8: Present Value Models
After completing this chapter, you should be able to: (1) identify the different kinds of present value (PV) models; (2) understand the types of questions different PV models can answer; and (3) construct PV models that represent defending and challenging investments.
To achieve your learning goals, you should complete the following objectives:
- Learn about the different kinds of PV models.
- Learn about the unique questions each kind of PV model can answer.
-
Learn how to construct the following PV models:
- net present value (NPV),
- internal rate of return (IRR),
- maximum (minimum) bid (sell),
- annuity equivalent (AE),
- break-even,
- optimal life, and
- payback.
- Learn how to represent the opportunity cost of the defender using its return on equity (ROE) or its return on assets (ROA).
Introduction
Picture yourself traveling across the country driving a luxury car or a tractor. Alternatively, picture yourself pulling a plow with a luxury car or a tractor. While both a luxury car and a tractor can provide transportation and pulling services, they are not equally suited for the two assignments.
One word economists use to describe activities a firm can perform but with unequal efficiency is comparative advantage. Luxury cars are perfectly suited for traveling across country but not for pulling a plow. A tractor is well designed for pulling a plow but not for long-distance travel.
Similarly, there are several different kinds of PV models, but they are designed for answering different kinds of questions. Each model has a comparative advantage for answering a particular kind of question. In what follows we examine some of the most important PV models and the questions they can be used to answer.
Net Present Value Models
In a static (timeless) model, profits describe the difference between costs and returns. In a PV model, an equivalent concept is called the Net Present Value (NPV). The NPV measures the difference in present dollars between cash outflows and cash inflows. NPV models reveal whether the present benefits of an investment outweigh its present costs. The important difference between NPV models and profit calculations is that profits may include non-cash returns and expenses such as increases in inventory and depreciation. The numbers that enter PV models besides the discount rate and the exponents on the discount rate of dollar are cash flow measures where the liquidation value of the investment at the end of its economic life is treated as though it were converted to cash.
Suppose an investment in a challenger requires an outflow of V0 dollars and generates a positive cash flow of R 1 dollars one period later plus the investment’s liquidation value S 1 . For now, assume that the discount rate, or the rate of exchange between present and future dollars for the defender, is r percent, and represents the opportunity cost of sacrificing the defender to invest in the challenger. To determine if the benefits of this investment outweigh the costs, we find the NPV defined as:
\[ \label{8.1} N P V=-V_{0}+\frac{R_{1}+S_{1}}{(1+r)}\]
Suppose a challenging investment costs $100 and returns R 1 = $100 and S 1 = $20 in period one while the defending investment could earn r equal to 10%. Then the NPV of this investment would be:
\[N P V=-100+\frac{\$ 100+\$ 20}{1.1}=\$ 9.09 \label{8.2}\]
For this investment, NPV is positive, and the present value of cash benefits outweighs the present value of cash costs. Another way to describe this result is that the challenger earned a higher rate of return than the defender. How much more? The exchange rate for the challenger is ($100 + $20)/$100 = 1.20 compared to $110/$100 = 1.10 for the defender.
But what does it mean to say that the challenger earned a higher rate of return than the defender? An investment is a commitment you make to a project. If the project pays you more than the rate of return on the defender, then the investment’s NPV is positive. In our example the project pays at the rate of 20%. In NPV models, think of the discount rate as the rate earned by the defending investment. And if NPV is positive, the challenger will earn a rate of return higher than could be earned by continuing to invest in the defender.
Of course, investments can be much more complicated than described above. For example, an investment may generate (positive and negative) cash flows for n periods ( R 1 , R 2 , ⋯, R n ) rather than just one period. Furthermore, most capital assets generate a positive or negative liquidation value ( S n ) at the end of its economic life which should be accounted for explicitly.
Using subscripts to indicate the time periods in which the cash flows occur, we write a more complete NPV model as:
\[N P V=-V_{0}+\frac{R_{1}}{(1+r)}+\frac{R_{2}}{(1+r)^{2}}++\frac{R_{n}}{(1+r)^{n}}+\frac{S_{n}}{(1+r)^{n}} \label{8.3}\]
Internal Rate of Return (IRR) Models
Assume that the cash flow represented in Equation \ref{8.3} are largely determined by the market or from other sources of information and are known. But where do we find r , the opportunity cost of capital, the rate of return earned on the defender? The answer is that we set the NPV of the cash flow that describes the defender equal to zero and solve for the discount rate which we then refer to as the defender’s internal rate of return or its IRR. In such a model, the IRR will be equal to the defender’s ROE or its ROA depending on the focus of the PV model. To make clear the distinction between cash flows associated with the challenging and defending investments we adopt the following notation: we superscript cash flow for the challenger with the lowercase letter “c” and cash flow for the defender with the lowercase letter “d”.
\[N P V^{d}=-V_{0}^{d}+\frac{R_{1}^{d}+S_{1}^{d}}{(1+r)} \label{8.4}\]
Now set NPV d equal to zero. Such a PV model in which NPV is set equal to zero is called an Internal Rate of Return (IRR) model, and r is the rate at which dollars are exchanged between time periods for the defender. Because NPV is zero in this model, r is exactly the rate of return earned on the defending investment, the rate at which future dollars earned on the investment are exchanged for present dollars. Finally, the internal rate of return earned on the defender is what is sacrificed to invest in the challenger, and equals the opportunity cost in the NPV model. (The calculation of the defender’s IRR in multi-period models can be much more complex.) To illustrate, we solve for the rate of return earned on the defender or the IRR in the previous model. The IRR equals:
\[r=\frac{R_{1}^{d}}{V_{0}^{d}}-1 \label{8.5}\]
Next substitute the defender’s IRR into the NPV model of the challenger:
\[N P V^{c}=-V_{0}^{c}+\frac{R_{1}^{c}+S_{1}^{c}}{(1+r)} \label{8.6}\]
If the NPV of the challenger is positive, the challenger is preferred to the defender. Why? Because exchanging future dollars for present dollars at the same rate of exchange as the defender leaves the investor with a positive net present value.
Maximum Bid (Minimum Sell) Models
Different kinds of investment questions created the need for different kinds of PV models. The maximum bid and minimum sell models assume r is known and NPV is zero. The models then solve for the purchase price of an investment in a maximum bid price model, or the sale price in a minimum sell price model. In a maximum bid model, the sale price that equates NPV to zero indicates how much the buyer can bid for the investment and still earn the IRR rate r on the defender. Or, from the seller’s perspective, the minimum sell price is the lowest price a seller can accept in exchange for the cash flow stream generated by the investment and still earn the IRR rate r earned on the defender.
To illustrate, begin by assuming that r , the IRR of the defender, is known as well as the cash flow that can be earned by the challenger. Now evaluate the challenger by setting NPV equal to zero and finding the maximum bid (minimum sell) price :
\[N P V^{c}=-V_{0}^{c}+\frac{R_{1}^{c}+S_{1}^{c}}{(1+r)}=0 \label{8.7}\]
The maximum bid (minimum sell) price is:
\[V_{0}^{c}=\frac{R_{1}^{c}+S_{1}^{c}}{(1+r)} \label{8.8}\]
The Break-even Model
Assume that the investor knows the investment cost , its liquidation value , and the IRR of the defender r but doesn’t know the amount of returns on the challenger that would be required if the investment were to earn a rate of return equal to that available on the defender. The firm can find the amount required to return break-even earnings by setting the NPV model to zero and solving for R 1 as follows:
\[ \label{8.9} N P V^{c}=-V_{0}^{c}+\frac{R_{1}^{c}+S_{1}^{c}}{(1+r)}=0\]
and the break-even earnings amount is:
\[R_{1}^{c}=(1+r) V_{0}^{c}-S_{1}^{c} \label{8.10}\]
If r were 10%, the liquidation value were zero, and the initial investment of were $100, then the break-even return would be $100(1.1) = $110, the amount the investment would be required to earn in order to break-even. Break-even here has a specific meaning which is to earn the defender’s IRR.
The Annuity Equivalent (AE) Model
An annuity is a financial product sold by financial institutions. The essence of an annuity is that an individual pays into a fund that is invested and grows until some point in time when the investment is paid back to the investor as a constant stream of payments for a specified period of time. In this book we define a related concept, an annuity equivalent. An annuity equivalent is a constant stream of payments whose present value is equivalent to some other stream of payments that may not be constant. The annuity equivalent model finds an annuity associated with an investment. An annuity is like a time adjusted average. Suppose we wished to find the annuity equivalent associated with the generalized NPV model below:
(8.11)
Now consider an alternative model in which the NPV A is equal to the NPV A in Equation \ref{8.11}:
(8.12)
The value for R in Equation \ref{8.12} is the annuity equivalent cash flow that yields NPV A in Equation \ref{8.11}.
