Primer

Over your first few years of life, you learned how to do a lot of things: how to speak, how to think, how to walk, how to grasp and manipulate objects, etc. But you learned most of these things implicitly—that is, without having words for what you were doing and how. For example, you learned many words before you knew the word “word,” and you learned how to put words together into sentences before you had the word “sentence” or words for the parts of speech (much less for the rules of grammar). Here’s another example: Many of you have probably never thought much about how to walk, but if you’ve ever had physical therapy to recover from an injury or taken a movement class as part of learning how to dance or to act, you may have learned to analyze the act of walking into component parts and this will have given you more control over the way you walk. Likewise, when athletes train, they often learn to analyze movements they learned as a child into simpler movements, and this process gives them finer grained control over their movements. It is possible to gain greater control of your thinking in the same way, by learning to analyzing the complex activity of thinking into the various mental acts that make it up. In this primer we’re going to focus on a part of our mental activity that looms large in our daily lives, in the sciences, and in philosophy: the activity of arguing. We will learn what an argument is, how to analyze an argument into its parts, and how to use this analysis to methodically assess an argument. In §1, I provide an overview of the whole process, and §2–5 deal in greater detail with different aspects of the process.

Propositions

Before we discuss arguments in earnest, it will be helpful to say a bit about propositions, which are the smaller units of thought from which arguments are composed. A proposition is the sort of thought that is capable of being true or false, believed or disbelieved, and asserted or denied. Such thoughts are asserted by declarative sentences. Here are some examples:

1. Healthy grass is green.

2. O. J. Simpson killed Nicole Brown.

3. Twice two is four.

4. Twice two is five.

5. Many diseases are caused by bacteria.

6. Stalin was evil.

7. Joe Biden is the 46th President of the United States.

8. Either a Republican or a Democrat will win the 2024 Presidential election.

9. The Senate should have convicted Donald Trump in both of his impeachment trials.

10. Hillary Clinton would have been an awful president.

The proposition is not the same thing as the sentence asserting it because the same thought can be asserted by different sentences. For example, here are several different ways of asserting Proposition 3 from the list above:

3a. Twice two is four.

3b. Two times two is four.

3c. 2 x 2 = 4

3d. Deux fois deux c'est quatre.

3e. 兩次兩次是四次。

Sentences 3a and 3b are two different ways of asserting the same proposition in English. 3c is a way of asserting it in mathematical notation, 3d is a way of asserting it in French, and 3e is a way of asserting it in Chinese.

Another reason why a proposition is not the same thing as a sentence, is that propositions can be asserted as parts of more complex sentences that include multiple propositions. For example, here’s a sentence that asserts both Propositions 9 and 10 from the list above: "Even though Hillary Clinton would have been an awful president, the Senate should have convicted Donald Trump at both of his impeachment trials."

Some of the propositions from the list above are uncontroversially true and others are uncontroversially false. I expect that everyone in the class will all agree that Propositions 1, 3, 5 and 7 are true and that Proposition 4 is false. We will probably disagree over some of the others—with some students thinking they’re true and others thinking they’re false.

Some of you may think that some of the propositions we may disagree over aren’t really the sort of things that can be true or false at all. In particular, some of you may think this about Propositions 6, 9, and 10, because these propositions express evaluations. Some people think that evaluations aren’t the sorts of things that can be true or false. But even if you think this, you’ll have noticed that many people believe or disbelieve each of these propositions (and many other evaluations) as though they were true or false, and they make arguments to try to convince other people that these propositions are true or falce. So, to understand their role in arguments you’ll have to treat them as the sort of thing that can be true or false.

The Anatomy of an Argument

As the term is used in philosophy, an argument is a set of related propositions (called premises) that is given as a reason for believing a further proposition (called the conclusion). Consider, for example, the following, simple argument:

Whoever murdered Carl had to have access to his rose garden at midnight, and the only person who did was Natalie, therefore Natalie must be the murderer.

Here the conclusion is that Natalie murdered Carl, and there are two premises: (1) that whoever murdered Carl had access to Carl’s rose garden at midnight, and (2) that only Natalie had such access.

