1. Introduction

Suppose there is good evidence for assuming that Mary has gone swimming (p): She has told me so yesterday. However, there is even better evidence that she has not gone swimming, but is in her office instead (q): The office door is wide open—as I can clearly see. But there is even better evidence that she is not in her office, but in the copying room (r): I have just met Jack, who never ever lies and who is very reliable; he tells me that he's just seen Mary at the copying machine.

Let us also assume that the quality of the evidence is the only thing that counts with respect to the rational acceptance of the three propositions in the example. We can then say that if the person has better evidence for one proposition (q) than another (p), then she has better reasons to accept q than to accept p. One could also say: She is better justified in accepting q than in accepting p. Let ‘ $R(q,\,p) $ ’ stand for ‘the person [e.g., me in the example above] has better reasons, all things considered, to accept q than to accept p.’ In our example the following is true:

There seems no doubt whatsoever that this implies $R(r,\,p) $ . Our example thus suggests a general principle about reasons (or justifications) for accepting propositions (or sets of propositions, or theories) as true:

(T) For all propositions (or sets of propositions, or theories) p, q, and r:

One can call this the principle of the ‘transitivity of reasons’.Footnote ¹

(T) seems to work in everyday cases, as well in cases of theory choice. In the latter case, R is a relation that holds between two theories A and B (R(A, B)), if and only if A is (overall) a better theory than B—taking all relevant criteria of theory evaluation into account. The relevance of those criteria is, of course, highly controversial; however, the argument in this paper does not presuppose any particular view on relevance; we can thus leave this topic aside here. The argument that follows is so general in nature that it holds (if, indeed, it does hold) no matter what one's particular views about rational acceptability of theories (R) are. We can thus also leave this topic aside here.

Suppose there are three competing theories under discussion in a scientific community. Let us assume for a short moment that the quality of the evidence that speaks in favor of a theory is the only thing that matters when it comes to rational acceptability of theories. After some time, the scientists come to the conclusion that theory 1 is better supported by the evidence than theory 2, and that theory 2 is better supported by the evidence than theory 3. It seems that there is no further discussion necessary about which theory to accept or choose: Theory 1 must also be better supported by the evidence than theory 3. Hence, it must be the best theory in this situation (given, of course, that only the quality of the evidence counts).

(T) looks like an essential aspect of the rational acceptance of the rational belief in propositions, sets of propositions, or theories. Apart from all that, (T) looks like an (almost) trivially true statement about the nature of rational epistemic choice. But, it is not. I will argue here that it is neither trivial nor true. This goes against a basic assumption of decision theoretic approaches to epistemic rationality.Footnote ² In the following, I will restrict myself to the case of theory choice.Footnote ³ I cannot present a case study here (this would require another paper), but instead, I only intend to present a general argument. It will proceed in two steps as detailed in Sections 2 and 3.

2. First Step: The Basic Idea

There is—let us assume—more than one criterion for the evaluation of scientific theories: support by empirical evidence counts a lot, but so does explanatory power, simplicity, and other factors.Footnote ⁴ Assume further, that the plurality of criteria is irreducible such that none of them can be explained in terms of one or more of the other criteria. Let us, for the sake of argument, also assume that the three criteria just mentioned are all the relevant criteria, and that they are independent in the sense that the ‘quality’ of a theory in one dimension does not imply anything (or at least not very much) about how good the theory is with respect to the other dimensions of evaluation.Footnote ⁵ Suppose that three theories—call them ‘A’, ‘B’, and ‘C’—are under discussion in a scientific community. Let us further suppose that the following is also true. A can clearly explain more than C, and it can explain everything C explains, but in a much more convincing way. On the other hand, C is clearly better supported by the data and not as complicated (a theory) as A. Which of the two theories should one accept?

A lot depends on how one weighs the different criteria against each other: How important, for instance, is explanatory power in comparison with support by data? One rather extreme view would have it that there is a lexical ordering of different criteria. This seems pretty unrealistic, though. Another extreme view would say that there is no way to compare the relative importance of the different evaluative criteria relevant for theory choice; this would lead to some kind of incommensurability.Footnote ⁶ However, we can resist this view. There might be no context-independent and general ways of weighing the different criteria against each other, but in at least many particular situations of scientific theory choice, scientists seem able to clearly weigh the criteria against each other (cf., e.g., Sankey Reference Sankey1994, 176). Only for the sake of simplicity, will I assume for a moment that in our example that the three criteria are equally important (nothing essential hinges on this assumption; I will later skip this assumption).Footnote ⁷ All this suggests is that the scientists have better reasons overall to prefer C to A. This might change over time, as new data come in, or more sophisticated versions of the original theories are developed; in the particular situation at hand, however, C seems to be the ‘better theory’ (as we can also say).Footnote ⁸

