Putting flesh on Stathis Psillos’ skeleton

“Verisimilitude”, “truthlikeness”, what makes one false statement more like or closer to the truth, is a bugger of a concept.  Since Popper’s first (Popper, 1972), and failed (Miller, 1974) (Tichý, 1974) attempt to elucidate the concept many different attempts with different approaches have been made. Many have had distinct advantages but none seem to capture what we mean when we say that, though theory x is false it is “near enough” to the truth and certainly nearer than that pile of rubbish theory y. 

After a review of theories of truthlikeness (Psillos, 1999) Stathis Psillos’ put forward an “intuitive” proposal.  Psillos described his proposal as a “skeleton”, a brief sketch used to show that, yes, truthlikeness exists (and, so, realism is saved!) rather than a formalized theory, still less a way of determining judgments of relative truthlikeness. I think there is more to Psillos’ proposal than this.  Let us take Psillos’ proposal a little more seriously than Psillos, himself, seems to take it and see where we get.

Consider the following statements:

1. The earth is flat, “FE”;
2. The earth is a sphere, “SE”;
3. The earth is an oblate spheroid, “OE”. (An oblate spheroid is a sphere that “bulges” at the equator.  It is a “squashed” ball; a tangerine rather than an orange).

None of the three is true. The closest description is, of course, the oblate spheroid and to describe “OE” as false appears to be an exercise in pedantry. Never-the-less, the earth is not quite an oblate spheroid.  The south pole is a little nearer the equator than the north pole and the crust “bulges” slightly more south of the equator than north of the equator. The earth is, ever so slightly, pear-shaped (Asimov, 1989).

The three statements then are false and that, as far as truth and falsity goes, is the end of the matter. The three statements, though, stand in relation to each other and to an unspoken true statement about the shape of the world. This relation can be described in terms of difference from that true statement. OE differs little from the true statement, so little that it seems “near enough” for most purposes. There is no quantification of this difference, beyond the vague predicates of “little”, “small”, “insignificant” and the like.  But the lack of quantification does not matter for describing the relation between the three statements. Whatever value of the difference between “OE” and the true statement it is smaller than the difference between “SE” and the true statement.  This, in turn, is smaller than the difference between “FE” and the true statement.  Taking the true statement to be “t” we can take the pairs of statement in brackets, indicate the difference with Δ and state: Δ(“FE”,”t”) >  Δ(“SE”,”t”) > Δ(“OE”,”t”). Alternatively, taking the arrow to indicate the relation, display it schematically:

Our task is to establish this relation a little more securely than appealing to “intuition” or just taking it to be obvious.

Psillos’ proposal is:

“A description D approximately fits a state S (i.e, D is approximately true of S) if there is another state S’ such that S and S’ are linked by specific conditions of approximation, and D fits S’ (D is true of S’).” (Psillos, 1999, p. 277)

It is a description which echoes the mathematical understanding of similarities of relations. From Russell:

“Let x and y be two terms having the relation P. Then there are to be two terms z, w, such that x has the relation S to w, y has the relation S to w, and z has the relation Q to w.” (Russell, 1920, p. 54)

Where this holds the relation P between x and y is similar to the relation Q between z and w.  Say x and y are part of a series of increasing numbers of oranges created by the relation P, which is adding one orange.  z, and w meanwhile are a series of numbers of apples with the relation S being the act of replacing oranges with apples.  It is clear that the relation Q is adding one apple.  If we add one orange (P) then replace the oranges with apples we end with the same number of apples (w) as when we convert x oranges to apples and then apply Q.  The series of increasing numbers of apples is analogous, or “mirrors” the series on increasing numbers of oranges.

In Psillos’ schema relation S is replaced by one of correspondence. 

The state that would be the case were D to be true can be denoted by removing the inverted commas from the statement.  Were “FE” true we would be presented with a flat earth: FE. Similarly “SE” describes a spherical earth (SE) and “OE” an oblate spheroid earth.  Where the true description, “t”, of the shape of the earth to be true, which it is, we would be presented with an earth the same shape, T, as it is.