There are a few ways in which we can represent an argument that make its structure clearer. One popular way is called standard form. Here’s what the argument we have been discussing looks like in standard form:

1. Whoever murdered Carl had access to his rose garden at midnight. 2. Only Natalie had access to Carl’s rose garden at midnight.

3. Natalie murdered Carl.

Each proposition has been written on its own line and labeled with a number. The premises are written first, and a line is drawn to separate them from the conclusion. This line represents the inference—the mental act of moving in thought from the premises to the conclusion.

You will occasionally find arguments written out in this way in some of the readings for this class. However, we will be making more use of a newer way of representing the structure of arguments, which is called “argument mapping.” Here’s a map of the argument we’ve been discussing.

Map 1:

In the map, each proposition is written in its own box and labeled with a number. The inference is represented by a green circle and labeled with a letter. Thin lines connect the circle to the boxes containing the premises, and a thicker line with an arrow at the end connects the circle to box containing the conclusion.1

Both of these ways of representing the argument are meant to make it easy to see that Propositions 1 and 2 combine to provide a reason to believe Proposition 3. Another way we can express this point is to say that Propositions 1 and 2, taken together, are meant to help us to tell that Proposition 3 is true.

Notice that neither of these propositions on its own provides any reason at all to think that Natalie murdered Carl. If someone had no idea that Natalie had access to Carl’s rose garden at midnight, then learning that the murderer had access to the garden at this time, wouldn’t give him any reason to suspect Natalie of the murder. Likewise, if someone had no idea that the murderer had access to the rose garden, then knowing that Natalie was the only one with access wouldn’t give him a reason to think she committed the murder. It’s only when we put the two propositions together that they give us a reason to believe the conclusion. The argument map represents this relationship by having the lines from the two premises meet at the circle representing the inference, and then having a single arrow go from the circle to the conclusion.

Uses of Argument

If you were trying to convince someone, perhaps a jury, of Natalie’s guilt, you might present them with the argument we discussed in the last section. Or, you might formulate this argument silently to yourself in the course of trying to figure out who killed Carl. In either case, we would say that you inferred that Natalie was Carl’s murderer from the premises that the murderer had access to the rose garden at midnight and that only Natalie had such access.

We tend to think of making arguments in situations where people disagree and are trying to convince the other (or to convince some third party). This is the case when people argue in court or around a dinner table; but this is not the only use of arguments, nor is it their fundamental use. Much of what we know is based on other knowledge and inference is the process by which we reach new knowledge from old. It is only because of arguments that we know many of the things we do, and it is only by further arguments that we can come to know many of the things we would like to learn.

Let’s survey some of the areas in which much of our knowledge is due to inference. Arguments like the one concerning Carl and Natalie enable us to know who committed crimes for which there were no eyewitnesses. In the natural sciences, almost everything we know beyond the basic data (consisting in observations and measurements) is reached by inference.2 So is much of our knowledge in higher mathematics. Our knowledge of the future is inferred from what we know about the past and present. And our knowledge of the distant past (beyond the scope of our memories) is also based on argument. Here, in some cases, we have firsthand accounts from people who claim to have observed or taken part in various historical events, but there are questions about the honesty and accuracy of these reports (especially when some contradict others), so we need arguments to determine which accounts we can rely on.

Inferring is a crucial part of how you as an individual come to know the world. It’s a process that’s fallible, and sometimes it results in errors—beliefs that aren’t really knowledge. When two people argue with one another—if they’re arguing honestly—they are sharing with some of the reasons they each have for believing the things they do. Each is trying to teach the other some of what he thinks he knows, and each can help the other identify errors in his own thinking.

Of course, people also often argue with one another dishonestly. Think of a conman who sells a bogus medicine and gives arguments to trick sick people into believing that his medicine will cure their illnesses. The conman knows that the conclusion he’s trying to convince them of is false—or, at least, he doesn’t really care if it is true. His goal is not to help them to tell what’s true, but to get them to believe a conclusion that he wants them to believe, regardless of whether it is true. It is possible to be dishonest with oneself in this same way. You may not be able to work to convince yourself of something you self-consciously know is a lie, but people often make arguments to convince themselves of things that they want to believe—or that they think they are supposed to believe—regardless of whether the things are really true. For example, someone who suspects her husband of having cheated may not want to believe it because she finds the idea too painful, or she may feel guilty for believing it because she thinks she is supposed to trust him. In either case, she may fish around for arguments to convince her that he is faithful.

