Inductive Validity

In the last chapter, we introduced the distinction between logical form and content.  The best kind of support in terms of logical form is deductive validity.  Valid arguments are ones in which the premises, if they were true, would guarantee the truth of the conclusion.  

There is another class of arguments we can identify that are not valid, but the logical structure still has some merit that could lend probabilistic support to their conclusions.  

Consider:

1. Most of the people in the crowd are wearing coats.

2. Smith is a person in the crowd.

_________________________________

3. Therefore, Smith is wearing a coat. 

or,

1.  99% of CSUS students own a smart phone.  

2.  Ramirez is a CSUS student

___________________________________

3.  Therefore, Ramirez owns a smart phone.  

What can we say about these arguments?  In the first, it seems like it's a pretty good bet that Smith is wearing a coat.  And in the second, if the premises are true, then it seems like Ramirez probably owns a smart phone.  But the arguments, strictly speaking, are invalid by our definition of deductive validity.  The premises, if they were true, would not guarantee the truth of the conclusion.  The premises in the first argument could be true and Smith is not wearing a coat.  And Ramirez might be in the 1% who do not own smart phones.  

So how should we characterize these arguments?  The premises do give us some substantial support for the conclusion.  So we need another category.  We will call these arguments cogent: 

Cogency defined:  an argument is cogent when both of these two conditions apply:

1.  It is invalid.

AND

2.  The premises, if they were true, would make the conclusion likely to be true.

The idea that cogent or inductively valid arguments have a logical structure such that the premises provide probabilistic grounds in support the conclusion.  Valid arguments have this property:  the premises, if they were true, would guarantee the truth of the conclusion.  But we need to be able to identify arguments that are not deductively valid, but their premises still support their conclusion by making it probably true.  Notice in the examples below that there is a general premise in every argument that has some language like, "most," "in almost every case," "vast majority," "probably," and so on.  These premises make a claim about the majority of a class or group of objects having a particular property, like "Most Americans are female."  "Most" means greater than 50%.  If these premises said "some," or "several," they would not provide support for their conclusions.  From the fact that "Some Americans are billionaires," for example, and the fact that "Matt McCormick is an American," we cannot reliably conclude that "McCormick is a billionaire.  But this argument is cogent and it does provide probabilistic support for its conclusion:

1.  Most Americans believe that the Earth is round.

2.  McCormick is an American.

__________________________

3.  Therefore, McCormick believes the Earth is round. 

The first premise says that the majority of a group of things (Americans) possess a property (believing that the Earth is round.).  So if we know that McCormick is in that group, as premise 2 asserts, then we can conclude that he probably believes the Earth is round.  It's not a guarantee.  He might be in the minority of people who do not believe it.  But if these premises were true, and we knew nothing else, we could conclude that 3 is probably true.  So that's a cogent argument. 

Induction:  Induction is complicated and there are many different kinds.  All of these are cases of induction:  

The sun has risen every day for billions of years.

Therefore, the sun will rise tomorrow.

In multiple controlled studies, smokers have much higher rates of lung cancer than non-smokers.

Therefore, smoking causes lung cancer.

Drug A and Drug B are chemically very similar and affect the same receptors.

Drug A causes liver toxicity.

Therefore, Drug B is likely to cause liver toxicity

The lawn is wet, the sky is overcast, and rain was forecast overnight.

Therefore, it probably rained last night.

This coin has landed heads 507 times out of 1,000 flips under fair conditions.

Therefore, the coin is approximately fair.

For the purposes of this class, we are going to simplify all of the different types of induction into one logical pattern.  Each of the kinds of induction above can be interpreted according to this form because there is a generalization like premise 1 being implicitly relied upon:

1. Most As are Bs.

2. X is an A.

     __________________

3.  Therefore, x is a B

We will understand “most” to mean “greater than 50%”.  And expressions like “Probably,” “In the vast majority of cases,” “usually,” and “predominately” as equivalent.  So these arguments fit our pattern and they are cogent:

