Part I: Propositional Logic
3. Good Arguments
3.1 A historical example
An important example of excellent reasoning can be found in the case of the medical advances of the Nineteenth Century physician, Ignaz Semmelweis. Semmelweis was an obstetrician at the Vienna General Hospital. Built on the foundation of a poor house, and opened in 1784, the General Hospital is still operating today. Semmelweis, during his tenure as assistant to the head of one of two maternity clinics, noticed something very disturbing. The hospital had two clinics, separated only by a shared anteroom, known as the First and the Second Clinics. The mortality rate for mothers delivering babies in the First Clinic, however, was nearly three times as bad as the mortality for mothers in the Second Clinic (9.9 % average versus 3.4% average). The same was true for the babies born in the clinics: the mortality rate in the First Clinic was 6.1% versus 2.1% at the Second Clinic.^{[5]} In nearly all these cases, the deaths were caused by what appeared to be the same illness, commonly called “childbed fever”. Worse, these numbers actually understated the mortality rate of the First Clinic, because sometimes very ill patients were transferred to the general treatment portion of the hospital, and when they died, their death was counted as part of the mortality rate of the general hospital, not of the First Clinic.
Semmelweis set about trying to determine why the First Clinic had the higher mortality rate. He considered a number of hypotheses, many of which were suggested by or believed by other doctors.
One hypothesis was that cosmic-atmospheric-terrestrial influences caused childbed fever. The idea here was that some kind of feature of the atmosphere would cause the disease. But, Semmelweis observed, the First and Second Clinics were very close to each other, had similar ventilation, and shared a common anteroom. So, they had similar atmospheric conditions. He reasoned: If childbed fever is caused by cosmic-atmospheric-terrestrial influences, then the mortality rate would be similar in the First and Second Clinics. But the mortality rate was not similar in the First and Second Clinics. So, the childbed fever was not caused by cosmic-atmospheric-terrestrial influences.
Another hypothesis was that overcrowding caused the childbed fever. But, if overcrowding caused the childbed fever, then the more crowded of the two clinics should have the higher mortality rate. But, the Second Clinic was more crowded (in part because, aware of its lower mortality rate, mothers fought desperately to be put there instead of in the First Clinic). It did not have a higher mortality rate. So, the childbed fever was not caused by overcrowding.
Another hypothesis was that fear caused the childbed fever. In the Second Clinic, the priest delivering last rites could walk directly to a dying patient’s room. For reasons of the layout of the rooms, the priest delivering last rites in the First Clinic walked by all the rooms, ringing a bell announcing his approach. This frightened patients; they could not tell if the priest was coming for them. Semmelweis arranged a different route for the priest and asked him to silence his bell. He reasoned: if the higher rate of childbed fever was caused by fear of death resulting from the priest’s approach, then the rate of childbed fever should decline if people could not tell when the priest was coming to the Clinic. But it was not the case that the rate of childbed fever declined when people could not tell if the priest was coming to the First Clinic. So, the higher rate of childbed fever in the First Clinic was not caused by fear of death resulting from the priest’s approach.
In the First Clinic, male doctors were trained; this was not true in the Second Clinic. These male doctors performed autopsies across the hall from the clinic, before delivering babies. Semmelweis knew of a doctor who cut himself while performing an autopsy, and who then died a terrible death not unlike that of the mothers who died of childbed fever. Semmelweis formed a hypothesis. The childbed fever was caused by something on the hands of the doctors, something that they picked up from corpses during autopsies, but that infected the women and infants. He reasoned that: if the fever was caused by cadaveric matter on the hands of the doctors, then the mortality rate would drop when doctors washed their hands with chlorinated water before delivering babies. He forced the doctors to do this. The result was that the mortality rate dropped to a rate below that even of the Second Clinic.
Semmelweis concluded that the best explanation of the higher mortality rate was this “cadaveric matter” on the hands of doctors. He was the first person to see that washing of hands with sterilizing cleaners would save thousands of lives. It is hard to overstate how important this contribution is to human well being. Semmelweis’s fine reasoning deserves our endless respect and gratitude.
But how can we be sure his reasoning was good? Semmelweis was essentially considering a series of arguments. Let us turn to the question: how shall we evaluate arguments?
3.2 Arguments
Our logical language now allows us to say conditional and negation statements. That may not seem like much, but our language is now complex enough for us to develop the idea of using our logic not just to describe things, but also to reason about those things.