This brings me to an important point of clarification. Earlier I said that an argument is meant to give us a “reason to believe” its conclusion. When we speak of the reasons we have for performing an action, we usually have in mind either some goal that we hope to achieve by performing the action, or some obligation that we think we have for taking it. These are called pragmatic or practical reasons. By contrast, we can call the reasons that arguments give epistemic reasons. They are the sort of reasons that help one to tell that a conclusion is true and thereby to put one in a position to know it.

The primary use of arguments is to tell what’s true. A secondary use is to persuade oneself or others that something is true. We can distinguish two forms of persuasion, which we might call "honest" and "dishonest." In Honest persuasion, you articulate your own actual reasons for thinking the conclusion is true, in order to persuade someone else of it or to make clearer in your own mind. In dishonest persuasion you come up with reasons to convince someone of something without regard for whether that thing is really true. Thus a dishonest lawyer or salesman, might try to persuade someone of innocence of his client or the quality of his wares by appealing to premises that he knows are false (or has no reason to think are true), but that he thinks his audience is likely to believe, or he may use arguments that seem strong at first glance, but don't stand up to scrutiny. Not all sorts of dishonest persuasion stem from trying to make money or advance one's career (as these two cases do). People also argue dishonestly to impress others or to feel important themselves.

It is possible to engage in dishonest persuasion with themselves as well as with others. This often happens when someone wants to believe something (or thinks that he is supposed to believe it). Thus someone in an unhappy marriage might try to convince himself that he and his spouse are happy together, despite the growing evidence that they are not, or someone who has grown up in a family that professes certain beliefs and feels that he's supposed to believe them, might try to convince himself that they're true. We can call this dishonest reasoning with oneself, rationalizing, or BSing oneself. It is often hard to tell whether someone is being honest or dishonest in his use of arguments, and it can sometimes even be difficult to tell in one's own case. However, the essential difference between the two ways of arguing is clear: Is one using argument in an attempt to tell what's true (and to share this with others) or is one using them to defend a position that one is committed to defending regardless of its actual merits? Genuine reasoning is the attempt to tell what's true, and dishonest uses of arguments are just attempts to fake (to oneself or others) that one is engaged in this process. It is for the sake of genuine reasoning that we really need arguments, so we will focus on honest reasoning in this primer.

Relations Between Arguments

Our thinking isn’t usually made up of single, isolated arguments, but of complex sets of interrelated arguments. And argument mapping helps us to visualize and understand these relations. In this section we’ll consider how the argument we looked at earlier concerning Natalie might be related to other arguments. Before we do that, let’s review the simple map of that argument and get clear on how we are going to refer to each of its parts.

Map 1

Each box contains a numbered proposition. We’ll refer to them as “Proposition 1,” “Proposition 2,” and “Proposition 3.” The green circle represents the act of inference, and we’ll refer to it as “Inference A.” We’ll use the phrase “Argument A” to refer to the whole argument, which includes the inference, along with its premises and the conclusion.

With this terminology under our belts we can look at and discuss more complex maps that show relations between arguments.

Arguments that share a conclusion

One way in which arguments can be related is by sharing the same conclusion. Here’s an example:

Map 2:

Argument B has the same conclusion as Argument A. They are two distinct arguments because they give separate reasons for believing the conclusion. To see that the reasons are separate, pick one of the premises from either argument, and think about what other premises you would need to combine it with before it would give you a reason to think that Natalie murdered Carl. You will see that in order for Proposition 1 to convince you that Natalie murdered Carl, you would need to also know Proposition 2, but that you wouldn’t need to know Propositions 4, 5, or 6. Likewise, in order for Proposition 5 to convince you, you’d need to also know Propositions 4 and 6, but not Propositions 1 or 2.

