The purpose of this course is to learn how to think and reason logically, or cogently. We don’t always want to think in this way—for instance, when we’re daydreaming, we might want to avoid logical thinking—but we often do want to think logically, and the purpose of this course is to learn how to do this well.
Thinking logically, or critically, involves lots of different of things, and it is useful in many different ways. It can involve taking some information and determining what else follows from it; thus, it can be a way of arriving at new information on some topic. It can also provide a way of figuring out what you ought to do in problematic situations. We very often find ourselves not knowing what to do in some situation; but when we slow down and reason through the situation, one course of action sometimes emerges as the most rational one. Critical thinking can also provide a way of determining when other people are saying things that we ought to believe and take seriously and when they are saying things that we ought not to believe or take seriously. For instance, suppose that somebody says that it is wrong to allow homosexuals into the military. If we are good critical thinkers, then we can ask whether this person has any good reason for saying this or whether he or she is just voicing an opinion that we have no good reason to accept. Moreover, if the person does offer a reason for accepting this claim, then again, if we are good critical thinkers, we can determine whether the reason given is a good one or a bad one (and hence, whether we ought to accept the person’s claim or not).
Critical thinking skills can also help us to evaluate our own beliefs and to form new ones. People very often believe things that they have accepted for years without thinking much about them. When we ask ourselves why we believe these things, we can sometimes produce good reasons for them and sometimes not. But people can easily deceive themselves in such settings, and so it is important to be able to think critically in order to be able to figure out when we have good reasons for our beliefs and when we do not. Finally, in forming new beliefs, i.e., in trying to decide what we ought to believe about various issues, it is important to be able to think things through in a clear and skilled way. For if we can, we will increase the likelihood of arriving at true or worthwhile beliefs.
This is just a small sample of why it is desirable to be able to think logically and critically—not all the time, but when it is called for.
The central concept of this course will be the concept of an argument. An argument is a written or spoken message that provides a reason to believe something. What we are going to do in this course is learn how to tell the difference between a good argument and a bad argument. This will have a dual purpose. First, it will enable us to evaluate the arguments of other people. For instance, we will be able to read newspaper editorials and critically analyze them to determine whether they provide good reasons to believe their conclusions or whether the reasons employed are unsound and, hence,
can be ignored. Second, by learning the difference between a good argument and a bad one, we will be improving our own arguing and reasoning skills, so that we will be able to construct cogent arguments for our own beliefs and, indeed, deduce new consequences from these beliefs that we hadn’t previously noticed. Another way to put this point is to say that we are going to improve our reasoning skills. Reasoning is the process of moving from a set of statements or beliefs to a new set of statements or beliefs; i.e., it is the process of noticing what follows from what. And what we’re going to do is learn how to use good arguments and avoid bad arguments in our reasoning; that is, we’re going to learn to reason correctly, that is, according to the rules of correct reasoning.
But let’s start at the beginning. The first thing we need to do is understand the concept of an argument. The word ‘argument’ actually has a few different meanings; for instance, it can be used to refer to a verbal dispute, or something like that. This is related to the meaning we will be concerned with, but it is not the same. The definition of ‘argument’ that we’ll be concerned with is as follows:
DEFINITIONS: An argument is a message that provides a reason to believe a given claim (or a sentence, or proposition, or whatever). More precisely, an argument consists of a group of one or more sentences that serve as reasons (or a single reason) to believe another sentence. The sentences that serve as reasons are called premises. And the sentence being argued for is called the conclusion. Thus, we can also think of an argument as a group of premises together with a conclusion that those premises are supposed to establish.
For example, one might say:
Palmer was buying drinks for everyone at the bar, so he must have won some money at the track today.
In this passage, the premise is: ‘Palmer was buying drinks for everyone at the bar’; and the conclusion is: ‘Palmer won some money at the track today’. For in this argument, the former is supposed to provide a reason to believe the latter.
Other arguments look different. For instance, we can have arguments with multiple premises, and the conclusion can be stated before the premises. Here’s an example:
Violet Beauregard would make a great president, because she’s cute, self-assured, and bossy.