1. Most cheap phone chargers break within a year. 

2. My phone charger is cheap.  

     _______________________________

3. Therefore, my phone charger will break within a year.  

1. Most students who skip lectures do worse on exams.

2. Alex skips lectures.

     _____________________

3. Therefore, Alex will do worse on exams.

1. In most of the cases where drug A and drug B are chemically very similar, they cause similar effects.  

2. Drug A causes liver toxicity and is very similar to drug B.  

     _________________________________________

3. There drug B is likely to cause liver toxicity

Arguments that are Not Cogent

Valid arguments are not cogent.  That is, by definition, a cogent argument is not valid.  In order to separate the concept of valid arguments from cogent arguments, valid arguments are defined as arguments in which the premises, if they were true, would guarantee the truth of the conclusion.  And cogent arguments are defined as arguments that are 1) invalid, and 2) the premises, if they were true, would make the conclusion probably true.  

Reversing the Generalization:  What about arguments like this:

1.  Most marathon runners are physically fit.  

2. Eddie Hall is physically fit.  

     __________________________

3.  Therefore, Eddie Hall is a marathon runner.  

Or, 

1. Most terrorists are Muslims

2. Saleem is a Muslim.

     _________________________

3.  Therefore, Saleem is a terrorist.  

It should be clear that while these arguments might appeal or they might even have true conclusions, they do not meet our criteria for cogent arguments.  They have the logical form of:

1.  Most As are Bs.

2. X is a B

     _________________

3.  Therefore, x is an A.  

This pattern is not cogent, it reverses the order of the second premise and the conclusion from our examples above.  The mistake here is a bit like the Affirming the Consequent mistake we considered in the last chapter.  The problem with the logical inference is clear in these cases:

1. 75% of the dogs in the pound have had their rabies shots.

2. Roscoe has had his rabies shot.

     __________________________

3. Therefore, Roscoe is a dog in the pound.                     [NOT COGENT]

If all you know is that Roscoe has had his rabies shot, you can’t infer that he’s probably in the pound.  There are many more dogs with shots out of the pound than in the pound.  We should not interpret the claims “Most As are Bs” to mean “Most Bs are As.”  “Most of the students at CSUS are women”  (54%) is not equivalent to “Most of the women (in the world) are students at CSUS.” There are over 6 billion people on the planet, and about 28,000 people at CSUS.  These arguments are not cogent:

1. Most of the students at CSUS are women.

2. Elizabeth is a woman.

      _____________________________

3. Therefore, Elizabeth is a CSUS student

"Most of a most" arguments are not cogent.  Arguments that compound the probability claims are not cogent.  Consider:  

1. Most CSUS students are women.  

2. Most women are liberal.  

3. Yin is a CSUS student.  

     _________________________

4.  Therefore, Yin is liberal.  

We might be tempted to treat the inference here as transitive, the way we did for hypothetical syllogism, but for compound probability claims, there is a particular kind of erosion of support.  

There could be cases where if these premises were true, they would make the conclusion likely to be true.  If the “Most” in premise 1 was in fact 99%.  And then if “most” in the second premise also meant “99%” then they would support the conclusion.  But if the probability is lower, the support for the conclusion quickly drops.  

In fact, only about 53% of CSUS students are women.  And the majority margin for women being liberal is thin.  Suppose it was only 52%.  In that case, the correct way to figure out the support would be to multiply .53 by .52, which is only .2756.  In general, a “most of a most,” argument is not cogent.  From these premises alone, we cannot infer that the conclusion is likely to be true when the premises are assumed to be true.   Here's why:

The multiplication rule in probability theory says that if the odds of an individual's having one property are, say, 60%, and the odds of having some other independent property are 60%, then you multiply the two probabilities to determine the likelihood that the individual has both:  .6 x .6 = .36 or 36%

So it is not probable that an individual has both properties, even it is probable that the individual has each of the properties separately. 

Some Invalid Arguments are Not Cogent

What about this argument? 

1. If a person inherits a huge fortune from a dead relative, then they will be rich.

2. Smith is rich. 

     ____________________

3. Therefore, Smith inherited a huge fortune from a dead relative.

This argument is invalid.  There are many ways to get rich, so if all we know about Smith is that she is rich, we can’t infer that she got it through inheritance.

