Chapter Six: Analyzing Arguments
As I have characterized it in this handbook, critical thinking is primarily about the evaluation of arguments. Evaluation of arguments requires that we have arguments to evaluate, and it has been the purpose of the preceding chapters to describe how we locate and represent such things. Once we have them, though, we can get down to business. Evaluation is, after all, the central activity of critical thinking, and it is in this chapter that we will focus on this activity. The goal is to provide the tools to use in detecting good arguments. This contrasts with the goal in the next chapter, which is to provide tools to use in detecting bad arguments. We begin this chapter by discussing the nature of argument evaluation, and the relationship between this stage of critical thinking and the stages we have discussed in earlier chapters.
As we observed in Chapter Three, arguments understood as products have both form and content. In light of this, it should come as no surprise that they can be evaluated in terms of their form and their content. Both types of evaluation will be discussed in this chapter. Evaluation of form, or formal analysis, falls within the province of logic. Logic, as a discipline, concerns the identification and development of formal patterns of reasoning (and related concepts) that meet certain standards, such as validity or non-deductive strength. After discussing the nature of argument analysis in general, we will turn to a development of the logical tools necessary to engage explicitly in the evaluation of argument form. Evaluation of content, or subject matter analysis, will vary from field to field. This sort of evaluation requires familiarity with specific disciplines, their facts, theories, and standards. As such, there is little we can say here about the specific character of this form of analysis, but there are some general considerations that I address in the penultimate section. The final section is devoted to the third stage of evaluation, context analysis. After assessing an argument for merit on its own, we must examine if it serves the purposes of the arguer. Does it advance her persuasive goals, and if so, how does it advance them?
II.1 A Worked Example
Consider the following argument:
Is this a good argument? Before answering this, we must determine exactly what the
argument is. The conclusion is that we must vote for someone other than Al Gore in the
upcoming Presidential election. The reasons offered turn on the effectiveness of
Democratic Presidents in the recent past. In particular, the argument contends that Gore
will be ineffective because he is a Democrat and the last two Democratic Presidents,
Carter and Clinton, were ineffective. The argument can be represented in standard form as
follows, bringing in a few implicit premises to fill it out:
Let's consider this first from the perspective of its form. As reconstructed, the argument is a combination of a couple of lines of reasoning. First, steps (1) through (3) supply what we will call inductive support for step (4), and then steps (4) through (6) supply what we will call deductive support for the final conclusion. To assess these, we assume that the supporting reasons are true and then ask whether they provide any support for their conclusions. The first line of reasoning is, from a formal perspective, rather weak for a couple of reasons. Any generalization that is based on two data points should be regarded with some suspicion--two instances rarely prove any rule. Second, while Gore is in the same party as Carter and Clinton, the nature of the party has changed considerably since Carter was President. Given this, it would appear as though only one data point, Clinton, is relevant to an assessment of a possible Gore Presidency, cutting the strength of an already weak inference in half. Turning to the second line of reasoning, we find an argument in good formal shape--if (4), (5), and (6) are true, then (7) is also true.
Formal analysis involves assessing whether the conclusions are supported by their reasons, assuming that the reasons are true, and this suggests that the first line of reasoning is weak, a fact that threatens to vitiate the argument as a whole. Content analysis involves asking whether the reasons are in fact true. It is here that the argument would really appear to founder on the rocks. Whatever you may think of Carter and Clinton, it isn't at all clear that if they are or were ineffective, this was due to the factors cited in the argument. While (3) is true, the formal concerns raised above should lead us to suspect the truth of (4)--we are supposed to believe this on the strength of (1) through (3), but even if those claims were true, which is in doubt, they would not supply much support for (4). In addition, it is not obvious that you must have an effective leader to have an effective government, contrary to what (5) asserts. And if (4) and (5) are not true, then the argument fails to supply a compelling reason to believe (7). On the whole, the claims that constitute this argument do not exactly inspire one to embrace the final conclusion.
Lastly, we evaluate the argument relative to the persuasive context in which it appears. This is difficult here, given that I've supplied no contextual information, but there are a couple of things that we can say. First, the argument would appear to be in the service of a practical agenda, viz., to convince the audience to vote for someone other than Gore. Previous analysis reveals that the argument is weak, a fact that makes it ill suited to advance the arguer's agenda.. Note, though, that even if it weren't so weak, it still might not serve the arguer well. In particular, this argument focuses on one generalization, and as such it ignores many factors that bear on an advance assessment of a Presidency. Thus, the consideration adduced by this argument may be outweighed by competing considerations, thereby undermining the arguer's goal.
II.2 Formal Analysis
When you analyze an argument, you determine whether the conclusion is supported by the premises. One part of this effort is formal analysis, or evaluation of the form of an argument. When engaged in this type of analysis, you hold the content of the argument constant and experiment with the form to see if it is well-suited to convey you from premises to the conclusion. As we saw with the example, this is done by assuming for the sake of argument that the premises are true and then assessing whether their truth gives one good reason to believe that the conclusion is true as well. Thought experimentation of this sort reveals if the argument would support the flow of truth from premises to conclusion, on the assumption that the premises are in fact true. (Compare: you would test whether a canal worked in the first instance by assessing the dry channel and asking whether it could convey water from one end to the other, if the water were released into it.)
Formal analysis is largely a structural matter. There should be connections of content between reasons in an argument, but these reasons must be structurally related in an appropriate way for them to provide support for the conclusion. The trick in teaching formal analysis is to train students to distinguish between appropriate and inappropriate structural relations. One way to do this would be to present them with a system of logic, deriving truth preserving structural relations from axioms and primitive inference rules. This is, however, not the tack I take in this handbook. Instead, I will introduce standards of appropriateness and then describe a number of common patterns of reasoning that meet these standards. Such patterns of reasoning come in all shapes and sizes. Here are three paradigmatic examples, organized according to types we will discuss in the next few sections:
Patterns such as these form the subject matter of logic, which studies the structure of
such patterns with a view to distinguishing those that are argumentatively strong from
those that are not. For our purposes, the leading idea is that there exists identifiable
principles which can be used to distinguish good arguments from bad ones. When one
"does logic," one investigates these principles by developing and exploring
systems of inference that they constrain. Critical thinking is not logic. While there is
overlap, they have different subject matters and have different goals. Critical thinking
is a discipline with a practical goal, viz., to produce people who are in control of their
own beliefs. Logic, by contrast, is a branch of mathematics and is concerned with deriving
results about axiomatic structures. Nevertheless, it is important to spend time with
logic, for several reasons:
You cannot be an effective critical thinker unless you are attuned to the form of arguments, and the best way to become attuned is to spend some time studying logic.
