Aronson, Harr´e and Way (henceforth AHW) (1994) and Psillos (1999), label the formal attempts by Tich´y (1976), Oddie (1986), and Niiniluoto (1987) as the ‘possible world approach’. Niiniluoto (1999) himself uses the name
‘similarity approach’, because he doesn’t actually rely on possible worlds talk.
In any event, in spite of terminological differences, the approaches by Tich´y, Oddie and Niiniluoto have important commonalities. What follows is a mixed presentation of the ‘possible worlds’ and ‘similarity’ aspects of the approach.
I mainly draw on Niiniluoto’s (1999) more sophisticated formal results when discussing the ability to solve real philosophical problem, but apply the handier formulas of Tich´y and Oddie to simple examples.
Truthlikeness`a la Tich´y and Oddie is characterized in terms of the distance between a possible world and the actual world. A theoryT picks out a set of
possible worlds from the set of all possible worlds. T is characterized in terms of a set of basic states it ascribes to the world. So, given nbasic states, there will be 2n possible worldsWi, defined through the conjunction:
Wi=
n
^
i=1
±hi,
where i = 1, . . . , n, and hi are sentences formulated in a semantically inter-preted languageL, characterizing basic states. The actual worldWAis among the possible worlds. Every consistent theory characterizes a possible world, while the actual world corresponds to the true theory. Possible worlds then correspond to all conceivable distributions of truth-values of hi. The set of statements corresponding to the basic states constitutes, in Niiniluoto’s terms, a cognitive problem:
B ={hi|i∈I}
The requirement is that the elements of B be mutually exclusive and jointly exhaustive:
` ¬(hi & hj) for all i6=j,i, j∈I, and
` _
i∈I
hi.
If the basic states of the actual world are unknown (as it is frequently the case in science), we have a cognitive problem B with the target h∗. Thus, the cognitive problem consists in identifying, among all hi’s which constitute the possible worlds Wi, the sentence h∗ true of the actual world. The above conditions guarantee that there is one and only one element h∗ ofB which is true in WL. Niiniluoto defines the statements hi in B as complete potential answers. Disjunctions of complete answers constitutepartial potential answers;
the latter belong to the disjunctive closure D(B) of B:
D(B) ={_
i∈J
hi|φ6=J ⊆I}
A real function is then introduced, in order to measure the distance between the elements of B:
∆ :B×B →R, ∆(hi, hj) = ∆ij,
where 0≤∆ij ≤1, and ∆ij = 0 iffi=j. ∆ needs to be specified in each epis-temic context. But, as Niiniluoto (1999: 69-71) shows, there are standard ways of doing this for specific problems. For example, in a mathematical problem dealing with real numbers, the distance between two points x = (x1, . . . , xn) andy = (y1, . . . , yn) in Rnis given by
v u u t
n
X
i=1
(xi−yi)2.
Next, he extends the definition of ∆ to a function B×D(B)→R, such that
∆(hi, g) measures the distance of partial answersg∈D(B) from the complete answers,hi ∈B. Ifg∈D(B) is a potential answer so that
`g= _
i∈Ig
hi,
where I ⊆ Ig, then g is true if and only if it includes h∗. At this point, Niiniluoto introduces the following measures:
∆min(hi, g) =minj∈Ig∆ij
∆sum(hi, g) = P
j∈Ig(∆ij) P
j∈I(∆ij)
∆γγms0(hi, g) =γ∆min(hi, g) +γ0∆sum(hi, g),
(γ >0, γ0 >0), with the following meanings: “∆min is the minimum distance from the allowed answers to the given answer, ∆sum is the normalized sum of these distances, and ∆ms is the weighted average of the min- and sum-factors.” (Niiniluoto 1999: 72). Quantitative definitions of approximate truth and, respectively,truthlikeness can now be offered:
AT(g, h∗) = 1−∆min(h∗, g), T r(g, h∗) = 1−∆γγms0(h∗, g).
The plausibility of these formulas is supported by some interesting properties.
gis approximately true when ∆minis sufficiently small. In the limit, if ∆min = 0,gis strictly true, that is, approximately true to the degree 1. Truthlikeness
is defined not only as closeness to truth, but also as information content (i.e.
as exclusion of falsity). So, Niiniluoto’s min-sum formula provides us with a method to achieve a trade-off between these desiderata: “the weightsγ and γ0 indicate our cognitive desire of finding truth and avoiding error, respectively.”
