3.1. Licensing by Uniformity

Norton makes liberal use of inferences that can be formalized as ‘A are generally B; c is A; therefore, c is B’ (Norton Reference Norton2014, 674). Such arguments enable us to infer claims about sample representativeness from local uniformity principles: given that elements are generally uniform in their melting points and bismuth is an element, we can infer that bismuth is uniform in its melting point, and therefore observed samples of bismuth are representative (with respect to their melting points) of bismuth in general.

However, despite its essential function in the MTI, Norton does little to justify this form of inference. Norton (Reference Norton2014, 674) briefly discusses these inferences but mainly to claim that ‘A are generally B’ functions in an analogous manner to ‘If A, then B’ claims in deductive logic. Norton says that, just as we are warranted in making a deductive inference from A to B when we accept that ‘If A, then B’, so we are warranted in inferring from A to B when we accept ‘A are generally B’.

Even when the local uniformity belief is added as a premise to an inductive argument, this form of reasoning is still deductively invalid: “the inference is ampliative, even with explicit adoption of the warranting fact as a premise” (Norton Reference Norton2014, 674). This entails that Norton cannot use the various justifications of deductive logical inferences, such as model-theoretic proofs of metalogical virtues like soundness and completeness.Footnote ³ These inferences are also not a subclass of inductive inferences, in the sense of ‘induction’ that Norton uses: ‘Ravens can generally fly, the next flock of birds I see will be ravens, therefore the next flock of birds I see will be able to fly’ includes a premise about ravens as a class, rather than a statement about observed ravens. Thus, their rationality is distinct from the issue of the rationality of induction, and consequently Norton’s arguments about the latter are not automatically applicable to the rationality of inferences from local uniformity principles to the hypothesis that particular samples are representative of the relevant target populations.

The mirror of this point is that a rich account of induction would give guidance about when it is irrational to accept local uniformity principles that can license an inductive inference. Imagine that you are exploring the Antarctic and discover a large underground ecosystem. Among the animals are a species of penguins. You observe that all of the dozens of penguins that you see are blind and have white feathers. Intuitively, you might reasonably infer that almost all members of this species are blind, because blindness might easily have evolved for them in their subterranean existence. It would also seem to be irrational to infer (without further evidence) that ‘All members of this species of penguins are white’, because you know that species of birds generally vary in the color of their plumage. However, suppose that you postulated that ‘Subterranean species of penguins are generally uniform in their color.’ Are you entitled to infer that all members of the species of penguins are white?

The answer seems to be negative: we cannot just invent local uniformity principles. Naturally, Norton would agree, but it would enrich his theory to have an account of when we can and cannot make such inferences. (This account need not be a formal theory, in his sense of ‘formal’.) It would also help Norton answer critics who have theories of induction that have more extensive accounts of this reasoning, such as Bayesians, who can just say that we should consult the relevant conditional probabilities in our distribution.

3.2. Differences of Support

Local uniformity principles seem to license inductive inferences to varying extents. Imagine that you know (a) that 99% of the 200 balls in urn 1 have the same color and (b) that 50.5% of the 200 balls in urn 2 have the same color. Suppose you reach into urn 1 and draw a gray ball and then reach into urn 2 and draw a gray ball. Assume that you have no background reason to think that the selected balls are unrepresentative (with respect to color) of the class of balls in their respective urns. You seem to have more license to infer that the next ball you draw from urn 1 is gray than that the next ball you draw from urn 2 is gray.

However, the MTI does not contain an account of how different sorts of evidence provide greater or lesser support for a given hypothesis, relative to different local uniformity principles. In the urns example, the differences in evidential support are obvious and might even seem to be quantitatively describable, but this transparency is not generally the case. This silence in the MTI is a problem, because we are not interested in just whether our inductive conclusions are supported by our evidence but also how well they are supported. Perhaps a quantitative theory of evidential support is impossible. Perhaps we can only achieve a comparative or qualitative analysis. Perhaps there are multiple dimensions of evidential support, such as confirmation by the evidence, the reliability of the evidence, and the quantity of total evidence; these dimensions might be incommensurable. We do not know except insofar as we try. Bayesians have extensive analyses of these issues, and if the MTI has no alternative, then this is a respect in which the MTI currently compares unfavorably to Bayesianism.

