Chapter 4: Deductively Strong, Inductively Strong, and Weak Arguments

Inductive Strength

Now let’s consider our second category of strong arguments.  We will define inductively strong arguments this way:  An argument is inductively strong for a person S when it meets three conditions:

1) it is cogent

AND

2)  the premises are reasonable for S to believe.

AND

3)  the argument is not defeated by S's total evidence.

So if you confront an argument that is cogent, and the premises are reasonable or true (you believe them), and if it is not defeated, then you should accept the conclusion as reasonable.  The argument is inductively strong and gives you good grounds for accepting the conclusion.  It won't guarantee the truth of the conclusion the way valid arguments will, but it does give good inductive, probabilistic justification for accepting the conclusion.  It should be clear from what we learned about cogent arguments in the earlier chapter that while they give logical support to their conclusions, it is probabilistic and not as good as a valid argument.  

What does it mean to say that an argument is defeated for a person S?  Consider this argument:  

1.  The vast majority of humans have not walked on the moon.  

2. Neil Armstrong is a human. 

_____________________________

3.  Therefore, Neil Armstrong has not walked on the moon.  

Depending on what you know about the background here, this argument may be defeated for you.  Neil Armstrong, famously, is the first person to walk on the moon in 1969.  An argument is defeated when it meets three conditions:

1. The argument is cogent

2. S believes that all of the premises are true.

3. But S has background information leading them to believe that the conclusion is false.  

We will be using the term “defeat” only in this technical sense; do not make the mistake of labeling an argument that has some other sort of problem like a false premise or being ill-formed as defeated.  Being defeated in another way that arguments on our approach to argument analysis can be weak.  

This version of the argument is inductively strong for you:

1. The vast majority of humans have not walked on the moon.  

2. Professor McCormick is a human.  

__________________________

3.  Therefore, Professor McCormick has not walked on the moon.  

This version is cogent, it has true premises, and it is not defeated.  So it meets the conditions of being inductively strong.  

Let’s consider some more cases to clarify the distinction between an inductively strong argument for S and one that is defeated and therefore weak.  Reflect on this argument:

1) In the vast majority of cases, when a woman takes a home pregnancy test and gets a positive result, she is pregnant. 

2) Caroline took a home pregnancy test and got a positive result.

__________________________________________________

3) Therefore, Caroline is pregnant. 

This argument is cogent; it follows the pattern we've been considering for cogent arguments.  Furthermore premise 1 appears to be true because home pregnancy tests are typically 98% or more accurate.  You probably believe premise 1.  And let’s stipulate that our hypothetical person, Caroline, also believes that premise 1 is true.  What about premise 2?  Suppose Caroline did take the test and she got a positive result, then she’d believe that premise 2 is true.  So condition 2 for an inductively strong argument would be met.  

Now let's consider two situations with regard to condition 3.  Suppose that Caroline is a normal, healthy woman with no known medical issues and she's of child bearing age.  She'd have no independent reasons to think, in that case, that she's in the 1-2% minority of cases where the test is not accurate or gives a false result.  So in that case, she'd have no background information that might defeat or undermine the conclusion.  So condition 3 would be met and this argument would be inductively strong for Caroline. 

But suppose Caroline had this defeating piece of information:  she also knows that A) she has an unusual medical condition that makes it impossible for her to get pregnant, and furthermore, one of the effects of her condition is that it makes her prone to test positive on pregnancy tests anyway.  If A) was true and Caroline knew it, then the argument would still be cogent, and the premises are still true, but A) defeats the argument.  In this case, the argument would not meet the third condition for being inductively strong.  We add condition 3) for inductive strength because at best, cogent arguments would make their conclusions likely to be true if their premises are true, but there could still be a possibility that the premises are true but the conclusion is false.  It can happen that the case being considered falls into the minority of cases not covered by the inductive generalization in premise 1.  And S might have information that indicates that this is what has happened.

Let’s consider one more case of an argument that is inductively strong for one person and defeated, and therefore weak, for another person.  Suppose that Ellen reads an article in a news sources she trusts that the article cites several reputable seeming studies that say most people in her neighborhood in Sacramento are Democrats. Then Ellen considers her neighbor Amy.  She doesn't have any particular reason to doubt that Amy is an exception, or in the non-Democrat minority.  So this argument:

1.  Most people who live in Sacramento are Democrats.

2. Amy lives in Sacramento.

____________________________

3. Therefore, Amy is a Democrat.

is inductively strong for Ellen.  It's cogent, it has true premises, and the argument is not defeated by Ellen's background information. 

