A Ranking-Theoretic Account of Ceteris Paribus Conditions Wolfgang Spohn Presentation at the Workshop Conditionals, Counterfactual and Causes In Uncertain Environments Düsseldorf, May 20 22, 2011
Contents Four Readings of Ceteris Paribus Conditions Four Problems for Explaining Normal Conditions Eight Interpretations of Normal Conditions (1) - (2) Going for the Epistemic Interpretation Very Brief Basics of Ranking Theory (1) - (5) Explicating Normal Conditions: the Set-up (1) - (2) Normal and Exceptional Conditions (1) - (3) The Final Explication (1) - (3) Concluding Remarks (1) - (3) Bibliographical Note Spohn, Ceteris Paribus Conditions 2
Four Readings of Ceteris Paribus Conditions That something a claim or assertion, a law, etc. holds ceteris paribus may mean four things: that it holds other things being equal (equal to what?); then it means something like the general law of causality: equal causes, equal effects, or equal conditions, equal outcomes whatever equal means here, other things being absent; then it assumes that other relevant variables have or take a natural zero value, other things being ideal; then the topic should rather be dealt with under the heading idealizations in science or other things being normal; this is perhaps the most common reading that finds the largest interest; in any case, it is the only one I will consider here; so, for the purposes of this talk ceteris paribus = normal conditions. Spohn, Ceteris Paribus Conditions 3
Four Problems for Explaining Normal Conditions Most explanations of cp (= normal) conditions founder at least at one of the following problems: that according to the explanation (1) a cp claim is too vague to be useful, (2) a cp claim is too easily true, (3) in particular, both cp, F s are G and cp, F s are non-g are true, (4) a cp claim is too easily false. Spohn, Ceteris Paribus Conditions 4
Eight Interpretations of Normal Conditions (1) That something a claim or assertion, a law, etc. holds given normal conditions has received at least the following eight interpretations: the natural interpretation: cp, F s are G means usually or mostly, F s are G. This has problem (1). the statistical interpretation: tries to replace the vague usually by a precise statistical probability. However, everybody rejects this idea; cp claims are not statistical. Moreover, usually means usually for us and is hence indexical and hardly amenable to statistics. the eliminativistic interpretation: cp, F s are G is to be replaced by under conditions C, all F s are G. This has problem (4); under all replacements actually proposed cp claims thereby turn out false. There is only a hope that one day one might find the right conditions. the trivial interpretation: cp, F s are G means F s are G, unless they aren t. This has problem (2). Spohn, Ceteris Paribus Conditions 5
Eight Interpretations of Normal Conditions (2) the existential interpretation: cp, F s are G means there are suitable conditions C under which all F s are G. Most efforts went into this interpretation and in particular into specifying suitable. However, all proposals have problems (2) and (3). the causal interpretation: cp, F s are G means being F is a partial cause of, or plays a causal role for, being G. Has again problems (2) and (3). Also, one might say, if that s the right understanding, then better stop trying to explaining cp and turn to causal analysis. the conditional interpretation: that interprets the conditional in cp, F s are G as some kind of non-monotonic conditional (there are various options). the epistemic interpretation: the unusual or exceptional is the unexpected, and the usual or normal is the not unexpected. Hence, cp, F s are G roughly means under conditions which are not expected to be violated, all F s are G. Spohn, Ceteris Paribus Conditions 6
Going for the Epistemic Interpretation The epistemic interpretation avoids problem (1), as we shall see, problem (3), obviously, and problems (2) and (4), though apparently in too radical a way, namely by refusing (objective) truth values to cp claims and rendering them as expressions of our beliefs or expectations. I shall discuss the point at the end of my talk. In the epistemic interpretation, cp clauses referring to normal conditions somehow express, and are tied to, conditional belief. Therefore we should first consider conditional belief. My quarrel with the conditional interpretation is this: it is burdened with, but remains indeterminate about the mysteries of the interpretation of the conditional, and if it determinately takes the epistemic side, it does not use the most preferable account of conditional belief. Spohn, Ceteris Paribus Conditions 7