From these gaps in Norton’s theory, it does not follow that any rivals of the MTI do better (their accounts might be very flawed), but it does follow that the MTI would be enriched by supplementing it with the means of answering these questions. I return to this issue in section 6, in which I argue for a research program for these additions.

4.1. The Problem and Norton’s Answer

There are many versions of the Problem of Induction; it would be more accurate to talk of the Problems of Induction. There is an interesting historical question of which version, if any, should be called the “Humean” Problem of Induction, but that question is beside the point of this article. Instead, I focus on the problem that Norton and his critics are discussing, which is interesting even if it is not Hume’s problem.

Philosophers have almost all assumed that the justification of induction must be inductive or deductive. However, this presents a dilemma: (1) if we appeal to induction’s past successes, then our justification is circular, since our justification involves an inductive inference, while (2) if we appeal to some implicit premises that render inductive arguments into deductively valid arguments, then our justification is too strong, because inductive inference is ampliative rather than deductive.Footnote ⁴ We have a paradox: we believe that induction is reliable, but this belief is very hard to justify (Norton Reference Norton2003, 666).

Norton’s answer is that each (rational) inductive inference is justified by a local uniformity principle, rather than the past successes of induction in general. For our actual inductive inferences, he argues that we should take horn 1 of the dilemma: our inductions are justified by past inductions, but these antecedent inductions were inferences of local uniformities. Obviously, there is a suspicion of a vicious regress. Norton acknowledges that there is a regress in his justification, but he denies that it is vicious, because there is no reason to suppose that its termination must be problematic. Norton argues that the “branching trees” of justification might end in “brute facts of experience that do not need further justification” (Reference Norton2003, 668). He later argues that the regress might produce a benign, coherent circle (Norton Reference Norton2014, 683–87).Footnote ⁵ Briefly, according to Norton, our inductions are justified by past inductions, but there does not need to be a vicious regress.

4.2. The Regress Problem

Philosophers of science have mostly been unsympathetic to Norton’s answer to the Problem of Induction. For example, Samir Okasha (Reference Okasha2005, 250), Thomas Kelly (Reference Kelly2010, 760–63), and John Worrall (Reference Worrall2010, 746) have all argued that the regress must be vicious. Their arguments assume a first inductive inference, before any nonobservational knowledge, to which all other inductive inferences have an epistemic link. (This was a fair assumption, because Norton [Reference Norton2003, 668] accepts the possibility of a first inductive inference.) Their implicit argument seems to be:

Therefore,

Therefore,

Therefore,

But,

Therefore,

If Norton’s critics are correct, then the MTI and some plausible auxiliary premises entail inductive skepticism. This entailment would not be entirely surprising: Okasha (Reference Okasha2005, 249) notes that Sober (Reference Sober1988, chap. 2) has a theory of induction that is similar to Norton’s, but he comes to a skeptical conclusion about induction. Since Norton denies inductive skepticism, he needs some way of avoiding C₄. Norton commits to P₁ from the outset of his primary article on the MTI (Reference Norton2003), and it captures much of the MTI’s normative core. All parties in the controversy seem to accept P₃. However, Norton does deny P₂ (Norton Reference Norton2014, 679–80), and P₄ could also be challenged.Footnote ⁶

Norton provides many interesting arguments against P₂, but they stray into controversial issues in the philosophy of science and general epistemology. In his fuller answer to the Problem of Induction, Norton increasingly relies on theses that are beyond the MTI (Norton Reference Norton2014), which raises the question whether the MTI really requires so many background assumptions to be tenable. Furthermore, his arguments that the MTI does not lead to some form of inductive skepticism have not convinced many philosophers.

In section 6, I argue that Norton has access to an answer to the Problem of Induction that requires only the MTI, a plausible emendation to P₃, and a theory of direct inference. Before I discuss this answer, I turn to one final outstanding issue for the MTI, partly because addressing this issue is a requirement of the answer in section 6.

6.1. The Problem of the Reference Class

The most important task for a theory of direct inference is answering the Problem of the Reference Class. However, the name of this problem is imperfect: it suggests that the theory should identify one statistical statement about one reference class, which can be rationally included as the statistical generalization in the direct inference. As an alternative, we might adopt a theory of direct inference in which a direct inference might involve a set of conflicting statistical statements as premises. To avoid begging this question, a good theory of direct inference would provide systematic answers to these questions:

1. 1. When should we exclude statistical statements from the premises? For example, in the bismuth example, there seems to be a rough intuition that we should ignore the statistical information that substances are not generally uniform in their melting points, in favor of the statistical information that elements are generally uniform in their melting points.

