Jonah Bossewitch	Prof. Johnson-Laird

Cog 200	11/5/93

Problem 3: If, and, or

This experiment explores the way in which people with no logical background interpret the logical connectives 'if', 'and', and 'or'. Subjects who know logic may have acquired a new and different meaning of the connectives since these terms are rigorously defined in logic. For this reason it is important to test subjects with little or no background in logic. In the experiment, the subject was shown four assertions (figure 1) that made claims about letters and numbers drawn on a theoretical blackboard. The subject was then asked to write down a list of the possible circumstances would make each assertion true. She was then asked to go through the assertions a second time and write down the possible circumstances which would make the assertion false. The subject's responses are shown in figure 2.

In order to assess the performance of the subject we need to distinguish between two types of errors that might result in answers that conflict with the correct logical answers. First, the subject may not be using the logical definition of the terms. This type of error is a not a mistake since consistent use of a sense, albeit not the "logical" one, is just an alternative interpretation. Second, the subject may make an error that is inconsistent with the terms as she interprets them. This class of errors includes not listing some of the situations that make the assertion true, and listing situations that do not falsify the assertion as falsifiers. If the process of generating situations is carried out by modeling the situation, a theory of interpretation of the connectives must also predict the second type of error, that make the answer wrong even with respect to the alternative meaning of the terms.

The subject's responses were consistent with the theory of modeling and in particular is demonstrated in the shift from specific instances to generality. When asked what situation would falsify assertion 1, she wrote 'an A and a 5' whereas by assertion 3 she wrote 'a D and not a 5'. The question that needs to be answered is weather situations that would have verified the assertions were omitted because those models had not been considered or because the words terms were interpreted non-logically. The subject consistently used 'if and only if...then' in every instance of 'if...then'. Although one might argue that the model of '2 and not A' or '3 and not C' were simply not considered, the subject was able to generate 'A and not 2' which completes the model of 'if and only if...then'. The interpretation of 'or' was controlled by the clause 'and both conditionals may be true' and this interpretation is demonstrated in the responses to assertion 3 and 4. The subject correctly listed the verifiers of assertions 3 and 4, however, when listing the falsifiers she treated the 'or' like an 'and'. The subject's interpretation of 'and' was not as consistent as her use of 'or'. When asked for the verifiers of assertion 1 she gave a response that treated the 'and' as an 'or'. However, when asked for the falsifiers of assertion 1, she treated the 'and' as a logical 'and'. Similarly, she gave falsifiers of assertion 3 that treat 'or' as 'and' and she even responded that assertion 3 is the same as assertion 1. However, a logical interpretation of 'and' is demonstrated in the correct responses to assertion 2.

Future experiments need to further explore the interpretations of 'and' and 'or'. It is possible that the interpretations of these terms are not constant over time, and that people mean different things in different contexts. The experiment should be repeated with different clauses more familiar to the subject. The interpretation of 'not' also needs to be explored, and although this experiment implied a 'not' in asking the subject to falsify the assertions, explicit uses of not should be made in the assertions. From the data collected it seems that the negation of the clauses was not done correctly (~(A and B) = (~A or ~B)). This would might explain the wrong results for falsifiers, and the ability to negate clauses should be tested explicitly. Finally, the complexity of the clauses also seems to affect the interpretation of the connective. 'And' was used correctly in the simple case of assertion 2, but it seemed to "get lost" in the complexity of assertion 1. It seems as if people tend to interpret connectives differently in different contexts and it will require further testing to determine in which contexts different interpretations are used.

Figure 1: Assertions (the subject was not shown the equivalent logical statement)

1. If there is a 'A' on the board then there is a '2',

and if there is a 'C' on the board then there is a '3'.

= ((~A or 2) and (~C or 3))

2. There is a 'G' on the board and there is an '8',

and there is a 'H' on the board and there is a '4'.

= ((G and 2) and (H and 4))

3. If there is a 'D' on the board then there is a '5',

or if there is a 'E' on the board then there is a '6',

and both conditionals may be true.

= ((~D or 5) or (~E or 6))

4. There is a 'J' on the board and there is a '9',

or there is a 'L' on the board and there is a '7',

and both conditionals may be true.

= ((J and 9) or (L or 7))

Figure 2: Results

The subject was Tammy Bossewitch, age 17, a senior in high school with no formal training in logic.

Under what circumstances would this assertion be true?

1. If I saw a 'A and 2' or a 'C and 3' or a 'A and 2 and C and 3'.

2. If I saw a 'G and 8 and H and 4'

3. If I saw a 'D and 5' or 'E and 6' or 'D and 5 and E and 6'.

4. If I saw a 'J and 9' or 'L and 7' or 'J and 9 and L and 7'.

Under what circumstances would this assertion be false?

1. If I saw a 'A and 5' or 'A and not a 2' or 'C and 2',

or 'A and C and 5 and 6'.

2. If none of them are 'G or 8 or H or 4'.

3. Same as 1. If I saw a 'D and not 5' or 'E and not 6'.

4. If I saw a 'J and not 9' or 'L and not 7'.