Because annuity equivalents appear in so many contexts, including constant payment loans, retirement benefit payouts, and others, we adopt a notation to describe them. Factoring R in Equation \ref{8.12} we write:
(8.13)
The notation equals the bracketed expression in Equation \ref{8.13} and stands for the sum of a uniform series of $1 payments discounted by a rate of r for n periods. Obviously, there is a direct correlation between NPVs and annuity equivalents, and both are used to rank investments.
To illustrate, suppose that NPV were $10,000, monthly installments were for 5 years, and the discount rate were 3 percent. The AE for this investment is equal to $179.69 and
Optimal Life Models
Optimal life models ask what is the optimal life of this investment? The optimal life model can be written as:
(8.14)
or in continuous time as:
(8.15)
The optimal solution has a specific meaning in the context of the optimal life model. It is that value of n that maximizes the NPV. Formally, the solution employs calculus to optimize the NPV. In the discrete models which are most often employed in practice, the optimal value is found through trial and error or through repeated calculations of alternative values for n .
A related optimal life model asks: what is the optimal value of n that maximizes NPV if there are replacements for the investment? In this case, NPV S is the sum of NPVs from individual investments. Such a replacement model is described below:
(8.16)
The Payback Model
While PV models are generally carefully deduced and the data required to solve them is explicit, sometimes decision makers just want a “ball park” estimate of the desirability of a financial strategy. In such cases, decision makers are willing to sacrifice rigor and precision for approximations. When this is the case, the payback model is often employed. To obtain an approximation, it assumes the discount rate in PV models is zero. In other words, the payback model assumes that present and future dollars are valued the same—a very unrealistic assumption. Then the payback model calculates the number of periods required to earn the investment’s present value. The number of periods required to earn the investment’s value is the payback period, the criterion used to rank investments. All cash flow after the payback period is assumed to have no influence on the criterion. To illustrate, n is the payback period in the payback model that follows.
(8.17)
And if periodic cash flows are constant we can express the payback period as:
To illustrate, consider an initial outflow of $5,000 with $1,000 cash inflows per month. In this case, the payback period would be 5 months. If the cash inflows were paid annually, then the result would be 5 years. More generally, cash flows will not be equal to one another. If $10,000 is the initial outflow investment, and the cash inflows are $1,000 at year one, $6,000 at year two, $3,000 at year three, and $5,000 at year four, then the payback period would be three years, as the first three years are equal to the initial outflow.
Despite its popularity, the payback model is not recommended for several reasons. Mainly, ignoring the time value of money basically treats an inter temporal investment as though it were a static profit problem. Furthermore, it treats cash flows earned after the payback period as though they were of no worth. In sum, in many respects it inadequately accounts for important elements of the investment problem.
Ranking Investments
Questions PV models answer. It may be helpful to categorize PV models into similar types by recognizing the questions they can answer. The first type answers the question: what is the break-even price such that NPV is zero. Included in this category are maximum bid (minimum sell) models, break-even models, and annuity-equivalent models. A second type of model answers the question: what is they optimal life of the investment. These models find the time period that maximizes NPV. The third type of PV model—and the most popular—are PV models that rank investments. They answer the question: which investment is preferred? The two most popular ranking PV models are the NPV and IRR models. In one survey, about 75% of chief financial officers of large corporations used IRR and NPV models nearly equally (Graham and Harvey, 2001). About 50% used payback models.
The advantage of the NPV model is that we can experiment with several different IRRs to find out how sensitive the NPV of the challenger is to alternative discount rates or IRRs. As we will discover in later chapters, consistency between NPV and IRR rankings exists only under homogeneous measures of size, term, taxes, and other measures.
There are alternative, easier ways of ranking investments such as the payback model. They are not recommended here because they fail to recognize a fundamental property of PV models: dollars in different time periods are valued differently and ignore cash flows after the payback period.
More complicated investment ranking problems. When we described the NPV model, the discount rate was equal to the defender’s IRR. When the firm is considering the adoption of two mutually exclusive investments, then which investment is the defender and which is the challenger is an arbitrary decision. If the capital budgeting decision is to replace an existing investment with one of two mutually exclusive investments, then typically the defender is the investment in place. Then, the investment problem becomes calculating the NPV of the two challenging investments and selecting the one with the greater NPV.
More complicated ranking problems involve replacing a defender with a newer version of the same investment. Such an investment might be when to replace a cohort of growing chickens or other livestock grown for slaughter with a new cohort. In this case, there are multiple defenders equal to the same investment at several different ages. This class of NPV models and their corresponding criterion is referred to as replacement models and is the focus of Chapter 10.
Consider the investment problem involving the least cost source of financing. We have decided on an investment, but we have alternative ways to fund the investment and we want to find the least cost source of funding. In this problem, we designate one of the funding sources as the challenger and the other as the defender. We then compare the IRRs of the two funding sources. In this case, the funding source with the smallest IRR is preferred and used as the discount rate in finding the NPV of the potential new investment.
One can easily imagine more complicated investment problems where the designation of the defender is not always clear. In these circumstances, we ask the question do the rankings depend on which investment is selected as the defender? The answer depends on whether or not the investments being ranked are compared using homogeneous units. If the investments are adjusted to make the units consistent, then ranking them by their NPVs will result in the same rankings, regardless of which investment is chosen as the defender.
IRR, ROA, and ROE
So far, the opportunity cost of capital has been introduced without specifying whose capital is being invested. Is it equity capital, debt capital, or a combination of debt and equity capital? Or does it matter? The short answer is that it does matter as we will demonstrate in Chapter 12. If the focus is on the return on equity, then the discount rate represents the return on equity (ROE). In this model, interest costs on debt are subtracted from the cash flow included in the model. If the focus is on the return on assets (ROA), then the cost of the investment is subtracted at the beginning of the model and returns reflect a return to the assets and interest costs are irrelevant since the asset is treated as though it were purchased at the beginning of the investment.
The two approaches, ROE versus ROA, represent two schools of thought in investment analysis. The ROA approach focuses on the returns generated by the firm’s assets. The ROE approach focuses on the returns generated by the firm’s equity invested net of any interest payments on debt.
The difference between the two approaches matter because most firms rely on a combination of debt and equity to fund assets. Reduced to its essence, the issue is whether the opportunity cost of capital reflects the rate of return on the firm’s assets or equity. We now describe the two approaches in more detail.
Return on Assets (ROA)
In the ROA approach, the cost of the investment is subtracted in the present period and net cash flow representing return to assets do not subtract debt or principal payments because the investment is paid for “up-front” or at the beginning of the investment. The advantage of this approach is that the analysis considers the rate of return on the entire investment made at the beginning of the period. The NPV for the ROA approach for the single period can be expressed as:
(8.18)
Note that in Equation \ref{8.18}, if NPV = 0 as it would in an IRR model, then (1 + ROA) V 0 is equal to R 1 + S 1 . If we replace V 0 with E 0 + D , we can write R 1 + S 1 = (1 + ROA)( E 0 + D ). This fact will be helpful as we connect ROA and ROE measures.
Return on Equity (ROE)
In the ROE approach, the analysis depends on how the asset is financed. In this approach, the cost of interest and debt payments is subtracted explicitly, and the initial investment is equal to the amount of equity invested, since the debt is paid directly to whoever supplies the investment. However, the debt D plus average interest costs charged at interest rate i ( iD ) are subtracted at the end of the period. The NPV for the ROE model is expressed as:
(8.19)
The fundamental rates of return identity. One of the issues to consider when comparing the ROE versus the ROA approach is that in equations (8.18) and (8.19), E 0 and V 0 are not equal because part of the acquisition of the investment is supported by debt, V 0 = D 0 + E 0 . The main point here is that the two approaches are not equivalent. We made this point when we derived Equation \ref{7.13} which is repeated here as Equation \ref{8.20}.
(8.20)
Notice that in Equation \ref{8.20}, if the cost of debt i is equal to the return on assets ROA, then ROA and ROE will be equal. But if i is less than the return on assets, then ROE > ROA. A homework problem at the end of this chapter asks you to explain this result. Equation (8.20) is so important in financial analysis that we give it a special name: the fundamental rates of return identity.