2. 2. If there are conflicting statistical statements and there are no grounds for ignoring some of them, then how should we combine them? Bayesians have an account: we should look at our full probability distributions and use Bayes’s theorem. Norton (Reference Norton, Bandyopadhyay and Forster2011a) has vigorously argued for rejecting Bayesianism as a general theory of the scientific method, but that still leaves non-Bayesians like Norton and myself requiring an answer.

One might wonder why such a theory must be systematic. Perhaps our intuitive sense of appropriate or inappropriate uses of reference class data is sufficient. However, people’s intuitions (including philosophers’ intuitions) can differ for particular cases of direct inference. Setting up clear systems of direct inference enables us to evaluate and constructively critique the systems that might embody these divergent intuitions, which shifts the dispute onto more promising ground than personal feelings about specific cases. Furthermore, we sometimes make bad direct inferences, due to prejudice, wishful thinking, fatigue, and so on. A systematic theory can help us identify these mistakes. Finally, as in deductive logic, there could be complex or difficult cases when our intuitions are unclear or absent, but a systematic theory can provide definite answers that we have deduced from plausible principles.

The mere fact that a systematic theory would be desirable does not entail that it is possible. Perhaps, as with induction, Norton might suspect that context is king: there is nothing to say about direct inference that is both very general and very detailed. Certainly, there can be more or less contextually focused theories of direct inference, but there do seem to be at least some general and plausible normative claims about direct inference. For example, many philosophers (like Reichenbach and Kyburg) have endorsed the principle that, if there is a conflict between two acceptable statistical statements for a target characteristic and a given individual, and one statement describes a reference class that is known to be a proper subset of another, then that statement about the proper subset should be favored over its competitor. That is far from a general theory of direct inference, but it at least provides some systematic guidance and implies that at least some degree of systematicity is achievable. Furthermore, the claim that induction is highly context dependent is logically independent of whether direct inference is highly context dependent, just as both are independent of whether deduction is highly context dependent.

How we should decide between rival theories of direct inference? Most of our classic metalogical criteria will not be helpful. For example, we know that direct inferences can have true premises and false conclusions, and so no theory of them can guarantee truth preservation. Philosophers who have developed theories of direct inference have taken different approaches to their evaluation. One promising example is the approach to theory evaluation developed by Kyburg and Teng (Reference Kyburg and Teng2001, chap. 10). They propose to evaluate formal theories of direct inference using criteria similar to those used for the evaluation of formal deductive logics, such as metalogical soundness. However, these metalogical standards are adapted to the specific ambitions we have for our direct inferences, in contrast to deductive inferences. For example, one test that Kyburg and Teng propose is that, if a system of direct inference says that a conclusion is highly supported given the premises, then the conclusion should hold in a high proportion of the models of the relevant premises. It seems that we can rationally assess systems of direct inference and choose between better or worse systems. I now turn to how such a systematic account of direct inference could improve the MTI.

6.2. Evidential Support

I now return to the two gaps concerning evidential support in the MTI that I identified in section 3. First, there was the issue of when we can use direct inferences. Sometimes, as in Norton’s bismuth case, our background knowledge seems to be suitable to infer a local uniformity from a claim like ‘Generally, elements have uniformities in their melting points’. Yet we do not want scientists to arbitrarily accept local uniformity data to license just any inductive inference. A theory of direct inference can aide our understanding of illicit inductive reasoning, by both (a) providing standards of when local uniformity data are acceptable as licensing background information for an inductive inference and (b) providing guidance for when epistemically acceptable local uniformity principles can be ignored for inferential purposes.

Second, there was the issue of degrees of support. Take the schema:

1. 1. To an extent of r, samples with characteristics Φ are representative (within a margin of error) of Ψ with respect to a characteristic F.

2. 2. x is a sample of Φ.

Therefore,

1. 3. x is a representative sample of Ψ with respect to F.

   (Where r is either (i) a real or interval value for a proportion or measure or (ii) some qualitative qualifier like ‘generally’ or ‘almost always’. “Representative” here refers to a variety of possible senses of the term, such as having a mean value of F that is no more than 3% from the population mean.)