But is it cogent?  That is, if these premises were true, would they make the conclusion likely to be true?  If you know that inheritance produces wealth and that Smith is wealthy, can you infer that Smith probably got her money from inheritance?

No, we can’t.  Think about it this way.  How many other ways are there to get rich?  Hundreds?  Thousands?  Of all the rich people in the world, what percentage of them got that way by inheritance?  Maybe a lot of them.  But this argument, especially premise one, does not give us that information.  It doesn’t tell us the rate at which rich people inherited their wealth.  If it said, “Most rich people get their money through inheritance.”  And then we added that Smith is rich, then we could infer cogently that Smith probably got her money through inheritance.  But the argument, as it stands, doesn’t give us that information.

The mere fact that an argument is invalid is not sufficient to indicate that it is cogent.  And the arguments that we've been considering with conditionals (If P then Q) claims in them, or universal generalizations (All As are Bs) generally won't be cogent arguments unless there are other premises that give us the probabilistic grounds, or "most," or "majority" language that we have seen is necessary for a cogent argument.  

Inductive Generalizations  Consider these three arguments, the generalizations that they employ in premise 1, and overall effect that has on the logical support relationship: 

1.  All Americans are above the poverty line.  

2. McCormick is an American.

     ___________________________________

3.  Therefore, McCormick is above the poverty line.  

1.  Most Americans are above the poverty line

2. McCormick is an American

     ___________________________________

3.  Therefore, McCormick is above the poverty line.  

1. Some Americans are above the poverty line

2. McCormick is an American

     ___________________________________

3.  Therefore, McCormick is above the poverty line.  

The first argument is valid (UA), the second argument is cogent, and the third argument is ill-formed.  An ill-formed argument is neither valid nor cogent.  The premises, even if they were true, wouldn’t guarantee the conclusion, nor would they make the conclusion probably true.  We will consider ill-formed arguments more at the end of the chapter.  

Let’s focus on the concept of a generalization here.  The first premise in each of the arguments above is a generalization.  They are sentences of the form:  

Quantifier of a class or category has a property.   

For inductive arguments, the primary logical pattern we will consider is an argument:

1.  Most As are Bs. 

2. X is an A.

     ____________________

3.  Therefore, X is a B.  

We will consider sentences such as “The vast majority of As are Bs,” or “If something is an A, then it is probably a B.” or “Probably, if A is true, then B is true,” etc. to be equivalent when we encounter them, and we will rewrite or reconstruct those arguments with the “Most As are Bs” form.  

Furthermore, the concept of a quantifier clears up an ambiguity problem that is often present when talk and think informally about the world.  Someone might say, “Basketball players are tall,” or “Women are short,” or “Philosophers are bad at math,” or “Politicians are liars.”  These kinds of claims are common and they often produce a lot of disagreement and discussion.  The disagreements will often take the form, “That’s not true, Muggsy Bogues and Spud Webb are both basketball players and they are short.”  or “Politicians aren’t liars; the mayor of Boston is virtuous and doesn’t lie.”  

We should notice a couple of things here.  First, the sentences are missing a quantifier and which one we use makes a lot of difference in whether the claim is true.  “All,” “Most,” "On average," “Some,” and “No.”  For all of these sentences, the “All” claim is certainly false.  For all of these sentences the “Some” claim is certainly true.  And the “No” claim is also false.  Whether or not the “Most” claim, where “most” is understood as “greater than 50%” is true will require some more thought, investigation, and evidence.  

______basketball players are tall.

______women are short.

______philosophers are bad at math

______politicians are liars.

A reasonable person who says something like “Basketball players are tall,” should be charitably interpreted as meaning “most,” and they would be correct.  And if that’s what they meant, then the objection of the form, “X is a basketball player and X isn’t tall,” misses the point.  If someone says that most As are Bs, or the some As are Bs, then an example of an A that is not B is logically consistent with their claim, and therefore isn’t an objection at all.  Objections, to be successful, need to deny the truth of the claim or at least raise doubts about it, and a short basketball player does neither.  