II.3 Subject Matter Analysis
The second stage of analysis is subject matter analysis. There are a number of differences between these stages of analysis. First, formal analysis requires us to consider whether we have good reason to believe a conclusion of an argument on the assumption that the premises are true. Subject matter analysis is devoted to determining whether those premises are in fact true. Second, formal analysis reveals patterns of reasoning that are found in many different domains, from religion to politics to medicine to sports. Subject matter analysis, by contrast, requires attending to facts and claims that are typically part of a single domain. To analyze the content of the argument at the beginning of this section, we must know something about the recent political history of the United States, for instance. Related to this is the third difference: logic is useful as a tool for formal analysis, but it is not for subject matter analysis unless logic is itself the subject matter of the argument in question.
It may seem that logic has more of a role to play in subject matter analysis than this. In particular, there will be occasions when you take a premise to be true because of an associated, supporting argument. While it is true that in such a case, you will likely avail yourself of logic to help in assessing the supporting argument, in the end you will need to ask of its premises whether they are true. Perhaps you will decide that they are, and perhaps this decision will be based on further arguments. But the premises of these must be evaluated for truth or falsity, and so on and so on. This process can take a good deal of time and energy, but eventually it must stop if you are going to pass judgment on the argument. Logic will be of assistance as you search for the truth values of the premises, but at some point you will need to ground your assessment in facts or intuitions whose truth is non-argumentatively established.
Subject matter analysis requires knowledge of the subject matter of the argument, and this is generally is non-logical. Thus, when teaching critical thinking, you must make sure to emphasize the importance of factual knowledge to the process. If you can establish that premises are true, based on independent knowledge of the content of those claims, and if you further know that the argument has a form that conveys truth from premises to conclusion, then you have good reason to believe in the conclusion. In either of these fail, i.e., if you conclude that a premise is false or that the form of the argument is weak, then while the conclusion may nevertheless be true, the argument does not give you reason to believe it.
II.4 Context Analysis
Only in the cold, antiseptic climate of the critical thinking classroom will you find arguments stripped from all context, served up without any introduction or background. Outside the doors of this classroom, arguments are almost always asked to serve some larger purpose, e.g., to convince others, to protect oneself, to establish credentials, etc. Evaluation of the relationship between an argument and these larger purposes forms the focus of our third stage of analysis, viz., context analysis. An argument might be strong when considered independently of context, but it may not fit well with the broader purposes of the arguer. For instance, it may be a strong argument for a conclusion that is unrelated to their broader goal or perhaps even, unbeknownst to the arguer, a conclusion that undermines their broader goal. On the other hand, an argument might turn out to be weak but nevertheless effective if the way in which it is delivered enables it to serve the goals of the arguer; in such a case, it might prove as effective as a stronger argument. In both cases, proper evaluation of the argument as a move in a persuasive discourse requires understanding it in its context.
This sort of analysis requires that one identify the broader practical goals of the arguer. What work is the arguer asking the particular argument to do? How does it relate to other things the arguer has said? Is it consistent with them? Inconsistent with them? If consistent, does it strengthen them or is it unrelated? An effective critical response to the argument requires knowing the big picture, and this typically requires that one engage in context analysis after having assessed the quality of the argument considered on its own.
Formal analysis reveals whether the form of a given argument is capable of supporting a justified conclusion. Here the analogy, mentioned above, of the canal is instructive. To convey water from one place to another, a canal must have the appropriate structure, e.g., it must lead from one point to the other, it must have a certain depth at all times, etc.; it might fail even if it does have the correct structure, but it will certainly fail if it lacks this structure. Consider also a building: if you wish to construct a building for a certain purpose, it must be structurally sound; it may fail to meet the specs for other reasons even if it has structural integrity, but it will certainly fail to meet them if it lacks it. Analogously, an argument must be of the appropriate form to support a justified conclusion if it is going to compel people to believe that conclusion. In particular, it must be of a form that conveys truth from premises offered to the conclusion. As with the other structures, a structurally solid argument may fail to yield a justified true conclusion--for instance, it may be the case that one or more of the premises are false--but it will certainly fail to yield a justified true conclusion if it lacks structural integrity.
In this section, I introduce the elements of formal analysis, elements that are rooted in logic. I separate these elements into two broad categories, viz., deductive analysis and non-deductive analysis, which I introduce in the next section. I then devote an extensive section to each category, each of which begins with a discussion of the standards that are applied when engaging in the relevant type analysis followed by a description of the different argument structures that measure up to these standards. I close each of these two sections by remarking on techniques for applying these standards to argument structures that are not discussed.
III.1 Deductive Validity and Non-deductive Strength
In what follows, it will be helpful to identify two broad types of arguments: those that are deductively valid and those that are non-deductively strong. Deductively valid arguments are those in which the conclusions must be true given that their premises are true. That is, if the premises of these arguments are true, then it is impossible for the conclusion to be false. Arguments are non-deductively strong when it is improbable that their conclusions are false given that their premises are true, where the degree of strength varies directly with the degree of improbability. Note that, as defined, arguments that are deductively valid are non-deductively strong; in fact, they are the most non-deductively strong arguments. For the sake of clarity and precision, we will require that non-deductively strong arguments are not deductively valid. Thus, we have the following definitions:
There are a number of things to be said about these properties. First, they are both conditional properties. That is, an argument can be deductively valid or non-deductively strong even if its premises are one and all false. What each definition supplies is a conditional property, i.e., an "if then" property. Consider deductive validity: IF the premises are all true, THEN the conclusion must be true; however, in the case that the premises are not all true, then we dont know anything about the truth of the conclusion. And then there is non-deductive strength: IF the premises are all true, THEN it is improbable that the conclusion is false; however, in the case that the premises are not all true, we do not know anything about the truth or falsity of the conclusion. Thus, you could have an argument that is deductively valid but that is nevertheless bad because it has a false premise. This fact brings out the fact that these are structural properties of arguments: the fact that an argument has one does not tell us if its premises represent the world correctly (i.e., if the premises get the facts correct), but it does tell us that if they represent the world correctly, then we are justified in believing its conclusion because the form of the argument conveys the argumentative virtue from premises to conclusion.