IRRs, ROAs, and ROEs
We earlier defined IRRs as the discount rate that corresponds to NPVs of zero. Now we illustrate how to find IRRs on assets and equity. The fundamental rates of return identity alerts us to the fact that IRRs calculated on equity invested in the project and assets invested in the project will not usually be equal. To find the IRRs on assets invested in the project, we charge the entire investment at the beginning of the period and include its liquidation value as a return at the end of the period. This approach ignores the fact that investments may be financed and paid for over the life of the investment and charging for the investment at the beginning of the project doesn’t accurately reflect its cash flow. This is the ROA approach. In this case, the ROA approach, we ignore financing because our interest is in the productive capacity of the long-term asset, independent of the terms under which it can be financed.
We illustrate how to find IRRs for assets (IRR A ) in a simple example. Suppose the firm’s defender is a $1,000, non-depreciable investment that will earn $100 for one period and then will be liquidated at its acquisition price. We find the IRR A associated with $1,000 of assets invested in the defender by setting its NPV equal to zero in Equation \ref{8.18} and solving for ROA.
(8.21)
We find that the IRR A of the defender is equal to 10%.
Now reconsider the same example, except that the $500, or half of the defender, is financed at 9%. The other half of the investment, $500, is financed by the firm’s equity. We continue to assume that, after one period, the investment is liquidated for its acquisition value, the loan of $500 is repaid, and the firm recovers its investment of $500. The firm also earns in one period, $100, the same as before. But now it has to pay a rental fee for the use of the loan’s funds of 9% times $500, or $45. By setting the NPV model of the defender in Equation \ref{8.19} equal to zero, we can find its IRR associated with the firm’s equity (IRR E ) in the project equal to.
(8.22)
Now the defender’s IRR on its equity is 11%. In this case, the firm gained access to the use of an asset because of financing. The gains from a lender providing the firm access to $500 of debt capital to acquire a $1000 investment using only $500 of its own money increased its earnings on its equity from 10% to 11%. Meanwhile the investment earned only 10%. The value of the financing increased the rate of return on equity by 1%.
ROE or ROA?
For a variety of reasons, financial managers may prefer to represent the defender’s IRR as the defender’s ROA or the defender’s ROE. However, this same manager must be careful to make sure that the cash flows associated with the challengers are consistent with the method used to find the IRR of the defender. If the IRR of the defender represents the defender’s ROE, then debt and interest costs should be accounted for explicitly. If the ROA of the defender is preferred, the cash flows associated with the challenger and the defender do not separate debt and interest costs from the calculations.
In practice, PV models appear to prefer the ROA approach, even though both approaches are valid and provide unique information. Nevertheless, the dominance of the ROA approach has resulted in the identification of ROAs as simply the IRR of the investment, a practice we will also adopt in the remainder of this book.
Summary and Conclusions
This chapter reviewed several different kinds of PV models. They differ because they are designed to answer different kinds of questions. Some PV models, NPV and IRR, help us rank alternative investments. Others like AE models can be used to find the optimal time for replacing an investment—or what our periodic loan payments will be. And still other PV models like the maximum bid (minimum sell) models help us know the maximum (minimum) we can offer to purchase (sell) an investment and still earn our defender’s IRR. Finally, still other PV models such as capitalization formulas and payback models offer at most a crude rule of thumb for evaluating and making investment decisions.
Regardless of the PV model type, they all have one measure in common: it is the opportunity cost of a defending investment represented by the discount rate. The discount rate is the ROA or the ROE of the defender that must be sacrificed to acquire the new or challenging investment. This construction of PV models emphasizes the important role of opportunity costs in the construction of PV models and reminds us that we focus on opportunity rather than direct costs when making applied economic decision.
Questions
- Explain the connection between investment questions and corresponding PV models.
- Suppose you found the maximum bid price for a purchase you are considering. What would you conclude from your calculations?
- Suppose that you found that an investment’s AE reached its maximum at year 10 while its NPV reached its maximum at 15 years. How would you interpret your results?
- When finding the ROE measure in Equation \ref{8.22}, we explicitly accounted for the debt used to finance the investment. However, when finding the ROA measure in Equation \ref{8.21}, we did not account for the debt used to finance the investment. Explain the difference between the two approaches.
- When comparing the ROAs and ROEs, we concluded that the ROE > ROA as long as the ROA was greater than the average interest rate on the debt used to finance an investment. Please explain these finding and provide an example on these results. Refer to the fundamental rates of return identity when formulating your answer.
- Provide numerical PV models in which you find an investment’s IRR and NPVs using ROA-IRR and ROE-IRR, AE, maximum bid price, and the investment’s payback period. Defend your choice of a discount rate.
- Explain how the “life of the investment” principle guarantees that all economic activity associated with an investment will be captured by its cash flow.
- Most of the time, we don’t identify discount rates in PV models as being either a ROE or a ROA. Instead we seem to prefer to identify the defender’s ROA with the letter “r”. Why do we tend to prefer ROA to ROE measures? Can you describe a case when it would be important to evaluate the projects using the defender’s ROE instead of its ROA?
- Assume you were considering one investment that could be financed from two different financial institutions. Thus the only difference between the projects were their cash flow associated with their use of debt capital. How would you proceed to rank the two investments?
- In a previous chapter, we calculated the rates of return on equity and rates of return on assets for HQN for the year 2018 for the entire firm. But this calculation was for only one year and included non-cash items in the calculation. What is different between IRRs calculated as ROA or ROE for the entire firm versus IRRs calculated as ROA or ROE in a PV model?
- Suppose you ranked two challengers using their IRRs and by finding their NPVs using the IRR of the defender. What would you conclude if the IRR and NPV ranking provided conflicting ranking?


14.4: Writing Process- Informing and Analyzing
By the end of this section, you will be able to:
- Successfully apply citation conventions to your writing, understanding the ideas of intellectual property that motivate their use.
- Evaluate research materials for credibility, sufficiency, accuracy, timeliness, and bias.
- Compose to discover and reconsider ideas.
- Compose an annotated bibliography that uses correct style conventions and integrates the writer’s ideas with ideas from related sources.
Now it is time to try your hand at creating your own annotated bibliography. As you work through identifying and writing about your sources, you may want to seek out additional ones to help support your argument or thesis. This recursive process is similar to the fine-tuning that occurs during the writing process, in which a writer moves fluidly among drafting, editing, and revising. You can return to your annotated bibliography during the course of research and writing to modify, further develop, analyze, and add to your sources. Remember, using a variety of sources can broaden your perspective about your topic by showing how different scholars and popular publications approach it.
Summary of Assingment: Annotated Bibliography
For this assignment, you will create an annotated bibliography based on your research for one of the writing assignments in this course, preferably Writing Process: Integrating Research. After collecting and choosing sources, you will write citations using MLA Documentation and Format and compose a one- or two-paragraph annotation for each source. Remember, the purpose of the annotation is to provide
- a brief summary that allows readers to understand the background of the source and its basic claims; and
- an evaluation and reflection on the source’s reliability, its usefulness, and the author’s or organization’s credibility.
While the format for your MLA citation should follow the prescriptive pattern from the Handbook, you will decide on the language and structure of the annotations you include. While your topic, personal style, and ideas may lend themselves to a more standardized format, it is possible that they may instead challenge codified conventions in favor of a style more authentic to you.
Another Lens 1. Consider alternatives to a formal annotated bibliography that complements differing learning styles or abilities. If your instructor is amenable, you may informally annotate sources in their margins or use an online annotation program such as Kami ( https://openstax.org/r/Kami ).
Another Lens 2. Another alternative is to discuss, with a partner or in a small group, each source’s credentials, key ideas, and usefulness to your project. In either case, using a graphic organizer similar to the one in the next section should help you organize your ideas.
Quick Launch: Organizing Information
Begin by collecting sources. A good way to start is to make a list of keywords, known authors, organizations, and previously identified sources related to your topic. For example, imagine you have been assigned an argument project and chosen artificial intelligence as the topic. Begin, then, with the keywords artificial intelligence . A Google search will reveal a wealth of information on the topic, likely too much to help you craft a meaningful argument. However, a basic search will allow you to
- effectively define artificial intelligence as “the simulation of human intelligence by machines programmed to mimic human thought and actions”;
- name four types of artificial intelligence: reactive machines, limited memory, theory of mind, and self-aware;
- identify leading industries in which artificial intelligence is already present, such as self-driving cars and virtual butlers; and
- recognize the existence of a spectrum of thought surrounding the ethics and governance of artificial intelligence.