Very Brief Basics of Ranking Theory (1) For, the most preferable account of conditional belief is offered by ranking theory. Here are the required basics: Ranking theory is a theory of degrees of belief and thus of inductive inference or defeasible reasoning. In contrast to probability theory it is able to adequately represent not only belief or acceptance, but also conditional belief or acceptance. Thereby it is able to provide a full dynamic theory of belief. These basic features have widely ramifying consequences, and one will be in the analysis of normal conditions, as I will be going to explain. Spohn, Ceteris Paribus Conditions 8
Very Brief Basics of Ranking Theory (2) Let W be a possibility space or a set of possible worlds and A be a complete algebra of propositions over W. κ is a (negative) ranking function iff κ is a function from W into N + = N { } such that κ(w) = 0 for some w W. Alternatively, we can define κ as a set function. Then κ is a (negative) ranking function for A iff κ is a function from A into N + such that the following axioms hold: (a) κ(w) = 0 and κ( ) =, (b) κ(a B) = min (κ(a), κ(b)), [the law of disjunction] (b') for any B A κ( B) = min {κ(a) A B} [the law of (infinite) disjunction], (c) κ(a) = 0 or κ(~a) = 0 [the law of negation]. If the point function is A-measurable, the two definitions come to the same via the relation: κ(a) = min {κ(w) w A} for all A A. Spohn, Ceteris Paribus Conditions 9
Very Brief Basics of Ranking Theory (3) Ranking functions are gradings of disbelief. κ(a) > 0 says that A is disbelieved or taken to be false (to some degree), κ(a) = 0 says that A is not disbelieved (and perhaps believed). κ(a) = 0 and κ(~a) = 0 represents suspense of judgment w.r.t. A. κ(~a) > 0 says that A is believed or taken to be true. Axioms (a) and (b) entail that belief is consistent and deductively closed. One might also define a positive ranking function β by β(a) = κ(~a) expressing belief directly. However, negative ranking functions are the theoretically most fruitful version. Spohn, Ceteris Paribus Conditions 10
Very Brief Basics of Ranking Theory (4) If κ(a) <, the conditional rank of w given A and of B given A is defined as κ(w A) = κ(w) κ(a) if w A and = otherwise, and as κ(b A) = min {κ(w A) w B} = κ(a B) κ(a). Thus, κ(~b A) > 0 represents the conditional belief in B given A. This definition is the crucial advance of ranking theory over its predecessors (such as Shackle s functions of potential surprise as also used by Levi or Cohen s Baconian probability) and in my view the adequate representation of conditional belief entailing all the further strengths of ranking theory. Spohn, Ceteris Paribus Conditions 11
Very Brief Basics of Ranking Theory (5) The definition of conditional ranks is equivalent to (d) κ(a B) = κ(a) + κ(b A) [the law of conjunction]. One might also axiomatize ranking theory by that law of conjunction and (e) κ(b A) = 0 or κ(~b A) = 0 [the conditional law of negation]. This shows that ranking theory assumes nothing but conditional consistency besides the definition of conditional ranks. If one recalls how successfully the dynamics of subjective probabilities can be explained on the basis of the notion of conditional probability, then it is plausible that an equally successful dynamics of belief can be explained on the basis of the notion of conditional ranks. Spohn, Ceteris Paribus Conditions 12