Second, they are related to one another on a scale of justification: a deductively valid argument can supply all the support a conclusion can get, whereas an non-deductively strong argument can supply only some large percentage of that support. The trick will be, of course, to determine what counts as a "large percentage" in the case of non-deductively strong arguments. As with most decisions of this nature, I believe that there is no pat answer that will work in all cases. It must be greater than chance, of course, but beyond that what counts as a "large percentage" will vary from case to case, depending on the role the argument plays in the discourse, the needs and purposes of the discourse participants, and the nature of the subject matter, to name a few considerations. There is no similar problem with deductively valid argumentswhen they are good, they supply 100% support, since the conclusion cannot but be true when the premises are.
Third, an argument is deductively valid just in case the conclusion does not say anything that is not already implicit in the premises. Consider an argument where the conclusion makes a claim that goes beyond what is claimed by the premises. For instance, consider a conclusion that makes a claim about all UI students that comes on the heels of premises that only talk about UI students in philosophy classes. In such an argument, the premises could all be true but nevertheless fail to guarantee that the conclusion is truewhat is true of all UI philosophy students might not be true of, say, a UI ag econ student, and if it isnt, then any conclusion which generalizes over all UI students will be false. Indeed, the only way to guarantee that the conclusion is true, given the truth of the premises, is to require that the claim made by the conclusion go no further than those claims made by the premises. In other words, the information contained in the premises must include the information that receives mention in the conclusion. This restriction, however, does not apply to non-deductively strong arguments. In fact, they are of interest precisely because their conclusions do go beyond their premises and do so in a way that is nevertheless justifiable.
III.2 Deductively Valid Arguments
In this sub-section, I begin by introducing the two principal standards associated with analysis of deductively valid arguments. Then, I consider two categories of deductively valid arguments, propositional arguments and categorical arguments, describing along the way common forms of propositional and categorical arguments. I close by describing techniques for determining the validity of arguments in general.
III.2.A Standards for Deductively Valid Arguments
Two standards are apropos to the analysis of deductively valid arguments. The first, validity, has been defined above with D1. An argument is valid just in case it is deductively valid. Note that the term valid is one that, in common parlance, serves a great many purposes. It is used so variously as to be almost colorless, similar in this way to other terms of positive appraisal, such as good and right. However, in formal analysis, valid is a technical term with a precise meaning, and should be introduced as such. (Consider that arguments can be good, in the sense of being non-deductively strong, even though they arent valid, thereby establishing that the two terms cannot be used interchangeably when doing formal analysis.)
The second standard relevant to assessment of deductively valid arguments is soundness. An argument is sound just in that case where it is valid and its premises are true. Thus, it includes the formal standard of validity as a part. This is the more important of the two standards, as it is not merely formal but concerns also the relationships between the constituent claims and the world they purport to represent. When we analyze these arguments in our daily lives, we really care about their soundness. However, since soundness involves a component that concerns the subject matter of the argument, it is not a standard that will receive attention here. We will attend to part of it in this section, viz., validity, and we will return to it when we take up subject matter analysis in the next section.
In sum, two standards are relevant to the analysis of deductive arguments:
III.2.B Types of Deductively Valid Arguments
We will study two forms of deductively valid arguments in this handbook: propositional arguments and categorical arguments. Propositional arguments are built up out of propositions, which are claims that can be evaluated for truth or falsity. Propositions can be either simple or complex. Simple propositions are claims like "Maggie is tall" or "The Bush/Cheney ticket is exciting." They consist of a single subject term and a predicate. Complex propositions are built up out of simple propositions with the help of sentential connectives, such as and, or, if ... then, and just in case. Propositional arguments turn on relationships among complex propositions that are grounded in the connectives out of which the complex propositions are constructed. For example, if one premise in the argument is a complex claim of the form P and Q, then you can validly assert either P or Q because if P and Q is true, then P is true and so is Q. Thus, an argument that contained P and Q as a premise and P as a conclusion would be valid, and it would be valid because of the relationship between the claims, a relationship grounded in the connective and and that appears in the premise.
Categorical arguments are also built up out of propositions, but some of these propositions are marked by the presence of terms like some, each, all, and every. These terms indicate a concern with some subset of a group of objects, such as people in "All people are mortal" or Republicans in "Some Republicans wanted Elizabeth Dole for Bushs running mate". As these examples illustrate, the groups are determined by properties associated with terms in the sentences, such as being a person, or being a Republican. Call these general propositions. Categorical arguments include at least one proposition that makes a claim about a category or group of objects. They turn on relationships of inclusion or exclusion among the categories, or among the categories and individuals, that are mentioned in the constituent claims., and these relationships are grounded in the categories introduced by the constituent general propositions. For example, consider the famous syllogism: All men are mortal; Socrates is a man; therefore, Socrates is mortal. This is a valid categorical argument because the individual named in the second premise, Socrates, is a part of a group, men, that is itself part of another group, mortals, and the conclusion asserts essentially this, leaving out the middle term. Thus, Socrates is included in the group of mortal things, and this relationship of inclusion is guaranteed by the fact that he is a man and that all men are mortal things.
In a course on symbolic logic, one would begin with basic elements, such as predicate symbols (i.e., symbols for those linguistic items that mention the properties that define the relevant categories in general propositions), propositional symbols, connectives, and quantifier terms (i.e., terms like some and all), and build a system that could be used to identify valid arguments in general. This, however, is not our goal here. Instead, what I want to do is introduce a number of valid propositional and categorical argument forms, and then identify what it is about the constituent propositions that identify them and that secure their validity. In addition, I discuss techniques for determining valid forms of propositional and categorical arguments in general. I do this in the next two sub-sections.
III.2.C Propositional Arguments
Propositional arguments are a type of deductively valid argument that depend for their structural characteristics on connectives and the complex propositions they support. Connectives are terms, such as and and or, that connect propositions together to form more complex propositions. For example, we can take the proposition Maggie likes pomegranates and the proposition Maggie likes persimmons and combine them with the sentential connective and to yield the complex proposition Maggie likes pomegranates and Maggie likes persimmons, or what amounts to the same complex proposition in a more compact form, Maggie likes pomegranates and persimmons. In addition to and and or, we will concern ourselves with the connectives not, if ... then, and just in case (i.e., if and only if).
Because propositional arguments depend for their validity on the complex propositions they comprise, it is worthwhile to devote a bit of time to the nature of these propositions. We will do this by considering the impact that each of our five connectives has on the propositions into which they figure.