We will think of reasoning as providing an argument. Here, we use the word “argument” not in the sense of two or more people criticizing each other, but rather in the sense we mean when we say, “Pythagoras’s argument”. In such a case, someone is using language to try to convince us that something is true. Our goal is to make this notion very precise, and then identify what makes an argument good.
We need to begin by making the notion of an argument precise. Our logical language so far contains only sentences. An argument will, therefore, consist of sentences. In a natural language, we use the term “argument” in a strong way, which includes the suggestion that the argument should be good. However, we want to separate the notion of a good argument from the notion of an argument, so we can identify what makes an argument good, and what makes an argument bad. To do this, we will start with a minimal notion of what an argument is. Here is the simplest, most minimal notion:
Argument: an ordered list of sentences; we call one of these sentences the “conclusion”, and we call the other sentences “premises”.
This is obviously very weak. (There is a famous Monty Python skit where one of the comedians ridicules the very idea that such a thing could be called an argument.) But for our purposes, this is a useful notion because it is very clearly defined, and we can now ask, what makes an argument good?
The everyday notion of an argument is that it is used to convince us to believe something. The thing that we are being encouraged to believe is the conclusion. Following our definition of “argument”, the reasons that the person gives will be what we are calling “premises”. But belief is a psychological notion. We instead are interested only in truth. So, we can reformulate this intuitive notion of what an argument should do, and think of an argument as being used to show that something is true. The premises of the argument are meant to show us that the conclusion is true.
What then should be this relation between the premises and the conclusion? Intuitive notions include that the premises should support the conclusion, or corroborate the conclusion, or make the conclusion true. But “support” and “corroborate” sound rather weak, and “make” is not very clear. What we can use in their place is a stronger standard: let us say as a first approximation that if the premises are true, the conclusion is true.
But even this seems weak, on reflection. For, the conclusion could be true by accident, for reasons unrelated to our premises. Remember that we define the conditional as true if the antecedent and consequent are true. But this could happen by accident. For example, suppose I say, “If Tom wears blue then he will get an A on the exam”. Suppose also that Tom both wears blue and Tom gets an A on the exam. This makes the conditional true, but (we hope) the color of his clothes really had nothing to do with his performance on the exam. Just so, we want our definition of “good argument” to be such that it cannot be an accident that the premises and conclusion are both true.
A better and stronger standard would be that, necessarily, given true premises, the conclusion is true.
This points us to our definition of a good argument. It is traditional to call a good argument “valid.”
Valid argument: an argument for which, necessarily, if the premises are true, then the conclusion is true.
This is the single most important principle in this book. Memorize it.
A bad argument is an argument that is not valid. Our name for this will be an “invalid argument”.
Sometimes, a dictionary or other book will define or describe a “valid argument” as an argument that follows the rules of logic. This is a hopeless way to define “valid”, because it is circular in a pernicious way: we are going to create the rules of our logic in order to ensure that they construct valid arguments. We cannot make rules of logical reasoning until we know what we want those rules to do, and what we want them to do is to create valid arguments. So “valid” must be defined before we can make our reasoning system.
Experience shows that if a student is to err in understanding this definition of “valid argument”, he or she will typically make the error of assuming that a valid argument has all true premises. This is not required. There are valid arguments with false premises and a false conclusion. Here’s one:
If Miami is the capital of Kansas, then Miami is in Canada. Miami is the capital of Kansas. Therefore, Miami is in Canada.
This argument has at least one false premise: Miami is not the capital of Kansas. And the conclusion is false: Miami is not in Canada. But the argument is valid: if the premises were both true, the conclusion would have to be true. (If that bothers you, hold on a while and we will convince you that this argument is valid because of its form alone. Also, keep in mind always that “if…then…” is interpreted as meaning the conditional.)
Similarly, there are invalid arguments with true premises, and with a true conclusion. Here’s one:
If Miami is the capital of Ontario, then Miami is in Canada. Miami is not the capital of Ontario. Therefore, Miami is not in Canada.
(If you find it confusing that this argument is invalid, look at it again after you finish reading this chapter.)
Validity is about the relationship between the sentences in the argument. It is not a claim that those sentences are true.