The conclusion of this argument is that Violet Beauregard would make a great president, and we are given three different premises, or reasons to believe the conclusion—namely, that she’s cute, self- assured, and bossy. (Of course, it may be that none of these premises gives us a good reason to believe the conclusion, but for now, we’re not concerned with this. We’ll turn to argument evaluation shortly; indeed, as I’ve said, that is going to be the main topic of this course. But for now, we are just trying to learn what an argument is.)
We can represent this argument as follows.
Violet Beauregard is cute
Violet Beauregard is self-assured Violet Beauregard is bossy
Violet Beauregard would make a great president
This is called standard notation. We list the premises first, and then put a line under them, and then write the symbol ‘’ (which means ‘therefore’) followed by the conclusion.
1.3 ognizing Arguments
Now that we know what an argument is, we have to learn how to recognize them. For when people give arguments in writing, they do not put them in standard notation for us. And when they give us arguments in speech, they do not say, “Here’s an argument: here are my premises…, and here is my conclusion….” Instead, people give their arguments in ordinary English, buried in things like political speeches and newspaper editorials, without any buzzers or bells to flag them. And they assume that the audience will recognize what they are doing. Thus, what we need to do is learn how to recognize arguments when we encounter them in speeches and essays and so on.
There is no sure-fire mechanical way to determine when a passage contains an argument. But one thing you can do here, which is often very helpful, is to look for logical indicators, or more specifically, premise indicators and conclusion indicators.
DEFINITIONS: A premise indicator is a word that tells us that a premise is about to follow; and a conclusion indicator is a word that tells us that a conclusion is about to follow.
Examples of conclusion indicators are: ‘therefore’, ‘thus’, ‘hence’, and ‘so’. Let ‘A’ and ‘B’ stand for any sentences whatsoever and suppose that somebody says: “A; therefore, B”; then we know that the person has given us an argument, because the word ‘therefore’ just means ‘and the conclusion of this is…’. Thus, the premise of the above argument is A, and the conclusion is B. We know this without even knowing what A and B say. (Indeed, this is why it is helpful to use ‘A’ and ‘B’ here; the point is that the above analysis holds no matter what sentences we plug in for A and B. This will occur very often in this course; various logical facts hold for all sentences, and we will express these facts by using letters like ‘A’ and ‘B’ as placeholders for any sentence at all.)
All of these remarks hold for ‘thus’, ‘hence’, and ‘so’, as well as for ‘therefore’.
We can also have multiple-word conclusion indicators, such as ‘it follows that’, or ‘this suggests that’, or ‘this leads me to believe that’. And from this small sample, it should already be apparent that there is no limit to the number of conclusion indicators that we can construct in English. We can make them up at will.
Examples of premise indicators, on the other hand, are ‘because’, ‘since’, and ‘for’. Thus, if somebody says: “A, since B”, then we know that we have an argument on our hands, and more specifically, we know that B is the premise and A is the conclusion. For in the present context, the word ‘since’ just means ‘the reason for this is…’. Also, as with conclusion indicators, we can construct premise indicators at will. Examples are: ‘I say this because’, ‘my reason for saying this is that’, ‘this is clear from the fact that’, and so on.
Students would be wise to simply memorize that ‘therefore’, ‘thus’, ‘hence’, and ‘so’ are conclusion indicators and that ‘because’, ‘since’, and ‘for’ are premise indicators. However, since you cannot memorize all of the logical indicators that we might encounter, you also have to learn how to recognize logical indicators when you see them. But this is fairly easily done. For a conclusion indicator is just a word or group of words that means something like ‘here comes the conclusion’, and a premise indicator is just a word or group of words that means something like ‘here comes the premise’ (or ‘here come the premises’).
In using logical indicators to recognize when a passage contains an argument, there are two distinct problems that can arise. They are as follows.
