Deductively Strong, Inductively Strong and Weak Arguments

Deductive Strength

Building on what we have learned about valid, cogent, and ill-formed arguments, in very general terms, we are now going to divide arguments into strong and weak categories. Very generally, strong arguments are ones whose conclusions we should accept because they support the conclusions well. Previously, we focused just on logical form of the arguments, but not the actual truth of the premises or conclusions. We saw that a valid argument is one where the premises, if they were true, would guarantee the truth of the conclusion. And a cogent argument is one that is invalid, and the premises, if they were true, would make the conclusion likely to be true. Ill-formed arguments are ones that are neither valid nor cogent.

Now we are going to add to the valid, cogent, and ill-formed concepts, by considering whether the premises are actually true. We saw earlier that for a claim to be true, what it asserts must match or reflect what is actually the case in the world. So the claim, "The Earth is round." is true because there is a planet, the Earth, that is round. (There are people who don't believe that this is true, but they are mistaken and they have a false belief.) Truth, we saw earlier, is objective and doesn't vary from person to person, although beliefs obviously can. When you reasonably believe a claim, then you think it is true. We are going to identify deductive arguments with valid arguments and inductive arguments with cogent arguments.

Here is the definition of a deductively strong argument:

Deductively strong arguments meet two conditions:

For an argument to be deductively strong for a person S, it must be

a) valid

AND

b) it is reasonable for S to believe the premises.

If both conditions are met for a person, then that argument is deductively strong for that person.

So while validity is an objective property of argument--arguments are either valid or invalid, whether or not S recognizes it--the reasonableness of the premises will, to some extent depend upon the other views that S has, S's background information, S's evidence, and so on.

If the premises are reasonable for S to believe, that means that S thinks they are true. And if the argument is valid, then it is an argument where the truth of the premises would guarantee the truth of the conclusion. So S is rationally committed to accepting the conclusion. The argument supports the conclusion for S. A deductively strong argument, then, is the best argument one can give for a conclusion. If you consider an argument that is valid and the premises are true, then you should accept the conclusion as true or reasonable.

What if the person considering the argument isn't sure whether one or more of the premises is reasonable? What if they suspend judgment about one or more of the premises? Are the conditions for a deductively strong argument met in that case? No. If you suspend judgment about a premise, then it wouldn't be true to say that it is reasonable for you to believe. You're not sure, so you wouldn't consider the premise and believe that it is true. So if a person suspends judgment about any of the premises, then the second condition for deductive strength is not met, and this argument cannot, therefore, be deductively strong. It will be weak in this case.

If S thinks that one or more of the premises are false, or even if S is not sure and suspends judgment about one or more of the premises, then the argument will not be strong for her. That is, the argument doesn't support the conclusion. The conclusion might be true, and the argument might be valid. It might even have all true premises. But unless S believes all the premises, they won't be adequate support to render the conclusion reasonable.

Previously we considered deductively valid arguments. The validity of an argument is a sort of hypothetical property; the premises, if they were true, would guarantee the truth of the conclusion. So a weird or silly argument like this one can be valid:

1. All inhabitants of the moon are Canadians.

2. Professor McCormick is an inhabitant of the moon.

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3. Therefore, Professor McCormick is a Canadian.

The premises are both actually false in this case; There are no inhabitants of the moon, Canadian or otherwise, so premise 1 is false. And Professor McCormick is not an inhabitant of the moon, so premise 2 is false. However, IF these premises were true, they would guarantee the truth of the conclusion. So this argument is valid. The concept of deductive strength now builds on the validity and asks the additional question, are the premises true? Do you find them reasonable? If an argument is valid and if the premises are true, then it is deductively valid. So this argument:

1. There are no Americans currently living on the moon.

2. Professor McCormick is an American.

______________________________

3. Therefore, Professor McCormick is not currently living on the moon.

is deductively strong. It meets both requirements for deductive strength. Any argument that fails one or the other or both of the requirements will be either inductively strong (see the next section for details) or it will be weak.

So consider John who believes, like all of us do, that "If the earth was flat, then ships sailing on the ocean would fall off." And John also believes that, "Ships sailing on the ocean do not fall off." So when John considers this argument:

1. If the earth was flat, then ships sailing on the ocean would fall off.

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

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3. Therefore, the earth is not flat.

it is deductively strong. It is valid, and S believes the premises, so the argument supports the conclusion and S should accept it (John does), or S is being irrational.

This argument is also deductively strong for John:

1. If my textbooks and teachers are reliable, then the earth is round.

2. My textbooks and teachers are reliable.

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3. Therefore, the earth is round.

This argument is weak for John, however:

1. The United States wins every war it engages in.

2. The United States is engaged in a war in Afghanistan.

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3. Therefore, the United States will win the war in Afghanistan.