Another variation of this confusion seems to arise when we forgot to think carefully about the conditional. The definition of valid is not “All the premises are true, so the conclusion is true.” If you don’t see the difference, consider the following two sentences. “If your house is on fire, then you should call the fire department.” In this sentence, there is no claim that your house is on fire. It is rather advice about what you should do if your house is on fire. In the same way, the definition of valid argument does not tell you that the premises are true. It tells you what follows if they are true. Contrast now, “Your house is on fire, so you should call the fire department”. This sentence delivers very bad news. It is not a conditional at all. What it really means is, “Your house is on fire and you should call the fire department”. Our definition of valid is not, “All the premises are true and the conclusion is true”.
Finally, another common mistake is to confuse true and valid. In the sense that we are using these terms in this book, only sentences can be true or false, and only arguments can be valid and invalid. When discussing and using our logical language, it is nonsense to say, “a true argument”, and it is nonsense to say, “a valid sentence”.
Someone new to logic might wonder, why would we want a definition of “good argument” that does not guarantee that our conclusion is true? The answer is that logic is an enormously powerful tool for checking arguments, and we want to be able to identify what the good arguments are, independently of the particular premises that we use in the argument. For example, there are infinitely many particular arguments that have the same form as the valid argument given above. There are infinitely many particular arguments that have the same form as the invalid argument given above. Logic lets us embrace all the former arguments at once, and reject all those bad ones at once.
Furthermore, our propositional logic will not be able to tell us whether an atomic sentence is true. If our argument is about rocks, we must ask the geologist if the premises are true. If our argument is about history, we must ask the historian if the premises are true. If our argument is about music, we must ask the music theorist if the premises are true. But the logician can tell the geologist, the historian, and the musicologist whether her arguments are good or bad, independent of the particular premises.
We do have a common term for a good argument that has true premises. This is called “sound”. It is a useful notion when we are applying our logic. Here is our definition:
Sound argument: a valid argument with true premises.
A sound argument must have a true conclusion, given the definition of “valid”.
3.3 Checking arguments semantically
Every element of our definition of “valid” is clear except for one. We know what “if…then…” means. We defined the semantics of the conditional in chapter 2. We have defined “argument”, “premise”, and “conclusion”. We take true and false as primitives. But what does “necessarily” mean?
We define a valid argument as one where, necessarily, if the premises are true, then the conclusion is true. It would seem the best way to understand this is to say, there is no situation in which the premises are true but the conclusion is false. But then, what are these “situations”? Fortunately, we already have a tool that looks like it could help us: the truth table.
Remember that in the truth table, we put on the bottom left side all the possible combinations of truth values of some set of atomic sentences. Each row of the table then represents a kind of way the world could be. Using this as a way to understand “necessarily”, we could rephrase our definition of valid to something like this, “In any kind of situation in which all the premises are true, the conclusion is true.”
Let’s try it out. We will need to use truth tables in a new way: to check an argument. That will require having not just one sentence, but several on the truth table. Consider an argument that looks like it should be valid.
If Jupiter is more massive than Earth, then Jupiter has a stronger gravitational field than Earth. Jupiter is more massive than Earth. In conclusion, Jupiter has a stronger gravitational field than Earth.
This looks like it has the form of a valid argument, and it looks like an astrophysicist would tell us it is sound. Let’s translate it to our logical language using the following translation key. (We’ve used up our letters, so I’m going to start over. We’ll do that often: assume we are starting a new language each time we translate a new set of problems or each time we consider a new example.)
P: Jupiter is more massive than Earth
Q: Jupiter has a stronger gravitational field than Earth.
This way of writing out sentences of logic and sentences of English we can call a “translation key”. We can use this format whenever we want to explain what our sentences mean in English.
Using this key, our argument would be formulated
(P→Q)
P
______
Q
That short line is not part of our language, but rather is a handy tradition. When quickly writing down arguments, we write the premises, and then write the conclusion last, and draw a short line above the conclusion.
This is an argument: it is an ordered list of sentences, the first two of which are premises and the last of which is the conclusion.
To make a truth table, we identify all the atomic sentences that constitute these sentences. These are P and Q. There are four possible kinds of ways the world could be that matter to us then:
P | Q | |||
---|---|---|---|---|
T | T | |||
T | F | |||
F | T | |||
F | F |
We’ll write out the sentences, in the order of premises and then conclusion.
premise | premise | conclusion | ||
P | Q | (P→Q) | P | Q |
T | T | |||
T | F | |||
F | T | |||
F | F |
Now we can fill in the columns for each sentence, identifying the truth value of the sentence for that kind of situation.
premise | premise | conclusion | ||
P | Q | (P→Q) | P | Q |
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | T |
F | F | T | F | F |
We know how to fill in the column for the conditional because we can refer back to the truth table used to define the conditional, to determine what its truth value is when the first part and second part are true; and so on. P is true in those kinds of situations where P is true, and P is false in those kinds of situations where P is false. And the same is so for Q.