A theory of direct inference should provide an account of when we have warrant to use the schema above. Perhaps the value of r provides the relevant information about the degree of confirmation for the hypothesis that the observed sample in our evidence is representative, given the information in the premises. To the degree that we have warrant to believe that our samples are representative, we are warranted in making inductive inferences from evidence about the properties of those samples to our target populations. Other forms of inductive inference, such as eliminative induction and predictions about particular unobserved individuals, would derive their warrant indirectly from these generalizations from samples.

Inevitably, my proposals for addressing these gaps in the MTI are sketchy because we do not have a generally accepted “off-the-shelf” theory of direct inference. However, if we could combine a systematic theory with the MTI, then the concomitant account of induction would be much richer and less vague.

6.3. The Problem of Induction

I noted in section 4 that Norton’s claim that his MTI dissolves the Problem of Induction was controversial. However, at the end of that section, I suggested that Norton was very close to a very promising answer. As a preliminary point, note that both Norton’s critics (Okasha, Kelly, Worrall, and others) have not raised any objections to his use of direct inferences in the MTI. Even Hume takes a view of direct inference that is excessively antiskeptical: “If you suppose a dye to have any bias, however small, to a particular side, this bias, though it may not appear in a few throws, will certainly prevail in a great number” (Reference Hume1793, 112). Of course, there is no guarantee that even a very large finite sample of die rolls will match the long-run actual frequency, hypothetical frequency, or propensity of that die. Nonetheless, the quote from Hume illustrates how even skeptics about induction may reject skepticism about direct inference. Thus, Norton, Hume, and contemporary critics of the MTI have common ground in their belief that direct inference can be rational. This agreement will be crucial in my arguments in this section.

Let us return to the original setup of the Problem of Induction that Norton and his critics use. There is a dilemma between a deductive justification of induction (which is too strong) and an inductive justification of induction (which is circular). Attention to direct inference suggests that this is a false dilemma because direct inference is neither deductive nor inductive.Footnote ⁹

There is no generally accepted name for justifying induction via direct inference, but I will call it the Combinatorial Justification of Induction (CJI), following Lange (Reference Lange, Gabbay, Hartmann and Woods2011, 83). This answer to the Problem of Induction was first proposed by Williams (Reference Williams1947). It has subsequently been defended by a number of philosophers of science, including David Stove (Reference Stove1986, chap. 6), Timothy McGrew (Reference McGrew2001), and Scott Campbell and James Franklin (Reference Campbell and Franklin2004).

Norton, his critics, and Hume all grant that, at least under some circumstances, contingent local uniformity principles can license inductive inferences via direct inference. This raises the possibility of whether there is a local uniformity principle, knowable a priori, that could license inductive inferences. That would entail that, even in a hyperaustere evidential state where we have only observational evidence and a priori evidence, there could still be justified inductions.

I begin by defining a key term:

From combinatorics, we know that the size of n puts a lower bound on the number of ε-representative n-fold subsets of any finite set.Footnote ¹⁰ To see the intuition in a simple example, imagine a million-fold set of soft toys in a glass vat. If all the toys are blue teddy bears, then every 1,000-fold subset of soft toys from the vat will be ε-representative with respect to being blue teddy bears, because the proportion of blue teddy bears in both the subsets and the superset will be 100%. If none of the toys are blue teddy bears, then once again every subset will be ε-representative, because both the subset and superset will lack blue teddy bears. If 99% or 1% of the toys are blue teddy bears, then there will be a significant proportion of 1,000-fold subsets that are not ε-representative, but the number of 1,000-fold subsets that are ε-representative will still be proportionately high. The proportion of subsets that are ε-representative reaches at a minimum if exactly 50% of the soft toys are blue teddy bears. Put another way, if the proportion of blue teddy bears among toys in the vat is precisely 50%, then there is the highest possible number of subsets of that vat that are not ε-representative.

Williams, who first developed the CJI, was struck by the combinatorial fact that, even when exactly 50% of a finite set have a characteristic Φ, the proportion of large subsets of that set which are ε-representative is still extremely high. To adapt an example from Stove, if we define ε as 3%, make the anti-inductive assumption that 50% of the soft toys in the vat are blue teddy bears, and use theorems of combinatorics to calculate the approximate proportion of ε-representative 3,000-fold subsets of the toys in the vat, the answer is ${10}^{\mathrm{8,867.9}-0.00087}÷{10}^{\mathrm{8,867.9}}$. This proportion is astonishingly high: it is over 99%. Even if the subsets have a cardinality in the hundreds, the minimum proportion of ε-representative subsets can still be over 50%.