You also may discover some alternative search terms to use, such as machine learning , data mining , robotics , and neuroscience . Although you may not ultimately use them in your argument, the sources that you find may point you toward a focused thesis.
When you finish your keyword search, the next step is to search academic databases. Use your library’s resources to find scholarly sources such as journal articles, books, textbooks, expert interviews, and reputable periodical pieces. Aim for five to seven sources to begin with, though you likely will add more as you work on your project. You may already have an idea of what claims you want to make in your argument, but the sources may shape them as well. As you review the sources you collect, choose those that give you a varied range of perspectives on the topic. Remember that you can return to this procedure at various points in your research, adding sources as your needs change and as you refine your claims. While you search for sources, keep in mind the specifics of your research project, and limit your search to sources that specifically support your argument, inform a counterargument, or otherwise add information that enhances your research.
Thesis
Before you choose sources, you must have an idea of what you want to say. If you’re creating an annotated bibliography for the argument described Writing Process: Integrating Research, you likely have already drafted a thesis statement. If you are creating an annotated bibliography for another topic, you will need to draft a working thesis. Your thesis will state your position on the topic. It is often a single clear, concise sentence that reveals your side in the argument. You will use your sources to support the thesis you draft.
These frames may be useful when you draft your working thesis:
- Because ________, [someone] should ________.
- ________ saves ________, reduces ________, and helps ________.
- The lack of ________ shows ________.
- ________ influences ________ and, by extension, ________.
- ________ accurately (inaccurately) portrays ________ because ________.
- ________ is a result of ________, ________, and ________.
- Although some argue that ________, a close examination shows that ________.
- ________ and ________ prove that ________.
Citations and Annotations
For each source you choose, first write the source citation in MLA format. An example of a citation for the research project on artificial intelligence might be:
Buiten, Miriam C. “Towards Intelligent Regulation of Artificial Intelligence.” European Journal of Risk Regulation , vol. 10, no. 1, 2019, pp. 41–59, www.cambridge.org/core/journals/european-journal-of-risk-regulation/article/towards-intelligent-regulation-of-artificial-intelligence/ AF1AD1940B70DB88D2B24202EE933F1B. Accessed 23 Jan. 2021.
After preparing citations for each of your sources, write the annotations. To do this, consider the following: • The purpose of the work
- For whom the work is written
- How to summarize the content
- The author/organization producing the work
- Distinctive or interesting features of the content (particularly in comparison with other sources)
- The relevance of the work to your topic
- Strengths, weaknesses, and bias present
- Conclusions and implications
To annotate, you will typically begin with a summary of one to three sentences. For longer or more complex sources, the summary may be longer. Then discuss unique features of the content, an evaluation of the source, and a reflection on how the source is useful to your research. For each source, use a graphic organizer such as Table \(14.3\). It will help you collect the needed information, organize your thoughts, and get started summarizing and analyzing your sources.
| Source Citation | |
| Intended Audience | |
| Main Ideas, Arguments, and Themes Present | |
| Author's Point of View, Bias, and Expertise | |
| Comparison with Other Sources on the Topic | |
| Evaluation of Source's Relevance to Topic | |
| Evaluation of Strengths and Weaknesses | |
| Conclusions Drawn by Author |
Drafting:
Creating an annotated bibliography requires you to read your sources critically. As you first collect your sources, briefly review and examine the information they contain, specifically through the lens of how each can add to your research. As you read more critically, choose those that represent different perspectives on your topic as well as those that have similar viewpoints but arrive at them in different ways or from various angles.
If you have been using a graphic organizer for each source, as suggested, the information you need and your ideas about the source will be right there for you. If you haven’t organized your research in this way, now is a good time to do it, when you do not yet need to structure good paragraphs. When it is time for you to write your annotations, having your thoughts already mapped out will make your work easier. For the hypothetical artificial intelligence project, you might organize information for the article cited earlier as shown in Table \(14.4\).
| Source Citation | Buiten, Miriam C. “Towards Intelligent Regulation of Artificial Intelligence.” European Journal of Risk Regulation , vol. 10, no. 1, 2019, pp. 41–59, www.cambridge.org/ core/journals/european-journal-of-risk-regulation/article/towards-intelligent-regulation-of-artificial-intelligence/AF1AD1940B70DB88D2B24202EE933F1B. Accessed 23 Jan. 2021. |
| Intendend Audience | The journal is published by Cambridge University Press. Therefore, it is peer reviewed and intended for an academic audience. |
| Main Ideas, Arguments, and Themes Present | Discusses the unpredictability and difficulty of controlling artificial intelligence and examines what, if anything, can be done to increase transparency, specifically as it relates to biases of algorithms |
| Author's Point of View, Bias, and Expertise | Buiten is a law and economics professor at the University of Mannheim. She is the author or coauthor of nine publications, and contributed to others, about law and technology and digitalization, competition law, and European law. |
| Comparison with Other Sources on the Topic | Other sources call for total transparency in legal matters regarding artificial intelligence. This article questions whether such transparency is useful and/or feasible on the basis of current laws |
| Evaluation of Source's Relevance to Topic | Focuses on legal transparency for artificial intelligence, related to the question of whether artificial intelligence is harmful or helpful to society |
| Evaluation of Strengths and Weaknesses | Well-researched; uses dozens of peer-reviewed sources |
| Conclusions Drawn by Author | Transparency for artificial intelligence will be difficult and expensive to comply with and should be better defined in legal contexts before it is required. |
For some sources, you may be unable to find information for each category. In particular, for sources that are very short or from which you use only one or two bits of information, your annotations will not be long or complex, and your graphic organizer may seem sparse. That is to be expected. What is important at this stage is to identify how you will use your resources as they relate to your argument or thesis.
After you have outlined your sources by using the graphic organizer or a similar method, it is time to start writing the actual annotations. Remember the three tasks in writing an annotation:
- Summary the central idea or scope of the source, particularly as it relates to your research project.
- Evaluation the source for authority, author’s perspective, reliability, validity, and bias.
- Reflect on how the source affects your research and your thinking.
Your annotated bibliography should include at least some of these functions and, depending on the source, may contain all of them. Below is an example of an annotated bibliography entry for the article on artificial intelligence. The entry begins with the correctly formatted citation, followed by two paragraphs summarizing, evaluating, and reflecting. Because you already have completed the graphic organizer, much of the analysis is already done there. Simply take that thinking and shape it into useful paragraphs, like those the writer created here.
Figure \(14.13\) Sample annotated bibliography entry (CC BY 4.0; Rice University & OpenStax)
You can break down this sample annotation, analyzing each part separately as shown in Figure \(14.13\).
Buiten, Miriam C. “Towards Intelligent Regulation of Artificial Intelligence.” European Journal of Risk Regulation , vol. 10, no. 1, 2019, pp. 41–59, www.cambridge.org/core/journals/european-journal-of-risk-regulation/article/towards-intelligent-regulation-of-artificial-intelligence/ AF1AD1940B70DB88D2B24202EE933F1B. Accessed 23 Jan. 2021.
Format. Notice the MLA format, including information listing the author, title, publishing journal, and DOI or link to the article. The reader will be able to find and read the article easily.
Authority: Miriam Buiten is a law and economics professor. She has authored or coauthored eight journal articles and one book, covering topics such as law, technology, and digitalization; competition law; and European law. This article appears in a journal published by Cambridge University Press. It is peer reviewed and intended for an academic audience. The author discusses the unpredictability and difficulty of controlling artificial intelligence and examines what, if anything, can be done to increase transparency, specifically as it relates to the biases present in artificial intelligence algorithms.
Authority. The author is named and introduced with background information establishing her authority in the field. The publishing organization is also named, establishing credibility as a peer-reviewed journal.
Summary. The contents of the text are summarized briefly, allowing readers to quickly understand the topic and scope of the article and to begin to piece together its relevance to the overall research project.
This source focuses on legal transparency for artificial intelligence, which is related to the question of whether artificial intelligence is harmful or helpful to society. The author argues that transparency for artificial intelligence will be difficult and expensive to comply with and should be better defined in legal contexts before it is required. Many other sources call for total transparency in legal matters regarding artificial intelligence. However, this article questions whether such transparency is useful and/or feasible based on current laws. The article is well researched, using dozens of peer-reviewed sources.
Evaluation. This paragraph includes an evaluative statement that shows the article’s validity.
This article aids the researcher in shaping her argument that oversight of artificial intelligence is a highly complex endeavor that, although important, will be difficult to achieve. It is useful to the argument that laws need to develop at a much quicker pace to keep up with rapidly developing technology.