With these in hand, we can proceed to a specification of argument forms involving complex propositions formed from these connectives. I will list common and important forms in association with the connectives that underpin them. In so doing, I will first introduce them in schematic form, using the proposition symbols P and Q, and then I will supply an illustration.
There are several points to make about these. First, it is not uncommon that you have to dig to uncover the argument form in a particular case. People will use different terms and phrases in structuring their arguments, so it is important that you become familiar with the different terms associated with the sentential connectives. Further, it isnt always the case that arguers mention the connectivesthey may use gestures, or rely on a look or a tone. Finally, there is some flexibility to argument form, as we saw in our discussion of argument reconstruction. Second, the symbolic expression of the forms reveals their formal nature. The fact that you can represent the argument forms in this way demonstrates that the actual content of the propositions involved, i.e., what they say, is irrelevant to their structure. P and Q can be replaced by any proposition and, so long as they are related in the indicated ways, the argument produced will have a valid form. Third, most propositional arguments introduced in conversation or in written texts are complicated mixtures of the various forms introduced above. In effect, these forms serve as inference rules, transmitting you from claims with a certain structure to a new claim; as such, they can be linked in sequence to form longer and more complex arguments.
Finally, a word or two about procedures for verifying that an argument form is valid. As I have indicated, the forms above are valid because of relationships grounded in the connectives. The connectives serve as the foundation for the relationships between propositions because they guarantee that the complex proposition of which they are a part will have a certain truth value given the truth values of the constituent propositions. Given this, if a complex proposition containing a connective is taken to be true, we can expect certain claims to follow in certain conditions. For example, if we take if P then Q to be true, then we can expect Q to be true when P is; in addition, we can expect P to be false when Q is false. These expectations are grounded in our understanding of the way that the truth value of the complex proposition depends on the truth values of its parts, a dependency that is secured by the sentential connective that connects these parts. In effect, sentential connectives are functions of truth valuesthey take as inputs the truth values of the constituent propositions and yield the truth value of the complex proposition as output. It is for this reason that sentential connectives are called truth-functional connectives. Thus, sentential connectives are the backbone of propositional argument forms, and when they are valid, these forms depend for their validity on their truth functional character.
This conclusion points to general method for assessing the validity of propositional argument forms. The truth-functional character of connectives can be represented with truth tables, which are tabular representations of input and output truth values. Truth tables are robust enough to represent entire arguments, and so can be used to reveal the truth functional relationships between the propositions in these arguments. Thus, they can be used to identify whether a given argument form is valid or invalid. Call this the Method of Truth Tables. For more on this method, see Lectures 9 and 10 and the handouts for Chapters 3 through 6 on the Philosophy 202 homepage.
III.2.D Categorical Arguments
Categorical arguments are a type of deductively valid argument that depend for their validity on the relationships between categories and individuals mentioned in their constituent propositions. In particular, these are relationships of category inclusion or exclusion. These categories are introduced by terms that express properties, terms like zebra and likes grass. These property terms can either appear in subject position in a sentence, joined to what are called quantifier terms, such as all, some, and none, or they can appear in predicate position. Quantifier terms indicate how much of the category is involved in the claim. The sentence, "All zebras like grass," includes a property term, zebras, in subject position, and a property term, likes grass, in predicate position. The subject phrase in this sentence expresses the property being a zebra, while the predicate expresses the property, being a grass liker. The quantifier term, all, indicates that every item that falls under the category termin this case every zebrais involved in the claim. This sentence, then, expresses the proposition that every thing that is a zebra is also a grass liker. That is, the proposition claims that the category zebra is included in the category grass liker. These sentences express what we will call general propositions, as they do not concern individual items but rather categories of objects.
Category terms that form part of subject phrases in conjunction with quantifiers are typically common nouns, such as plant and coffee cup. The categories that these terms contribute to the claim can be delimited by modifying the terms with adjectives, such as wilted plant and chipped coffee cup, or with prepositional phrases. Two quantifiers contribute to these phrases, viz., universal quantification and existential quantification. Universal quantifier terms include all, every, each, and any, while existential quantifier terms include some, a few, many, a, an, and the (as in "The tiger is a dangerous predator"). Terms like no and none are also used as quantifiers, functioning very much like a universal quantifier since they are used to assert something negative about an entire category. Predicates can also introduce categories into a claim. Predicates are verb phrases that typically combine a verb with a modifier, such as an adverb or adjective, such as runs quickly or tastes sour. They can also explicitly express category containment with the copula followed by another common noun or noun phrase, are animals or is a disease. (Note that a as it appears in the second example is considered a quantifier, akin to some.)
Categorical arguments typically include sentences containing these terms and phrases
that express general propositions, or propositions that relate one category to another.
There are many such propositions, but the categorical relationships they express are
variations on four forms, known variously as the categorical forms or the Aristotelian
forms, as these were the focus of Aristotles syllogistic logic. These forms,
listed in association with their traditional letters, are as follows, with S
and P appearing in the position of category terms:
There are several things worth noting about these forms. First, the A and E forms are universal, in the sense that they apply to every member of the indicated category, whereas the I and O forms are existential, applying to some but (perhaps) not all of the members of the relevant category. The A and I forms are obvious, but the presence of negation in the E and O forms renders their status somewhat more uncertain. Nevertheless, an E proposition makes a claim about all members of class S, viz., that they do not have the property expressed by P, while an O proposition makes the same claim but only about some members of the class S. Recognition of this connection between E and O propositions reveals another pattern in the forms: A and I are what we can call affirmative forms, whereas E and O are negative forms. Thus, there are two generalizations that cut across our four forms, a fact that can be represented with the following table:
This exhaustively represents all of the relationships that can obtain between the categories S and P. Third, these forms can be realized by sentences that do not come close to matching the examples given above. As with most of the elements of critical thinking so far discussed, life in the real world of argumentative give-and-take is not so clean and neatsentences are put to uses for which they are not well-suited, and eloquence can create complexity that conceals the basic message. There will come many a time when you must decide whether a lengthy and overwrought sentence expresses a short and simple categorical form, and you will do this by first looking for the relevant categories and then asking whether the sentence expresses a proposition that relates these categories by inclusion or exclusion. If the answer to this question is affirmative, you have yourself a categorical form. Fourth and finally, it is worth calling attention to the fact that the logical nature of these forms remains a topic of heated debate amongst linguists and philosophers. For instance, the logical nature of the definite article, the, has been a flash point for a century, with some arguing that noun phrases formed with the article (e.g., the sentence youre reading, the shortest spy) are referring expressions, like names, that pick out particular individuals, while others argue that the is a quantifier and these phrases are existential. While these debates rage on, most of the live rounds are discharged in academic journals and do not threaten those who teach or learn critical thinking skills.