- Some logical indicators—e.g., ‘since’, ‘for’, and ‘so’—have other meanings. Thus, these words often appear in passages that do not contain arguments. For instance, if someone asks me how I’m doing and I say, “I’ve been very intelligent since last Wednesday, thank you,” I have not given an argument, because I am not using the word ‘since’ as a premise indicator. I am, rather, using it as a temporal indicator, or something like that. Likewise, the sentence ‘Twanda bought a twinkie for Aunt Tilly’ contains the word ‘for’, but it does not contain an argument. And the sentence ‘Hermie is so monstrous’ contains the word ‘so’, but it doesn’t contain an argument either.
- We just saw that we can have words that often serve as logical indicators without having an argument. But the opposite can also happen: we can have an argument without any logical indicators. Consider, for instance, the following passage:
Ralph will succeed at whatever he does. He’s smart, hard-working, and extremely ambitious.
It seems pretty clear that this passage contains an argument; in particular, the second sentence provides three different reasons for believing the first sentence. But there are no premise or conclusion indicators in this passage. In such cases, the only way to recognize that we’ve got an argument on our hands is to understand the passage and see that some of the statements are being used as reasons to believe another statement. (NOTE: When this happens, it is often the case that the first statement is the conclusion and the rest of the statements are reasons to believe the first one. This is just a fact about the way people actually speak; if they formulate their conclusion after their premises, they are more likely to use a logical indicator.)
In addition to these two problems, there are other problems that we might encounter in learning how to recognize which passages contain arguments and which do not. Here are three such problems…
- Students often confuse bold or controversial claims with arguments. If I say, “Same-sex marriages are wrong”, I have not given an argument. I’ve merely made a claim. To make an argument, I have to give reasons to believe something. For instance, I’d have to say something like “Same-sex marriages are wrong because they upset God”.
- Sometimes, people leave their conclusions unspoken. For instance, suppose that someone says: “If Smith wins the next election, then we’ll fall into a depression of Biblical proportions. And Smith is going to win the next election.” This person has not explicitly said that we are going to fall into a depression of Biblical proportions, but it is an obvious conclusion of what he or she has said, and it seems pretty clear that the person intends us to draw this conclusion on our own; thus, we can say that the person has given an argument—one with the conclusion left implicit.
- It is important to note the difference between an argument on the one hand and a conditional statement on the other. ‘If A, then B’ is a conditional statement, not an argument. Thus, it is very different from ‘A; therefore B’. One way to appreciate the difference is to notice that in uttering ‘If A, then B’, we do not commit to the truth of either A or B. For instance, if I say “If it rains, then the game will be canceled”, I have not said that I think it is going to rain or that the game will be canceled. In contrast, if I say “It’s going to rain; therefore, the game will be canceled”, then I have said that I think that it will rain and that the game will be canceled.
Another point worth noting in this connection is that ‘If A, then B’ is just one statement—a conditional statement—whereas ‘A; therefore B’ contains two statements. Thus, a conditional can function as a premise or a conclusion. For instance, one could argue as follows:
If A, then B If B, then C
If A, then C
All three statements here are conditionals. Together, they form an argument. But by themselves, none of these conditional statements is an argument.
In sum, logical indicators can help us determine which passages contain arguments and which do not, but in the end, the only fool-proof method here is to understand the given passage: if reasons are being given to believe something, then we have an argument; and if no reasons are being given, then we do not have an argument.
For each of the following passages, indicate whether it is an argument or not an argument.
- Since Tuesday, John has believed in a new world order.
- Four out of five dentists surveyed recommend sugarless gum for their patients who chew gum. The other one recommends that you chew gum with sugar in it. He also recommends that you eat as much of this sugary gum as you possibly can and that you refrain from brushing your teeth or flossing. Finally, he urges you to avoid bathing.
- Four out of five dentists surveyed recommend sugarless gum for their patients who chew gum. But I don’t chew gum, so presumably, they wouldn’t recommend sugarless gum for me.
- Imitation butter is better than real butter, because it spreads very evenly over popcorn.
- Imitation butter is very dear to me. Indeed, I often spread it over my entire body and muse, “You are the only one for me. You are my lotion, my oil, my destiny.”
- Real butter is very bad for you, but imitation butter is good for you.
- If marijuana isn’t any stronger than alcohol, then it shouldn’t be illegal.