Reflection. This part of the paragraph reveals how the source fits into the research puzzle, noting that it indirectly supports the claim that developing oversight for artificial intelligence is not a simple task and would require the law to evolve quickly.
Style. The annotation is written in third person, referring to “the researcher” and “her,” as opposed to using first-person pronouns such as me and I
Formatting
The format of an annotated bibliography can vary depending on the discipline and purpose. Therefore, be clear at the outset which style you need to use. In academic writing, that information often will come from your instructor. Generally, annotated bibliographies will be written in MLA Documentation and Format, APA Documentation and Format, or Chicago style.
This text uses MLA style. MLA source citations follow several principles in place of specific rules. The style has been adapted in recent years to respond to the evolving nature of text in an increasingly digital world. Thus, the MLA Handbook is organized by the process of citation, rather than listing rules for every type of source. However, certain overall guidelines apply. When you cite a source, first identify core elements that are present. Remember from earlier in this chapter that these are author(s), title of source, title of container, version, number, publisher, publication date, and location.
Buiten, Miriam C. “Towards Intelligent Regulation of Artificial Intelligence.” European Journal of Risk Regulation , vol. 10, no. 1, 2019, pp. 41–59, www.cambridge.org/core/journals/european-journal-of-risk-regulation/article/towards-intelligent-regulation-of-artificial-intelligence/ AF1AD1940B70DB88D2B24202EE933F1B. Accessed 23 Jan. 2021.
Author. The entry with gold highlighting begins with the author’s last name, followed by a comma and the remainder of the name. The author’s name is followed by a period.
Title of Source. The title of the source follows the author’s name in purple highlighting. The title is either italicized or placed within quotation marks, depending on the type of source, and followed by a period. Book titles are italicized, while print or online articles are placed in quotation marks, as shown.
Title of Container. The container, highlighted in teal, is the larger work to which a source belongs. An article may belong to a website or journal, a song to an album, or a video to a video-sharing site. This container comes next in the citation. It is generally italicized and followed by a comma. However, some sources do not have containers; for example, a book (versus a chapter in a book) or an entire website (versus a single page on a website) is self-contained and thus has no container to cite.
Version. Next, the version is listed, if there is one. For example, a textbook edition or version of a text would appear here, followed by a comma. This citation example doesn’t have a version, so that information is skipped.
Number. Some sources, especially academic journals, are part of a numbered sequence as shown in green highlighting. Journals usually have both volume and issue numbers; include both in your citation, separated by commas.
Publisher. The next element in the citation is the publisher, followed by a comma. The publisher does not have to be listed for some sources, including periodicals, works published by the author or editor, websites with the same name as the publisher, or websites that host works but do not actually publish them. Because this sample source is a periodical, no publisher is listed in the citation.
Publication Date. List the most recent date of publication available for the version of the source you used. The date is followed by a comma as shown in red highlighting.
Location. Location, shown in dark and light gray highlighting, refers to where in the source you found the information, including page numbers and URLs. Be as specific as possible, as this information allows readers to return to your source to read it for themselves. When listing a URL, remove the beginning tag of http:// or https://.
Optional elements
You may want to add optional elements to help readers identify the source more easily. These can include the following:
- City of publication, generally only for books that were published before 1900 or whose publishers either have offices in more than one country or are unknown in North America (MLA 8)
- Access date, which is present in the sample annotation
- DOI (digital object identifier), which may be used in place of a URL if available
- Original publication date, if it differs from the publication date used for the citation
Alphabetizing and Indenting
Other rules also apply to MLA citations in a research project. Most of these are formatting and style rules that add to a polished final product. Remember to list sources in alphabetical order according to the author’s last name. If the source has more than one author, list it according to the last name of the first author mentioned. If the source has no author named, insert it into your alphabetical list according to the first word in the title. For example, if Miriam C. Buiten’s name were not mentioned, you would enter the item under T, the first letter of the first word in the title, Towards. In your bibliography, double-space the citation, and do not leave a space between entries.
In an annotated bibliography, indent the entire annotation in the same manner as the source citation after the first line. In most word processing programs, you can create this formatting by highlighting the citation and annotation paragraphs and then creating a hanging indent. In Microsoft Word, open the Paragraph Settings icon on the Home tab. Under the tab that reads Indents and Spacing, find the section labeled Indentation. On the right side of that section is the label Special. Click the drop-down menu, and choose Hanging. Different word processing programs may require you to create hanging indentations in another way. Consult an MLA guide often to ensure that your citations are correct.
Annotations
Annotations are the most important part of an annotated bibliography. Although there is no set format for writing annotations, remember to write them in paragraph form and to summarize, evaluate, and/or reflect. Annotation lengths will vary depending on the length of the source, how it is used in your project, and how much analysis you do within the annotation. However, a general length of one to two paragraphs consisting of 100 to 200 words is roughly the standard.
Descriptive/Informative and Analytical/Critical Annotations
There are two major types of annotated bibliographies. The first is descriptive annotations , or informative, summarizing the material and explaining why the source is useful for your topic. This type of annotated bibliography also points out special features of the text, including any data, graphics, or other characteristics. Although you discuss the author’s main arguments and conclusions, you do not analyze or evaluate them. Descriptive annotations are useful for helping readers understand an author’s main ideas but less helpful for showing how the source has influenced the research project as a whole.
The other type of annotated bibliography is analytical bibliography , or critical. This type includes all the features of a descriptive annotated bibliography, including a summary of the material. In addition, it includes your analysis of the information in each source and explanation of how the source has influenced the development of your research. With analytical or critical annotations, you examine the strengths, weaknesses, and biases present in the author’s information and explain how the author’s work and conclusions apply to your research.
In projects for this course, especially those using arguments, you will almost always create analytical or critical annotated bibliographies, though the detail in which you analyze may vary from source to source. The detail in these bibliographies will contribute to your knowledge base and provide information for others.
Further Reading
To learn more about creating an annotated bibliography, consider the following sources:
“Annotated Bibliographies.” OWL: The Purdue Online Writing Lab , Purdue U, owl.purdue.edu/owl/ general_writing/common_writing_assignments/annotated_bibliographies/index.html.
Beatty, Luke, and Cynthia A. Cochran. Writing the Annotated Bibliography: A Guide for Students and Researchers . Routledge, 2020.
“How to Prepare an Annotated Bibliography: The Annotated Bibliography.” Cornell University Library , Cornell U, 5 May 2021, guides.library.cornell.edu/annotatedbibliography.
Works Cited
Krause, Steven D. The Process of Research Writing . 2007, www.stevendkrause.com/tprw/.
“MLA Citation Guide (8th Edition): Annotated Bibliography.” LibGuides at Columbia College (BC) , LibGuides, 16 June 2021, columbiacollege-ca.libguides.com/mla/annot_bib.
“MLA Formatting and Style Guide.” OWL: The Purdue Online Writing Lab , Purdue U, 2021, owl.purdue.edu/owl/ research_and_citation/mla_style/mla_formatting_and_style_guide/mla_general_format.html.


1.2.E: Problems on Relations and Mappings (Exercises)
For the relations specified in Problem 7 of §§1-3, find \(D_{R}, D_{R}^{\prime},\) and \(R^{-1}\). Also, find \(R[A]\) and \(R^{-1}[A]\) if
\[
\begin{array}{ll}{\text { (a) } A=\left\{\frac{1}{2}\right\} ;} & {\text { (b) } A=\{1\}} \\ {\text { (c) } A=\{0\} ;} & {\text { (d) } A=\emptyset \text { ; }} \\ {\text { (e) } A=\{0,3,-15\} ;} & {\text { (f) } A=\{3,4,7,0,-1,6\}} \\ {\text { (g) } A=\{x |-20<x<5\}}\end{array}
\]
Prove that if \(A \subseteq B\), then \(R[A] \subseteq R[B] .\) Disprove the converse by a counterexample.
Prove that
(i) \(R[A \cup B]=R[A] \cup R[B]\);
(ii) \(R[A \cap B] \subseteq R[A] \cap R[B]\);
(iii) \(R[A-B] \supseteq R[A]-R[B]\).
Disprove reverse inclusions in (ii) and (iii) by examples. Do (i) and (ii) with \(A, B\) replaced by an arbitrary set family \(\left\{A_{i} | i \in I\right\}\).
