguments, the concepts are strength and cogency.

A strong argument is one in which the premises are true and the conclusion is probably true. That is, there is a good chance the conclusion follows from the premises.

Here is an example of a strong argument:

The sun has risen every day.
It will rise tomorrow.

Weak arguments are those whose conclusion is not likely to be true. The problem is that, even when the premises are true, they just don't give much support to the conclusion. Based on the premises, it is not likely that the conclusion is true. Weak arguments act like their name: the evidence to support the conclusion is weak.

Here is an example of a weak argument:

Hillary looks so much like my favorite movie star.
I bet they're related.

People who look similar to each other are not necessarily related.

Let's leverage our understanding of analogical arguments by considering the basic criteria for evaluating an analogical inference. First, the number and relevance of the similarities is important to the quality of the reasoning. The greater the number and relevance of the similarities between the two objects or events, the more probable is the conclusion. Second, the nature and degree of disanalogy – the dissimilarity between the elements – is also important to the quality of the reasoning. Serious relevant disanalogies will undermine the likelihood that the conclusion is true. Lastly, the more specific the conclusion, the stronger the analogy must be. After all, the more specific the conclusion, the easier it is to falsify, which weakens the reasoning. So, a specific conclusion requires a diverse number of relevant similarities, and only a small number of relevant dissimilarities – and these cannot be compelling enough to outweigh the similarities. The ways in which the analogues are not alike can bear on the strength of the inference. The key characteristic(s) inferred about the secondary analogue can be undermined by these disanalogies. If you think an analogical argument is weak, list as many disanalogies as possible. The idea is that the difference between the compared items shows that the characteristic intended to be attached to the secondary analogue – the target in the conclusion – is not likely transferrable on the basis of the comparison.

Let's consider another argument from analogy, and then a series of disanalogies intended to demonstrate the original reasoning's weakness:

Analogical Reasoning

A car is like a cat. For example, a car has a hood, and a cat has a head. A car has an engine, and a cat has a heart. Since a car requires energy to run – fuel or battery power – it follows that a cat requires energy to run.

Disanalogies

Cars are made out of metal, steel, rubber, and other non-biological elements. Cats are made out of biological elements. Moreover, cars require regular maintenance in order to run, while cats do not – even a check-up by a veterinarian is not similar to processes such as "tune-ups".

Another way to show that an analogical argument is weak is to construct a counter-analogy. Recall that analogical reasoning generally follows this pattern:

(Primary analogue) A, and (secondary or target analogue) B, share characteristics, p, q, r, and s.
A also has characteristic t.
B also has characteristic t.

A counter-analogy will propose that the target analogue is not like A, but is actually more like something else, e.g., C. Moreover, the counter-analogy typically leads to a conclusion that contradicts the original analogical reasoning.

Finally, pointing out unintended consequences reveals that an analogical inference is weak. An unintended consequence of an analogical argument is one that the person who mounted that argument would not want to accept. As a consequence, the original argument would be revised or altogether abandoned. Hume, once again, provides an example of using unintended consequences to show that the design argument, as presented, is not as strong as it may seem at first.

If we consider the fact that increasingly complicated machines tend to require multiple makers, it should follow that the universe, which is extraordinarily complex, also requires more than one creator. Even if we make a distinction between a designer and a maker, and argue that, while it may take multiple builders to construct a home, for example, it takes only one architect. This objection is met, however, if we consider the variety of technical expertise required to complete a design, from engineering to electrical to plumbing, and so forth.

As we know, an argument's conclusion has to be relevant to the premise(s). In analogical reasoning, this relevance is determined by the characteristic(s) inferred about the secondary analogue – the target item in the conclusion – from the comparison to the primary analogue – the item to which the target is being compared. The more relevant similarities there are, the more likely it will be that the secondary analogue will be like the primary analogue's further characteristic. Provided that the key feature of the primary analogue transferred to the secondary analogue in the conclusion cannot be accounted for in other ways, the inference is strong.

To review, see Good Arguments.

Unit 2 Vocabulary

This vocabulary list includes the terms listed above that you will need to know to successfully complete the final exam.

• premises
• conclusion
• valid argument
• invalid argument
• sound argument
• standard form
• counterexample
• truth value
• modus ponens
• modus tollens
• hypothetical syllogism
• disjunctive syllogism
• constructive dilemma
• hidden/implicit assumption
• deductive argument
• inductive argument
• probability
• inductive reasoning
• analogical reasoning
• argument map
• strong argument
• weak argument
• disanalogy
• counter-analogy
• unintended consequences