Chains of Argument

A second way arguments can be related is that a premise of one argument can be the conclusion of another, forming a chain. Here’s an example of that:

Map 3:

Notice that Argument A (from Map 1) is part of this larger map. One of its premises is now the conclusion of another argument, labeled C, and one of C's premises is a premise for a further argument, labeled D. So, we can think of D, C, and A as a chain of arguments, with Proposition 3 as their ultimate conclusion. (Or, to put it another way, we can think of D, C, and A as making up a three-step argument for Proposition 3. And, we can think of Propositions 2 and 8 as intermediate conclusions of this larger argument.)

Complex Maps

Now let’s consider a larger map that includes all the arguments we’ve looked at plus three others.

Map 4:

Argument E is a third argument for Proposition 3. I expect you’ll agree that it a bad argument. The map shows two objections to this argument, each symbolized by a red octagon. Objections are (in effect) arguments that something is wrong with the argument they're objecting too.

Objection H objects to Proposition 11, which is a premise of Argument E. When someone objects to a premise, he is arguing that it is not a suitable premise. This could be because the premise us false (as is the case with Proposition 11), but there are other reasons why a premise might not be suitable. For example, we might not be in a position to tell whether it's true, which would make it useless as a premise. Or we might only have reason to think that it's true if we already believe the conclusion, in which case it would be a case of circular reasoning. In addition to objections to premises, there are objections to inferences. These are arguments that the inference is faulty.

Objection I objects to Inference E. To object to an inference is to argue that it's not a good inference—that, even we knew the premises to be true, the inference wouldn't give us a reason to believe the conclusion. We'll discuss the factors that make for a good or bad inferences later.

Inference F is represented by a red inference symbol instead of the usual green one because it represents a counterargument. Counterarguments are an arguments against conclusions, rather than for them. Counterargument F gives us a reason to think that its conclusion, Proposition 3 is false. If we wanted to treat F as a regular argument (represented by a green symbol), then it's conclusion would be that “Natalie did not murder Carl.” We could write this conclusion out as a distinct proposition, in its own box, but since it is equivalent to saying that Proposition 3 is false, it’s nice to have a way to relate F to Proposition 3 on the map, so that we could see a glance that F is arguing against the very same conclusion that Arguments A, B, and E are arguing for. This is the purpose of the red counterargument symbol.

Argument G supports Proposition 14, which is a premise for Counterargument F.

The point of an argument is to help us tell what’s true. So, if we end up in a situation where we have arguments with opposite conclusions as we do in Map 4, then there must be an error somewhere. In this map, Objections H and I, indicate what's wrong with Argument E. But even if we eliminate Argument E, we have two other arguments for Proposition 3, and we have a counterargument against this same proposition. Proposition 3 cannot be both true and false, so something must be wrong either with Arguments A and B or with Counterargument F. Let’s turn now to the things that can go wrong with arguments, and why some arguments are better than others.

Why Some Arguments are Stronger than Others

Mapping an argument is a way of analyzing it—breaking it up into its parts and showing how those parts fit together into a whole. The point of analyzing an argument is that it makes it easier for us to then assess the argument. When we assess an argument, we are asking how strongly it supports the conclusion—how good a reason it gives us to think that the conclusion is true. The strongest arguments enable us to tell that their conclusions are true. In doing so, they establish the conclusions as knowledge.

Map of epistemic status

If an argument supports its conclusion strongly enough to establish it as knowledge, the argument is called conclusive and is said to be a proof or to prove the conclusion.3 These arguments are especially valuable. On the opposite extreme would be an argument that gives us no reason to believe the conclusion and so leaves us no closer to knowing it than we were before. There is no special name for such arguments, but let’s call them worthless because they don’t do any of what an argument should do. Many arguments fall in between these two extremes. That’s because knowledge isn’t an all-or-nothing affair. There is a whole spectrum between really knowing a proposition and being entirely ignorant of it. We can call this spectrum the proposition’s epistemic status.