But Michael, who also knows Amy, and who knows that Amy lives in Sacramento, has listened to lots of anti-Democrat tirades from Amy.  He's heard her say that she's a Republican and that she wouldn't be a Democrat if you held a gun to her head.  And he thinks she's sincere. So the same argument is weak for Michael.  The premises are true and that they are reasonable for Michael to believe; the argument is not weak because he rejects a premise.  Furthermore, it’s cogent.  Bu it is weak because it is defeated by his total evidence.

So, once again, an argument is defeated only under these circumstances:  An argument is defeated for S if and only if, the argument is cogent, the premises are reasonable for S to believe, but S believes the conclusion is false because it conflicts with S's total background evidence.

And an argument is inductively strong for S when it meets these three conditions:

1. The argument is cogent

2. S believes each premise is true.  

3. The argument is not defeated by S’s total background information.  

Weak Arguments

Any argument that is not deductively strong or inductively strong is weak in our analysis.  That is, the three categories are exhaustive.  We’ve seen the ways that arguments can be weak along the way, but let’s enumerate the three ways arguments can be weak again for clarity.  In general, a weak argument is one that fails to convince of us the conclusion.  The conclusion might be correct, we might believe it.  But the argument has some problem or problems that prevent it from giving good rational support.  

There are three ways arguments can be weak:  (and they can be weak for one or more of these reasons) 

1)  S believes one or more premises are false, or S suspends judgment about one or more premises.  

That is, if you consider the premises and you think any of them are false (or more than one of them) or if you suspend judgment about any of them (or more than one of them), then the argument is weak.  

Consider this argument:  

1) All CSUS students are male. 

2) Arnold Schwarzenegger is a CSUS student.

____________________________________

3) Therefore, Arnold Schwarzenegger is male.

A moment’s reflection should make it clear that premise 1 is false.  Lots of CSUS students are female.  If you are a female CSUS students, or you know one, then that’s a counter example to premise 1 proving it false.  For universal generalizations of the form All As are Bs, a single case of an A that is not a B is counter example and decisively disproves it.  

Furthermore, you might have a couple of different perspectives on premise 2).  If you know who Arnold Schwarzenegger is (the famous action movie star, and former governor of CA), you probably doubt that 2) is true. You might not have as much conviction that 2) is false as your conviction that not all CSUS students are male, but you disbelieve it nevertheless. It seems unlikely that he’d be pursuing a degree at all, or at CSUS.  

If you don’t know who Arnold Schwarzenegger is, then you should consider 2) and suspend judgment.  If you don’t have any information about the claim one way or another, then the rational thing to do is to neither believe it nor disbelieve it.  Remain open to be convinced by the evidence.  But even in that case, this argument is now weak for you because you have suspended judgment about a premise.  

So this argument is weak for you because it seeks to convince you that its conclusion is true on the basis of premises that you think are false or that you suspend judgment about.  That’s not compelling grounds for the conclusion, even though the argument happens to be valid.  False claims should not convince any reasonable person of anything.  

2)  An argument is weak if it is ill-formed.  An ill-formed argument is neither valid nor cogent.  So consider this example, and notice the difference from the previous argument:

1.  All CSUS students are male. 

2.   Arnold Schwarzenegger is male.

__________________________________

3. Therefore, Arnold Schwarzenegger is a CSUS student.

This argument is invalid.  Even if the premises were true, the conclusion would not be guaranteed to be true.  Suppose all CSUS students were male, and that Schwarzenegger is a male.  But those claims alone don’t support the inference that he's a student at CSUS.  In fact, there are far more non-CSUS males than CSUS males on the planet, so 2. doesn't support the conclusion at all.  It's an instance of affirming the consequent, a logical fallacy.  Consider this parallel case to see the deep logical flaw in the structure of the argument:

1.  All bears are mammals. 

2.  Dogs are mammals.

3.  Therefore, dogs are bears. 

In this case, the premises are in fact true.  But the conclusion is false.  This argument contains a serious logical error because of the relationships between the concepts.  