It is important to note that it is not the relative size of the subset to the superset that drives this result. Instead, the lower bound for the proportion of subsets that are not ε-representative is determined by (1) the absolute size of the subsets and (2) the fact that the superset is finite. (Infinite supersets might not even have a well-defined mean—e.g., if its limiting frequency distribution is a Cauchy distribution—so Williams’s reasoning does not apply to all of them.) Thus, these combinatoric facts are also true of 3,000-fold subsets of a googolplex-fold superset, just as they apply to 3,000-fold subsets of a million-fold superset.Footnote ¹¹

The CJI proceeds as follows: suppose that you are inquiring about a set Z, where Z is defined to be a finite set. (To use an extreme example, ‘The nearest group of toys near to me that has a cardinality no greater than the square of Graham’s Number’. The entire observable universe is far too small to digitally represent the maximum possible cardinality of this set, although the set is still finite.) Suppose that all you know about an observed sample S of Z is that it is a large (such as 500-fold or 3,000-fold) subset of Z. By direct inference:

1. 1. Generally, large subsets of Z are ε-representative.

2. 2. S is a large subset of Z.

Therefore,

1. 3. S is ε-representative.

If you justifiably believe that S is ε-representative of Z, then you can use the observed characteristics of S to infer the characteristics of Z, with a margin of error of ε. (Obviously, background knowledge can defeat this inference.) Furthermore, the larger the cardinality of S, ceteris paribus, the stronger the qualifier that can be inserted in 1. Depending on the sample S’s cardinality, one could replace the qualifier “generally” by ‘almost all’, ‘over 80%’, ‘over 99%’, and so on. Even for small samples, we have the result that as n increases from 1 toward infinity, the subset becomes a member of a class of subsets with higher and higher proportions of representative subsets of any finite set to which they belong. Therefore, even relative to ultra-exiguous background information, one can justify some inductive inferences and identify some cases of inductive evidential support.

The formal similarity to Norton’s bismuth argument is striking. Moreover, direct inference and combinatorics are very solid foundations for induction. However, if Norton were to adopt this approach to the Problem of Induction, he would have to give up an assumption he (apparently) makes, which is that the licensing local uniformity principles are always contingent. The combinatorial principles that Williams and his philosophical descendants use are mathematical and therefore noncontingent according to most philosophies of mathematics. However, they are still local uniformity principles, in the sense that they describe uniformities (the proportion of ε-representative samples) in local domains of inquiry (the target populations of the inductive inferences). Yet, since both Norton and his critics accept direct inference, under appropriate circumstances, it is hard to see why Norton would reject direct inference in this case.

Of course, there are dozens of good extant objections to the CJI. A sharp reader has probably already formulated at least three. Some of these objections have proven relatively easy for defenders of the CJI to address. For instance, it is tempting to assert (although it is very hard to argue rigorously) that samples must be random before we can make an inductive inference, but there are detailed responses to this objection (McGrew Reference McGrew2001; Campbell and Franklin Reference Campbell and Franklin2004). The persuasiveness of this objection seems to come from a confusion of sets and samples. This confusion is encouraged by many proponents of the CJI, who use statistical expressions like “samples” and “populations” rather than set-theoretic expressions like “sets” and “subsets.” Talking about samples and populations naturally raises concerns about sample selections in general and can even lead to unwarranted confidence regarding sampling procedures, but it is inessential to the CJI.

The most difficult objection to the CJI seems to be the Problem of the Reference Class. We are never in the position where ‘Generally, large subsets of Z are ε-representative’ is our only information about a sample S. Even to distinguish S and its members, we must know some characteristics that differentiate it from other things that we have experienced: we need a definite description for our sample to individuate it, such as ‘That subset of soft toys from the vat that I have in front of me right now’. Therefore, if the CJI has any epistemological use, we would also need some means of picking out a proposition of the form ‘Generally, large subsets of Z are ε-representative’ as a suitable local uniformity principle for a direct inference of an observed sample’s ε-representativeness.