- If marijuana isn’t any stronger than alcohol, then it shouldn’t be illegal. But it isn’t any stronger than alcohol.
- Abortion should be illegal. It’s selfish, immoral, and when you get right down to it, a form of murder.
- Abortion is a form of murder and should not be legal.
- Because the work is starting to pile up, we shouldn’t do any of it.
1.4 ain Teaser Regarding the Definition of ‘Argument’
In section 1.2, we defined an argument as a message that provides a reason to believe a given claim. It is worth noting, however, that there is a problem with this definition. To appreciate this, consider the following passage:
(I) Violet would make a good president, because she is cute.
Intuitively, it seems that this passage is an argument but that it’s not a good argument. Moreover, our definition of ‘argument’ seems to gets this right: (I) is an argument, according to our definition, because it provides a reason to believe something, even though it is a bad reason.
But now consider this passage:
(II) Violet would make a good president; she is cute.
Intuitively, it seems that this is not an argument at all. But our definition of ‘argument’ seems to get this wrong. For we have already said that ‘Violet is cute’ provides a reason to believe ‘Violet would make a good president’. We had to say this in connection with (I) to get the right answer there—i.e., to obtain the result that (I) is an argument. But if ‘Violet is cute’ provides a reason to believe ‘Violet would make a good president’ in connection with (I), then it ought to do this in connection with (II) as well. The only difference between (I) and (II) is that the former contains the word ‘because’. But surely this cannot be what makes (I) an argument and (II) not an argument, because we’ve already seen that a passage needn’t have a ‘because’ here in order to be an argument. (For instance, it would seem that the passage “Abortion should be illegal; it is a form of murder” can correctly be called an argument.)
How can we solve this problem? Well, one thing we could do here is define ‘argument’ like this:
A passage is an argument if the intention of the author is to provide a reason (or reasons) to believe a conclusion.
This seems to get everything right; for presumably, the intention of the author of (II) is not to provide a reason to believe anything, whereas the intention of the author of (I) is to provide a reason to believe something. (And, presumably, the intention of the author of “Abortion should be illegal; it is a form of murder” is also to provide a reason to believe something, so definition #2 seems to get things right here as well.) But there are problems with definition #2. For instance, suppose that a lunatic stands up and says: “My mind is a bowl of oatmeal”; and suppose that in so doing, this lunatic intends to convince his or her audience that homosexuals should not be allowed to marry one another because they don’t have good family values. Then according to definition #2, this passage is an argument, since that is the author’s intention. But this just seems wrong; surely, this lunatic has not provided an argument here.
So definition #2 seems problematic. But our original definition is also problematic. For unless we can come up with a principled way of distinguishing passages that provide bad reasons from those that provide no reasons at all, we will be in trouble. Indeed, if there is no principled way of drawing this distinction, then it seems that our original definition of ‘argument’ will commit us to saying that every passage containing two or more statements is an argument and that only some of these are good arguments. But this is surely wrong.
For the purposes of this course, we do not need to figure out what the correct definition of ‘argument’ is. All we need to do is get clear on the basic intuitive idea of what an argument is, and hopefully, we have already done that.
By the way, it may be that it is impossible to find a definition of ‘argument’ that gets everything right. We have intuitions about which passages contain arguments and which do not; e.g., intuition dictates that (I) is an argument and (II) isn’t, and that “Abortion should be illegal; it is a form of murder” is an argument, while “My mind is a bowl of oatmeal” isn’t. What we want is a definition of ‘argument’ that agrees with our intuitions in all cases. But this may not be possible, for it may be that our intuitions aren’t as principled as we would like to think. On the other hand, it may be that we can find a good definition here. Perhaps you can figure it out.
COGENCY: GOOD VS. BAD ARGUMENTS
Now that we know how to recognize arguments, we are ready to move on to our main topic, namely, argument evaluation, i.e., deciding which arguments are good and which are bad. As was mentioned earlier, the reason we want to do this is two-fold: we want to be able to listen to arguments from other people and decide whether they are giving us good reasons to believe what they’re saying; and we want to improve our own reasoning skills.
DEFINITION: An argument is good, or as we will say, cogent, if and only if the premises really do provide good reason to believe the conclusion.
