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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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