# Philosophy, Computing, and Artificial Intelligence

Computational Logic and Human Thinking
Chapter 16 (217-231)

## How to Explain the Results in the Wason Selection

Kowalski argues that the hypothesis that humans do not naturally develop a capacity for logical deduction is not the only way to account for the results of the Wason Selection Task.

In the Wason Selection Task, the experimenters give the subjects a "task" to perform. The task is to make a certain selection among presented options. The experimenters pose this task in terms of cards the subject can see. There are four cards. The cards have letters on one side and numbers on the other. They are lying on a table so that one side of each card is showing.

The subject is asked to select those and only those cards that must be turned over to determine whether the following statement is true:

(*) if a card X has letter d on the letter side, then the card X has number 3 on the number side

### The Expected Results

The experimenters expect the subjects to check two cards:

modus ponens:
from observation φ ("d"), check for ψ ("3") on the number side.
modus tollens:
from observation ¬ψ ("7), check for ¬φ ("not d") on the letter side

### The Observed Results

All most all subjects perform modus ponens but fail to perform modus tollens. In addition, many subjects perform the fallacy (from a classical logic point of view) of affirming the consequent:

affirm the consequent:
from observation ψ ("3"), check for φ ("d") on the letter side

## Kowalski's Explanation of the Observed Results

Kowalski stresses that to understand the observed results, it is necessary to appreciate that subjects interpret the words they hear in ways the experimenters do not expect. He summarizes this idea in the equation "natural language understanding = translation into logical form + general purpose reasoning" (Robert Kowalski, Computational Logic and Human Thinking, 217).

The experimenters assume that (*) is a material conditional (φ → ψ), but Kowalski suggests that not all of the subjects understand the sentence in simply this way.

How, then, do they understand the sentence (or "translate it into logical form")?

"In our agent model, the agent’s response depends upon whether the agent interprets the conditional as a goal or as a belief" (Computational Logic and Human Thinking, 218).

### The conditional as a belief: why subjects perform modus ponens

"Modus Ponens. In Computational Logic, conditional beliefs are used to reason both backwards and forwards. In particular, given a (passive) observation of a positive predicate P, forward reasoning with the conditional if P then Q derives the positive conclusion Q. This is a classically correct application of modus ponens..." (Robert Kowalski, Computational Logic and Human Thinking, 221). On the logic programming/agent model, agents use their beliefs in forward reasoning from observations. This, Kowalski seems to claim, explains why subjects perform modus ponens (select the first card). If they incorporate the conditional as a belief, it is natural to fall into forward reasoning when they observe an instance of the antecedent of this conditional.

This explanation is initially plausible, but it is not complete. In the task for the first card, the subjects need to check for a "3" on the number side of the card. The reasoning in this investigation is not part of the logic programming/agent model.

Suppose the KB contains the observation "d" and the conditional (*) because the agent incorporated it as a belief. Suppose the query is 3 on the number side of the card. This query will be answered positively because it is a logical consequence of the KB. So the agent concludes that there is a "3" on the number side of the card. This is not the same as concluding that for (*) to be true, the first card must be turned to check whether it shows a "3."

Further, it is unclear why Kowalski says the reasoning is backward reasoning.

### The conditional as a belief: why subjects affirm the consequent

"Affirmation of the Consequent. In Computational Logic, conditionals are also used to explain observations. Given an observation of Q, backward reasoning derives P as a candidate explanation of Q. This derivation can be viewed ... as abduction with the conditional if P then Q ..." (Robert Kowalski, Computational Logic and Human Thinking, 221). On the logic programming/agent model, agents use their beliefs in forward reasoning but also in abductive reasoning. In abductive reasoning, they use their beliefs to explain their observations. This, Kowalski seems to claim, explains why subjects affirm the consequent (select the second card). If they incorporate the conditional as a belief, it is natural to fall into abductive reasoning when they observe an instance of the consequent of this conditional.