This definition will not be very helpful to students who want to learn how to distinguish good arguments from bad ones, for presumably, they don’t know how to recognize when an argument provides good reason for its conclusion. What student need to learn are the criteria for goodness in arguments. These criteria can be stated very easily. In order to be good, or cogent, an argument must satisfy two constraints:
(1) of its premises have to be true
(2) premises have to support the conclusion.
Thus, an argument can be bad for two different reasons. First, it can have a premise that is false. For instance, consider the following argument:
If California is a city, then it’s the most populous city in the world California is a city
California is the most populous city in the world
This argument is not cogent, because its second premise is false. And second, an argument can be bad, or not cogent, because its premises do not support its conclusion. For instance, consider this argument:
Snow is white
Grass is green
This argument is not cogent, because its premise doesn’t support its conclusion. In this case, it’s easy to see why the premise doesn’t support the conclusion; it’s because the premise is totally irrelevant to the conclusion. That is, the fact that snow is white simply doesn’t have anything to do with the question of whether or not grass is green, and so, obviously, the claim that snow is white couldn’t provide a good reason to believe that grass is green. But sometimes a premise can be relevant to its conclusion and still fail to support it. For instance, consider the following argument:
We polled 11 people outside of a Wayne Newton show in Las Vegas and 55% say they will vote for Wayne Newton in the next presidential election
Wayne Newton will be our next president
This argument is clearly not good, or cogent, even if the premise is true. The premise is relevant to the conclusion, in some sense, but it is simply too weak to support the conclusion.
These examples should give you a rough intuitive idea of what support is all about, but we are going to go into this in much more detail. The question of when a set of premises support their conclusion is a very complicated one, and it is going to be the central question of this course. We will spend a bit of time on the truth and falsity of premises, but we are going to spend the bulk of our time learning how to determine when the premises of an argument support their conclusion.
DEDUCTIVE VS. INDUCTIVE ARGUMENTS
3.1 A Rough-and-Ready Distinction
Before we say any more about the notion of support, we need to distinguish two different kinds of arguments, namely, deductive arguments and inductive arguments. The reason we need to do this now is that there are different standards for what counts as adequate support in connection with these two different kinds of arguments. In order for a deductive argument to provide adequate support for its conclusion, it needs to have a certain property that inductive arguments do not need to have in order to provide adequate support for their conclusions.
The basic intuitive idea behind the distinction between inductive and deductive arguments is very easy to grasp. We will see later that it is rather difficult to come up with a precise way of drawing the distinction that gets everything right. But this won’t matter for our purposes. Our aim will simply be to grasp the rough intuitive distinction.
The rough intuitive distinction between deductive and inductive arguments has to do with the question of whether an argument provides complete support for its conclusion or merely partial support. Those arguments that provide complete support (or in other words, that necessitate their conclusions) are deductive; and those that provide only partial support (or in other words, that merely make their conclusions seem likely, or probable) are inductive.
For instance, the argument
If Wilma is a swan, then Wilma is white Wilma is a swan
Wilma is white
is deductive, because if its premises are true, then it’s conclusion must be true. That is, the premises necessitate the conclusion, i.e., they force the conclusion on us, or provide total support for the conclusion.
The same goes for the argument All poodles are dogs
Fido is a poodle
Fido is a dog
This is a deductive argument, because if its premises are true, then its conclusion must be true.
On the other hand, the argument
We’ve polled 10,000 people at random and 70% say they will vote for Smith in the next presidential election
Smith will be the next president
does not necessitate its conclusion; it only makes its conclusion likely, or probable. We can imagine the premises being true and the conclusion false here. But this does not make it a bad argument.
Indeed, assuming that the poll was done in a reputable way, this is a good argument. That is, if its premise is true, then it makes the conclusion very likely, or probable. But, again, this argument is not fool-proof: its conclusion could be false, even if its premise is true. Thus, we can say that this argument provides inductive support for its conclusion, rather than deductive support.
