Transitive fish with people

A report on some work in progress, with the starting point being asking whether fish are sentient beings, are conscious beings. At least, it is work still in progress here in Epsom, although it may well have been brought to a satisfactory conclusion elsewhere.

We are not concerned here with what might follow from establishing that fish were conscious. What might follow, for example, for fish on their passage through a factory farm, on their way to our tables. Whether, for example, we need to clean up our act.

Binary relations

Let S be a set. Then a binary relation is a function which maps ordered pairs of members of S onto true and false. That is to say for any ordered pair, say (A, B), we can say whether that pair is true or false under our relation. Note that we do not allow unknown or not applicable, both categories which are important in real world surveys; just true or false. You can read all about these relations at reference 5.

If S is a set of people then an example of a binary relation would be ‘being an older full sibling of’. Then (A, B) is true if and only if it is true that A and B have the same two parents and A is older than B. We might also say that (A, A) is always false and that (A, B) is false when A and B have the same two parents but were born at the same time. Or perhaps within an hour of each other, this last to avoid splitting hairs.

Binary relations can have various properties. So it may be the case that if we have (A, B) then we always have (B, A) – which is clearly not the case in the example just given. It may be the case that we always have (A, B) or (B, A) – but not both – which is clearly not the case for the relation ‘is an ancestor of’, at least not for sensible sets of people. The property of present interest is transitivity, which says that if we have (A, B) and (B, C) then we always have (A, C). As it happens, both the examples just given do have this property.

An example of a relation which does not have this property is the ‘like’ relation among people. It does not follow from the facts that A likes B and B likes C that A likes C. This being an important source of plots for the writers of romances for the ladies’ market. One’s food and drink preferences might not have this property either.

Experiments

Suppose we have a set containing 5 objects, A, B, C, D and E. Perhaps biscuits with distinctive colours and shapes.

Suppose we have trained our subject to know that (A, B), (B, C), (C, D) and (D, E) are all true. Our subject shows his knowledge by, for example, picking A when you present him with the unordered pair {A, B}. A is in some sense the winner. A test which does not need to involve language.

Note that we are saying here that if we present an unordered pair {A, B}, then either (A, B) or (B, A) will be true. We do not allow neither – which would happen, being technical, if our relation was drawn from a partial order rather than a total order.

Then, what would our subject make of (A, E) or (B, D)?

The first pair is supposed to be easy because A wins in the only relation in which it appears (that is to say, (A, B)) and E loses in the only relation in which it appears (that is to say, (D, E)). So A winning in (A, E) is a good bet.

But this argument does not run with (B, D) as both B and D appear in both winning and losing situations. So maybe our subject has to guess or otherwise hypothesise that our relation is transitive, that in some sense it can be said that A > B > C > D > E.

This experiment can readily be generalised to a set with 6 objects.

Experiments of this sort can be done with intelligent animals and with very young humans as well as with adult humans. 

Evidence

Reference 1 argues for the possibility of consciousness in fishes, a possibility which seems to have a fair number of advocates, with the lady mentioned at reference 6 possibly being one such. One of the lines of the present argument is that fishes can be shown to exhibit transitive inference.

Reference 2 argues for transitive inference working better when human subjects are consciously aware of the relation being tested being transitive. Note that the subjects are not told about transitivity: they have to work it out for themselves – which some of the won’t manage, thus giving us the necessary two conditions.

Reference 3 reports on work on transitive inference with monkeys, work which is concerned with the role of the monkey version of the hippocampus (known to be heavily involved in relevant kinds of memory) in same. The story here being, very roughly speaking, that monkeys whose hippocampus has been destroyed can no longer do transitive inference.

While reference 4 suggests that consciousness of transitivity does not bear on the success or otherwise of transitive reasoning in humans. Reference 2 being a rebuttal of reference 4 in this regard.

More on binary relations

A different way to look at binary relations is in the form of a directed graph, as above. Where we have it that (B, D) is true in the case that there is an arrow from B to D, which there is not in this case. 

Presented in this way, it looks less rather than more likely that a relation would be transitive. It is natural enough to travel from G to B via D and there is no need for a direct route, no need for by-passes.

And even less likely that for any pair, say A and F, that there will always be an arrow from A to F or one from F to A.

 And then, if presented with evidence for the scenario at the top, what then? The facts that C is next in line after B and that D is next in line after C do not imply that D is next in line after B. Then how can we be sure that we don’t have something like the scenario bottom left, in the way of the phases of the moon, something which one might have thought was quite important in the past, at the time that our brains were still evolving. What would make us go for the scenario bottom right, where each member of our set is, in effect, assigned a number, perhaps height, weight or temperature. With the understanding that it is OK to rank members of the set according to those numbers, complete with transitivity. So D is heavier than A.

Where have we got to?

It is being suggested that in order to get (B, D) right, our human subject needs to have worked out that the evidence presented suggests A > B > C > D > E, which permits inference about all the combinations, in particular (A, E) and (B, D). Furthermore, that if asked, our subject will be able to explain this, at least after a fashion, to the experimenter. Does this amount to being conscious of this transitivity, at the time of deciding what to do about (B, D)?

To me, all this seems a bit unlikely. We can do all kinds of clever stuff in the unconscious, particularly with practise. Why should transitivity be such a problem? Why should understanding transitivity be a touchstone for consciousness?

I worry about sample sizes in these experiments, usually small, say of the order of 10-20 subjects.

I also worry about repeatability. Why do references 2 and 4 come up with different results? Then one cannot do the same experiment twice with the same subjects because they will have learned all the answers. And what happens if one’s subjects are mathematicians who know all about binary relations anyway?

So, work in progress indeed.

References

Reference 1: Consciousness in teleosts: There is something it feels like to be a fish - Michael L. Woodruff – 2017. 

Reference 2: Declarative memory, awareness, and transitive inference – Smith, C. and Squire, L. R. – 2005.

Reference 3: Entorhinal cortex lesions disrupt the relational organization of memory in monkeys – Buckmaster CA, Eichenbaum H, Amaral DG, Suzuki WA, Rapp PR – 2004.

Reference 4: Relational learning with and without awareness: transitive inference using nonverbal stimuli in humans - Greene AJ, Spellman BA, Dusek JA, Eichenbaum HB, Levy WB - 2001. 

Reference 5: https://en.wikipedia.org/wiki/Binary_relation.  

Reference 6: https://pumpkinstrokemarrow.blogspot.com/2010/06/uyuz.html.