John doesn't believe premise 1. Vietnam was a war that the United States engaged in, but did not win. So Vietnam is a counterexample to 1. Premise 2 is true. And the conclusion may turn out to be true (John doubts it.) But this argument is not good support for its conclusion.

The same argument could be deductively strong for someone else, however. Suppose that Marion believed premise 1. Suppose that she had received a poor education and lots of people she trusts and believes, and who are normally good sources of information either never mentioned Vietnam, or they were mistaken about Vietnam too. So she believes 1. falsely. And she believes 2. And the argument is valid. So it meets the conditions of deductive validity for Marion. But if Marion were to learn about Vietnam, and become convinced that that was a war that the United States lost, then the argument would be weak for Marion too.

More examples:

1. All Americans are Republicans.

2. Professor McCormick is an American.

________________________________

3. Therefore, Professor McCormick is a Republican.

This argument is not deductively strong (it is weak) because premise 1 is false.

1. All Americans are humans.

2. Professor McCormick is a human.

_______________________________

3. Therefore, Professor McCormick is an American.

This argument is not deductively strong (it is weak) because it is ill-formed. The logical structure is neither valid nor cogent.

1. All Americans are humans.

2. Professor McCormick is an American.

___________________________

3. Therefore, Professor McCormick is human.

This argument is deductively strong. It meets both conditions: it is valid (the premises, if they were true, would guarantee the truth of the conclusion) and the premises are true.

Inductive Strength

We will define inductively strong arguments this way: An argument is inductively strong for a person S when it meets three conditions:

1) 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, 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. We will consider what defeat is next. So consider 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.

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3) Therefore, Caroline is pregnant.

This argument is cogent; it follows the pattern we've been considering for cogent arguments. Premise 1 appears to be true because home pregnancy tests are typically 98% or more accurate. And if Caroline took the test and got a positive result, then premise 2 is true. So condition 2 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. Suppose D) is true: Caroline has an unusual medical condition that makes it impossible for her to get pregnant, and furthermore, one of the effects of the condition is that it makes her prone to test positive on pregnancy tests anyway. If D was true and Caroline knew it, then the argument would still be cogent, and the premises are still true, but D undermines the conclusion. D is a defeating piece of information that leads us to not accept 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. And S might have information that indicates that this is what has happened.

So consider Ellen who has read a New York Times article that cites several reputable studies that have found that most people in her neighborhood in Brooklyn 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 Brooklyn are Democrats.

2. Amy lives in Brooklyn.

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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 Brooklyn, 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 held a gun to her head. And he thinks she's sincere. So the same argument is weak for Michael. Notice that the premises are true and that they are reasonable for Michael to believe; the argument is not weak because he rejects a premise. It is weak because it is defeated by his total evidence.

For our purposes, 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.

Weak Arguments

The final and third category that arguments can fall into is when they are weak. All arguments will be either deductively strong, inductively strong, or weak. A weak argument is one that should fail to convince us of its conclusion. Given the way we are analyzing arguments, there are three ways that they can be weak.

1) An argument can be weak for a person when she considers the premises and she believes that one or more premises is false, or she suspends judgment about one or more of the premises. That is, if she thinks any premise is false, or if she suspends judgments about any premise, then that argument is weak. So consider this argument:

1) All CSUS students are male.

2) Arnold Schwarzenegger is a CSUS student.

3. Therefore, Arnold Schwarzenegger is male.

If I'm evaluating the strength of this argument, I'd have to point out that premises 1 is false because there are many CSUS students who are female. In fact, according to the last demographic information I saw on this from the administration, most CSUS students are female. I also believe that premise 2 is false. I don't think Schwarzenegger is a student at CSUS. He's a rich movie star. I don't think he is seeking a degree, and certainly not at CSUS. So the argument is weak because of both premises are false, I do not find them reasonable. If an argument seeks to convince that its conclusion on the basis of evidence or premises that I think are false, then they won't provide me with compelling grounds to accept the conclusion. False claims should not convince any reasonable person of anything.

2) An argument might also be weak because is logical form is bad. In our formal terms, this means that an argument can be weak because it is ill-formed. An ill-formed argument is neither valid nor cogent. So consider this example:

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. Schwarzenegger is a male. But that fact alone isn't enough to suggest 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. 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) The third and final way that an argument can be weak is because it is defeated. Defeat happens in the specific circumstances listed above under 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. it is an argument that is 1) cogent, 2) it has reasonable premises, but you know or reasonably believe the conclusion is false because of background information you possess. So consider this example:

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. So this argument meets the specific conditions for being weak by being defeated.

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.

A Weak argument is one that is neither deductively nor inductively strong.

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: