Closer to the truth (2): The ugly duckling problem

As we saw in my last entry, Karl Popper suggested around 1960 a way of formalising the idea that science can proceed towards the (full) truth through a series of theories or hypotheses all of which are literally false. Basically, his proposal was to define the relation of ‘theory A being at least as close to the full truth than theory B’ as meaning that all true consequences of B are consequences of A, and that all false consequences of A are consequences of B. We also mentioned the proof, a few years later, of a simple logical theorem showing that this formal relation cannot hold if at least one of the two theories, A or B, is false, hence destroying the very raison d´être of Popper’s precise definition. Sir Karl’s disappointing reaction to the discovery of the theorem was just saying that a precise logico-mathematical definition of the notion of verisimilitude was not necessary for his view of science, but that only an ‘intuitive’ idea of ‘false theories being progressively closer and closer to the truth’ was enough.

Many philosophers, of course, didn’t think the same: if a claim as essential to the notion of scientific realism as that of ‘closeness to the truth’ can only be understood as a vague, pre-theoretical concept, then the position of scientific realists was in real philosophical danger. Hence, it is natural that a number of (I would say, ‘formalism-friendly’) philosophers of science put themselves to the job of assembling some logical explication of the concept of verisimilitude (or truthlikeness, as it soon started to be more commonly known, instead of by the other, more ambiguous Popperian term), one explication that didn’t suffer the kind of problems Popper’s naive definition had.

Basically all the many approaches that erupted in the truthlikeness literature, mostly within the last quarter of the past century, can be grouped into two main categories. On the one hand there is what has been called the consequence approach, which was more faithful to Popper’s original idea: keeping the general view that a proposition A is closer to the truth than another proposition B if all the true consequences of B also follow from A, and all the false consequences of A also follow from B, but simultaneously trying to solve the logical inconsistency of Popper’s definition by restricting the set of consequences that are relevant. The idea, hence, is not considering all the propositions that logically follow from theories A and B in order to determine if one of them is closer to the truth, but only a proper subset of those propositions. A big family of proposals fell under this approach, some of them restricting the comparison between A and B to the ‘axioms’ that defined each theory (i.e., understanding scientific theories as axiomatic systems, like Newton dynamics can be understood as the conjunction of Newton’s three force laws plus the gravity law and some more specific propositions for specific cases, or as Maxwell electrodynamics as the conjunction of Maxwell laws, etc.), others restricting the formal complexity of the consequences (of theorems) derivable from the axioms of A or B, others restricting the types of inferences that are relevant (using, for example, what is known as relevance logics). I cite in the references the papers by Miller ¹ and Cevolani & Festa ² just as some prominent examples.

The other big path chosen to define a notion of truthlikeness was… well, what became known as the likeness approach (or similarity approach). In this case, the conceptual elements employed to work out the definition were not propositions, but what we can call possible states of the world (or, more simply, ‘possible worlds’: descriptions of the world as complete and exhaustive as our language -or ‘expressive capacity’- allows; of course, each individual ‘possible world’ is incompatible with any other one, whereas some arbitrary propositions can be mutually compatible). Under this framework, a proposition is not directly understood as a linguistic item, but as a collection of ‘possible worlds’, i.e., those set of ‘worlds’ that are such that, if some of them happened to be the real (or true) state of the world, then that proposition would be true. Under this paradigm, the notion of the whole truth is replaced, not by the logical conjunction of all the propositions that happen to be true, but as the one and only possible world in which all those propositions are simultaneously valid. I shall call it the real world, which is a point in the logical space composed of all the possible worlds (describable with a given formal language).

This ‘possible worlds’ framework had been devised in the sixties and seventies as a semantics for modal logic, and as a way of understanding the meaning of counterfactual conditionals (like, ‘if the moon were twice as big as it is, the tides would be four times as high as they are’: the meaning of such statement might be understood as something like ‘amongst all the possible worlds in which it is true that the moon is twice as big as it is in the real world, the possible world which is most similar to the real world is such that in it the proposition that says that the tides are four times as high as they are in the real world is true’). The application of this idea to the problem of verisimilitude was apparently straightforward: a scientific theory can be represented as a set of possible worlds, and hence, theory A being closer to the truth than theory B can ‘simply’ be understood as something like ‘the possible worlds in which A is true are more similar to the real world than the possible worlds in which B is true’. Ilkka Niiniluoto ³ and Graham Oddie ⁴ were the main champions of this approach.

Complications soon started to grow, however. First of all, even if we had a clear way of determining when is an individual possible world ‘more similar’ to another as a third one (which, as we shall immediately see, is not easy at all), how are we to define whether a set of worlds is more or less similar? Oddie’s idea was simply to use the average similarity to the real world of the members of that set. But Niiniluto argued that this unjustly favoured theories that were true in many worlds (if these happened to be closer to the truth), instead of following the Popperian maxim of favouring theories that were ‘bold’, in the sense of discarding many possible situations, and hence being more falsifiable. Niiniluoto, hence, preferred to define the truthlikeness of a theory A as a function of two things: the minimal distance to the real world amongst the possible worlds compatible with A, and the sum of the distances to the real world of all the possible worlds compatible with A (so that, the greater that sum, the lower A’s truthlikeness). This philosopher also introduced the (epistemic) notion of estimated truthlikeness of a theory, as the statistical average truthlikeness a theory would have when each possible world is considered with the probability of being the real world our empirical evidence attributes to it. Unfortunately, no significant progress was made in the task of finding out ways of actually calibrating the estimated truthlikeness of real scientific theories, nor of proving at least some methodological theorems that might assert something like ‘under such and such circumstances, theory A would have a higher estimated truthlikeness than theory B’.

But, as I said, a deeper problem stood on the way of the likeness approach, referring to the very notion of similarity between individual possible worlds. This relates to what in mathematics was known as the ugly duckling theorem ⁵ stating more or less than a swan is as similar as a duck as any duck is. Imagine we only have two properties to describe a system: P and Q. An object can be described in four different and incompatible ways: it is P and Q, or P and not-Q, or not-P and Q, or not-P and not-Q. Imagine we have three objects, X, Y and Z, such that X is P and Q, Y is P and not-Q, and Z is not-P and not-Q. Obviously, we’d say Y is more similar to X than Z, for X and Y share one property, but X and Z don’t share any. But this depends on our having chosen P and Q as ‘basic’ properties. From a logical point of view, we might have employed a different set of ‘basic properties’: just let Q as it is, and substitute P for R (defined as ‘being either P and Q, or not-P and not-Q’). A little formal tinkering shows that in this case, our object X has properties R and Q, Y has properties not-R and not-Q, and Z has properties R and not-Q… and hence Z is now more similar to X than Y! Our small swan has by magic become more similar to one pretty duckling than another (not exactly a spitting image copy of the first) duckling. This result can be generalised to any set of properties or relations, showing that judgments of similarity necessarily include some subjective or ‘arbitrary’ element, and cannot be defined by just logical, formal or mathematical relations. Hence, to anyone claiming that scientific theories have increasingly become more and more similar to the truth, the ugly duckling theorem would allow responding something as simple as: “Well, that’s what it seems like to you.”