Later we will learn about understanding these sorts of claims as correlations, such as “Being tall is positively correlated with being a basketball player,” or “Being a politician is negatively correlated with telling the truth.”  To assert that there’s a correlation between A and B is to say that the rate of As that are Bs is higher (or lower).  

It will be important for critical reasoning and being rational to understand what correlations are, what evidence would prove them, and what evidence might disprove them.  

Possible vs. Probable

We should contrast events that are naturally impossible, or ruled out by natural law, with events that are statistically improbable. If we dropped a million dice onto a parking lot and all of them came up with a 1 on top, it would be statistically improbable, but it is not ruled out by the laws of physics. The odds of winning the lottery are millions to one, and if your friend said he had the winning ticket, you might say, "That's impossible!" But it violates no logical or natural laws. It's merely statistically improbable. We might also think about it this way: the laws of natural, such as the ones governing the periodic table of elements, constrain the regular behavior of matter. They also make some events improbable. Snow is Sacramento is rare because of the typical weather conditions in the winter. But the natural conditions can happen. On Feb. 5, 1976, two inches of snow accumulated in Sacramento. Such an event is clearly logically possible, and it is naturally possible. But it naturally improbable. Having it snow in Sacramento on 10 consecutive days this winter is exceedingly unlikely. We might even say that it is impossible because it is so improbable. But on the definitions we are using of natural and logically impossibility, this would be a misnomer. Every year, approximately 1 person in 500 million is eaten by sharks. So the odds of your being eaten by sharks, all other things being equal, are about 99.9999998%. It is not logically impossible; clearly there is no logical contradiction in the scenario the way there is with a four sided triangle. But it is naturally possible. There is nothing about the laws of nature that prevent such events from occurring. But is it likely? No. Is it probable? No. Is it reasonable to believe?

In probability theory, odds are depicted on a range from 0 to 1, with 1 being certain. The threshold where the odds begin to favor belief is .5. As evidence accumulates in favor of a claim and the odds increase towards 1, the claim becomes more and more reasonable to believe. The conviction that you have about the truth of a claim should be proportional to the quality and quantity of the evidence you have concerning it.

The mere logical possibility that a claim is true should not be enough to elevate it across the .5 threshold and make it reasonable. That is, possible does not imply probable. While is is possible that you will be eaten by sharks today, as we saw above, it is not probable. It is so unlikely, you should believe with a great deal of conviction that you will not be eaten by sharks. Here are more examples illustrating the point that possible does not equal probable:

It is possible that the Holocaust didn’t happen.

It is possible that wearing a raw steak hat wards off disease.

It is possible that even though you are taking birth control pills exactly as prescribed everyday you are pregnant.

It is possible that the government is watching everything you do and hiding it very well.

It is possible that Christopher Marlowe wrote all of Shakespeare’s plays.

It is possible that having sex with a virgin cures HIV.

It is possible that eating the flesh of your enemies gives you power.

It is possible that birth defects are caused by wickedness from a past life.

It is possible that the Detroit Lions could win the Super bowl.

It is possible that fever is caused by demon possession.

It is possible that the earth rests on the back of a (invisible) turtle.

It is possible that lightening is thrown by an angry Zeus.

It is possible that the moon landing in 1969 was faked on a secret Hollywood set by NASA.

It is possible that wishful thinking can help you win the lottery

It is possible that wearing your lucky underwear will help you win the basketball game

It is possible that Santa exists.

It is possible that there are still dinosaurs.

It is possible that if you concentrate you can levitate.

It is possible that tossing spilled salt over your shoulder improves luck.

It is possible that opening an umbrella indoors or breaking a mirror is bad luck.

It is possible that conceiving in the spring produces boy babies.

It is possible that swinging a wedding ring on a string in front of a pregnant woman's stomach will reveal the sex of the baby.

Some of these will seem outrageous to you, and you won't be tempted to believe them, or even argue that we should be agnostic about them. But some of these might strike you as being more plausible than others.  In cogent arguments, we are looking for grounds to think that the conclusion is probable, not merely possible.   