This explanation, again, is initially plausible but incomplete.

Suppose the KB contains the observation "3" and the conditional (*) because the agent incorporated it as a belief. Suppose the agent engages in abductive reasoning to explain the observation. Given the KB, the explanation is that there is a "d" on the letter side of the card. At this point, in the logic programming/agent model, it is unclear what happens. The agent could add the explanation to the KB or pose the explanation as a query. In neither case does the agent conclude that the second card must be turned to check whether it shows a "d."

Further, given Kowalski's explanation of why the agents perform modus ponens and affirm the consequent, it seems that subjects should perform them in equal numbers. This, however, is not what happens in the experiment. More subjects perform modus ponens.

### The conditional as a belief: why subjects fail to perform modus tollens

With respect to whether to select the fourth card, what the subjects observe is "7." From this observation, in order to perform modus tollens with

(*) if a card X has letter d on the letter side, then the card X has number 3 on the number side

the subject must first know that the negative statement

it is not the case that the fourth card has number 3 on the number side

is a consequence of the observation ("7"). This, however, according to Kowalski, is not easy for the subject to know because the observation ("7") does not identify the the query

the fourth card has number 3 on the number side

the subject (on the logic programming/agent model) must use in reasoning to know the negative statement is a consequence of the observation ("7").

Kowalski says that this explains why so many subjects fail to perform modus tollens.

This explanation seems intially plausible too, but notice that even if the agent does identify the correct query (3 on the number side), the logic programming/agent model only gives the agent the ability to ask whether this query is a logical consequence of the KB. What the agent needs is a more sophisticated ability. It needs to know that it should pose the query only after it has turned the fourth card and observed what it is on this side of the card.

### The conditional as a goal: why subjects perform modus tollens

Kowalski also thinks that the logic programming/agent model explains why subjects do better with "meaningful" conditionals, such as

If a person is drinking in a bar, then the person is at least twenty-one years of old

"In this book, we have seen a variety of uses for an agent's conditional goals. Their primary use is [as a maintenance goal] to help the agent maintain a harmonious personal relationship with the changing state of the world. However, conditional goals can also serve a secondary function of helping to maintain harmony in a society of agents as a whole. In both cases, conditional goals regulate the behaviour of agents, both generating and preventing actions that change the state of the world" (Robert Kowalski, Computational Logic and Human Thinking, 225). According to Kowalski, it is natural for the subject to treat this sentence as a conditional goal he is tasked to enforce (similar to an integrity constraint on the world or on society).

This, Kowalksi claims, explains why subjects perform modus tollens with these "meaningful" conditionals. If the agent observes an underage person, the agent will need to make sure the antecedent fails. So he will attempt to observe whether the person is drinking in a bar. If the person is drinking in a bar, then the rule the agent is trying to enforce is violated.

In this way, the subjects interpret

If a person is drinking in a bar, then the person is at least twenty-one years of old

as an integrity constraint on society

If X is drinking alcohol in a bar and X is under twenty-one years of old, then false

Once the agent observes an underage person, he or she will need to make sure that the person is not drinking. Otherwise there is violation of "integrity."

### The conditional as a goal: why subjects fail to perform affirm the consequent

This also helps explains why the subjects do not affirm the consequent. The observation of the truth of the consequent of the rule the agent is trying to enforce does not trigger reasoning.

## What we have accomplished in this lecture

We considered Kowalski's explanation of the results of the Wason Selection Task. He argues that the logic programming/agent model accounts for the observed results. We saw that on the logic programming/agent model, there are different ways subjects can understand the conditional (*). This provides the potential to explain the experimental results in the Wason Selection Task and also why subjects tend to reason better with more meaningful conditionals. We saw, though, that this potential is not very clearly realized because the logic programming/agent model is too underdeveloped. It is not set out in enough detail to model the kind of intelligence required to complete the Wason Selection Task.