Consider the proposition “Natalie murdered Carl,” which was the conclusion of many of the arguments in the maps we looked at earlier. Suppose that this proposition is true and that you’re a detective investigating Carl’s murder. When you begin the case, you probably don’t know the proposition at all. At this point you have no idea who killed Carl, and no reason to suspect Natalie; you might not even know who Natalie is. So, the proposition would be at the extreme left end of the scale picture above. But the investigation proceeds and you learn more about Carl’s life and death, at some point you formulate the theory that Natalie murdered Carl because there is some evidence pointing to her. Perhaps at this point, it’s not much evidence. We certainly wouldn’t say that you know she killed him or even that you believe it (or have reason to believe it), but you now suspect her, so we wouldn’t say that you’re totally ignorant of her having murdered him either. The proposition is now somewhere on the scale to the right of ignorance but to the left of the half-way mark.

Eventually, as you accumulate more evidence, she becomes the lead suspect. Now if you had to bet, you’d say that she did it, and you’d have good reasons to support your bet. We might say you believed she did it, but you don’t know it yet. Finally, at a certain point you might get enough evidence to really be certain that she did it. Now the proposition has progressed to the right-most region on the scale—you know it.

All the evidence that you accumulated along the way could be spelled out as arguments. And we can think of what arguments do as helping us advance along the scale from ignorance to knowledge. An argument that’s strong enough to take us all the way to knowledge is a conclusive argument or proof. An argument that doesn’t take us any of the way is worthless. But many arguments take us part of the way—they give us some reason to believe the conclusion without giving us conclusive reason.

Scale of epistemic status

Notice that in the scale for epistemic status, knowledge is represented by a range and not by a point. This is because, even among the things we know, we think of ourselves as knowing somethings better than others. For example, you probably think you know both that Trump is the 45th President of the United States and that twice two equals 4. But you might think that you know the second of these propositions better than the first, since you can probably imagine some bizarre scenario in which Trump isn’t really the president and you’re the victim of an elaborate hoax, but it’s hard to imagine any scenario in which you can be mistaken that twice two is four. Perhaps some of you think that you don’t really know that Trump is the 45th President because you think you can’t totally rule out this hoax scenario. We’ll discuss these sorts of skeptical worries later in the course. For now, my point is just that to saying that you know something is not to rule out the possibility that there are other things that you know even better. That’s why I’m representing knowledge as a range rather than as a point. Similarly, to say that an argument is conclusive is just to say that it’s enough to establish its conclusion as knowledge. It is not to say that there cannot be some other argument that is even stronger.

The two factors that contribute to the strength of an argument are its premises and its inferences. So, to assess an argument we need to assess each premise and each inference.

The strongest premises are ones that we know to be true independent of knowing the conclusion. In order for an argument to prove its conclusion all of its premises must be like this. On the other extreme if we have no reason at all to think that a premise is true (or if we know that it is false), then it will make any argument it is part of worthless. If, on the other hand, the premise has an intermediate epistemic status, an argument containing it could still support the conclusion to some extent, without proving it. This is illustrated in the map below.

Map 1a

The scales placed in the boxed for Propositions 1 and 2 indicate the epistemic status of those propositions. The scale in the box for Proposition 1 has a mark in the rightmost region indicating that that proposition is known to be true. Proposition 2’s scale shows that the proposition isn’t quite known, though there is some reason to believe it. The scale drawn above the green circle indicates the strength of Argument A as a whole. We see that it is no stronger than the weakest premise, which is Proposition 2.

Since the case of Natalie and Carl is fictitious, there’s no way for us to actually assess the premises. The epistemic statuses in the map above are made up (as is everything else in the example). In §3, below, we’ll discuss how to assess actual premises of actual arguments.

In addition to assessing premises, we need to assess inferences. In order for an argument to support its conclusion, the premises and conclusion need to be related in such a way that it is unlikely for the conclusion to be false if the premises are true. The more unlikely it is for the conclusion to be false if the premises are true, the stronger the inference. Sometimes the premises and conclusion are related in such a manner that one would be caught in a contradiction if one held that the premises were true, but the conclusion was false. These inferences are called deduction and are said to necessitate their conclusions. Deductions are as strong as it is possible for an inference to be, so if we have a scale assessing the strength of an inference, we should represent deduction not as a range, but as a point at the end of the scale. (We will discuss how deductions work in §4.1, below.)