We are now positioned to consider several common categorical argument forms. As before,
I offer only a sampling of these forms, but the ones I offer are common and important.
Nevertheless, there are other common and important forms that will remain unmentioned
because of limits of spacea text on critical thinking or classical logic will supply
many more for further consideration. As with the propositional argument forms, I introduce
these first symbolically and then by example.
Once again, a few notes before moving on. First, as can be seen from these examples, categorical arguments do not contain only general propositions, and the general propositions they do contain need not be premises in the arguments. A categorical argument can maintain the inclusion or exclusion of individuals in categories, as well as categories in categories. The critical feature is that whereas propositional arguments are valid or not independently of any concern over category inclusion, categorical arguments essentially depend on this concern. Second, as with propositional arguments, it is often difficult to spot these straightaway, and they are often combined in complicated ways. In general, pulling an argument from a text is a risky business, and categorical arguments are no different. Third, the symbolic representations once again establish that form contributes substantially to good argumentation. The examples given above share their valid forms with countless other instances of impeccable argumentation. Of course, it is also true that countless bad arguments have valid forms (e.g., replace P with students, Q with aliens, and R with cats in (1) above), which just goes to show that form isnt everything.
I conclude this survey of categorical arguments with a few words about procedures for determining validity. As should be clear by now, categorical arguments do not rely on truth-functional connectives for their validity, so the truth table method introduced above will be of no use to us here. Instead, we must introduce a method that captures the categorical relations of inclusion and exclusion that obtain among categories and individuals. The method to us involves Venn diagrams. These interlocking circles can be used to express relations between categories, and we can also introduce other mechanisms to extend the reach of the method, such as an X for an individual and shading for the absence of individuals. Call this the Method of Venn Diagrams. For more on this method, see Lecture 6 on the homepage for Philosophy 404, or Chapter zxcv in Fogelin and Sinnott-Armstrong's book Understanding Arguments, which is mentioned at the end of the Applications section for this chapter.
III.3 Non-deductively Strong Arguments
I begin this subsection with a brief discussion of the general standards associated with non-deductively strong arguments and then turn to four important argument types: induction, abduction (i.e., inference to the best explanation), argument by analogy, and confirmation. I illustrate each of these types of argument and then speak to the standards of evaluation appropriate to each.
III.3.A Standards for Non-deductively Strong Arguments
Recall that non-deductively strong arguments are arguments that meet the following conditions: (a) the truth of their premises make it highly improbable that their conclusion is false, and (b) they are not deductively valid. Without the second condition, we would be forced to classify deductively valid arguments as non-deductively strong, since their conclusions must be true when their premises are, making it not only improbable but impossible that their conclusion is false when their premises are true. However, the second condition is not included as a way of deflecting attention to some weaker forms of argument; rather, it marks a difference in kind and not degree between arguments that deserve our attention. As I noted above, the conclusions of deductively valid arguments do not outstrip their premisesthese conclusions present information that is contained implicitly within the information expressed by their premises. It is for this reason that they cannot fail but be true when the premises are true; after all, they dont really say anything new. Conclusions of non-deductively strong arguments, by contrast, do outstrip their premises. They include generalizations or explanations that are only suggested by the premises, or they offer predictions that extend the information expressed by the premises into new and different quarters. Thus, they promise to tell us something that we dont already know, at least implicitly, and thereby extend our knowledge. Painting with a broad brush for a moment, we can say that deductively valid arguments are the medium of conceptual analysis, exposing to the sunlight what lies hidden in the folds of a theory, whereas non-deductively strong arguments are the medium of empirical investigation, extending knowledge and understanding into new and uncharted territories. Given this role, it is crucial that we make sense out of what it is that renders an argument non-deductively strong, and this amounts to identifying the standards we apply when assessing it.
The place to start this search is the first part of the definition of non-deductive strength. Two notions stand out as central to this: the improbability of the conclusion, and the fact that we are given that the premises are true. Lets begin with the second of these. An argument should be non-deductively strong because the premises support the conclusion; that is, the truth of the premises are what account for the fact that the conclusion is likely true. Thus, there must be some connection between the truth of the premises and the likelihood of the conclusion, and so "given that" cannot be read "when" or "while". Consider this argument: I passed a coffee shop on my way to work today, just as I did yesterday, and the day before that, and so on and so on; therefore, Bill Clinton is from Arkansas. If we were to read "given that" as "when"and so take non-deductive strength to consist in the likelihood of the conclusion when the premises are truethen this would count as an non-deductively strong argument; after all, the conclusion is true as a matter of fact, and so is quite likely to be true when the premises are. The problem with this argument is that there is no connection between premises and conclusionthe premises do no work in transforming the conclusion into a highly probably truth. Therefore, we must take the truth of the premises to stand in some content relation with the conclusion, such that their truth explains our assessment of the conclusion. When we are given that the premises are true, we must evaluate the conclusion in light of that fact and not independently of it.
Second, there is the matter of probability to consider. It is this notion that separates the good from the bad. There are a lot of arguments out there that share their form with non-deductively strong arguments but are nevertheless putrid examples of reasoning. In the good arguments, it is highly improbable that the conclusions are false when the premises are true, but it may not even be better than chance that they are false in bad arguments of the same form. Thus, probability is at the core of our standards of non-deductive strength. But how are we to interpret probability here? And what counts as highly improbable? With respect to the first question, would that we were in a better position to say. While deductively valid arguments have an associated logic that describes their structure and underwrites their standards, there is no consensus choice for a logic of non-deductively strong arguments. In fact, there are serious problems with inductive probability and with the notion of strength that undermine general agreement about such logics. Nevertheless, for the purposes of critical assessment, we can rely on a standard model of probability to guide us in identifying principles that distinguish non-deductively strong arguments from those that are non-deductively weak. As for the second question, what counts as strength will depend on the interests and purposes of those engaged in the discourse. In scientific circles, there are statistical conventions in place to determine what counts as adequate strengthfailure to meet these conventions will make it difficult to win grants and publish papers. Outside of these circles, things are much fuzzier, and this fuzziness vitiates any systematic attempt at interpretation. In lieu of such an attempt, I will introduce and justify context sensitive principles associated with the four forms of non-deductively strong arguments discussed below. In general, these principles require that the evidence cited in support of a generalization, an explanation, or a prediction be broad, diverse, extensive, and relevantly representativein other words, that it reduce the probability of the risk taken in reaching beyond the premises by ensuring that the conclusion is as much like the premises as possible.