As a general rule, inductive arguments are what scientists use to argue for their hypotheses. For instance, zoologists might claim that all swans are white, and their argument for this conclusion might go something like this:
We have observed 10,000 swans, and they have all been white
All swans are white
In contrast to this, mathematicians and logicians usually use deductive arguments; they construct proofs for their claims, and these necessitate their conclusions. For instance, we can prove that all even numbers greater than two are non-prime by looking at what it means to be even and what it means to be prime. We don’t have to consider a bunch of examples of even numbers greater than two, as we do in the swan example.
3.2 Trying to Define ‘Deductive’ and ‘Inductive’
Simple as the distinction between deduction and induction is, it is surprisingly hard to define the two sorts of arguments in an acceptable way. It might seem that we could simply say this:
A deductive argument is an argument that provides complete support for its conclusion, whereas an inductive argument is one that provides merely partial support for its conclusion.
But this will not work, for it does not take account of bad arguments. We saw above that the argument
All poodles are dogs Fido is a poodle
Fido is a dog
is a deductive argument, because its premises (if true) force the conclusion on us. But now let’s
compare this argument to the following argument:
All poodles are dogs Fido is a dog
Fido is a poodle
This is clearly not a good argument, for Fido might be a golden retriever. Thus, even if the premises of this argument are true, they do not force the conclusion on us. Does this mean that the argument isn’t deductive? No. It seems that it is a deductive argument but that it’s simply a bad deductive argument. But according to definition #1, this not a deductive argument, for it doesn’t provide complete support for its conclusion.
The same sort of thing arises in connection with inductive arguments. Consider the argument
We polled 11 people outside of a Wayne Newton show in Las Vegas and 55% say they will vote for Wayne Newton in the next presidential election
Wayne Newton will be our next president
This is clearly not a good argument, because the sample is too small and not chosen at random. Thus, the argument doesn’t even provide partial support for its conclusion,1 and so, according to definition #1, it is not an inductive argument. But this seems wrong; it seems that this is an inductive argument and that it is simply a bad inductive argument.
In a nutshell, the problem with definition #1 is that it only works for good arguments. It dictates that bad arguments are neither deductive arguments nor inductive arguments.
To solve this problem, one might try to draw the deductive/inductive distinction like this:
A deductive argument is an argument in which the author’s intention is to provide complete support for the conclusion, whereas an inductive argument is one in which the author’s intention is to provide merely partial support.
1 One might claim that the premise of this argument does provide partial support for the conclusion; for one might say that it provides a very small amount of support (although surely not enough support to motivate the conclusion). But I would argue that the premise of this argument provides no more support for the conclusion of this argument than does the statement ‘Snow is white’. Thus, I think it provides no support at all. (It should be noted in this context that to say that a premise p provides “no support at all” for a conclusion c does not mean that if p is true, then the probability that c is true is 0, or anything like this. I will say a bit about what “no support” means in chapter 11.)
This definition seems to do well in connection with all of the above cases. E.g., it predicts that
All poodles are dogs Fido is a dog
Fido is a poodle
is a deductive argument and that
We polled 11 people outside of a Wayne Newton show in Las Vegas and 55% say they will vote for Wayne Newton in the next presidential election
Wayne Newton will be our next president
is an inductive argument. For clearly anyone who used the former would be attempting to give a deductive argument, and anyone who used the latter would be attempting to give an inductive argument.
But there are problems with definition #2. For instance, suppose that someone uttered All poodles are dogs
Fido is a poodle
Fido is a dog
and suppose that in so doing, this person intended to provide merely partial support for the conclusion; that is, suppose that the author thinks that if the premises of this argument are true, then they provide partial, but not total, support for the conclusion. Then according to definition #2, we have an inductive argument here. But this seems wrong. It seems that regardless of what anyone intends to do with this argument, it is deductive. Perhaps this problem could be solved, but it is not obvious how it could be solved.
Another way that people have tried to draw the distinction is like this:
Deductive arguments proceed from general premises to particular conclusions, whereas inductive arguments proceed from particular premises to general conclusions.
We can see why one might think this definition is right by considering arguments like the following.