Let us now assume that the following is true about theories A, B, and C. With respect to explanatory power, A is better than B, and B is better than C. With respect to support by data, B is better than C, and C is better than A. With respect to simplicity, C is better than A, and A is better than B. We can illustrate this in the following way (the first, second, and third row gives us the best, second best, and worst theory, as evaluated by the relevant criterion):

Theory Ranking	Explanatory Power	Support by Data	Simplicity
Best	A	B	C
Second best	B	C	A
Worst	C	A	B

Given our assumptions, we would have to say that all things considered, C is a better theory than A (it ‘beats’ A 2:1 on the relevant criteria). Also, A is better than B, and B is better than C. In other words:

This, however, clearly goes against (T). Since it is hard to see why there should be something wrong with the individual evaluations, it seems to follow that the relation is a better theory than is not transitive (with ‘better’ understood as ‘better all things considered’). There is no best theory in this context. Whichever theory one chooses, there always seems to be a better theory on offer. I have assumed pluralism in the sense that there is an irreducible plurality of criteria without lexical ordering. If one wants to keep the question of pluralism open, one would have to replace the thesis above by a weaker, conditional thesis: If pluralism is true, then rational theory choice does not obey the transitivity rule.

3. Second Step: Further Aspects and Some Objections

One could object that the above comparison of the three theories neglects the possibility that, e.g., the difference in explanatory power between A and C might be much ‘bigger’ or should weigh much more than their difference with respect to support by empirical data. Should one not take such differences into account? It is not clear whether this makes sense at all: Can we, for instance, measure different degrees of explanatory power of scientific theories, and compare it with differences in simplicity? Let us, for the sake of the argument (or, for the sake of the objection) assume that we can (more or less roughly) measure different degrees to which a theory can meet a given criterion. Let us further—and also for the sake of the argument—assume that there are just two relevant criteria: explanatory power and simplicity; we can assume that our three theories do not differ with respect to how well they are supported by evidence. Would transitivity of rational choice always hold under these conditions? I will argue that the answer is ‘no’.

Suppose there are three competing theories: D, E, and F. D explains things much better than E, but is also more complicated. However, the difference in simplicity is not so big that it really counts much, whereas the difference in explanatory power really matters. Hence, all things considered, D is better than E. Similar things hold for E and F: E is a bit more complicated, but explains much better than F. Hence, E is better than F. What about D and F? Suppose F is clearly and remarkably simpler than D—so much simpler that it really matters. It matters more than the difference in explanatory power. Hence, F is better than D, all things considered. Again, we have an intransitive ranking among our three theories. There is no best theory, and for any given theory from our three competing theories, there is always a better theory. Again, there is no reason to believe that there is something wrong with the different evaluations of the theory.Footnote ⁹ Or, to put it in terms of our weaker, conditional thesis: If pluralism is true, then there is rational acceptability of intransitive rankings of theories.

One might want to propose the following way out of intransitive circles. In a first step, one determines for each of the different criteria how ‘good’ a given theory is with respect to that criterion. One could do this in a quantitative way and assign real numbers that measure the ‘goodness’ of a theory in one dimension (defined by one of the criteria of evaluation).Footnote ¹⁰ Assuming that the different criteria constitute different dimensions of one and the same quality of ‘goodness’ of a theory, one can then compute the overall ‘goodness’ of a theory with respect to the different criteria. Since one can do all this for each of the theories under review one can in the end compare the (absolute) ‘goodness’ of different theories in quantitative terms.

There are many problems with this proposal. One is quite simple: It assumes monism, that is, the idea that in the end there is only one criterion that matters, namely ‘goodness’. Apart from the doubts concerning monism mentioned above, there is an additional problem with this proposal: It is not clear at all what ‘goodness’ is supposed to be (if it does not reduce to the different aspects of evaluation captured by the different criteria). It does not seem to be an independent, substantial criterion at all. Whenever we say that a given theory is a good one, we have to specify criteria of evaluation with respect to what makes it a good one. But, then the alleged way out of intransitivity leads us back to where we started.