III.3.B Types of Non-deductively Strong Arguments
In this section, we consider four types of arguments that can be non-deductively strong. The first type, inductive arguments, is marked by a conclusion that generalizes more specific evidence cited in the premises. For instance, concluding that all John Waters films are quirky and bizarre on the basis of having seen a few of them is an inductive argument. Secondly, we have abductive arguments, or arguments that involve an inference in the conclusion to what one takes to be the best explanation of facts cited in the premises. Sherlock Holmes did a fair bit of abduction, and so have you if youve ever played the game Clue. So did I the other day when I inferred that I had left my paycheck stub in my jeans when I washed them from all the water-soaked paycheck confetti that was stuck to the entire load after the last spin cycle. Third, there are arguments by analogy. These arguments press conclusions about one matter that are motivated by the similarity between that matter and something else. When my friend argues that he will be a good father because he is good to his dog, he argues by analogy. Finally, there are confirmation arguments, or the testing of hypotheses. These arguments typically assert the truth of a theoretical claim on the basis of observational evidence that purportedly confirms the claim. These tend to start with conditional claims that include the theoretical claim as the antecedent and an observation as the consequent. Consider this instance: if exposure to the sun causes skin cancer, then we should find a higher incidence of skin cancer among those who spend a great deal of time in the sun; we do find such an incidence; therefore, exposure to the sun must cause skin cancer. Notice that in all of these cases, the conclusion reaches out beyond the premises, and therefore, it is possible that the conclusion is wrong in each case even given the truth of the premises. Such is life when you run with the non-deductively strong.
III.3.C Inductive Arguments
Inductive arguments begin with observations about specific instances of a particular type and then infer from these a more general claim about instances of that type. For example, on a hike at Robinson Park, I see only robins for the first 15 minutes, and this leads me to conclude that I will only see robins on my hike. There are several elements of this characterization that require elaboration. First, there is the nature of the type. In our illustration, the type is quite simple, viz., bird species spotted on my hike. However, the type in question can be complex, as when the observations concern a correlational relationship between events. For example, I have experienced countless events involving the opening of kitchen faucets, followed by the flow of water. This leads me to expect water when I turn a kitchen faucet on, and this expectation is grounded in the inductive generalization from my past experiences to my future experiences. The correlation in this case is causal, but it neednt be, as when we correlate being good at the piano with being good at mathematics. Second, there is the nature of the generalization that is based on the particular observations. In the first example, I generalize to a conclusion about all the birds in my type, i.e., all the bird species Ill spot on my hike. However, I might choose to weaken the generalization by commenting only on most of the bird species that Ill see. Similarly, the faucet inference could be to a conclusion about all faucet opening events, but instead concerns only those that will figure into my future experience. (I might be suspicious about the overall generality of this claim.)
Arguments of this type are not valid, since the conclusion will cover instances of the type that you have not observed and so will extend beyond the premises, leaving open the possibility that they are true while the conclusion is false. I might see only robins for all but the last few seconds of my hike, when a flock of cedar waxwings catches my eye, or I might go to a friends house and find beer flowing from the tap. However, arguments of this sort can be strong nevertheless. Several standards bear on evaluations of this type of argument. They are:
III.3.D Abductive Arguments
An abductive argument is an inference from a set of considerations, such as facts or clues about the matter at hand, to a conclusion that is deemed "best" of the available alternatives. This evaluation of the conclusion is made relative to a goal that motivates the arguer to serve up the argument. Two aspects of this argument type merit close scrutiny: the goal and the conclusion. First, the goal. For an example of one goal, we can turn to inference to the best explanation, where one is presented with a data set and infers from that a claim which does the best job of accounting for the data in the set. The goal in this case is explanation, and the conclusion is presented as the best available explanation of the data. If youve ever played Clue, or read a good mystery novel, then you have engaged in inference to the best explanationin each of these cases, you collect together a set of clues and seek to determine whodunit, where identification of the criminal will best explain the available clues. (Heres another example: "There is dirt on the floor by the plants and one of the stuffed animals is ripped up on the floorthe dog must have gotten loose again.") Abductive arguments neednt be inferences to the best explanation; they might instead be inferences to the best judgment, or the best policy, or the best plan. (In fact, it would not be inappropriate, given our definitions, to consider inductive arguments to be a type of abductive argument, viz., inference to the best generalization.) In each of these cases, the goal motivating the arguer differs. As an example, consider how a university might determine a policy for dealing with hate speech: they could collect judicial opinions about First Amendment protections, advice from staff attorneys, views of students, faculty, and alumni, and precedents from other universities; once all of this information is collected, they would examine it to determine if an overriding policy direction emerges, and this might then become the policy they adopt. In such a situation, we have an inference to the best policy, where the considerations include the evaluative opinions of other experts and institutions.
And then there is the conclusion. To begin, it is important to note that these arguments typically do not purport to identify the best possible way to achieve the goal; rather, they claim only to identify the best available way. The alternative conclusions available will vary dramatically, often as a function of awareness and imagination, to name two important factors. Second, there will be nothing about the considerations you adduce that force you to adopt one of the available conclusions over the other alternativesit could always turn out to be the case that one of the other conclusions is in fact superior. This points to the invalid nature of the inference; since there is nothing about the considerations that are mentioned in the premises that force the conclusion to be true, the argument is not deductively valid. (In fact, if you run an argument like this and you find one conclusion forced on you, it may well be better to represent it as a deductively valid argument.)
But once again, even though this form of argument is not deductively valid, it can nevertheless be strong. The strength of the argument turns on two issues: (a) whether the conclusion is really the best (or is at least debatably the best) of the available alternatives, and (b) whether the set of alternatives includes legitimate and plausible members. The standards one must impose in analyzing these arguments are keyed to these two issues, and so we will organize the standards accordingly.