Ill-formed arguments are neither valid nor cogent.  This argument is not cogent either.  A cogent argument is one that is 1) invalid, and 2) the premises, if they were true, would make the conclusion probably true.  The premises here don't make the conclusion probable.  In the first case, consider the class of all males and the tiny subset of that group that is the CSUS students who are males.  The non-CSUS student males vastly outnumber the CSUS student males.  So knowing that Arnold is a male doesn't make it likely that he is a CSUS student male.  In fact, knowing that he is a male makes it unlikely that he's a CSUS student male because they are such a small percentage of over 3 billion males on the planet.  Likewise with the second example.  The non-bear mammals far outweigh the bear mammals.  So knowing that dogs are mammals doesn't make it likely that they are bears, it makes it improbable.  So these arguments are the opposite of cogent. 

So an argument that is neither valid nor cogent is ill-formed.  And being ill-formed is one of the three ways that an argument can be weak--by having bad logical form. 

3)  An argument is weak if it is defeated.  Defeat happens in the specific circumstances listed above under the discussion of Inductively Strong arguments.  That is, we have a specific, formal definition for the concept of defeat.  It is not merely a bad argument or one that you don't accept.  

An argument is defeated for S when:  

1. It is cogent

2. S believes that all of the premises are true.

3. But S has background information leading them to believe that the conclusion is false.  

This example is defeated for you:  

1)  Most citizens of the United States do not live in the White House. 

2)  The President of the United States is a citizen of the United States. 

3)  Therefore, the President of the United States does not live in the White House.

In this argument, the logical form is cogent.  It follows the "1) Most As are Bs. 2) x is an A, 3) Therefore, x is a B," form that we have considered before for cogent arguments.  And the premises are both true.  There are over 330 million people in the United States, but only one of them lives in the White House.  So most American citizens do not live in the White House.  And we know that the constitution requires that the President be a citizen, so premise 2 must be true as well.  However, I know that 3 must be false because of other information I possess.  I know from watching the news, from my high school government class, and from sources I trust that the President is the one person who does live in the White House.  And it’s probably defeated for you for the same reasons.  

Critically Evaluating Arguments:

Now that we have this system for analyzing arguments and separating strong ones from weak one, we can say more about how to critically evaluate arguments.  Strong arguments are ones that should convince a rational person to believe their conclusion.  We now have a clear method for identifying deductively and inductively strong arguments.  We also now have several specific ways to identify weak arguments and terminology for identifying their problems.  

When we decide an argument is weak, and when we analyze them, we should:

• Be clear and specific about exactly where the argument goes wrong.  Is it the logical form (ill-formed) or do you doubt or reject one or more premises?

• If it is ill formed, explain why; give another similarly structured argument that illustrates the mistake or fallacy.  What is the logical problem?  

• If it’s a logical fallacy, then identify it specifically:  affirming the consequent, denying the antecedent, confirmation bias, ….

• If you suspend judgment about a premise, or think a premise is false, then say which premise, say whether you suspend judgment or whether you think it is false, and say specifically why you have doubts.  Give your reasons for withholding your assent.  Give examples, illustrate, explain.  

• Make the objection strong so that it shows the premise is mistaken.  Recall the law of non-contradiction. 

• Counter examples.  Recall that for a claim like All As are Bs, disproof comes from even a single instance of an A that is not a B.  A single woman who is a good driver disproves that All women are bad drivers.  

• Disproving Most As are Bs.  In order to disprove Most As are not Bs,  you need to give convincing evidence that actually greater than 50% of As are not Bs.  Polling data that shows that most college students are not Democrats, for example, shows that it’s not true that most college students are Democrats.  

• Defeated arguments have true premises, and they are cogent, but there is some crucial piece of information that shows that the conclusion does not follow.  Even though most humans have not walked on the moon, Neil Armstrong is one of the minority who have.  