Under which conditions are the following statements true?
\[
\begin{array}{cc} \text{(i) } R[x] = \emptyset ; & \text{(ii) } R^{-1}[x] = \emptyset ; \\ \text{(iii) } R[A] = \emptyset ; & \text{(iv) } R^{-1}[A] = \emptyset ; \end{array}
\]
Let \(f : N \rightarrow N(N=\{\text { naturals }\}) .\) For each of the following functions, specify \(f[N],\) i.e., \(D_{f}^{\prime},\) and determine whether \(f\) is one to one and onto \(N,\) given that for all \(x \in N\),
\[
\begin{array}{ll}{\text { (i) } f(x)=x^{3} ;} & {\text { (ii) } f(x)=1 ; \quad \text { (iii) } f(x)=|x|+3} \\ {\text { (iv) } f(x)=x^{2} ;} & {(\mathrm{v}) f(x)=4 x+5}\end{array}
\]
Do all this also if \(N\) denotes
(a) the set of all integers;
(b) the set of all reals.
Prove that for any mapping \(f\) and any sets \(A, B, A_{i}(i \in I)\),
(a) \(f^{-1}[A \cup B]=f^{-1}[A] \cup f^{-1}[B]\);
(b) \(f^{-1}[A \cap B]=f^{-1}[A] \cap f^{-1}[B]\);
(c) \(f^{-1}[A-B]=f^{-1}[A]-f^{-1}[B]\);
(d) \(f^{-1}\left[\bigcup_{i} A_{i}\right]=\bigcup_{i} f^{-1}\left[A_{i}\right]\);
(e) \(f^{-1}\left[\bigcap_{i} A_{i}\right]=\bigcap_{i} f^{-1}\left[A_{i}\right]\).
Compare with Problem 3.
[Hint: First verify that \(x \in f^{-1}[A]\) iff \(x \in D_{f}\) and \(f(x) \in A . ]\)
Let \(f\) be a map. Prove that
(a) \(f\left[f^{-1}[A]\right] \subseteq A\);
(b) \(f\left[f^{-1}[A]\right]=A\) if \(A \subseteq D_{f}^{\prime}\);
(c) if \(A \subseteq D_{f}\) and \(f\) is one to one, \(A=f^{-1}[f[A]]\)/
Is \(f[A] \cap B \subseteq f\left[A \cap f^{-1}[B]\right] ?\)
Is \(R\) an equivalence relation on the set \(J\) of all integers, and, if so, what are the \(R\) -classes, if
(a) \(R=\{(x, y) | x-y \text { is divisible by a fixed } n\}\);
(b) \(R=\{(x, y) | x-y \text { is odd }\}\);
(c) \(R=\{(x, y) | x-y \text { is a prime }\}\).
\((x, y, n \text { denote integers.) }\)
Is any relation in Problem 7 of §§1-3 reflexive? Symmetric? Transitive?
10. Show by examples that \(R\) may be
(a) reflexive and symmetric, without being transitive;
(b) reflexive and transitive without being symmetric.
Does symmetry plus transitivity imply reflexivity? Give a proof or counterexample.


1.4: Arguments
When we give reasons for holding a view or reaching a conclusion, the process is intrinsically articulate. Giving reasons requires language, concepts and skill with words, not just a simple ability to talk. For two (or more) people to share their reasoning requires a kind of reflective articulateness, an ability to express reasoning in a way that is intelligible to a particular audience. . . . What we will call “argument” here is meant to capture everything we make explicit when we give reasons for a view or proposition. . . . Much of the difficulty in reasoning together comes from simply not understanding each other’s arguments—from not understanding the significance of the words to express reasons and the views they support.
—LARRY WRIGHT1
The Importance of Arguments
I am in an unusual career, where arguments dominate my professional life. I am paid to teach my students what philosophers have had to say on a great array of topics. Almost always, these philosophers, whether they are the historical “biggies” or contemporary thinkers, support their theories with arguments. Obviously, if I am going to do my job, I need to help my students sort out the good arguments from the bad. As a professor, I also participate in what is called “shared governance,” and as a consequence, I am required to vote and help decide policies, curricula, and other matters of university importance. Administrators and my colleagues consistently defend positions of great significance to the institution with impassioned arguments. Since I take this part of my profession very seriously, I find it essential to decide which of these arguments I find most persuasive.
Most of us are not simply consumers of arguments, we are producers as well. In my own professional work, I present and defend theories about privacy, the Constitution, the death penalty, the existence of God, and a number of other philosophically, politically, and legally controversial topics. As an active participant in shared governance, I take sides, advocate positions, and occasionally lead the charge for particular causes. All this demands that I defend my views. In these cases, it is my name on the argument, and there is a heightened sense not just of ownership but of personal and professional responsibility. I need my arguments to be as strong as they can be not just convincing but plausible and hopefully pointing in the right direction.
Your life may not be as argument-intensive as mine, but if you stop to reflect a bit, I think you’ll find arguments all around you. Advertisers make arguments for why you should buy their products and politicians for why they deserve your vote. You may need the advice from accountants and economists, hopefully backed up with arguments, to plan for a major business investment or your retirement. And you are a producer of arguments as well. That memo you wrote to your boss for a change in the way things are done or the case you just made to your partner about the need to buy a new car is an argument.
This book is about arguments and a suggested technique for distinguishing good arguments from bad ones. These general hints are intended to be of use both when you find yourself in the position of the consumer of an argument and when you must make some decision about the quality of its evidence as well as when you are the producer of the argument and desire to present the strongest evidence you can. I don’t claim to have a magic bullet that will automatically show the truth about complicated issues. But I think you will be pleasantly surprised at how often this technique proves useful for thinking through these issues, finding out where you stand, and even beginning the process of formulating your own arguments about them.
What Is an Argument?
One potentially misleading aspect in some of my previous examples is that when it comes to controversial issues such as abortion or the death penalty, tempers can be high. And trust me, debates about curriculum or university policy can be just as emotionally explosive. There is a perfectly fine use of the word argument that basically means a verbal fight. Joe and Sally got into a terrible argument about his failure to do his share of the house cleaning. That is not what we will mean by the term, however. Sure, there will be times when arguments are very important, and disagreements about their strength or weakness will touch our emotions as much as our reason. There will be many other times, however, when arguments are simply there for our consideration, and we can assess them free of any passion or personal commitment. Indeed, as much as is possible, I would recommend adopting the more dispassionate approach, even when you feel strongly about what is at issue.
It is useful to see an argument as a complex arrangement of three quite different things. There will be what logicians call a conclusion —some theory, hypothesis, or position that the argument seeks to defend. There will be premises —facts, data, or evidence that the argument uses to support the conclusion. And there will be a relationship between the premises and conclusion whereby the conclusion follows from the premises. We can schematically represent an argument as follows:
e 1 . Premise
e 2 . Premise
e 3 . Premise
. . .
e n . Premise
t 0 . Conclusion
Let’s begin at the bottom. Every argument will have a conclusion —that’s part of the definition of an argument. When we put an argument in what we will be calling its schematic form , it will always come at the end, under the big, heavy line. But in the real world of arguments, we should treat the term conclusion as technical jargon. Conclusions don’t always come at the conclusion of a person’s argument. Sometimes they come at the beginning.
Dick’s cheating on Jane. He told her he had to work late, but Sally saw his car at Joe’s Bar. Not only that, he leers at other women, and the last three times she called him, he didn’t answer.
Sometimes they come in the middle.
Charlie’s take-home exam was word-for-word identical to Sarah’s. Clearly, Charlie copied it from Sarah. The guy’s a loser, never comes to class, and doesn’t know how to write very well.
And, of course, some of the time, they are at the end.
The light from virtually every galaxy is “red-shifted.” This shows that every galaxy is moving away from every other galaxy. Therefore, the physical universe is expanding.
I have used the lowercase letter t in my schematic representation to stand for theory . The subscript “0” is used to do two jobs. Although there is only one theory defended in the argument’s conclusion (though that single conclusion can be complicated and composed of many parts—“therefore, Jake did it or helped plan it, or someone read his diary”), we will need to keep track of other possible theories besides the one defended in the argument. So “0” can be understood as the number zero and starting a sequence of numbered theories. But the “0” can also be read as the letter o and standing for original —the original theory or conclusion in the argument.
To standardize things, we will use the lowercase letter e to stand for an individual bit of evidence . There are no set numbers of premises, or pieces of the evidence, in an argument. Sometimes there will be just a single datum, and sometimes, there will be quite a bit of supporting data. The previous examples illustrate not just that conclusions can come in many places in the statement of an argument but that the same holds true the statements of the evidence.