Place of deduction on the epistemic scale

Inference A is a deduction. If whoever murdered Carl had access to his rose garden at midnight, and Natalie was the only person who had access then, then Natalie has to be the murderer. If we said she wasn’t we would be saying that the murderer was someone other than Natalie who according to Proposition 1 had access to Carl’s rose garden at midnight, but Proposition 2 tells us that only Natalie had access to the rose garden at midnight. So, to hold both premises and deny the conclusion would be to say that Natalie was and wasn’t the only person with access to the rose garden at midnight, and that’s a contradiction. If the premises are true, the conclusion has to be. The only way to consistently deny the conclusion is to deny one of the premises. So, Inference A is as strong as an inference can be.

We can add this assessment into our map of Argument A as follows.

The scale below the green circle represents our assessment of Inference A, and it is marked at the rightmost point to show that the inference is a deduction. The scale above the circle represents our assessment of Argument A as a whole. Notice that, even though the inference is a deduction, the argument taken as a whole isn’t conclusive, because we don’t know that Premise 2 is true. The strength of the argument as a whole depends on the strength of all its elements, whereas the strength of the inference is independent of the strength of the premises. That’s illustrated by the following map:

This map contains two separate arguments (Arguments F and G). Argument F has two awful premises—premises that no one has any reason to believe and that we all know to be false. But Inference F is as strong as can be; it’s a deduction. Propositions 13 and 14 are obviously false, but if they were true, then Proposition 15 would have to be true also. Nevertheless, Argument F is worthless, because its premises are so bad. Argument G is also worthless, but for an opposite reason. We know that both of its premises are true, but the premises aren’t related to one another and to the conclusion such that their being true gives us any reason to think that the conclusion is true as well. The problem here is with the inference. You can’t infer anything about Carl and Natalie from premises about birds and insects. Inferences this bad are called non-sequiturs.

Some inferences are extremely strong without being deductions. Consider the following argument:

Propositions 18 and 19 do not necessitate Proposition 20. But knowing them would give us an extremely strong reason to believe Proposition 20. The reason is so strong that in most context we would say that Argument H would establish Proposition 20 as knowledge. Let’s use the word compelling for inferences that are strong enough to establish their conclusions as knowledge, if their premises are true. If so, here’s what our scale of inference strength looks like:

Even inferences that aren’t compelling can be useful. Consider the following argument:

Inference I isn’t compelling, but it’s not bad, either. If you knew the premises to be true, this argument wouldn’t put you in a position to know the conclusion, but (unless you had other relevant knowledge of Estelle), it ought to lead you to expect that she can speak English.

In §3 below, we’ll discuss how to identify and assess different sorts of inference. The examples in this section are intended just to give you a sense that some are stronger than others.

To review then, arguments range in strength from worthless to conclusive, with conclusive arguments being the ones that are strong enough to establish their conclusions as knowledge. The strength of an argument is determined by the strength of its premises and of its inference. The strength of a premise is its epistemic status. The highest epistemic status is knowledge, and the lowest is that of a proposition one is either wholly ignorant of or knows to be false. The strength of an inference is determined by the relationship between the premises and the conclusion, and this is separate from whether the premises are true. The weakest inferences are called non-sequiturs and the strongest are called compelling.

An argument as a whole can be no stronger than its weakest element (premise or inference). And in an argument the weaknesses compound, so if multiple elements have weaknesses, the whole will be weaker than the weakest part.

How to Assess Arguments

Once you have mapped an argument in order to assess it you have to first identify all of the unsupported premises—the propositions that serve as premises in arguments, without themselves being conclusions of other arguments. On the map, these will be all the boxes that do not have arrows pointing to them. You then must assess each unsupported premise and each inference. To assess a premise is to determine its epistemic status. Since the examples in this primer are fictitious, the assessments of the premises are fictitious as well. See section §3, below, on how to assess actual premises.