Flow Chart:  And here is all of the information so far represented in a flow chart.  Imagine asking these questions, from left to right, of an argument.  Follow all of the branches to conclude whether it is deductively strong, inductively strong, or weak:

Argument Analyses:

In order to master the concepts and the argument analysis approach we've built so far, we will do the sort of argument analyses shown below.  For each argument we will identify and label all of the features.  Different people will have different background information, so part of these analyses will depend on the individual.  When confronted with the same premise, two people might differ on whether they think it is true, false, or they suspend judgment.  Consider polling questions that ask for your view of, "I approve overall of the job the President is doing,"  or "In general, vaccines are safe and effective at preventing disease."  Our immediate goal in these argument evaluations is not to resolve disagreement between people about whether a premises is true or false.  Here we only want to figure out its status for a person.  And whether you agree or disagree is subjective and varies from person to person because belief and disbelief vary and are subjective.  Therefore, the T, F, and SJ aspects of these argument evaluations are "no fault."  There is no wrong answer, as it were, for individuals indicating whether they agree, disagree, or suspend judgment about a claim.  Consider an example:

1.  There earth is flat.  

2. If the earth is flat, then the scientific establishment has been misleading us about the truth.  

______________________________________________

3.  Thererfore, the scientific establishment has been misleading us about the truth.  

Argument analysis: 

a. Is the argument Valid, Cogent, or Ill-formed?   __V___  This argument is valid; it follows the Modus Ponens pattern we've been studying.

b.  Do you think the premises are True, False, or do you Suspend Judgment?  1.__F__    2.__T___  

Most people will recognize that the first premise is false.  However, there could be someone with a very different set of background beliefs and information than you who considers premise 1 to be true.  Recall the medieval peasant from the 13th century we considered in the first chapter.  He might believe that 1 is clearly true because the earth looks flat, he can't conceive of any alternative, everyone he knows and trusts says that it is flat, and it fits with the rest of what he knows about the world.  

c. Is the argument Deductively Strong, Inductively Strong, or Weak?  ___W______

d. If it is weak, why? ___This argument is weak because premise 1 is false.  We have evidence that the earth is round.  Astronauts have flown around the earth.  We have pictures of it from space.  It's possible to get on a plane and fly around it.  Experts, teachers, scientists, and everyone whose authority I trust agrees that it is round.  So I reject premse 1.  ____

But the argument analysis for our hypothetical medieval peasant would be:  

a. Is the argument Valid, Cogent, or Ill-formed?   __V___  This argument is valid; it follows the Modus Ponens pattern we've been studying.

b.  Do you think the premises are True, False, or do you Suspend Judgment?  1.__T__    2.__T___  

c. Is the argument Deductively Strong, Inductively Strong, or Weak?  ___DS_____

d. If it is weak, why? _______

So the same argument is weak for one person and deductively strong for another.  That is, both are correct answers the way they've been presented.  But whether an argument is valid, cogent, or ill-formed, by contrast, is not dependent on the person and not subjective.  Those are objective features of these arguments.  And whether an argument is DS, IS, or Weak, cannot vary given the definitions of those terms.  DS arguments must meet all of the conditions for DS arguments as we've defined them, and the same for IS or Weak arguments.    

The AI tutor for this chapter will train you how to do a particular sort of argument analysis based on what we’ve learned so far.  Here are a few examples of how we can apply these concepts to arguments.  For our tests and for the course, you’ll need to demonstrate your mastery of all of the above concepts by being able to correctly do analyses like the ones below, where you decide whether you believe (T), disbelieve (F), or suspend judgment (SJ) about the premises:  

Covid Vaccine Argument

1.  If a vaccine is approved by the FDA, then it has been shown to be safe and effective in controlled clinical trials.

2. The Covid vaccine has been approved by the FDA

____________________________

3.  Therefore, the Covid vaccine has been shown to be safe and effective in controlled clinical trials.   

Example Argument analysis for Smith: 

a. Is the argument Valid, Cogent, or Ill-formed?  ___V______

b. Do you (or the person in the example) think the premises are True, False, or do you Suspend Judgment?  1.__SJ__    2.__T___  

c, Is the argument Deductively Strong, Inductively Strong, or Weak?  ___W______

d. If it is weak, why? __I'm suspending judgment about premise 1 because I've come to doubt whether the FDA actually thoroughly and carefully checks these things, particularly when there is a lot of money to be made.  __  (We can imagine someone who might think this about premise 1).  

Example Argument analysis for Jones: 

a. Is the argument Valid, Cogent, or Ill-formed?  ___V______

b. Do you (or the person in the example) think the premises are True, False, or do you Suspend Judgment?  1.__T__    2.__T___  

c, Is the argument Deductively Strong, Inductively Strong, or Weak?  ___DS______

d. If it is weak, why? ____  (We can imagine someone who might think this about premise 1).  