All swans are white Ralph is a swan
Ralph is white
We have observed 10,000 swans, and they have all been white
All swans are white
The first argument here is clearly deductive and the second is clearly inductive, for the premises of the former (if true) force the conclusion on us, whereas the premises of the latter (if true) only make the conclusion likely, or probable. Moreover, the former argument proceeds from a general law about all swans to a conclusion about a particular swan, namely, Ralph, whereas the latter argument proceeds from claims about particular observed swans to a general law about all swans. Thus, it’s easy to see why someone might think that deductive arguments proceed from the general to the particular, whereas inductive arguments proceed from the particular to the general.
But this is an overgeneralization, and we can appreciate this by looking at arguments like this: If Wilma is smart, then she will pass the test
Wilma is smart
Wilma will pass the test
This argument is deductive, but it is concerned with particular facts only. And even trickier is the argument
If Wilma is smart, then everyone is Wilma is smart
Everyone is smart
This seems to proceed from the particular to the general, in some sense, but despite this, it is a deductive argument, for its premises necessitate its conclusion.
Finally, some inductive arguments do not proceed from particular premises to general conclusions. For instance, consider this argument:
Wilma is smart, and she passed the test Ralph is also smart
Ralph will also pass the test
This is an inductive argument, but it proceeds from particulars to another particular.
All of this suggests that definition #3 is not a good definition. But it is worth noting that there is a grain of truth in definition #3 and that students can use this to help them distinguish deductive arguments from inductive arguments. For instance, it is true that any argument of the form
All A’s are B’s x is an A
x is a B
is deductive. And it is also true that any argument of the form
All observed A’s are B’s
All A’s are B’s
is inductive. And more generally, any argument of the form x % of observed A’s are B’s
x % of all A’s are B’s
is inductive. In short, whenever we proceed in this fashion from a premise about a sample of a population to a conclusion about the entire population, we are arguing inductively.
Thus, again, while definition #3 is not a good definition, there is some truth to it, and we can sometimes use this in determining whether certain arguments are deductive or inductive.
So it is not obvious how, precisely, we ought to distinguish deductive arguments from inductive arguments. However, in the present context, we do not need a precise distinction here. Our aim is simply to grasp the intuitive idea behind the distinction.
Probably the best of the above definitions, from a pragmatic point of view, is definition #2. Thus, students can use that definition.
(It should be noted, by the way, that some people think that there is no good way to draw the distinction between deduction and induction in a precise way; that is, they think there is no way to define ‘deductive argument’ and ‘inductive argument’ in a simple way that captures our intuitions here. Some of these people think that we can define ‘good deductive argument’ and ‘good inductive argument’—they would do this by using something like definition #1 above—but they doubt that we can provide definitions here that work for bad arguments as well as for good ones.)
3.3 Some Hints for Learning How to Distinguish Deductive Arguments From Inductive Arguments
How can we tell if a given argument is deductive or inductive? Well, the best way is simply to read the argument and understand it and decide whether it is supposed to provide total support for its conclusion or merely partial support. But there is also a sort of “trick” that we can use here. We can look for what might be called modal words. By this, I mean words that can be used to indicate the strength of support that an argument is claiming for its conclusion. An example is the word ‘necessarily’. Suppose that someone says something of the following form:
A; therefore, necessarily, B.
Whatever A and B are, this argument is presumably deductive, because in this context, the word ‘necessarily’ just means something like ‘this provides complete support for’. Likewise, if we replaced ‘necessarily’ with ‘probably’ here, we would presumably have an inductive argument, because in this context, the word ‘probably’ tells us that we do not have complete support here.
In addition to ‘necessarily’, another word that can function as a sort of deductive indicator is ‘must’; if we replaced ‘necessarily’ in the above argument form with ‘it must be the case that’, the result would be the same. A other words that can function as inductive indicators, in addition to ‘probably’, are words like ‘suggests’ and ‘likely’; if we replaced ‘necessarily’ in the above argument form with ‘this suggests that’, or ‘it seems likely that’, this would be more or less equivalent to replacing it with ‘probably’. There are also other words that can play these roles.
