Informatics 1 CG – Tutorial 6
Chris Lucas
Week 7
1. Category-based induction
Single-premise inferences
Consider the following premise/conclusion pairs, in the syntax used by Osherson
et al. and others.
A. Owls have protein P
-------------------Hawks have protein P
B. Owls have protein P
---------------------Rabbits have protein P
C. Owls have protein P
-------------------Birds have protein P
D. Birds have protein P
-------------------Owls have protein P
E. Owls have protein P
---------------------Animals have protein P
F. Owls have protein P
-------------------------Sea sponges have protein P
G. Owls have protein P
-----------------------------------Mathilda the pygmy owl has protein P
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Exercises
1. For each of the premise/conclusion pairs above, how strongly does the
premise support the conclusion? Rank each premise/conclusion pair according to your own intuitions.
a. Do your rankings align with the phenomena summarised in Osherson
et al. (1990)? Were there any surprising or interesting points of
disagreement, or aspects Osherson et al. didn’t address?
The pairs above have been given letters A-G for convenience.
Concerning Osherson et al.’s phenomena, as they relate to our examples,
there are different possible orderings, but here are a few relevant phenomena:
• Pairs G and D are cases of “premise-conclusion inclusion”, which
Osherson et al. take as indications of “perfectly strong” arguments.
• Pair A involves two examples birds of prey and has a narrower conclusion that pair C, making it a stronger argument than C. In turn,
pair C has a narrower conclusion than E.
• The inclusion fallacy suggests that people might take sea sponges to
be a more weakly supported conclusion than animals, despite their
being animals.
The points above suggest the ordering below:
{GD} > A > C > E > B > F
b. How did you interpret the premises and the conclusions? For example,
do you take “Owls have protein P” to mean that every owl must have
protein P? If so, how and when are these premise/conclusion pairs
useful – are they relevant to our everyday intuitions about animals and
their characteristics? If not, explain how you interpret a conclusion
like “Owls have protein P.”
If we interpret a statement like “Owls have protein P” as a logical implication, then the premises are all fairly extreme statements, and perhaps a
bit strange. How often is a protein necessarily present in all owns? For
many proteins that are typically present in owls, one might imagine an
animal that looks and acts like an owl and has a genome that is typical of
an owl’s, with the exception that it doesn’t express protein P but rather a
protein Q that has a similar causal role as protein P. We’d then have a
creature that most people would agree is an owl but doesn’t have protein
P, even if P is present is all living owls in the wild.
Nonetheless, we might imagine that there are certain proteins that
an owl must have, e.g., proteins that are essential for the survival of
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eukaryotes (for some possible genes that encode such proteins, see
(https://nar.oxfordjournals.org/content/37/suppl_1/D455.full). Without
such a protein the cells in our would-be owl wouldn’t function, so perhaps
it couldn’t be an owl. That said, if a human being suddenly loses all of
her DNA, does she cease to be human in the remaining hours of her life?
See [https://whatif.xkcd.com/book/] for discussion of this scenario.
In everyday life, truly necessary features of categories are rare, if they exist
at all. If they were common, then the definitional theory of concepts might
have been more successful. In general, a logic-based view of category-based
induction is unlikely to be useful outside of very specific categories, e.g.,
mathematical ones.
As an alternative to a logical view, we could take “Owls have protein P” as
a statement of probability, e.g., that an owl selected at random will have
protein P with probability 1 − , for a small value of .
2. How might you use a similarity-based model to make judgments about a
conclusion given its premises? There are several possible answers to this;
sketch out an approach that makes sense to you.
NB: This is a difficult question, and the best approach is still a subject of
scientific debate. Students in inf1-cg aren’t expected to give more than a
high-level gist of their approach.
Suppose we’re talking about pair C above – we’re making and inference
about birds based on knowledge about owls. We can use a statement of
probability, as above, for both the premise and the conclusion: what is
the probability a bird selected at random has protein P, supposing owls
selected at random have a probability 1 − of having the protein. Under
an exemplar model, we can frame this more specifically as:
N
1 X
P (hi )
N i=1
where N is the set of all birds in our exemplars, i is the ith bird, and
hi is the event of the bird having protein P. If the bird is an owl, then
P (hi ) = 1 − . If the bird isn’t an owl, we might just say there’s some
unknown probability that it has protein P. Our final probability is a
weighted combination that depends on how many birds are owls and how
plausible it is that birds have protein P. Unfortunately, this approach
neglects the fact that learning something about owls tells us about other
birds. To handle that, we need some way of connecting the features
of superordinate categories to subordinate ones. That’s tricky, but one
approach is given in “Structured Statistical Models of Inductive Reasoning”
by Kemp and Tenenbaum (2009).
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Knowledge effects
Consider the premise/conclusion pairs below, filling in X as appropriate (see the
exercises for details).
H. Sea eagles X
-------------Owls X
I. Rabbits X
----------Owls X
J. Ostriches X
-------------Owls X
K. Kittens X
----------Owls X
Exercises
1. For each of the premise/conclusion pairs above, how strongly does the
premise support the conclusion? Rank them, filling in X in the following
ways:
a. What if X is “carry the parasite Q”?
b. What if X is “produce the hormone H”?
c. What if X is “consume 10-20 units of vitamin B12 on an average
day”?
2. Did your ranks differ for different X? Why or why not?
Many rankings are sensible, but the point of this exercise is that there’s
reason to have different rankings for different kinds of properties. For
(a), some people might say that rabbits and owls are especially likely to
share a parasite because owls eat rabbits; many parasites have lifecycles
that include traveling from prey to predator. For (b), we might assign
the highest rank to I, because sea eagles and owls the most taxonomically
similar pairs here. For (c), we might expect owls and kittens to consume
similar amounts of a vitamin B12 because they’re both carnivores and
many owls weight about as much as a kitten, whereas sea eagles tend to
be larger and have a somewhat different diet.
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2. Theory of mind
Exercises
1. Give an example of a scenario where two different people might disagree
about what a particular behaviour reveals about a third person’s mental
states, where the behaviour is:
a. A choice among different options, e.g., on a restaurant menu, among
different means of transportation, or when purchasing a retail good.
b. A statement in natural language.
c. A physical action, e.g., walking to a location, looking at something,
or a gesture.
2. For each of the above examples, suggest a piece of evidence that would
reduce the ambiguity or potential for different interpretations.
For instance:
(a) Given two choices of boxed lunches, a person might choose an option that
has a cheese sandwich and a cookie over one with an eggplant sandwich. One
person might infer that the chooser dislikes eggplant sandwiches, whereas
another person might think the chooser wants a dessert. Emotional facial
expressions directed at the sandwich or the cookie would be informative,
here.
(b) Cultural differences are fertile ground for this kind of example. The table
listed at (http://languagelog.ldc.upenn.edu/nll/?p=3154) asserts that an
American who hears “That is a very brave proposal” implies admiration or
a positive assessment on the part of the speaker, whereas a Briton takes the
statement as a sign of negative affect. In this case, the tone of voice might
be informative, as would whether the speaker follows up with additional
supportive or discouraging remarks.
(c) If a person looks in a cupboard, two observers might disagree about what
the person was looking for. Whether the person retrieves and item – and
what that item is – might resolve disagreements. The Sally-Anne task
is one variation on this scenario, and it’s worth thinking about different
explanations for an adult looking in the “task-failing” location, including
knowledge of Anne’s propensities, a hidden camera, or incidental curiosity
about what’s in the cupboard.
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