To assess an inference, assume that you knew all of its premises are true, and then ask yourself how strong a reason they would give you to believe that the conclusion is also true. How to assess different sort of inferences is discussed in §4, below, but you should be able to get an intuitive sense of how strong an inference is just by asking yourself if the premises were true how strong a reason would they give you to believe the conclusion.

Once you have assessed all the premises and inferences, you can then assess each argument as a whole. The argument can be no stronger than its weakest element (premise or inference). Weaknesses within an argument compound, so if there are weaknesses in more than one element, the argument will be weaker than its weakest element.

For some sorts of premises and arguments, there are precise mathematical ways to evaluate their strength, but that sort of precision is not always possible. It is enough for our purposes to place premises, inferences, and arguments in rough regions of the scales that we are using to evaluate them.

Let’s try this process, with Map 3 from above. Here the map is again:

Map 3:

The first step is to identify all the inferences and unsupported premises. There are two inferences (A and C) and three unsupported premises (Propositions 1, 7, and 8). Proposition 3 isn’t a premise at all, and Proposition 2 is a premise for Argument A, but it isn’t unsupported, because it is the conclusion of Argument C. So, we will need to assess the premises and the inferences. In the map below I’ve added blank scales for the elements we will need to assess.

Let’s assess the premises first. If this were a real-life argument, we would have to reflect on how strong a reason we have for believing Propositions 1, 7 and 8, but since the example is fictitious, we’ll have to make up their epistemic statuses as well. I’ve done that in the map below:

As the premises are assessed on this map, we know Propositions 1 and 7, but we don’t quite know 8. Perhaps we think there’s some possibility that a second key was made or that Natalie’s key was stolen from her.

Now that the premises have been assessed we’ll turn to assessing the two inferences.

We’ve already seen that Inference A is a deduction, so it is as strong as can be. What about Inference C? It is definitely not a deduction, there is no contradiction involved in holding that someone other than Natalie had access to the garden at midnight, even though it is locked after 10pm, and Natalie had the only key. It’s just unlikely that someone else had access, since keys are the normal way of accessing locked places, and locks are designed to keep people without keys out. Still it’s possible for people to enter locked places without keys—locks can be picked, and presumably the rose garden has walls that can be climbed. To determine how strong Inference C is we’d need to think about how plausible these alternative routes of access are, and that would require some background knowledge. Assessing non-deductive inferences is more difficult than assessing deductions because it requires making use of such knowledge. In this case, since the example is fictitious, there is no background knowledge to rely on, so we’ll have to make up more about the example. If we took it for granted that the lock is of a kind that is almost impossible to pick without leaving marks (that weren’t found), that picking it would have taken time in which someone doing it would likely have been observed, and that the walls of the garden couldn’t be scaled without sounding an alarm (that didn’t sound), then I think this inference would be compelling. But in that case, it would be a lot clearer if the person making the argument had made these assumptions explicit by including them as premises. In any case, for the sake of the example, let’s assume that we aren’t in a position to quite rule out lock-picking and that the inference is strong but not compelling. That’s how I marked it on the map above.

Once we have assessed all the unsupported premises and inferences, we can go on to assess the arguments as wholes. The map below incorporates all the assessments we’ve discussed so far, plus blank scales for assessing the two arguments.

If any of an argument’s premises are themselves supported by other arguments, we need to assess those other arguments as wholes before assessing the initial argument. So, in this case, we’ll need to assess Argument C before we assess Argument A.

Argument C has one premise (Proposition 7) that is known, but it has two weaknesses. Proposition 8 is not (quite) known to be true, and Inference C is not compelling. Either of these weaknesses taken on its own is sufficient to prevent the argument from being conclusive. The argument as a whole can be no stronger than its weakest element, which (as we’ve filled out the scales above) is Inference C. But since the weaknesses in an argument compound, and inference C is not the only weak point, in this argument, the argument as a whole is weaker than Inference C. We can represent this on the map below by putting a mark on the scale for Argument C a bit to the left of the mark on the scale for Inference C.