This problem is more pressing, a fortiori, if the CJI is to justify of actual science. A defender of the CJI would have to combine it with an account of how the beautifully complex methods of science can be justified by the CJI and the gigantic corpus of scientific knowledge justified in turn by these methods. Perhaps these methodological accounts are possible (Williams [Reference Williams1947], chap. 5, provides a brief and telegraphic sketch), but in any detailed methodological account, there will be bodies of evidence that provide huge masses of statistical data with conflicting messages about members of multiple reference classes. Therefore, a defender of the CJI must reckon with the Problem of the Reference Class.

However, if we could have a theory of direct inference that could answer the Problem of the Reference Class, and if we could answer the other objections to the CJI, then Norton could deny P₃ in the regress argument, that ‘Local uniformity principles can only license inductive inferences if these principles are justified by antecedent justified inductions’. I do not know any good arguments for P₃, but it seems that philosophers of science have often assumed the truth of P₃ or something close to it. In practice, almost all good inductions would involve antecedently induced local uniformity principles, but there could be exceptions. Thus, Norton can adopt the following consistent group of claims:

1. 1. All justified inductive inferences must be licensed by local uniformity principles (P₁ of the regress argument).

2. 2. All local uniformity principles must be justified by earlier inferences (P₂ of the regress argument).

3. 3. Local uniformity principles can only license inductive inferences if these principles are justified by antecedent justified inductions or by direct inferences (the revised version of P₃ in the regress argument).

4. 4. If there are any justified inductive inferences, then there must have been a first justified inductive inference (P₄ of the regress argument).

5. 5. There are some justified inductive inferences (the denial of inductive skepticism).

A skeptic might object that the CJI involves such artificially exiguous background knowledge that any relation to real inductive reasoning is absent. In practice, we often have reasons (beyond combinatorial considerations) to suspect that our samples are unrepresentative: for example, I know that an opinion poll on the Dodd-Frank financial regulations, using a sample consisting only of stock traders at the New York Stock Exchange, will almost certainly be unrepresentative of US public opinion in general. Yet one of the insights from Norton’s studies of induction is that more realistic scenarios can be more favorable to inductive reasoning, because we can have background reasons (beyond combinatoric reasoning) to think that in some cases our samples are representative, as well as evidence to suspect them in other cases. For example, we might have an opinion poll that has been carefully weighted using pollsters’ background knowledge of response rates and other relevant information, such that it is likely to be representative. The CJI would be useful for the MTI because it would prove that even if such favorable background knowledge were absent, there could still be justified inductive reasoning.

Thus, direct inference is very promising against inductive skepticism in both (1) the epistemically rich contexts Norton analyses, where the MTI provides a rationale of inductions’ justifications, and (2) the exiguous contexts that philosophers like Williams contemplated, where the CJI might provide the rationale. However, all this assumes that direct inference is not itself as troubled as induction. That is why a theory of direct inference has such value.

Would Norton consider my arguments to be helpful support in his battle against inductive skepticism? The CJI avoids the abstract and sweeping formalistic claims that Norton rejects (it uses defeasible direct inferences and the local properties of specific populations) while also providing a means of ending the regress of local uniformity principles in some contexts. This last qualification is important, because the CJI’s dependence on direct inference entails that it is defeasible: additional background information can block the relevance of purely combinatoric considerations. However, in the hyperexiguous contexts that the critics of Norton employ in their skeptical arguments, it is supposed (by design) that there is no background information available (beyond what is knowable via mathematics, logic, and other conceptual reasoning) since his critics aim to bar any appeal to contingent background information about samples’ representativeness. Ironically, it is the very austerity of the skeptical scenario that makes the CJI so promising for a benign termination of the regress in the MTI. The CJI’s usefulness for answering the Problem of Induction will depend on how we conceptualize that problem; my point is that even if we formulate the challenge in the manner of Norton’s critics, the CJI is a promising option within the MTI.

Additionally, amending P₃ as I have suggested would put the MTI in a strictly stronger dialectical position with respect to the Problem of Induction, because Norton’s present responses to his critics (as well as other criticisms of P₂, P₃, and P₄) would still be available, but the CJI would also be available. The CJI might not be viable: as noted, there are many objections to it, especially regarding the finer points of direct inference. Still, there is nothing in the letter of the MTI that is inconsistent with the CJI and much that could be gained from their concatenation.