Explicating Normal Conditions: the Set-up (1) Although cp talk seems to speak about open-ended possibilities consisting of don t know-whats, we need to refer to fixed conceptual frame, a fixed set W of possibilities (= small worlds) and a well-defined algebra A over W; otherwise theorizing cannot start. More precisely, let A be generated by (random) variables on W, as is familiar from probability theory. Specifically, we will have to refer to two target variables X and Y and one (comprehensive) background variable Z (that may decompose in many component variables). So, we may conceive of small worlds as triples x, y, z of possible values of X, Y and Z. X, Y, and Z are only about a given single case, a specific application. Of course, we can generalize by repeating the set-up (just as in probability theory, where we generalize from a single throw of a die to an infinite series of throws). Here, we may focus only on the single case. Spohn, Ceteris Paribus Conditions 13
Explicating Normal Conditions: the Set-up (2) We have, i.e., (tentatively) believe in a hypothesis H (a law) about the relation between the target variables X and Y. That is, H is a proposition that is X-Y-measurable (i.e. only about X and Y), but neither X- nor Y-measurable (i.e. it makes no claims about single variables). We also think that the relation between X and Y somehow depends on the background variable Z. We will have to study how. In any case, our beliefs or rather our doxastic attitude concerning this set-up are represented by some ranking function ξ for A. That the hypothesis H is conjectured according to the ranking function ξ means that H is the strongest proposition about X and Y believed according to ξ, i.e., ξ(~h) > 0 and for no X-Y-measurable H' H ξ(~h') > 0. Spohn, Ceteris Paribus Conditions 14
Normal and Exceptional Conditions (1) The basic observation now is this: Even if the hypothesis H is unconditionally believed in ξ, i.e. ξ(~h) > 0, it need not be believed under all conditions. There will be a background condition, i.e., a Z-measurable proposition N that is unsurprising or not excluded; that is, ξ(n) = 0. In this case, we still have ξ(~h N) > 0; the hypothesis H is still held under such a condition N, which we may provisionally call a normal condition. However, there might also be a condition E about the background Z, given which the hypothesis H about X and Y is no longer held, i.e., such that ξ(~h E) = 0. This entails ξ(e) = ξ(~h E) ξ(~h); i.e., such a background E must be at least as disbelieved as the violation of H. We may therefore provisionally call E an exceptional condition with which we do not reckon according to ξ. Spohn, Ceteris Paribus Conditions 15
Normal and Exceptional Conditions (2) So far I have suggested: (1) The condition N for the variable Z is normal iff ξ(n) = 0, (2) the condition E on Z is exceptional iff ξ(~h E) = 0. One should think that E and ~E cannot both be exceptional. This is implied by (2). One should allow that N and ~N are both normal. (1) does this. One should think that E is exceptional iff ~E is normal. (1) and (2) do not satisfy this. Therefore one might suggest to replace (1) by: (3) The condition N on Z is normal iff ξ(~h N) > 0. However, this closely resembles the trivial interpretation of normal conditions. Spohn, Ceteris Paribus Conditions 16
Normal and Exceptional Conditions (3) Moreover, we are used to speaking of the normal conditions and the exceptional conditions. This might mean, respectively, the weakest normal and the weakest exceptional condition. Here, we find that (2) and (3) imply: (4) if N and N' are normal, N N' is normal, too, (5) if E and E' are both exceptional, then E E' is exceptional, too. It might also mean, respectively, the strongest normal and the strongest exceptional condition. However, (2) and (3) do not guarantee that: if N and N' are normal, N N' is normal, too, if E and E' are both exceptional, then E E' is exceptional, too. Rather, we might have ξ(~h E E') > 0. Then, E E' would be normal according to (3), while intuition would like to call E E' doubly exceptional, because the judgment about H is reversed twice. Spohn, Ceteris Paribus Conditions 17
The Final Explication (1) These considerations allow me to introduce the explication I want to finally endorse: First, we may use (5) to define: (6) E 1 = {E E is exceptional} as the (weakest) exceptional condition, and correspondingly, (7) N 0 = ~E 1 as the normal condition (still to be justified). Next, I have suggested: (8) E E 1 is doubly exceptional iff ξ(~h E) > 0, i.e., iff the hypothesis H is in turn supported by E. Then, we may use (4) to define: (9) E 2 = {E E 1 E is doubly exceptional} as the (weakest) doubly exceptional condition. And so on. Spohn, Ceteris Paribus Conditions 18