III.3.E Argument by Analogy
An argument by analogy consists of two salient parts, viz., an analogy and an argument. First, there is the analogy. This involves the telling of a story that is intended to parallel some take on the topic in question in certain salient respects. For instance, if in a conversation about cats a friend draws an analogy between caring for a cat and caring for a teenager, the analogy will be suggestive only if there are significant similarities that shine through the many differences. In this case, there are, since both involve taking care of another organism that doesnt seem to want much to do with you. Here the care of cats is the topic and the care of a teenager is the story. Note that the similarities are primarily structural, while the differences are material, i.e., the topic and story are similar at the more abstract level of the relationship between care giver and cared for, with the differences entering in as these roles are occupied in each case. Good analogies have that character: they involve a story that is structurally similar to the topic, even though the specific content of each may differ dramatically.
Second, there is the argument. One may introduce analogies for all manner of reasons, among them amusement, intellectual exercise, and literary flourish. One significant role analogies are asked to play is that of motivating a conclusion. As such, they serve as reasons for the conclusion, and so are at the heart of the argument. When an analogy is put to argumentative work, it is typically done in the following way: once the story is introduced, the arguer derives an inference from it, perhaps to a moral; then, because of the structural similarity between story and topic, the arguer suggests that the topic motivates a similar conclusion, differing only in the content-related ways that story and topic differ. In the story, a certain conclusion follows, so because the story and the topic have the same structure, the same conclusion (mutatis mutandis) should follow from the topic. (Consider: "My son said something like that to my daughter when he was trying to trick her into giving him her allowance. Are you trying to swindle me?") This is a very suggestive and powerful way to argue. Among other virtues, it can keep a highly controversial topic on the table by retaining its structure through changes in detail that are meant to defuse the passion and controversy that frustrate straightforward attempts to address the topic. A classic argument found in discussions of abortion exemplifies this: consider that a fetus is to a person as an acorn is to an oak tree; note that when you kill an acorn, you are not killing an oak tree; therefore, when you kill a fetus, you are not killing a person. Without passing judgment on the merits of this argument, it does demonstrate how an analogy motivates transferring a conclusion from one an uncontroversial context to a more controversial context. However, this form of argument is obviously invalid, since an analogous story must be different from the topic in several, if not many, respects, and so there is quite a bit of room for falsehood to creep in. Thus, the analogy may be a good one, but the differences that exist between story and topic might be enough to undermine the inference from the topic to the desired conclusion.
Even so, some arguments by analogy are better than others, and we can identify standards that help distinguish the good from the bad. Some important standards that apply are as follows:
III.3.F Confirmation Arguments
The last non-deductive argument well consider is the confirmation argument. This type of argument, one begins with a theory, or perhaps just a hunch, and reasons that if this were true, then so would some additional claim; upon further investigation, one establishes that this additional claim is indeed true, and this supplies a certain degree of confirmation for the theory. The argument is a report of this process, beginning with an assertion of the starting point and the claim it implies, followed by the announcement that this claim is in fact true, establishing that at least on this point, the starting point is confirmed. This type of argument is at the heart of the experimental method in the sciencesan experiment is a controlled test of an implied claim (i.e., a hypothesis) designed to determine if the theory that implies this hypothesis is deserving of credence. Thus, science is home to many instances of this type of argument, but it is also instantiated in much less controlled and elaborate settings. Consider: "Look, if you are who you say you are, then you should remember the time we went out to the dam, back after high school graduation, and ... you do remember! It must be you!."
This argument type has a number of elements that deserve further elaboration. First, there is the starting point. As noted, this could be a full-blown theory, or it could be something as sketchy and provisional as a hunch. In any case, it will be some sort of representation the truth of which is in question. The confirmation argument is designed to buttress the claim this representation has on truth, and it does this by focusing on a claim this representation implies. Thus, the representation must be robust enough to imply substantive claims that can themselves be tested for truth. The relation between representation and the implied claim is the second element. This could be deductive implication, but it neednt beit could, for instance, be some sort of non-deductive implication, such as an abduction. The key to this sort of argument is that the truth of the representation increase the likelihood of the implication, so that if the implication is found to be true it casts a positive light back on the representation. The more the implication depends on the representation, the greater the degree of confirmation it can supply. Third, there is the nature of the implied claim. The crucial thing about the implied claim is that it must be testableif it isnt, then it will not contribute to a confirmation argument. In scientific cases, it will typically be an observable claim about the physical world. (There are cases of implied claims that cannot be tested as yet, or perhaps can never be tested; these are claims that, while implied, are not part of any confirmation arguments.) In the more mundane cases, the claim will still be testable in some fashion, and in most cases the test will involve observation in some central way. Finally, there is the nature of the test. In arguments intended to establish conclusions with scientific precision, the test could be some sort of controlled experiment. Alternatively, the test might involve the type of discovery that field work or just dumb luck could supply. In less constrained cases, the test might involve simple observation. In all cases, however, the test involves assessing the implied claim for truth, and how this is done will vary with from claim to claim, depending on whether it is more observational or theoretical in character.
An example of a very well-developed account of this type of argument is found in Hempel 1966. In this book, Hempel details what has come to be called the hypothetico-deductive method for determining assessing hypotheses and, ultimately, the theories from which they derive. According to this account, scientific labor involves (a) the identification of hypotheses, or candidate explanations of phenomena, that flow deductively from background theories, and (b) the empirical and often experimental evaluation of these hypotheses to determine their truth. If they are revealed as true by their evaluation, then they offer a certain measure of confirmation for the theory that implied them; however, if they are shown to be false, then deductive logic takes over and conveys the falsity back to the theory, in conformity with modus tollens. Of course, the actual situation is much more complicated than this simple sketch suggestsfalsity is often difficult to determine, for sure, and the falsity conveyed back to the theory may only reveal the inadequacy of some auxiliary assumption that can be rejected without damage to the background theory. But even so, the method supports either confirmation or falsification, depending on the evaluation of the hypothesis.
Like the other forms of non-deductive inference, confirmation arguments are invalid when they supply confirmation for theories. In fact, when the hypothesis is deductively inferred from the background claims, the argument shares its structure with a famous fallacy that we will study in the next chapter, viz., affirming the consequent. Even so, this sort of argument captures the importance we typically assign to consequencesif the consequences of some set of claims are themselves valuable according to some measure, then this value redounds to the credit of the set of claims. (This sort of reasoning is especially prevalent in moral theory.) However, a given hypothesis designed to lend such support may wind up serving as a part of a deductively valid modus tollens argument for the falsehood of a theory. The centrality in this discussion of confirmation and falsification of theories by the hypotheses they imply exposes the important connection between this type of argument and prediction. One desideratum of a theory in any domain is that its predictions be realized, and confirmation arguments derive their force from value ascribed to predictive success. This can be contrasted with, for example, abductive arguments, which are keyed to a different desideratum, viz., explanation. Together, these argument types underwrite the rational attempts we make to understand the world of our experience.