When you do an argument analysis for this course, you must decide whether you believe, disbelieve, or suspend judgment about each premise, and then you must explain why.  

Flat Earth Argument

1. If the Earth were flat, then ships sailing on the ocean would fall off.

2. Ships sailing on the ocean do not fall off.

––––––––––––––––––––––––––

3.  Therefore, the Earth is not flat.

Example Argument analysis: 

a. Is the argument Valid, Cogent, or Ill-formed?  ___V______

b. Do you (or the person in the example) think the premises are True, False, or do you Suspend Judgment?  1.__T__    2.__T___  

c. Is the argument Deductively Strong, Inductively Strong, or Weak?  ___DS______

d.  If it is weak, why? _______________________________________

Astrology Argument: 

1. The predictions in my astrological forecast are accurate some days.  

2. If the predictions in my astrological forecast are accurate some days, then astrology works.  

3. If astrology works, then I should consult my astrology forecast when I make big decisions.  

__________________________________

4. Therefore, I should consult my astrology forecast when I make big decisions.  

Example Argument analysis: 

a. Is the argument Valid, Cogent, or Ill-formed?  __V_______

b. Do you (or the person in the example) think the premises are True, False, or do you Suspend Judgment?  1.__T__    2.__F__  3.__T___ 

c. Is the argument Deductively Strong, Inductively Strong, or Weak?  __W______

d. If it is weak, why? __I don't accept premise 2.  Astrology forecasts are vague, general, and they throw a lot of different predictions out there.  Getting the "right" answer a few times or on some days alone isn't enough to convince me that it works.   A broken clock gets the time right twice a day.  Astrology forecasts would need to be very often accurate, precise, detailed, and reliable before I'd conclude that they work.  ______________________

Undocumented Immigrants Argument

1. A high rate of undocumented immigrants are coming through the American border illegally.  

2. If a high rate of undocumented immigrants are coming through the American border illegally, then the U.S. is failing to maintain its border.  

3. Any country that fails to maintain its border will have its social, legal, and political structure ruined.  

______________________

4. Therefore, the U.S. will have its social, legal, and political structure ruined.  

Example Argument analysis:  

a. Is the argument Valid, Cogent, or Ill-formed?  ___V______

b. Do you (or the person in the example) think the premises are True, False, or do you Suspend Judgment?  1.__SJ__    2.__T___  3.__SJ___ 

c. Is the argument Deductively Strong, Inductively Strong, or Weak?  __W_______

d.  If it is weak, why? _I'm suspending judgment about premise 1.  I just don't have enough information or evidnece to know whether the rate is high now, whether it's higher or lower than it has been in the past, whether it's going up or down.  So I'm not sure.  I'm also suspending judgment about premise 3.  The claim seems strong; some border compromise surely won't ruin a country.  Borders are difficult to maintain, and social, legal, and political structures can withstand some illegal immigration. So I'm not sure about this claim too.  _________

In the AI tutor, in our class discussions, in our quizzes, and in our tests, when we do an argument analysis, these examples show how to do it.  There are some features of the arguments that are objective and some parts, the T, F, and SJ concerning the premises that will depend on your view.  Study these examples, practice them with the AI tutor, and master all of the concepts involved in deductively strong, inductively strong, and weak arguments.  

Summary: 

Here, again, are all of the important definitions so far:

• A Valid argument is one where the premises, if they were true, would guarantee the truth of the conclusion.

• A Cogent argument is 1) invalid, and 2) the premises, if they were true, would make the conclusion likely to be true.

• An Ill-formed argument is one that is neither valid nor cogent.

• A Deductively Strong argument is one that is 1) valid and 2) the premises are reasonable for you to believe.

• An Inductively Strong argument is one that is 1) cogent, 2) the premises are reasonable for you to believe, and 3) it is not defeated by your total evidence.

• Arguments can be weak  three different ways:  

  •  S believes one or more premises are false, or S suspends judgment about one or more premises.

  • The argument is ill-formed

  • The argument is defeated  An argument is defeated when: 

    • It is cogent

    • S believes that all of the premises are true.

    • But S has background information leading them to believe that the conclusion is false.