(1999: 73). He further notes that “if we favored only the truth (γ0 = 0), then nothing would be better than a trivial tautology, and if we favored only information content (γ = 0), then nothing would be better than a logical contradiction.” (1999: 73)
A.2.1 The criticism of the possible world/similarity approach Among the critics of the possible world/similarity of truthlikeness account are AHW (1994) and Psillos (1999). In spite of relative clarity in their recon-struction of the approach, both AHW and Psillos – who shares virtually all the views of the former – commit certain simplifications which put into ques-tion their proper understanding of the matter. For one thing, they use the concepts of truthlikeness and verisimilitude interchangeably in the context of Niiniluoto’s theory, which is simply flawed. Moreover, they ascribe to truth-likeness the definition which Niiniluoto actually reserves for approximate truth, a fact indicating a further confusion: the one between approximate truth and truthlikeness. We have just seen that approximate truth and truthlikeness have different definitions: AT = 1−∆min, while T r = 1−∆γγms0. Concern-ing verisimilitude, it also incorporates an element of epistemic probability, P(hi/e), understood as the rational degree of belief in the truth of hi given the empirical evidence e. The expected degree of verisimilitude of g ∈ D(B) given evidence eis then
ver(g/e) =X
i∈I
P(hi/e)T r(g, hi).
Notwithstanding these inexactitudes in Psillos’s description, his criticism of the possible worlds theory of truthlikeness draws upon two reputable objections raised by Miller (1976) and, respectively, Aronson (1990) and AHW (1994).
(a) Miller constructs two weather-predicates: ‘is Minnesotan’ and ‘is Arizo-nan’ out of three natural weather predicates ‘hot’, ‘rainy’, and ‘windy’: a type of weather is Minnesotan if and only if it is either hot and rainy or cold and dry:
m=df (h∧r)∨(¬h∧ ¬r);
a type of weather is Arizonan if and only if it is either hot and windy or cold and still:
a=df (h∧w)∨(¬h∧ ¬w).
Given these definitions, the following equivalence can be easily obtained:
h∧r∧w≡h∧m∧a.
Thus, the target problem can be formulated either in terms of {h, r, w}, or alternatively, in terms of {h, m, a}. A problem arises from the following fact:
if the target theory is h∧r ∧w, then the statement ¬h∧m∧a, which is logically equivalent to¬h∧r∧w, proves to beless truthlike than the statement
¬h∧ ¬m∧ ¬a(which is logically equivalent to ¬h∧ ¬r∧ ¬w). In other words, according to Niiniluoto’s definition of truthlikeness, while it is obvious that
T r(¬h∧r∧w)> T r(¬h∧ ¬r∧ ¬w),
the logical equivalents stay in a reversed truthlikeness relationship:
T r(¬h∧m∧a)< T r(¬h∧ ¬m∧ ¬a).
The problem seems indeed to be a serious one. Some philosophers (Urbach 1983; Barnes 1991) have concluded that truthlikeness`a laNiiniluoto is a shaky concept.
Nonetheless, Miller’s objection is itself problematic. It assumes complete liberty in choosing the language in which to formulate the cognitive theory.
That is, it assumes that the{h, r, w}–language and the{h, m, a}–language are equally entitled to serve for the formulation of target theories. When a target theory is being formulated in one language and thereafter mapped onto an-other language, what happens is that the metric ∆ is in general not preserved.
Consequently, such mappings can dramatically distort the cognitive problem.
Of course, in scientific practice the difficulty is avoided by the fact that only one language – namely, the one in fact used by the scientific community – is in use.
But is this methodological fact metaphysically arbitrary? Can we dispose at libitumof the linguistic framework in which to formulate our cognitive prob-lem? I do not believe so. There is a straightforward sense in which ‘hot’, ‘rainy’, and ‘windy’ are more fundamental than ’is Minnesotan’ and ‘is Arizonan’. The former are ‘natural kind’ predicates and serve as constitutive elements for the latter. It suffices now to argue that I take them to be ‘natural’ not in the sense of any essentialist metaphysics, but in the sense of their belonging to the most adequate linguistic framework, given the constraints set by the outside world.
A language is adequate if it contains concepts lawfully related to the quantities involved in the target theory. In this sense, as Niiniluoto states, “it may be possible to find, for each cognitive problem, a practically ideal language (or a sequence of more and more adequate languages).” (1999: 77).
The adequation of the language to particular cognitive interests brings a methodological or pragmatic dimension to the concept of truthlikeness. In order to see whether objects are similar by measuring the distance between their basic states, we use a class of relevant features, as well as certain weights for these features. It follows that truthlikeness is relative to the cognitive problems. Yet, pragmatic interests by no means exclude epistemic objectivity.
Niiniluoto is explicit about the ways in which truthlikeness depends on our cognitive interests:
(i) We measure the distance from truth relative to the target theoryh∗, not relative to the whole world.
(ii) The choice of the metric ∆ involves dimensions and weights correspond-ing to specific cognitive interests.
(iii) The weighted average of themin andsum factors directly expresses our cognitive interests in finding truth (γ) and shunning error (γ0). These two parameters “point in opposite directions, and the balance between them is a context-sensitive methodological decision, and cannot be effected in purely logical grounds.” (1999: 77)
I believe these considerations can do justice to the scruples legitimately voiced by Wolfgang Spohn:
An aprioristic procedure seems questionable also on general grounds.
Analogy, in its full sense only meagerly captured in formal models..., is a highly a-posterior matter; it is concerned with often rather vague considerations (or should we say: feelings?) of how theories in one empir-ical field might be carried over to another field; and passable intuitions about concrete analogies only evolve after a thorough-going examination of the subject at hand. (Spohn 1981: 51)
Moreover, Niiniluoto’s above remarks are also cogent with respect to a second interesting objection raised against the similarity theory of truthlikeness.
(b) Aronson (1990) and AHW (1994) have raised a different criticism, which they deem even more devastating, based on the following two intuitions:
...first of all, no false statement can be equally true or truer than the truth; and, secondly, the number of basic states in the universe should
not, in itself, affect the verisimilitude of a proposition. The second can be put another way. Theories carve out chunks of the world which are semantically independent of one another. For example, the fact that there are one billion Chinese should not affect the truth or verisimilitude of the special theory of relativity unless the latter somehow entail the former.
(AHW 1994: 118)
First, AHW have noticed that by adding new items of information to the de-scription of the world, the truthlikeness of extant propositions changes. Recall that in the weather-predicates model, the target theory given the three basic states ish∧r∧w. Let us see what happens to hafter adding new basic states.
Initially, the truthlikeness ofhalone isT r(h) = 0.67. However, after the addi-tion of a fourth predicate, say ‘cloudy’, the Oddie-Tich´y truthlikeness measure ofh decreases toT r(h) = 0.625. A fifth weather-predicate diminishes it even further: T r(h) = 0.6. The trend continues further, so that the truthlikeness ofh given nbasic states is given by the formula
T r(h) = n−1 2n .
Meanwhile, the truthlikeness of false propositions follows the opposite trend:
by adding a fourth basic state, the truthlikeness of¬hincreases fromT r(¬h) = 0.33 to 0.35, and goes to 0.375 for five predicates. The evolution is given by the formula
T r(¬h) = n+ 1 2n . Obviously,
n→∞lim T r(h) = lim
n→∞T r(¬h) = 1 2.
This indicates that as the number of basic states tends to infinity, a false proposition has the same truthlikeness as a true one, a result which AHW rightly find aberrant.
Second, AHW have noticed that the truthlikeness ofhis also altered by the addition of further basic states which have nothing at all to do withh’s con-tent: “his about the weather while the 100th state might be about something entirely unrelated to the weather: say, the average height of the mountains on the moon.” (AHW 1994: 119). Of course, it isprima facie utterly counterintu-itive that the verisimilitude of a contingent proposition is senscounterintu-itive to addition of propositions which have nothing to do withh’s content. So, AHW conclude, a “pernicious holism” is being entailed by this version of truthlikeness.
For these reasons AHW deem the possible world/similarity approach to truthlikeness to be insuperably flawed and propose instead their own account of verisimilitude: the so-called ‘type-hierarchies’ approach, which will be briefly discussed below. For his part, Psillos loses faith in the possibility of reaching a capable formalized account of truthlikeness and opts for an intuitive concept.
Nonetheless, their conclusions are way too hasty. For one thing, I do not agree that their objection is that damaging to Niiniluoto’s account. For another, as can here be elucidated only in passage, their own proposals do not fare any better.
It does indeed follow from Niiniluoto’s min-sum formula that false proposi-tions can be more truthlike than true ones. But truthlikeness has two dimen-sions: (i) acquiring truth, and (ii) shunning error. The trade off between them is reached by the weights respectively assigned to γ and γ0. If one values ex-clusively the truth-finding aspect (i) (i.e.,γ = 1), then any tautology is better than the best established statements of our best science. However, we know that it cannot be so. We value scientific statements also for being informative.
Consequently, we must accept that some false statements, if sufficiently close to truth, are better than some true ones.2
As to the other aspect of the criticism, we have already insisted that truth-likeness is defined for specific target problems; it is relative to specific cognitive interests and to specific epistemic contexts. When the information about a target problem changes, the target itself changes, so it’s unsurprising that the distance of a given statement from the target changes as well. The point is illustrated by Psillos, based on a personal communication from Niiniluoto:
Suppose, for instance, that you are asked to tell the color of Professor Niiniluoto’s eyes, and that you have a theory h which says (correctly) that they are blue. But now suppose you give the same answer in the context of a question that concerns the color of his eyes, hair and skin. In this context the answer his less verisimilar than it was in the context of the previous question, because it gives much less information about the relevant truth. (Psillos 1999: 269)
Psillos accepts this as a fair point but still displays an uneasiness arising from the contextual character of truthlikeness judgements. Now, it has been suffi-ciently argued that the dependency of truthlikeness on our cognitive interests does not make the measurements of the distances to the truth less objec-tive. This feature is also important in responding the accusation of ‘pernicious
2To have a more precise notion of how close must the false statement be to the truth, we can impose, in specific epistemic contexts, threshold values which set constraints on the acceptable values of ∆sum.
holism’. A specific epistemic context excludes addition of propositions corre-sponding to basic states of the world which are irrelevant to the target theory.