The Final Explication (2) So, the general proposal is this: (10) We set E 0 = W, the whole possibility space considered. For odd n we define E n = {E E n-1 ξ(~h E) = 0}, for even n we set E n = {E E n-1 ξ(~h E) > 0}, and we call E n the exceptional condition of degree n. (11) E n = E n E n+1. Thus, N 0 defined in (7) is the same as E 0. The sequence E 0 = N 0, E 1, E 2, forms a partition of W. Let [z] denote the proposition that the background variable Z takes the specific value z. [z] may be called a maximal condition. The sequence thus provides a classification of maximal conditions: the maximal condition [z] is exceptional to degree n (and normal if n = 0) iff [z] E n. Spohn, Ceteris Paribus Conditions 19
The Final Explication (3) Thereby, we finally arrive at a classification of all background conditions: (12) A proposition or condition E about Z is exceptional to degree n iff n is the minimal degree of exceptionality among the maximal conditions [z] E. If n is even, we have ξ(~h E) > 0, and if n is odd, we have ξ(~h E) = 0. A condition N that is exceptional to degree 0 is normal. Thus, the normal condition N 0 is believed in any case, while a normal condition N may even be disbelieved. Moreover, we have: (13) [z] N 0 iff ξ([z]) < ξ(~h). (14) 0 = ξ(e 0 ) ξ(e 1 ) ξ(e 2 )..., and for even n: ξ(e n+1 ) > ξ(e n ), for odd n: ξ(e n+1 ) max {ξ([z] [z] E n } ξ(e n ). Spohn, Ceteris Paribus Conditions 20
Concluding Remarks (1) This proposal is an explication of the epistemic interpretation of ceteris paribus or normal conditions. It escapes problem (1), i.e., it is precise, because the underlying formal epistemology is precise. It avoids problem (3), since the hypothesis H and its negation cannot be held true at the same time. It avoids the trivial interpretation. The normal condition N 0 does not simply contain all the maximal conditions [z] under which H is held true. It is stronger by excluding the doubly, fourfold etc. exceptional maximal conditions under which H is also held true. Indeed, this proposal is, as far as I know, the only one that can deal with the phenomenon of multiply exceptional conditions. Spohn, Ceteris Paribus Conditions 21
Concluding Remarks (2) If we would refer the other interpretations, in particular the trivial, the eliminativistic, the existential, or the causal one, to a fixed setup, i.e., a fixed background Z, their inadequacy is salient. They derive their prima facie plausibility from their reference to open background conditions, which, however, are theoretically unmanageable. By contrast, the proposed epistemic interpretation has something substantial and reasonable to say already given any fixed background (that may then be assumed to hold for open backgrounds as well). The epistemic interpretation entails that a ceteris paribus law does not make a specific claim, i.e. has no objective truth condition; the notion of normality it involves is epistemically relativized. This need not make it disrespectable. Indeed, I think this is how it is with laws in general (but this is another issue). Spohn, Ceteris Paribus Conditions 22
Concluding Remarks (3) Still, there are certain beliefs associated with a ceteris paribus law or with normal conditions. In the set-up X, Y, Z, ξ, we first have ξ(~n 0 ) > 0, i.e., the normal condition is held true. The second characteristic belief is ξ(~h N 0 ) > 0, i.e., the conditional belief in the hypothesis given the normal condition. Both entail the belief in H & N 0. Moreover, the belief in N 0 can be true or false, the normal condition may actually obtain or not. The conditional belief in H given N 0 is usually not something that can be true or false. There is, however, an objectification theory for ranking functions, which explains the extent to which conditional beliefs can be (objectively) true or false. Hence, the proposed account of ceteris paribus laws need not be stuck in the epistemic relativization. Spohn, Ceteris Paribus Conditions 23
Bibliographical Note The talk is taken from chapter 13 of my book The Laws of Belief. Ranking Theory and Its Philosophical Applications (ca. 770 pp.), to appear at OUP. Spohn, Ceteris Paribus Conditions 24