Let’s recast our schematized argument in terms of evidence for a theory:
e 1 . Evidence (datum)
e 2 . Evidence (another datum)
e 3 . Evidence (another datum)
. . .
e n . Evidence (another datum)
t 0 . Theory
Logical Connection
We’ve said a bit about the top and the bottom in our schematic representation of an argument. What about that conspicuous, big, fat line? In good arguments, the conclusion follows from the premises; the evidence supports the theory. What exactly is this relationship of support or following from? That turns out to be a very controversial issue in both philosophy and mathematical logic.
In some cases, the relationship is semantic . If we just understood enough about the meanings of all the words in the premises, we would see that the conclusion has to be true. Often the examples are pretty trivial.
e 1 . The number is even.
e 2 . The number is greater than seventeen.
t 0 . The number is not prime.
 The figure is a plane triangle.
t 0 . The interior angles of the figure equal exactly 180°.
Arguments of the previous type have a technical name. They are called deductive arguments . In a successful deductive argument, the relationship between the premises and conclusion (it’s artificial here to call them evidence and theory) is a very special one. Logicians call it validity . Valid arguments are ones where if the premises are true, the conclusion has to be true . Many colleges and universities have whole courses on deductive (or symbolic) logic. Very sophisticated techniques are developed for determining validity. We will not spend time reviewing this material because as interesting (and just plain fun) as it is, one almost never finds deductive arguments being put forward outside of academic philosophy and mathematics.
A second way of connecting premises to conclusions relies on the technical fields of mathematics and statistics. We cannot as conveniently ignore these arguments, since they play huge roles in contemporary science. Our approach to them, however, will be a little indirect. Rather than going through the basics of probability theory and then developing statistical tests for making sense of numerical data, we will treat these arguments as special cases of inductive arguments . This latter jargon simply means that the argument claims that the conclusion follows from the premises but not deductively—that is, it is possible for the premises to be true, yet the conclusion turns out to be false. Now, of course, it should be relatively rare that in good inductive arguments, the premises would be true and the conclusion false; otherwise these arguments will not be very useful. It is a matter of great controversy in logic, philosophy, and even the sciences as to how we describe this relationship between evidence and theories. The rest of this book is devoted to showing you one way of characterizing this relationship.
Inference to the Best Explanation
Consider the three short examples mentioned previously. We had purported evidence that Dick was unfaithful—the leering, the excuse about being sick, the car outside the bar, and the missed phone calls. We had purported evidence about the copied take-home exam—the word-for-word identical submissions, Charlie’s chronic absences, and his failures as a writer. Finally, we had the evidence about the expanding universe—the red-shifted light from distant galaxies. In each of these cases, the suggested theory explains significant parts of our evidence. Charlie being a cheater doesn’t explain his bad writing, but it sure helps us understand how the two exams ended up being the same. Dick’s cheating (in a very different way) would explain why he was at the bar when he said he was sick. And an expanding universe explains the Doppler shift we observe in the light from galaxies.
This suggests a generalization. Suppose we treat the theory being defended in an inductive argument as an explanation of the data (at least some of the data) contained in the evidence. We get the following very symmetrical picture of an argument:
Inference to the best explanation assumes this general picture of inductive arguments. The relationship of support or following from becomes one of good explanation. Evidence for a theory is strong, or good or sound, if and only if, the theory best explains the relevant data that is being offered as evidence. This definition of good evidence gives us a very useful device for testing the quality of purported evidence.
In the next chapter, I intend to lay out a kind of practical test for answering questions about the strength of arguments, about the quality of evidence. Does Connie have good evidence that her boyfriend was smooching Mary Jane during his absence at the record hop? Is there a strong argument that Charlie copied the take-home exam? Do astronomers really know that the physical universe is expanding? We can only begin to answer these questions when we are in absolute agreement about what the argument is in the first place.
A Couple of Arguments from Sherlock Holmes
Let’s look at a couple of examples of evidence that lead to some conclusions for Sherlock Holmes.
Here are the missing links of the very simple chain: 1. You had chalk between your left finger and thumb when you returned from the club last night. 2. You put chalk there when you play billiards, to steady the cue. 3. You never play billiards except with Thurston. 4. You told me, four weeks ago, that Thurston had an option on some South African property which would expire in a month, and which he desired you to share with him. 5. Your check book is locked in my drawer, and you have not asked for the key. 6. You do not propose to invest your money in this manner. 2
The beginning of The Adventure of the Dancing Men begins with a little case study in Sherlock Holmes’s “deductive” method. Holmes’s method, of course, is not deductive in the formal logician’s sense but inductive, or better, abductive. It is an inference to the best explanation. Holmes possesses a fair amount of data.
e 1 . Watson had chalk between his left finger and thumb.
e 2 . He uses the chalk when he plays billiards.
e 3 . He only plays billiards with Thurston.
e 4 . He told Holmes four weeks ago that Thurston had an option on some South African property, which would expire in a month.
e 5 . Watson’s checkbook is locked in Holmes’s drawer.
e 6 . Watson has not asked for the key.
Holmes explains all this with the hypothesis that Watson has decided against the investment. Holmes goes on to explicate his reasoning with the metaphor of a chain.
You see, my dear Watson . . . it is not really difficult to construct a series of inferences, each dependent on its predecessor and each simple in itself. If, after doing so, one simply knocks out all the central inferences and presents one’s audience with the starting-point and the conclusion, one may produce a startling, though possibly a meretricious, effect. Now, it was not really difficult, by an inspection of the groove between your left forefinger and thumb, to feel sure that you did not propose to invest your small capital in the gold fields. 3
Although I think it is clear that Holmes exaggerates when he claims that the inferences follow in sequential lockstep, the insight that explanatory reasoning often proceeds in steps is important. Here is how I would schematize Holmes’s inference.
e 1 . Watson had chalk between his left finger and thumb.
e 2 . He uses the chalk when he plays billiards.
e 3 . He only plays billiards with Thurston.
t′ 0 . Watson played billiards with Thurston last night.
e 4 . He told Holmes four weeks ago that Thurston had an option on some South African property, which would expire in a month.
e 5 . Watson’s checkbook is locked in Holmes’s drawer.
e 6 . Watson has not asked for the key.
t″ 0 . Watson has decided against the investment.
Each of these inferences is to an “alleged” best explanation. t′ 0 explains the chalk on his hand and is consistent with Holmes’s background knowledge of Watson’s preferences in playing partners. t″ 0 explains the lack of a request for the key and is consistent with Holmes’s knowledge of what Watson told him four weeks ago and the location of the checkbook.
But The Adventure of the Dancing Men is not really about chalk and South African investments; it’s ultimately about murder and a couple of other mysteries that lead to the murder and its solution.
You’ll enjoy the story, and it would be great practice if you paused here for a while and tried your hand at schematizing Holmes’s argument for what happened at Ridling Thorpe Manor.
Consider, first, the mystery that gives the story its name. Holmes shows Watson a piece of paper with the above pencil markings.4
This is our first piece of evidence.
e 1 . The dancing men document
Watson immediately offers an account: “Why, Holmes, it is a child’s drawing.” Holmes thinks otherwise. The dancing men drawings “have a meaning,” but they might be “arbitrary” (think “one if by land and two if by sea”) or they might be “systematic” (a cipher). Holmes bets on the latter. His client, Mr. Cubitt, has provided more data:
e 2 . Mr. Cubitt’s wife, Elsie, received a letter from America that very much upset her.
e 3 . Other dancing men inscriptions, on both paper and chalk applied to buildings, have turned up at Ridling Thorpe Manor.
e 4 . Holmes has knowledge of the relative frequency of letters in the English language.
t′ 0 . The dancing men figures stand for letters. And the messages are in English.
Holmes breaks the code. He is now able to translate the different messages that have turned up.
This leads to the second mystery: Why is Elsie so reticent to tell her husband about her past? Why is she so upset and frightened by the dancing men messages? Why did she stop her husband from confronting the stranger they caught in the act of scrawling one of the messages? Holmes is in a position to answer many of these questions simply by decoding the messages he has. In addition, Holmes explains Elsie’s reticence and fear in terms of some connection to criminal activities in her past, and he knows that the correspondent is named Abe Slaney. A cable to a colleague in the States confirms that Slaney is “the most dangerous crook in Chicago.” Slaney is imploring Elsie to return to him and finally threatening her. Holmes and Watson have plenty of evidence that necessitates hurrying to the Norfolk countryside.
e 5 . The contents of the messages
e 6 . Slaney is the most dangerous crook in Chicago.
e 7 . Slaney’s threat to Elsie
t″ 0 . Elsie, and perhaps Mr. Cubitt’s, lives are in danger.