Now, consider all those kinds of ways the world could be such that all the premises are true. Only the first row of the truth table is one where all the premises are true. Note that the conclusion is true in that row. That means, in any kind of situation in which all the premises are true, the conclusion will be true. Or, equivalently: necessarily, if all the premises are true, then the conclusion is true.
premise | premise | conclusion | ||
P | Q | (P→Q) | P | Q |
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | T |
F | F | T | F | F |
Consider in contrast the second argument above, the invalid argument with all true premises and a true conclusion. We’ll use the following translation key.
R: Miami is the capital of Ontario
S: Miami is in Canada
And our argument is thus
(R→S)
¬R
_____
¬S
Here is the truth table.
premise | premise | conclusion | ||
R | S | (R→S) | ¬R | ¬S |
T | T | T | F | F |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | T | T |
Note that there are two kinds of ways that the world could be in which all of our premises are true. These correspond to the third and fourth row of the truth table. But for the third row of the truth table, the premises are true but the conclusion is false. Yes, there is a kind of way the world could be in which all the premises are true and the conclusion is true; that is shown in the fourth row of the truth table. But we are not interested in identifying arguments that will have true conclusions if we are lucky. We are interested in valid arguments. This argument is invalid. There is a kind of way the world could be such that all the premises are true and the conclusion is false. We can highlight this.
premise | premise | conclusion | ||
R | S | (R→S) | ¬R | ¬S |
T | T | T | F | F |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | T | T |
Hopefully it becomes clear why we care about validity. Any argument of the form, (P→Q) and P, therefore Q, is valid. We do not have to know what P and Q mean to determine this. Similarly, any argument of the form, (R→S) and ¬R, therefore ¬S, is invalid. We do not have to know what R and S mean to determine this. So logic can be of equal use to the astronomer and the financier, the computer scientist or the sociologist.
3.4 Returning to our historical example
We described some (not all) of the hypotheses that Semmelweis tested when he tried to identify the cause of childbed fever, so that he could save thousands of women and infants. Let us symbolize these and consider his reasoning.
The first case we considered was one where he reasoned: If childbed fever is caused by cosmic-atmospheric-terrestrial influences, then the mortality rate would be similar in the First and Second Clinics. But the mortality rate was not similar in the First and Second Clinics. So, the childbed fever is not caused by cosmic-atmospheric-terrestrial influences.
Here is a key to symbolize the argument.
T: Childbed fever is caused by cosmic-atmospheric-terrestrial influences.
U: The mortality rate is similar in the First and Second Clinics.
This would mean the argument is:
(T→U)
¬U
_____
¬T
Is this argument valid? We can check using a truth table.
premise | premise | conclusion | ||
T | U | (T→U) | ¬U | ¬T |
T | T | T | F | F |
T | F | F | T | F |
F | T | T | F | T |
F | F | T | T | T |
The last row is the only row where all the premises are true. For this row, the conclusion is true. Thus, for all the kinds of ways the world could be in which the premises are true, the conclusion is also true. This is a valid argument. If we accept his premises, then we should accept that childbed fever was not caused by cosmic-atmospheric-terrestrial influences.
The second argument we considered was the concern that fear caused the higher mortality rates, particularly the fear of the priest coming to deliver last rites. Semmelweis reasoned that if the higher rate of childbed fever is caused by fear of death resulting from the priest’s approach, then the rate of childbed fever should decline if people cannot discern when the priest is coming to the Clinic. Here is a key:
V: the higher rate of childbed fever is caused by fear of death resulting from the priest’s approach.
W: the rate of childbed fever will decline if people cannot discern when the priest is coming to the Clinic.
But when Semmelweis had the priest silence his bell, and take a different route, so that patients could not discern that he was coming to the First Clinic, he found no difference in the mortality rate; the First Clinic remained far worse than the second clinic. He concluded that the higher rate of childbed fever was not caused by fear of death resulting from the priest’s approach.