Another way to avoid intransitivity of rational theory choice would be to introduce some kind of ‘metaevaluation’ of the criteria of theory choice. I have assumed here that no criteria are mistaken in any way. Metaevaluation could then only be about the relative importance of the different criteria. However (as should be plausible by now), the argument presented here does not only not exclude or ignore this aspect, but rather takes it into account from the start (see above). Larry Laudan (cf. Reference Laudan1987, Reference Laudan1990) defends a ‘normative naturalism’ according to which we can resolve deep methodological disputes by engaging in some metaevaluation of the different methods used. According to Laudan, one method is better than another one, if it does a better job in reaching its cognitive aim. Be it as it may—Laudan's point concerns the evaluation of methods of inquiry, and not of criteria of evaluation of theories (which is something different). Hence, it does not seem to make sense to extend Laudan's metamethodology to the evaluation of criteria. Apart from that: What constitutes a valuable cognitive aim depends on the correct criteria of evaluation. Laudan's metamethodology takes the latter as given—which we cannot do here if we want to engage in the proposed metaevaluation of criteria.Footnote ¹¹

What now could explain the (rational) possibility of intransitive rankings of theories? Consider our initial example about Mary: It does not speak against (T). We can now see why: There was only one criterion of evaluation. In the case of theories D, E, and F, however, there are two independent criteria: simplicity and explanatory power. Furthermore, at least one of these criteria (simplicity) is ‘nonlinear’ in the sense that small differences (say between theories F and E, or E and D above) do not matter much (or not at all) to the chooser, but as they add up (see the case of theories F and D above), their importance for theory choice might suddenly be much bigger.Footnote ¹² There might be thresholds of complexity: Small differences of complexity might be completely negligible up to a certain point; if they are even slightly greater than the threshold value, then they might suddenly make all the difference (or at least a huge difference). But even without such thresholds, the function that maps degrees of complexity onto degrees of ‘goodness’ of a theory, with respect to simplicity, might not be linear.Footnote ¹³

In other words: If there are two or more relevant criteria for the evaluation of theories, and if at least one of those criteria is nonlinear, then transitivity of rational theory choice (i.e., (T)) might be violated. Multiplicity of criteria is necessary, but not sufficient, for the rational acceptability of some intransitive preferences among theories. The same holds for nonlinearity: It is necessary for it, but not sufficient. Pluralism and nonlinearity are individually necessary, and jointly sufficient conditions, for the rational acceptability of some intransitive preferences among theories. I am not saying that rational theory choice will always violate transitivity (according to (T)) if those two conditions are given; the point is, rather, that it can (but need not) be rationally acceptable to violate transitivity, given those conditions. It is very realistic to assume that there is more than just one independent criterion for the evaluation of scientific theories and that there is no lexical ordering of criteria. It is also realistic to assume that at least some of those criteria would be nonlinear. Hence, we have good reasons to doubt and reject (T). Or, if one wants to keep the question of pluralism open: If pluralism is true, then rational theory choice does not obey the transitivity axiom (T).

The upshot is this: The violation of transitivity, as explained by (T), might be a common phenomenon in science—whenever there is a choice between more than two competing theories. More important even: Such transitivity might be violated in such a way that there is no reason at all to say that there is something wrong or irrational about the way the scientists evaluate the different theories. The life of science could just be like that. In some cases (not all, of course), rational choice between theories violates the transitivity assumption. And, similar things can presumably be said about cases from everyday life, at least if they show a certain complexity and are not as simple as the initial example about Mary.Footnote ¹⁴

One final objection: Have I not completely ignored the normative question of how someone who chooses among several theories ought to choose? Should I not say something about how a rational chooser ought to weigh the different criteria of theory choice? Isn't there, for instance, a normative question about whether a given criterion ought to be linear or not (and if not: in what ways it ought to be nonlinear)? I have indeed not dealt with this whole normative dimension. Fortunately, this does not do any damage: Take any normative view about theory choice, and the same problem with intransitivity will come up within that framework (given criteria pluralism). Hence, we can leave the whole normative dimension aside here.

4. Conclusion

But how is rational comparison between theories or between propositions possible if all that is true? If the possibility of a rational answer to the question, ‘Which theory should we choose?’ presupposes that there is always a best theory (or a group of equally good theories such that no theory is better), then there cannot always be a ‘rational’ answer in that sense. However, if what I have said above is true, then we should not be too obsessed with the idea that there must always be a best theory. What matters is that the theories we choose from are good enough. Furthermore, the phenomenon of intransitivity might be a passing one: As new additional evidence comes up—to mention just one dimension of evaluation—the intransitivity might disappear. Intransitivity might be an unstable phenomenon; it is hard to think of a case in the history of science in which intransitivity did not disappear after a relatively short time.

All this does not lead to Kuhnian incommensurability. It could still be true that for all theories (or, alternatively, propositions) A and B:

where E stands for ‘is equally good a theory as’. But in another respect, one would probably be tempted to agree with Kuhn: Theory choice—or, more generally, the acceptance of certain propositions—is not (always) a purely rational process, or just the application of some algorithm. It also includes more ‘pragmatic’ elements (not just epistemic criteria): what appeals to a group of researchers at a time, and what they can manage to persuade each other of.