Bayesian Belief and Updating

Up to this point, we have said repeatedly that a rational person should proportion belief to the strength of the evidence. We have treated probability as a continuum from 0 to 1 and suggested that when the evidence tips a claim above the .5 threshold, belief becomes reasonable. But that leaves an important question unanswered: how exactly should new evidence change what we believe? If you begin somewhat skeptical of a claim and then encounter new evidence in its favor, how much more confident should you become? If you begin highly confident and encounter contrary evidence, how much should your confidence decrease? These questions arise constantly in medicine, law, science, politics, and everyday life. Bayesian reasoning is an attempt to answer them carefully and systematically. You do not need advanced mathematics to understand the basic idea. The core principle is intuitive: new evidence should change your belief in proportion to how strongly that evidence supports or undermines the claim. Bayesian reasoning simply makes that principle explicit.

Whenever you consider a claim, you do not begin from zero. You already have background knowledge about how the world works, and that background knowledge gives you an initial level of confidence before you examine new evidence. This starting level of confidence is called your prior probability. Suppose someone tells you, “There is a tiger loose in this building.” Before you look for evidence, your confidence in that claim is already extremely low. It is not logically impossible, but given what you know about zoos, building security, and everyday life, it is highly improbable. Now compare that to the claim, “There is a dog loose in this building.” Your prior probability here is much higher. Dogs escape occasionally. That fits more comfortably with your background understanding of the world. The prior probability represents everything you reasonably believe before encountering the new piece of evidence, and it matters. The same evidence can move you a great deal or only slightly depending on where you started.

Now suppose you hear loud growling and see deep claw marks on the hallway door. How much should that increase your confidence that a tiger is present? That depends on two questions. First, how likely would this evidence be if the claim were true? If there really were a tiger in the building, growling and claw marks would be expected. Second, how likely would this evidence be if the claim were false? Could the noise be something else? Could the marks be a prank? Could they be from construction equipment? Evidence supports a claim when it would be much more likely if the claim were true than if it were false. If the evidence would be equally likely either way, it tells you very little. If it would be extremely unlikely unless the claim were true, then it should move your confidence significantly. The interaction between your starting confidence and the strength of the evidence is the heart of Bayesian reasoning.

A medical example makes this clearer. Suppose a disease affects 1 in 1,000 people. That means before testing, your prior probability of having the disease is 0.1%. Now suppose the test is 99% accurate. If you have the disease, it returns positive 99% of the time. If you do not have the disease, it returns positive only 1% of the time. You test positive. Should you now be 99% confident that you have the disease? No. Out of 1,000 people, about 1 actually has the disease and about 999 do not. Of those 999 healthy people, about 1%—roughly 10 people—will falsely test positive. So among the roughly 11 people who test positive, only 1 actually has the disease. That means your probability of truly having it, given a positive result, is closer to 9% than 99%. The test dramatically increased your belief—from 0.1% to roughly 9%—but it did not make the disease likely. The reason is the low prior probability. Rare conditions require very strong evidence to become probable. This example illustrates why ignoring base rates is such a serious reasoning error. We often focus on the accuracy of the test and forget how uncommon the condition is in the first place.

Bayesian reasoning tells us that rational belief updating depends on two factors: how confident you were before and how strongly the evidence favors the claim over its alternatives. If your prior probability is extremely low, weak or moderate evidence will not move you very far. If your prior is moderate, the same evidence may move you substantially. If your prior is high and you encounter strong contrary evidence, your confidence should decrease. Beliefs are not all-or-nothing. They should move in degrees, and evidence pushes them up or down by different amounts depending on its strength.

Why Priors Matter: A Medical Example

Consider a medical test.  Suppose a disease affects 1 in 1,000 people.  That means your prior probability of having the disease, before testing, is 0.001 (0.1%).  

Now suppose the test is very accurate:  If you have the disease, it is positive 99% of the time.  If you do not have the disease, it is positive 1% of the time.  And you test positive.

Should you now be 99% confident you have the disease?  No.  Why not?  Because the disease is very rare.  Out of 1,000 people, only 1 person has the disease, and 999 do not.  Of the 999 healthy people, about 1% (roughly 10 people) will falsely test positive since the test is 99% accurate.  

