# Transitive inference

"Transitive inference (TI) is a form of logical reasoning that involves using known relationships to infer unknown relationships
(A > B; B > C; then A > C).

“TI has been found in a wide range of vertebrates” … and now in certain insects.

Tibbetts Elizabeth A., Agudelo Jorge, Pandit Sohini and Riojas Jessica. (2019). Transitive inference in Polistes paper wasps. 15. Biol. Lett.
http://doi.org/10.1098/rsbl.2019.0015

Published:08 May 2019https://doi.org/10.1098/rsbl.2019.0015

https://royalsocietypublishing.org/doi/10.1098/rsbl.2019.0015

This form of reasoning can be coded as a program, e.g.

if p(A,B) & p(B,C)
then p(A,C)

where p is any of a set of relational properties (larger than, stronger than, etc.)

Tibbetts is quoted in the NYT:
“They are organizing all those pairs into a linear hierarchy".

The paired items are relationships. They may be called dominance relationships, or it may be that something like perception of threat is involved. If wasp A has beaten wasp B, and wasp B has beaten wasp C, then those wasps perceive that wasp A can beat wasp C and will do so if the contest is tried.

This might be a perception of relationship among relationship perceptions. We already know that this appearance of recursion is possible. At a lower level, for example, we may perceive a configuration that is made up of configurations.

``````TTTTTTTTTTTT
TTT
TTT
TTT
TTT
TTT
``````

Transitive inference may also predicated upon set membership, as may be expressed by formulations such as this:

if a ∈ {A} & {A} ∈ {B}
then a ∈ {B}

Returning to the report by Tibbets et al., the set of wasps that A can beat includes the sets of wasps that they each can beat.

The set of wasps that A can beat includes the sets of wasps that any of them can beat.

These sets look like Bill’s notion of Category perceptions. A set or category may be defined criterially (x is a member of {A} if certain perceptual input functions fire [in sufficient number or strength] when the perception x synapses with them) or it may be defined by exemplars (p, q, and r are definitely members of {A}, and sufficient number/strength of the same perceptual input functions which fire upon perceiving one of the exemplars also fire upon perceiving x). Defining by inclusion criteria is generally thought to be formally different from defining by by similarity to central exemplars , but, as you can see, when we put it in terms of perceptual signals these formally different ways of defining set membership amount to the same thing.

Now, it is a different question entirely whether or not there may be a distinct order or level of perceptual input functions which receive those criterial (or exemplar-like) perceptual signals and produce for each set or category a single perception that “some member of {A} is present” (for each set {A}). Personally, I suspect that, rather than higher-level comparators receiving Category perceptions, the ‘categorizing’ is done by the perceptual input functions for those higher-level comparators. Those that control by means of some of the same perceptual inputs are thereby associated with each other. Since memory is local to each synapse, they are associated in memory. And that may be sufficient basis for ‘transitive inference’.

Just because a control phenomenon sounds like logic or reasoning and can be coded as a program, that does not signify that it involves perceptual control at a Program level.

Looks like an interesting study. I liked this:

They put the wasps in a series of bi-colored chambers. If they moved toward the wrong color, they got a mild electric shock. In this way, “you train them that blue is better than green, and once they learn that, you train them that green is better than purple,” said Dr. Tibbetts. They did this for four pairs of five colors.
After the wasps had been trained, they were put in a new chamber. This one had colors they were familiar with from the training, but hadn’t been paired up before. About 67 percent of the time, the wasps successfully chose the “better” color — the one less likely to shock them.
“They are organizing all those pairs into a linear hierarchy in their head,” said Dr. Tibbetts. Next, she plans to study how the wasps actually use this ability during social interactions.

Does look like a linear hierarchy (sequence?) but doesn’t look super strong, judging by the 67%.

I remember reading an old study on wasps being unable to break a ‘behavior sequence’. If the researchers would break the sequence so that the wasp perceives the first element of the sequence, it would repeat all the other elements. I’ll try to find the study, I think it was something about storing live prey into holes.

In any case, it looked as that sequence level was quite high in the hierarchy of the wasp brain.

I am not convinced that transitive inference has been demonstrated in this experiment. I did a little analysis of their data and found this:

For all wasps, the average proportion correct on both training and test trials is close to .67 (the average deviation from ,67 is very small,.02).

I presume the researchers take this as evidence of transitive inference because the wasps got “feedback” in the training trials, in the form of a shock, if they made the wrong response and this is what presumably allowed them to do significantly better than chance (.5 correct). Since there was no shock in the test trials, the wasps’ ability to do better than chance was presumably because they were making transitive inferences.

But it seems to me like an odd coincidence that the presumed transitive inference in the test trials allows the wasps to perform at almost exactly the same level as they did with feedback in the training trials. My guess is that there was something about the apparatus – possibly always putting the “correct” stimulus on the same side of the apparatus-- that allowed the wasps to average .67 correct.

What is needed is data on how often wasps make the “correct” choice when first placed in the apparatus and given no shock. That is, put the wasps in the the same no-feedback situation with the training stimuli as they were with the test stimuli. If the proportion correct in that situation comes out to average .5, then there would be some evince that wasps do something like transitive inference.

I agree with Rick that there does not seem to be any indication of transitive inference in the data. I downloaded the data file and did two simple calculations. First, I computed the variance of the number correct data (2.67) and compared it with the theoretical variance of a binomial distribution with the same mean success rate (2.15). There may be some indication of initial learning, as judged by the mean success rates, and there may be a tiny indication of individual differences in the excess variance of observed versus the theoretically expected data if all wasps were the same. But a day-to-day learning trend could also account for the slight discrepancy.

If there is transitive inference, one would expect the wasps that learned best in training would also show best in the transfer test. If they did, there would be a clear correlation between their training and test success rates. For Dominula (I didn’t do Metricus), that correlation is r=0.02. I see absolutely no evidence that these wasps show transitive inference.

One could test Rick’s hypothesis that they don’t show training effects either by looking for a day-to-day trend in the probability correct during training. I didn’t look for that, but such a trend might account for the slight excess of observed variance over the theoretically expected binomial variance of the success rates.