(V→W)
¬W
_____
¬V
Is this argument valid? We can check using a truth table.
premise | premise | conclusion | ||
V | W | (V→W) | ¬W | ¬V |
T | T | T | F | F |
T | F | F | T | F |
F | T | T | F | T |
F | F | T | T | T |
Again, we see that Semmelweis’s reasoning was good. He showed that it was not the case that the higher rate of childbed fever was caused by fear of death resulting from the Priest’s approach.
What about Semmelweis’s positive conclusion, that the higher mortality rate was caused by some contaminant from the corpses that doctors had autopsied just before they assisted in a delivery? To understand this step in his method, we need to reflect a moment on the scientific method and its relation to logic.
3.5 Other kinds of arguments 1: Scientific reasoning
Valid arguments, and the methods that we are developing, are sometimes called “deductive reasoning”. This is the kind of reasoning in which necessarily our conclusions is true if our premises are true; these arguments can be shown to be good by way of our logical reasoning alone. There are other kinds of reasoning, and understanding this may help clarify the relation of logic to other endeavors. Two important, and closely related, alternatives to deductive reasoning are scientific reasoning and statistical generalizations. We’ll discuss statistical generalizations in the next section.
Scientific method relies upon logic, but science is not reducible to logic: scientists do empirical research. That is, they examine and test phenomena in the world. This is a very important difference from pure logic. To understand how this difference results in a distinct method, let us review Semmelweis’s important discovery.
The details and nature of scientific reasoning are somewhat controversial. I am going to provide here a basic—many philosophers would say, oversimplified—account of scientific reasoning. My goal is to indicate the relation between logic and the kind of reasoning Semmelweis may have used.
As we noted, Semmelweis learned about the death of a colleague, Professor Jakob Kolletschka. Kolletschka had been performing an autopsy, and he cut his finger. Shortly thereafter, Kolletschka died with symptoms like those of childbed fever. Semmelweis reasoned that something on the corpse caused the disease; he called this “cadaveric matter”. In the First Clinic, where the mortality rate of women and babies was high, doctors were doing autopsies and then delivering babies immediately after. If he could get this cadaveric matter off the hands of the doctors, the rate of childbed fever should fall.
So, he reasoned thus: if the fever is caused by cadaveric matter on the hands of the doctors, then the mortality rate will drop when doctors wash their hands with chlorinated water before delivering babies. He forced the doctors to do this. The result was that the mortality rate dropped a very great deal, at times to below 1%.
Here is a key:
P: The fever is caused by cadaveric matter on the hands of the doctors.
Q: The mortality rate will drop when doctors wash their hands with chlorinated water before delivering babies.
And the argument appears to be something like this (as we will see, this isn’t quite the right way to put it, but for now…):
(P→Q)
Q
_____
P
Is this argument valid? We can check using a truth table.
premise | premise | conclusion | ||
P | Q | (P→Q) | Q | P |
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | F |
From this, it looks like Semmelweis has used an invalid argument!
However, an important feature of scientific reasoning must be kept in mind. There is some controversy over the details of the scientific method, but the most basic view goes something like this. Scientists formulate hypotheses about the possible causes or features of a phenomenon. They make predictions based on these hypotheses, and then they perform experiments to test those predictions. The reasoning here uses the conditional: if the hypotheses is true, then the particular prediction will be true. If the experiment shows that the prediction is false, then the scientist rejects the hypothesis.^{[6]} But if the prediction proved to be true, then the scientist has shown that the hypothesis may be true—at least, given the information we glean from the conditional and the consequent alone.
This is very important. Scientific conclusions are about the physical world, they are not about logic. This means that scientific claims are not necessarily true, in the sense of “necessarily” that we used in our definition of “valid”. Instead, science identifies claims that may be true, or (after some progress) are very likely to be true, or (after very much progress) are true.
Scientists keep testing their hypotheses, using different predictions and experiments. Very often, they have several competing hypotheses that have, so far, survived testing. To decide between these, they can use a range of criteria. In order of their importance, these include: choose the hypothesis with the most predictive power (the one that correctly predicts more kinds of phenomena); choose the hypothesis that will be most productive of other scientific theories; choose the hypothesis consistent with your other accepted hypotheses; choose the simplest hypothesis.