In addition to adding the assessment of Argument C to Map 3d, you’ll see that I added a scale for Proposition 2 and marked it with the same assessment. This is because this map shows Argument C as our reason for believing Proposition 2. The reason for having separate scales for Argument C and Proposition 2 is that sometimes we will have multiple arguments for the same proposition. We’ll discuss a case like this in a moment. Before we do, let’s finish assessing the arguments in this map.

What remains is to assess Argument A. Here there is only one weak element. One of its premises is known the be true and the inference is a deduction, so the argument as a whole will be as strong as the remaining element, Proposition 2. The dot indicating our evaluation of Argument A is therefore placed in the same spot on its scale as we placed the dot on Proposition 2’s scale.

If Argument A were the only reason given to accept Proposition 3, then we would give Proposition 3 the same epistemic status we’ve given Argument A. However, as we’ve already mentioned, conclusions are often supported by many, separate lines of reasoning. If that were the case, we would have to assess Proposition 3 in light of both the strength of Argument A and the strength of the additional arguments supporting it. This is shown on the map below (2a) which includes Argument A along with another argument (Argument B) for Proposition 3. (I’m omitting Argument C from this map to save space.)

Since Proposition 3 is supported (in this map) by two different arguments, to establish its epistemic status, we need to take both into account. Argument A is fairly strong, though not compelling.

But as I’ve represented it here, Argument B is very weak. Inference B is a deduction, but the epistemic statuses I’ve given to its premises are much weaker than those of the premises of Argument A. (Recall that since these are fictitious arguments about fictitious people, the epistemic statuses are also fictitious.) Proposition 6 is approximately as strong as Proposition 2, but Propositions 4 and 5 are considerably weaker, and in an argument the weaknesses compound, so the argument as a whole is considerably weaker than its weakest premise. It’s not quite worthless, but it’s not worth very much. At best it could give one reason to suspect Natalie of the murder.

So, what epistemic status does Proposition 3 have based on these two arguments? Argument A gives us a pretty strong (though not conclusive) reason. Argument B doesn’t add much to it, but a weak argument doesn’t take away from the reasons given by a strong one, so over all we have about as much reason to believe the conclusion as Argument A gives us. It’s not knowledge, but we should regard it as something that’s probably true.

An argument is at least as weak as its weakest element, and if an argument has multiple weaknesses the weaknesses compound. But a conclusion is as at least as strong as its strongest argument, and if there are multiple arguments, their strength can compound.

There is one important caveat to this claim that a conclusion is as strong as its strongest argument. A conclusion’s strength can be diminished if you have an argument against it. (An argument against a proposition is often called an objection or counterargument.) Consider the map below: Natalie murdered Carl.

Argument E is an argument against Proposition 3, so whatever strength it has is going to counteract the strength of Argument A. Like Argument A, it is not conclusive. Its inference isn’t quite a deduction, but it is compelling. Proposition 12 is known to be true, so the argument would be conclusive, if proposition 11 was also known. But it isn’t, so the argument is correspondingly weaker. However, it is still pretty strong. Taken on its own it would lead us to think that Natalie probably didn’t murder Carl (since to murder him she would have had to have used a crossbow, and she probably doesn’t know how to use one). Taking it in the context of the whole map, it considerably reduces the epistemic status that Proposition 3 would otherwise have due to Argument A. That’s why I marked it as just about in the middle in the map above.

This raises an interesting question. What would happen, if both Arguments A and E had been conclusive? That would mean that Argument A would establish that Proposition A is true, while Argument E would establish that it is false. But it can’t be both true and false. So, if we find we’re in that situation we know we’ve made a mistake somewhere in our assessment. As Ayn Rand puts it: “Contradictions do not exist. Whenever you think that you are facing one, check your premises. You will find that one of them is wrong.”4 I’ll add that you should also check your inferences, since you may have misevaluated one of them. We will discuss how to check your premises in §3 below, as part of our wider discussion of assessing premises. We’ll discuss how to assess different types of inferences in §4, and in §5 we’ll revisit the issue of how to assess a conclusion in light of multiple arguments for and against it.

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