Note that arguments don’t always have modal words in them. Thus, in such cases, if you want to decide whether a given argument is deductive or inductive, the only thing you can do is read the argument, and understand it, and decide whether it is supposed to provide complete support for its conclusion or merely partial support.
Another problem is that modal words can lead us astray. Consider, for instance, the argument
You shouldn’t hire him, because stupid people are usually bad employees, and he must be the stupidest person alive.
Despite the presence of ‘must’, this is an inductive argument; what’s going on here is that ‘must’ is part of one of the premises; it has nothing to do with the relationship between the premises and the conclusion. Thus, it is not relevant to deciding whether the argument is deductive or inductive.
Other problems arise here because people often use modal words inappropriately. Thus, for instance, people often say things like this:
Almost all somnambulists are severely confused, and Floyd is a somnambulist, so he must be confused.
This is a very common way of using the word ‘must’, but strictly speaking, it is an incorrect use of that term. What’s going on here is that the speaker has overstated the case for the conclusion. The premises do not necessitate the conclusion, and the speaker probably knows this. He or she just got carried away and used ‘must’ while stating an inductive argument. In any event, this is the most charitable reading of this argument. It would very uncharitable to read the author of this argument as attempting to provide a deductive argument; for if we read this argument as a deductive argument, it is a very bad argument.
Homework Set II: Indicate whether each of the following arguments is inductive or deductive.
- My mother is insane, so your mother probably is too.
- I have been exposed to asbestos, so there’s a pretty good chance that I am going to die of cancer.
- All cats are republicans, and all republicans are chickens. Therefore, all cats are chickens.
- In this study, we examined hundreds of different samples of lettuce, and for each of these samples, we found that it was mostly water—indeed, they were all 96% water. Therefore, we conclude that this is simply the nature of lettuce: it is 96% water and only 4% lettuce.
- Every time we drop a physical object and measure how quickly it descends to earth, we get the same answer. No matter how much these objects have weighed—whether they were pennies or pick- up trucks—they fell at the same rate, so long as we dropped them from the same height. Therefore, it seems clear that all objects fall to earth at the same rate.
- 34 is even, so it can’t be prime.
- Bill Clinton is a democrat, so he can’t be conservative.
- Bill Clinton is a democrat, so he probably isn’t conservative.
- All androids are emotionless and Spock is emotionless, so he must be an android.
- I took sixteen different samples of water from the pond, from various different areas, and all of them were contaminated. So I think we can safely conclude that the whole pond is contaminated.
PART II DEDUCTIVE ARGUMENTS
VALIDITY AND THE IMAGINATION METHOD
4.1 Argument Evaluation
We want to learn how to distinguish good arguments from bad ones. This involves
(1) determining whether all of the premises of a given argument are true
(2) determining whether the premises of a given argument support the conclusion.
Now, in order to perform task (2) in connection with a given argument, we first have to know whether the argument is deductive or inductive. This is because ‘support’ means something different in connection with the two different kinds of arguments. With deductive arguments, it means total support, and with inductive arguments, it means mere partial support. Because of this, the task of evaluating a deductive argument—i.e., of determining whether or not it is good, or cogent—is completely different from the task of evaluating an inductive argument. We are going to begin by learning how to evaluate deductive arguments. This will occupy us for the whole of Part II. We will return to inductive arguments in Part III.
Let us turn, then, to the evaluation of deductive arguments.
4.2 Validity and Soundness
Like any argument, a deductive argument needs two things in order to be good, or cogent: (1) all of its premises need to be true, and (2) its premises need to support its conclusion. Let us begin with the second idea here, i.e., support.
When the premises of a deductive argument support the conclusion, we say that the argument is valid. We can define this notion as follows:
DEFINITIONS: An argument is deductively valid if and only if it is impossible for all of the premises of the argument to be true simultaneously with the conclusion being false. And an argument is deductively invalid if and only if it is possible for all of the premises to be true simultaneously with the conclusion being false.
This is one way to define ‘valid’. Here’s another (equivalent) way:
DEFINITION: An argument is deductively valid if and only if it has the following property: if its premises are true, then its conclusion must also be true; i.e., the conclusion couldn’t possibly be
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