What Semmelweis showed was that it could be true that cadaveric matter caused the childbed fever. This hypothesis predicted more than any other hypothesis that the doctors had, and so for that reason alone this was the very best hypothesis. “But,” you might reason, “doesn’t that mean his conclusion was true? And don’t we know now, given all that we’ve learned, that his conclusion must be true?” No. He was far ahead of other doctors, and his deep insights were of great service to all of humankind. But the scientific method continued to refine Semmelweis’s ideas. For example, later doctors introduced the idea of microorganisms as the cause of childbed fever, and this refined and improved Semmelweis’s insights: it was not because the cadaveric matter came from corpses that it caused the disease; it was because the cadaveric matter contained particular micro-organisms that it caused the disease. So, further scientific progress showed his hypothesis could be revised and improved.
To review and summarize, with the scientific method:
- We develop a hypothesis about the causes or nature of a phenomenon.
- We predict what (hopefully unexpected) effects are a consequence of this hypothesis.
- We check with experiments to see if these predictions come true:
- If the predictions prove false, we reject the hypothesis;^{[7]}
- If the predictions prove true, we conclude that the hypothesis could be true. We continue to test the hypothesis by making other predictions (that is, we return to step 2).
This means that a hypothesis that does not make testable predictions (that is, a hypothesis that cannot possibly be proven false) is not a scientific hypothesis. Such a hypothesis is called “unfalsifiable” and we reject it as unscientific.
This method can result in more than one hypothesis being shown to be possibly true. Then, we chose between competing hypotheses by using criteria like the following (here ordered by their relative importance; “theory” can be taken to mean a collection of one or more hypotheses):
- Predictive power: the more that a hypothesis can successfully predict, the better it is.
- Productivity: a hypothesis that suggests more new directions for research is to be preferred.
- Coherence with Existing Theory: if two hypotheses predict the same amount and are equally productive, then the hypothesis that coheres with (does not contradict) other successful theories is preferable to one that does contradict them.
- Simplicity: if two hypotheses are equally predictive, productive, and coherent with existing theories, then the simpler hypothesis is preferable.
Out of respect to Ignaz Semmelweis we should tell the rest of his story, although it means we must end on a sad note. Semmelweis’s great accomplishment was not respected by his colleagues, who resented being told that their lack of hygiene was causing deaths. He lost his position at the First Clinic, and his successors stopped the program of washing hands in chlorinated water. The mortality rate leapt back to its catastrophically high levels. Countless women and children died. Semmelweis continued to promote his ideas, and this caused growing resentment. Eventually, several doctors in Vienna—not one of them a psychiatrist—secretly signed papers declaring Semmelweis insane. We do not know whether Semmelweis was mentally ill at this time. These doctors took him to an asylum on the pretense of having him visit in his capacity as a doctor; when he arrived, the guards seized Semmelweis. He struggled, and the guards at the asylum beat him severely, put him in a straightjacket, and left him alone in a locked room. Neglected in isolation, the wounds from his beating became infected and he died a week later.
It was years before Semmelweis’s views became widely accepted and his accomplishment properly recognized. His life teaches many lessons, including unfortunately that even the most educated among us can be evil, petty, and willfully ignorant. Let us repay Semmelweis, as those in his own time did not, by remembering and praising his scientific acumen and humanity.
3.6 Other kinds of arguments 2: Statistical reasoning
Here we can say a few words about statistical generalizations—our goal being only to provide a contrast with deductive reasoning.
In one kind of statistical generalization, we have a population of some kind that we want to make general claims about. A population could be objects or events. So, a population can be a group of organisms, or a group of weather events. “Population” just means all the events or all the things we want to make a generalization about. Often however it is impossible to examine every object or event in the population, so what we do is gather a sample. A sample is some portion of the population. Our hope is that the sample is representative of the population: that whatever traits are shared by the members of the sample are also shared by the members of the population.
For a sample to representative, it must be random and large enough. “Random” in this context means that the sample was not chosen in any way that might distinguish members of the sample from the population, other than being members of the population. In other words, every member of the population was equally likely to be in the sample. “Large enough” is harder to define. Statisticians have formal models describing this, but suffice to say we should not generalize about a whole population using just a few members.
Here’s an example. We wonder if all domestic dogs are descended from wolves. Suppose we have some genetic test to identify if an organism was a descendent of wolves. We cannot give the test to all domestic dogs—this would be impractical and costly and unnecessary. We pick a random sample of domestic dogs that is large enough, and we test them. For the sample to be random, we need to select it without allowing any bias to influence our selection; all that should matter is that these are domestic dogs, and each member of the population must have an equal chance of being in the sample. Consider the alternative: if we just tested one family of dogs—say, dogs that are large—we might end up selecting dogs that differed from others in a way that matters to our test. For example, maybe large dogs are descended from wolves, but small dogs are not. Other kinds of bias can creep in less obviously. We might just sample dogs in our local community, and it might just be that people in our community prefer large dogs, and again we would have a sample bias. So, we randomly select dogs, and give them the genetic test.