So among all the people who test positive, only 1 truly has the disease, and 9 do not.  That means your probability of actually having the disease, given a positive test, is closer to 1 in 10 — about 9%.  The test increases your belief significantly — from 0.1% to about 9% — but it does not make it near certain. 

Even though you tested positive, the low prior probability or low base rate weakens the test as an indicatore.  Or another way to think about it, is that the disease is more rare than the test is accurate. This example shows why ignoring base rates leads to serious reasoning mistakes.

Priors and Defeat

This framework also clarifies our earlier discussion of defeat. An argument can be cogent and have true premises, yet still fail to move you because your total background evidence strongly supports the opposite conclusion. If your prior probability for the conclusion is already very low due to strong counterevidence, modest new evidence will not be enough to cross the threshold for belief. Bayesian reasoning explains why this is not stubbornness but rational resistance to weak updating.

Notice how this connects to our account of inductive strength. When we say, “Most As are Bs; X is an A; therefore X is probably a B,” we are implicitly describing a belief update. If 85% of As are Bs, learning that X is an A should significantly increase your confidence that X is a B. If only 51% of As are Bs, the update should be small. Stronger statistical evidence justifies larger shifts in confidence. Bayesian reasoning makes explicit what we have been doing informally all along.

You do not need to memorize formulas in order to reason well. What you should cultivate is the habit of asking the right questions. How likely was this claim before I saw this evidence? Would I expect this evidence if the claim were false? How much should this really change my confidence? Without this discipline, we overreact to vivid cases, misinterpret test results, ignore base rates, and allow weak evidence to produce strong conviction. Bayesian reasoning is simply a precise way of expressing something we have already endorsed throughout this course: rational belief is belief that adjusts carefully and proportionally to the evidence. Beliefs should move when the evidence moves, and they should move by the right amount. That is Bayesian rationality in its most basic and practical form.

Ill formed arguments  

We can now consider a final category of logical forms of arguments.  Arguments that are neither valid nor cogent are ill-formed.  Consider: 

1. If you are a politician, then you are a liar.

2. Smith is a a liar.

     _________________

3. Therefore, Smith is a politician.

This argument is ill formed;  it is neither valid nor cogent.  It follows an invalid, fallacious argument pattern that we have seen before.  And it also does not provide probabilistic grounds to support its conclusion.  These premises don't guarantee this conclusion, nor do they make the conclusion likely.  The same goes for:

1. Most politicians cannot be trusted.  

2. McCormick cannot be trusted.  

     ___________________________

3. Therefore, McCormick is a politician. 

The class of “cannot be trusted” is bigger than politicians.  If all we know is that McCormick cannot be trusted, it doesn’t follow that he’s probably a politician.  If premise 1 was, “Most people who cannot be trusted are politicians,” then this would be cogent.  

And arguments that don’t even resemble a valid or cogent argument, such as this, are ill-formed.  The general notion is that the premises, even if they were true, wouldn’t provide us with grounds for thinking that the conclusion is true:

1. My roommate said that Airborne prevents colds.  

2. The label on the box says that it supports the immune system.  

     ___________________________________

3.  Therefore, Airborne prevents colds.  

Ill-formed arguments are just bad arguments logically.  

Summary

Cogent or Inductively Valid Arguments:

• Cogency defined:  an argument is cogent when both of these two conditions apply: 1.  It is invalid, and 2.  The premises, if they were true, would make the conclusion likely to be true.

Arguments that are Not Cogent

• Reversing the Generalization:  Arguments of the form “Most As are Bs,” “x is a B,” “therefore, x is an A” are not cogent.

• “Most of a most” arguments are not cogent because of the multiplication rule 

• Some Invalid Arguments are Not Cogent

Inductive Generalizations have this form:  Most As are Bs.

• Generalizations have this form:  Quantifier of a class or category has a property.   

• “Possible” means that a claim is not logically contradictory, or that it is logically consistent, or that there is a non-zero chance that it is true.  

• “Probable” means that there is a greater than 50% chance that it is true.  

Ill formed arguments  are neither valid nor cogent are ill-formed.