By hypothesis, the relation between “FE”, “SE” and “OE” is one of decreasing difference from “t”. This suggest that a similar relation subsists between FE, SE and OE.

“Measuring” the difference

Psillos sees this relation as one of approximation and idealization.  I think the relation is more like…er...”like”.  I suggest we can see what is going on if we look at “identical” twins: a paradigm of likeness.

Tweedledum (TM), Tweedledee (TE) and Alice (AL) provide us with three pairings: (TM,TE), (TM,AL) and (TE,AL). As Tweedledum and Tweedledee are “identical” twins we can expect the differences (TM,TE) are smaller than the differences in either of the other two parings. Now Tweedledum and Tweedledee are not, actually, identical.  Close family and friends (CFF) can distinguish the two, as is usual with “identical” twins.  It is other people (OP) who have the difficulty.  OP have no difficulty distinguishing either Tweedledum or Tweedledee from Alice.  Of course CFF have no difficulty here, either.

Consider the set {CFF, OP} to be a set of distinguishers.   It is the differences in appearance that enable, were they do so enable, the members of that set to make the distinction.  The distinguishers “fire” depending on the differences within the pairings.  And they “fire” differentially.  Not only differentially but the distinguishers that “fire” when presented by Tweedledum and Tweedledee, {CFF}, is a proper subset of those that “fire” when presented with either of the other two pairings.  From the set of distinguishers {CFF, OP} the differences (TM, TE) receive a smaller response than either of the other two pairings. The differences (TM,TE) are, therefore, smaller.

What means do we have to differentiate between, say, FE and T?  From the schema we see that the relation between FE and T is similar to that between the statements “FE” and “t” and we need not worry unduly about differentiating FE and T or “FE” and “t”.  And we have a ready means to differentiate between “FE” and “t”: those things that enable us to tell that “FE” is false.  The ancients provided us with some of these:

a.      a.       The shape of the shadow cast by the earth on the moon.  This is always the same.  Were the earth a disk then, at times, the sun’s rays would hit the earth side on resulting in a line or eliptic shadow.

b.     b.       That ships masts are the last to disappear as a ship travels over the horizon and the first to appear when a ship returns.

c.       c.       Stars disappear (and appear) over the horizon at differing latitudes. (Aristotle via Azimov , 1989)

Distinguishing “SE” from the “t” required:

a.       d.       Observations of other astronomical objects and

b.       e.       The theoretical predictions of Newtonian physics

c.        f.       18^th century surveying technology

Whilst distinguishing “OE” from the truth required:

g.        g.       Detailed measurements from satellites

Of course the detailed measurements from satellites (g) also showed “SE” and “FE” to be false.  And d, e and f falsified “FE” no less than it falsified “SE”.   The evidences that falsify the shape of the earth theories are, then:

FE: Set of distinguishers of (“FE”, “t”): {a, b, c, d, e, f, g}

SE: Set of distinguishers of (“SE”, “t”): {d, e, f, g}

OE: Set of distinguishers of (“OE”, “t”): {g}

Now OE is a proper subset of SE which is a proper subset of FE.  OE is smaller than SE, which is smaller than OE:

Now we can draw fig.1, this time supplying robust reasons, for the statements positions: place them in order of truthlikeness in inverse order of the set of distinguishers that successfully distinguish them from the truth.

Asimov, I., 1989. The Relativity of Wrong. The Skeptical Inquirer, Fall, 14(1), pp. 35-44.

Miller, D., 1974. Popper's Qualitative Theory of Verisimilitude. British Journal for the Philosophy of Science, June, 25(2), pp. 166-177.

Popper, K., 1972. Conjectures and Refutations. 4th ed. London: Routledge & Kegan Paul.

Psillos, S., 1999. Scientific Realism. Abingdon, Oxon: Routledge.

Russell, B., 1920. Introduction to Mathematical Philosophy. 2nd ed. London: George Allen and Unwin.

Tichý, P., 1974. On Popper's Definitions of Verisimilitude. The British Journal for the Philosophy of Science, June, 25(2), pp. 155-160.