Suppose the results were positive. We reason that if all the members of the randomly selected and large enough sample (the tested dogs) have the trait, then it is very likely that all the members of the population (all dogs) have the trait. Thus: we could say that it appears very likely that all dogs have the trait. (This likelihood can be estimated, so that we can also sometimes say how likely it is that all members of the population have the trait.)
This kind of reasoning obviously differs from a deductive argument very substantially. It is a method of testing claims about the world, it requires observations, and its conclusion is likely instead of being certain.
But such reasoning is not unrelated to logic. Deductive reasoning is the foundation of these and all other forms of reasoning. If one must reason using statistics in this way, one relies upon deductive methods always at some point in one’s arguments. There was a conditional at the penultimate step of our reasoning, for example (we said “if all the members of the randomly selected and large enough sample have the trait, then it is very likely that all the members of the population have the trait”). Furthermore, the foundations of these methods (the most fundamental descriptions of what these methods are) are given using logic and mathematics. Logic, therefore, can be seen as the study of the most fundamental form of reasoning, which will be used in turn by all other forms of reasoning, including scientific and statistical reasoning.
3.7 Problems
- Make truth tables to show that the following arguments are valid. Circle or highlight the rows of the truth table that show the argument is valid (that is, all the rows where all the premises are true). Note that you will need eight rows in the truth table for problems d-f, and sixteen rows in the truth table for problems g and h.
- Premises: (P→Q), ¬Q. Conclusion: ¬P.
- Premises: ¬P. Conclusion: (P→Q).
- Premises: Q. Conclusion: (P→Q).
- Premises: (P→Q), (Q→R). Conclusion: (P→R).
- Premises: (P→Q), (Q→R), P. Conclusion: R.
- Premises: (P→Q), (Q→R), ¬R. Conclusion: ¬P.
- Premises: (P→Q), (Q→R), (R→S), P. Conclusion: S.
- Premises: (P→Q), (Q→R), (R→S). Conclusion: (P→S).
- Make truth tables to show the following arguments are invalid. Circle or highlight the rows of the truth table that show the argument is invalid (that is, any row where all the premises are true but the conclusion is false).
- Premises: (P→Q). Conclusion: P.
- Premises: (P→Q). Conclusion: Q.
- Premises: P. Conclusion: (P→Q).
- Premises: (P→Q), Q. Conclusion: P.
- Premises: ¬Q. Conclusion: (P→Q).
- Premises: (P→Q). Conclusion: (Q→P).
- Premises: (P→Q), (Q→R), ¬P. Conclusion: ¬R.
- Premises: (P→Q), (Q→R), R. Conclusion: P.
- Premises: (P→Q), (Q→R). Conclusion: (R→P).
- Premises: (P→Q), (Q→R), (R→S). Conclusion: (S→P).
- In normal colloquial English, write your own valid argument with at least two premises. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like logic). Translate it into propositional logic and use a truth table to show it is valid.
- In normal colloquial English, write your own invalid argument with at least two premises. Translate it into propositional logic and use a truth table to show it is invalid.
- For each of the following, state whether the argument described could be: valid, invalid, sound, unsound.
- An argument with false premises and a false conclusion.
- An argument with true premises and a false conclusion.
- An argument with false premises and a true conclusion.
- An argument with true premises and a true conclusion.
[5] All the data cited here comes from Carter (1983) and additional biographical information comes from Carter and Carter (2008). These books are highly recommended to anyone interested in the history of science or medicine.
[6] It would be more accurate to say, if the prediction proves false, the scientist must reject either the hypothesis or some other premise of her reasoning. For example, her argument may include the implicit premise that her scientific instruments were operating correctly. She might instead reject this premise that her instruments are working correctly, change one of her instruments, and try again to test the hypothesis. See Duhem (1991). Or, to return to the case of Semmelweis, he might wonder whether he sufficiently established that there were no differences in the atmosphere between the two clinics; or he might wonder whether he sufficiently muffled the Priest’s approach; or whether he recorded his results accurately; and so on. As noted, my account of scientific reasoning here is simplified.
[7] Or, as noted in note 6, we reject some other premise of the argument.