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In this short paper I describe the research project I am currently work- ing on. I also look at recent research regarding the role of logical implica- tion in mathematical thought and suggest that Sperber & Wilson’s (1986) relevance theory may shed light on the cognitive processes involved in un- derstanding and interpreting conditional statements.
For my Ph.D. research I am looking at how mathematicians, both successful and unsuccessful, use and understand logical implication. Logic has always been considered an important part of mathematics but, perhaps strangely, there has been little research into how mathematicians use formal logic in their reason- ing. Currently I am planning a research project that looks at the relationship between the formal concept definition of logical implication as it is taught at university (the material conditional) and the (syntactic) concept image that mathematicians use on a day to day basis.
Initial investigations using the Wason selection task seem to indicate that there is a significant difference between these two constructs (Inglis & Simpson, 2004). If this is the case it raises questions about the value of teaching the material implication in first year university courses.
Whilst investigating students’ understanding of logical implication, several au- thors (e.g. Rogalski & Rogalski, 2001; Durand-Guerrier, 2003) have used what has become known as the Durand-Guerrier maze task (see figure 1). Participants in this task are told that a person (named X) passes through the maze without using the same door twice. They are then asked to evaluate the truth/falsity of a series of statements about the situation. For each statement participants are given three options: true, false or can’t tell. For example, ‘X crossed P ’ is a false statement, whereas the correct answer to ‘X crossed M ’ is can’t tell.
1The ideas expressed in the paper were developed jointly with my supervisor, Adrian Figure 1: The Durand-Guerrier Maze Task.
One of the questions on the task is to evaluate the truth/falsity of ‘if X crossed L then X crossed K’. Durand-Guerrier (2003) reports that the original teachers who authored the task believed the correct answer was ‘false’, whereas “pupils, especially those deemed good at mathematics” (p.9) thought the answer was ‘can’t tell’. Durand-Guerrier explains that those who answer ‘false’ see the statement as a generalised conditional – one of the form ‘for all X, if X crossed L then X crossed K’2 – whereas those who answer ‘can’t tell’ view the conditional as a material conditional, a straight ‘if X crossed L then X crossed K’ with no It is interesting to note that the teachers who smuggle in an illicit ‘∀’ into the last question do not do the same for a statement without a conditional.
For example, ‘for all X, X crossed M ’ is false, but those who answer ‘false’ to question 6, answer ‘can’t tell’ to ‘X crossed M ’. For some reason, the presence of the ‘if. . . then’ implication seems to lead some to add a ‘∀’ at the start of the statement. Why? Why did these mathematics teachers3 infer a generalised conditional from a material conditional? I suggest that to answer this question 2Although they are defined differently, to a large extent this conception of the conditional overlaps with Deloustal-Jorrand’s (2002) notion of “causal implication”.
3My own exploratory investigations suggest that this behaviour is also exhibited by a large number of successful (postgraduate and postdoctoral) mathematicians. This leads meto doubt Durand-Guerrier’s claim that “those deemed good at mathematics” tend to answer‘can’t tell’.
we need to investigate the general principles behind communication.
Relevance theory (Sperber & Wilson, 1986) is an inferential theory of commu- nication. Such theories postulate that communication takes place via a process where the audience infers the meaning the communicator intended using the evidence available to them.4 By defining the notion of relevance, Sperber & Wilson can begin to explain how this inferential process takes place.