Therefore, there is no reason to alter, say, a target problem in quantum physics on the grounds of information regarding the Chinese population.
There is no purely semantic theory of truthlikeness, strictly in terms of distances to the truth. Truthlikeness also incorporates an essential pragmatic or methodological component. As a matter of fact, the latter is also present in AHW’s (1994) account of verisimilitude: the ‘type-hierarchies approach’.
They construe theories as networks of concepts (nodes) related by links repre-senting the relations between concepts. The higher nodes in the network stand for higher types, while the lower nodes stand for subordinate types. Links are then relations of instantiation between nodes in the hierarchy. AHW’s account is based on the idea of similarity within type-hierarchies. Two types are said to be similar when they are represented as subtypes of the same type. For example, both ‘whale’ and ‘dog’ are subtypes of the type ‘mammal’. Further-more, among the subtypes of ‘mammal’, some are more similar to one another than others. For instance, ‘whale’ is more similar to ‘dolphin’ than it is to
‘dog’. AHW borrow a distance function from Amos Tversky (1977) in order to measure degrees of similarity.
The question to be raised then is, what determines a type-hierarchy in the first place, if not prior judgements of similarity? How do we establish the framework in which to make quantitative measurements? Obviously, the an-swer lies in the particular cognitive interests of the scientific community. The prior configuration of the network corresponds to pragmatic and methodolog-ical interests. Thus, AHW’s approach is by no means less context dependent than the possible worlds/similarity approach. It can be therefore concluded that the similarity approach of truthlikeness manages to overcome the criti-cisms raised by Miller and by AHW. Moreover, the approach has the ability to deal with a multitude of other major challenges in the dynamics of the-ories: approximation, idealization, meaning variance, conceptual enrichment, reduction, etc. (cf. Niiniluoto 1999).
A.3 Anti-truthlikeness: Giere’s constructive-realist proposal
Many realist philosophers have lost optimism in the prospects of a formalized account of increasing truthlikeness. Devitt (1984) is one of them. Yet, he doesn’t think that problems with the doctrine of truthlikeness are problems for scientific realism. He defends the concept that scientific realism is not necessarily related to the doctrine of convergence, as it is not necessarily related
to any theory of truth (Devitt 1984: 114–5). For the purposes of scientific realism, disquotationalism with respect to the usage of ‘refer’, ‘truth’, and
‘approximate truth’ is as good as a robust theory (Devitt 2003). However, as already pointed out, there are reasons why straight talk of approximate truth is more help that its disquotation (see chapter 2).
Psillos (1999) is also skeptic about the prospects of a formalized account of truthlikeness. However, he deems the concept to be indispensable to scientific realism and believes that an intuitive account will suffice. But we saw that there are situations related to the dynamics of theories that repel any vague-ness in the truthlikevague-ness measures, that is, situations demanding quantitative measurements of the distances to the truth.
There are also realists who sanguinely reject any approach of verisimilitude as misguided. Richard Giere (1988, 1999), for example, thinks that philosophy of science should do away with ‘the bastard semantic relationship of approx-imate truth’ (1988: 106). This claim ought to be understood as part of his conviction that the philosophy of science should be freed from general ques-tions about language (1999: 176). He opposes the propositional view of theories – which he deems a vestige of logical empiricism – and proposes instead the so-called semantic view of theories – which construed theories as sets ofmodels.
According to Giere, models are non-linguistic representational devices satisfy-ing certain theoretical definitions – usually, sets of mathematical equations.
For example,
A one-dimensional linear harmonic oscillator is a system consisting of a single mass constrained to move in one dimension only. Taking its rest position as origin, the total energy of the system is,
H =T+V = p2 2m+1
2kx2 where
p=mdx dt.
The development of the system in time is given by solutions to the fol-lowing equations of motion:
dx dt = ∂H
∂p
dp
dt =−∂H
∂x. (Giere 1999: 175)
What is the relationship between this theoretical model and the physical world?
Giere chargestheoretical hypotheses with the task of making this link. The idea is to define theoretical systems so as to be faithful replicas of real systems. Of course, no real linear oscillator (e.g., a bouncing spring) can be replicated in all detail by the above model. However, as Giere suggests, this can be done in specified respects and to specified degrees:
I propose we take theoretical hypotheses to have the following general form:
The designated real system is similar to the proposed model
The designated real system is similar to the proposed model