Given the importance of confirmation arguments to the construction of a rationally defensible worldview, it is incumbent on us to identify the standards that divide the good from the bad. These are grounded in the structure of the inference type, the elements of which were identified above. Thus, we have the following standards:
IV. Subject Matter Analysis
Formal analysis is devoted to evaluating the structure of arguments. An argument that is poorly structured will be ill equipped to compel one to embrace the conclusion, given the truth of the premises. But while form can break an argument, it cannot make it. An argument with unimpeachable form must still pass muster at two additional levels of analysis, viz., subject matter and context, if it is to be rationally compelling in most episodes. In this section, I focus on subject matter analysis.
Subject matter analysis concerns what the argument is about, where this is determined by its constituent claims. Each premise, along with the conclusion, will make a claim that is either true or false of the world, broadly construed, and the claim made by each of these counts as its content; the content of the claims taken together counts as the content of the argument. An analysis of the subject matter of an argument, then, is an analysis of its content, and this resolves into an analysis of the content of its constituent claims.
We analyze the content of the constituent claims in two ways. First, we need to be sure that the set of claims is thematically coherent. Second, we need to be sure that the claims themselves are true. We begin with a few remarks about coherence. A necessary condition on compelling arguments is that their conclusions concern the same subject matter as their premises. One can have arguments that are formally impeccable even though they fail to satisfy the thematic coherence standard. For instance, consider this argument:
Since there is no way the conclusion of this argument can be false when the premises are true, this is a deductively valid argument by our lights; however, the argument is nevertheless in bad shape, due to its failure to satisfy the standard of thematic coherence. In a case such as this one it is easy to detect a failure to satisfy the standard, but it is not easy to make precise exactly what it takes to meet it. Fortunately, our purposes in this handbook do not require a precise definition of the standard. The crucial thing to determine for any argument is whether the conclusion and premises either share predicates and/or names of items, or contain predicates and/or names that stand in semantical relationships that are relevantly similar, given the nature of the argument. This is altogether too vague, of course, and it isnt clear that the context couldnt justify violation of this. (Consider: the above argument failed to meet the standard, but it did the job of illustrating the point I wanted to make and so was perfectly legitimate.) In most episodes, however, sensitivity to this rough and ready rule of thumb will ensure that a violation of thematic coherence is duly noted.
Given that the steps of an argument are thematically coherent, we are then set to determine whether the claims are true. Recall that the two principal formal standards, deductive validity and non-deductive strength, both specified the formal strength of arguments in terms of truth: in arguments that meet these standards, we have good reason to believe that the conclusion is true given that the premises are true. The conditional nature of both standards is tied to the flow of truthwhen a certain condition, identified in terms of truth, is satisfied, then we have good reason to believe that truth flows to the conclusion. Thus, full assessment of an argument requires determining whether the premises are in fact true so that our hypothetical appreciation for the conclusion can become categorical.
Premises come in one of two forms: either they are obviously true, or they are not. If they are obviously true, they are either true because they are a first principle and so make a claim that we all find intuitively true in virtue of our status as rational beings, or they are true because they are trivial. (Perhaps there is not a real difference between these?) First principles are hard to come byDescartes thought he found one, but there are many who believe that he should have thought harder. As for the second sort, we have claims like the conclusion above, "The handbook is here or it isnt here." These claims are known as tautologies, and the truth of these claims is guaranteed by their form, regardless of their specific subject matter. As a result, they are trivial claims that add nothing to arguments that contain them. (Of course, tautologies in the wild are almost never as obvious as this, even though they are no more informative or any less trivial.) Thus, most of the premises that really pack a punch will be claims that are not obviously true. These will come in many forms and will concern as many things as concern us, and a good many of them will find their support in additional argumentation. To be sure, we will be able to read the truth values of some of these off of the world directly, via observation, but often premises in one argument will be conclusions of supporting arguments and will depend on those supporting arguments for their force. As a result, the task of determining the truth of such a premise will depend on the nature of the subject matter it concerns. It is at this point that logic and formal methods give way to the facts and methods characteristic of specific subject areas. For instance, a full assessment of an argument in organic chemistry requires familiarity with the details of organic chemistry, assuming of course that the form of the argument withstands scrutiny. Thus, in addition to formal facility, critical thinking requires subject knowledge, and the range of contexts in which one can apply critical thinking skills will increase in direct proportion with the number of subjects one knows.
If the premises of a formally strong argument emerge as true, the argument must be regarded as establishing its conclusion. If it is deductive, we say that it is sound in this case, where soundness requires both validity and true premises. There is no corresponding term for the non-deductive case, although most arguers will accept compelling in its place.
But wait! The analysis is not yet complete. It is an accomplishment to craft an argument that measures up to both formal and subject matter analysis, but such an argument may nevertheless fail to have the intended effect. Outside of critical thinking classrooms, people do not argue in a vacuumarguments are typically adduced in the service of achieving broader goals, e.g., creating consensus, advancing ones research agenda, creating community, getting ones way, etc. Given this, there remains one level of analysis to discuss, viz., context analysis. How does the argument fit into its context? If it fails to fit the context, it likely will not compel even though it is unobjectionable when considered by itself. If it fits its context, then it should have its intended effect, assuming a certain level of rationality on the part of the audience.
What counts as context is a vexed philosophical question, and it seems likely that there are no necessary and sufficient conditions for delimiting it. The relevant frame for conducting this sort of analysis of an argument will change given the subject matter, the participants, the location of the argument, the surrounding political climate, etc. etc. We are quite adept at intuitively recognizing what is relevant and what is not in specific circumstances, but identifying a small set of generalizations that systematize these elements is a very tall order, and one I am in no position to fill. Nevertheless, there are a few common contextual dimensions that recur regularly and warrant mention here. They correspond to elements in a typical argument episode, and are listed here in connection with those.
There are other dimensions, but these three stand out as more or less general and, where they are found, critical. An argument that runs the gauntlet of analysis, avoiding pitfalls of form, subject matter, and context, will stand out as compelling and will have a deserved claim on the beliefs of those who value rational consistency.