Sadly, they are too late. Immediately upon their arrival at the train station, they learn of the tragedy.
“It’s a terrible business,” said the stationmaster. “They are shot, both Mr. Hilton Cubitt and his wife. She shot him and then herself—so the servants say. He’s dead and her life is despaired of.” 5
The stationmaster has, of course, not only introduced crucial new evidence but also offered an explanation. Holmes proceeds to the murder scene, interviews two servants, and examines the room where the bodies have been found.
e 8 . Cubitt is shot dead, and Elsie is gravely wounded with a gunshot to the head.
e 9 . A revolver is found, still containing four bullets.
e 10 . The servants report several things:
- • Both were awakened by the sound of a loud gunshot.
- • A short while later, they heard a second shot.
- • The victims were downstairs in the study.
- • A strong smell of gun smoke in the study.
- • The study window was shut and fastened.
- • The house was locked from the inside, and no one could have left.
- • They were both conscious of the smell of powder from the time they awakened upstairs.
There are still several pages to go in The Adventures of the Dancing Men , but Holmes has now basically solved the case. His reasoning is classic inference to the best explanation (IBE). How and why did Mr. Cubitt and Elsie come to be shot? Why were the servants conscious of the smell of powder while upstairs? And what about the mysterious Slaney?
Holmes is particularly fascinated by the powder aroma upstairs—“I commend that fact very carefully to your attention.”6 Holmes’s provisional hypothesis is that the window must have been open and the draft carried the smell. Careful examination of the study produces one last crucial bit of evidence.
e 11 . A third bullet is discovered in the window sash.
A third bullet! Someone else is involved! They examine outside the window and discover important new data.
e 12 . Trampled flowers, large masculine footprints, and a spent cartridge
But why only the sound of two gunshots? The first one was extremely loud. Might it “have been two shots fired almost at the same instant”?7 Holmes concludes “it was undoubtedly so.”8
Holmes proceeds to entrap Slaney, and he confirms that he was one of the authors of the dancing men messages and that he and Mr. Cubitt exchanged gunshots through the window. The story never tells us precisely what happened to Elsie, but we know, just as Holmes knew.
t 0 . Slaney sought to win Elsie back and was refused. He came to the manor and exchanged almost simultaneous gunshots with Cubitt. Elsie closed the window and either heartbroken at the death of her husband, guilt-ridden and feeling partial responsibility, or misguidedly seeking to preserve her husband’s good name, shot herself in the head.
Schematizing the Argument
Let me state something explicitly. The single hardest part of argument analysis or the IBE recipe may often be simply identifying what the argument is in the first place . There are a number of reasons for this. First and foremost, people aren’t always as clear as the might be when they state their arguments. But there are other complicating factors as well. My guess is that Conan Doyle would have said he wasn’t presenting an argument at all but simply telling a story. Still, I think it’s clear that the story is about Holmes’s following the evidence and coming to a conclusion about what happened. Add to all that some arguments touch on deeply divisive moral and political issues and few of us read them and set our personal politics to the side. These unavoidable biases that we all carry with us will often tempt us to simply misread what the argument is. Finally, as we get a hint of in this short story but becomes daunting when an argument is developed over the course of a whole book, the sheer number of words, thoughts, and sentences makes it extremely challenging to keep the structure of the argument clearly in mind.
Granted all this, the first step in the IBE procedure that we will develop in the next chapter is not only the most difficult; it is the most important. If we misrepresent what the argument is, then all our work in analyzing it will be a waste of time. Who cares if you show “the argument” to be a spectacular success or a dismal failure if it wasn’t the real argument in the first place?
Useful schematization requires three virtues, all of which defy simple characterization. First and foremost, as we have just emphasized, you should strive for copy fidelity . Your task is to characterize “the other person’s argument,” a representation of his or her evidence. You may think of better ways to make the argument, or you may even think that the evidence points in a different direction. That’s all fine and good and will be useful in later steps. Right now, however, your job is to faithfully represent the argument as it was stated. You want to also strive for brevity . We just saw how an argument might take up several pages of a short story but just imagine when we look in a later chapter at Darwin’s “abstract” of his theory in On the Origin of Species 9 and try to keep straight all the evidence presented in more than four hundred pages. In order for your schematization to be useful to you, you will need to keep your representation of the evidence down to, say, no more than a page. Finally, and most difficult of all, you should strive for charity in your schematized arguments. You want to present the argument in the strongest form you can. This is not because you are being nice or discounting the previous virtue of copy fidelity. It is because you want to avoid at all costs weakening the evidence in the way you choose to schematize it. This is particularly important when you are dealing with arguments with which you disagree. If you come to the judgment that the evidence is weak, you need to make darn sure that you’ve given the evidence its best shot.
Start at the Bottom (Find the Conclusion)
We’ve already discussed the fact that conclusions may come anywhere in a statement of an argument. Still, in the schematic form I am urging on you, they always come at the bottom; they are always are identified as t 0 (“0” to start a sequence of explanations and o as a reminder that this explanation is the original one), and they are always explanations of the data, not simply statements of the data. I strongly suggest that you begin your schematizing of the argument by trying to identify its conclusion.
Oftentimes, you will find hints in the statement that will guide you to the argument’s conclusion. There are many words and phrases that are commonly used to alert readers or listeners that an inference is being drawn. Some of the classics you will find in any introductory logic book are as follows: “therefore,” “hence,” “so,” “it follows that,” and many others. But at other times, you are simply expected to pick out what the theory is that is supposed to be supported by the evidence. The best advice in these latter cases is simply to ask yourself something very general and vague such as “What’s the point of all this?” or, as suggested previously, “What the heck is going on?” Once you have a candidate, now see whether it explains some of the data in the argument. If it doesn’t seem to, you might want to look for another candidate as the argument’s conclusion.
Two other general comments are appropriate here. First, don’t get discouraged. This is hard stuff. It will get easier and more natural as you get more experience using the recipe. And second, there will be times when you fail to discover a conclusion to begin your schematization because the passage of prose in front of you is not an argument in the first place. We obviously use language do lots of things—make simple assertions, push people’s buttons, or simply vent—stating an argument is only one use of language.
Find the Relevant Evidence
In The Adventure of the Dancing Men , we learned a lot of stuff.  And that Inspector Martin was a “dapper little man, with a quick, alert manner and a waxed moustache.” This is just good literary technique. There’s also the important information at the end of the story when Slaney is arrested and confesses. But these data are only tangential to solving the murder. The data in e 1 through e 12 , however, are crucial to understanding what happened, and all of it should be included in a complete schematization of the argument.
Another Brokenhearted Teenager
Here’s a sad story.
Lyrics to “It’s My Party” can be found here: https://genius.com/Lesley-gore-its-my-party-lyrics , and Lesley Gore performing her song can be found here: https://youtu.be/acRMALrg1t4 .
Lesley is in tears, and she assures us we would be too if something similar happened to us. So what the heck happened? She doesn’t spell it out, but she knows and so do we. Take some time, now, and schematize Lesley’s argument. What is her evidence that Johnny is no longer hers and has taken up with Judy?
EXERCISES
- 1. What is an argument? What are the three components of any argument?
- 2. How can you have a big argument, with lots of name-calling, without either party presenting an argument for why they feel aggrieved?
- 3. What do you think of the quality of Holmes’s evidence for his theory that Watson had decided not to go ahead with the investment? Why?
QUIZ FOUR
The following article comes from the New York Times . Your task is to schematize the argument for the conclusion that William Henry Harrison died from enteric fever. The article “What Really Killed William Henry Harrison?” by Jan McHugh and Philip A. Mackowiak, March 31, 2014, is available here: https://www.nytimes.com/2014/04/01/science/what-really-killed-william-henry-harrison.html
Notes
1. Larry Wright, Critical Thinking (New York: Oxford University Press, 2001), ix.
2. Sir Arthur Conan Doyle, “The Adventure of the Dancing Men,” in The Return of Sherlock Holmes (New York: W. R. Caldwell, 1905), http://www.gutenberg.org/files/108/108-h/108-h.htm .
3. Doyle.
4. Doyle.
5. Doyle.
6. Doyle.
7. Doyle.
8. Doyle.
9. Charles Darwin, On the Origin of Species: A Facsimile of the First Edition (Cambridge, MA: Harvard University Press, 2001).