The key feature of the theory is the concept of relevance: “an input is relevant to an individual when its processing in a context of available assumptions yields a positive cognitive effect” (Wilson & Sperber, 2004, p.251). When the cognitive effect of an input (a communication, be it verbal or written) increases, so does its relevance; however, when the effort needed to process an input increases, its relevance decreases. The classic example from the literature is of telling a friend what time their train is. Saying “the train is at ten past three” is more relevant than either (i) “the train leaves after two o’clock” or (ii) “the train leaves 960 seconds after 2:54”. Sentence (i) yields less of a cognitive effect and sentence (ii), whilst yielding the same cognitive effect, requires more processing to reach Having defined what relevance is, the theory attempts to explain human communication with two general principles. The cognitive principle of relevance claims that human cognition “tends to be geared to the maximisation of rel- evance” (Wilson & Sperber, 2004, p.254) and the communicative principle of relevance states that every communication “conveys a presumption of its own optimal relevance” (p. 256). This means two things. Firstly that every com- municator believes that their communication is relevant enough to be worth the audience’s processing effort. And secondly, that the audience assumes the communication is the most relevant one compatible with the communicator’s So, if a friend tells you that “the train leaves after three o’clock”, following the communicative principle, you infer that either they don’t know the exact 4Inferential theories stand in contrast to the so-called code models which suggest that communication is merely a matter of encoding and decoding thoughts.
time the train leaves (their abilities prevent them from telling you and therefore increasing relevance) or, possibly, that they don’t want you to catch it (their personal preferences prevent them from being more relevant).
How, then, can relevance theory help to explain why experienced and success- ful mathematicians often infer generalised conditionals when actually all that is written (or spoken) is a material conditional? The answer is now clear: gen- eralised conditionals are more relevant than their material counterparts. They provide a greater cognitive effect – you can ‘do’ more with them.
Every communication conveys with it the presumption of optimal relevance.
Why would someone say “if X crossed L then X crossed K” if this wasn’t true for all routes X? It would be more relevant (as it would be easier to process) to say either “X crossed both L and K” or “X didn’t cross L”. The very fact that the communicator has chosen not to say these things leads the audience, following the communicative principle, to search for a more relevant interpretation. As a result, they infer that the communicator in fact means something more. This explanation accounts for why the phenomena only occurs when a conditional is present. When someone says “X crossed M ”, there is no more relevant way to convey the information that X crossed M . The expected level of relevance is met and the audience doesn’t search for another interpretation.
In mathematics almost all conditionals met in textbooks and lectures are generalised, and even when they are not they are often inferred as if they were.
This is because they are more relevant – they provide a greater cognitive effect.
Relevance theory has had a wide variety of applications. Here I have suggested that the reason why mathematicians (even very successful ones) often read more into conditionals than has been explicitly written is because of relevance the- oretic considerations. When seeing a conditional the audience searches out an interpretation of that meets their expectations of relevance. Inferring a material conditional very rarely manages to do this. There is almost always a more relev- ant way – a way that requires less processing – of stating the same information that a material conditional provides. Given this, the audience interprets the conditional as generalised. Now it provides a cognitive effect that better reflects the processing required to achieve it: it meets the audience’s expectations of Researchers into students’ understanding of logical implication have a habit of throwing their hands up and despairing at the supposed logical ineptitude of their subjects. I suggest that not always producing a answer that fits with formal logical theory is behaviour that is also exhibited by very successful math- ematicians. A large part of any experiment such as the maze task is compre- hending what the question is trying to communicate. With an understanding of the processes behind how communication ‘works’, students’ answers to logical tasks may not be as perverse as they first seem.
Deloustal-Jorrand, V. (2002). Implication and mathematical reasoning, PME26, Vol. 2, Norwich, East Anglia, pp. 281–288.
Which notion of implication is the right one? From logical considerations to a didactic perspective, Educational Studies Inglis, M. & Simpson, A. (2004). Mathematicians and the selection task, to URL: http://www.warwick.ac.uk/∼edryar/files/pme2004 small.pdf Rogalski, J. & Rogalski, M. (2001). How do graduate mathematics students evaluate assertations with a false premise?, PME25, Vol. 4, Utrecht, Hol- Sperber, D. & Wilson, D. (1986). Relevance: Communication and Cognition, Wilson, D. & Sperber, D. (2004). Relevance theory, UCL Working Papers in

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