Cheating and Logical Types

Thinking a bit about the challenge put by Dave in rhizo14 (see his unguide to the thing), this is what I came up with.

I’m taking a clue from Gregory Bateson, who applied this kind of reasoning a lot in his thinking, and specially in his reflections about play (I don’t have the references right now, but you can find something about this in his Steps to an Ecology of Mind, and also in his Mind in Nature).

Dave’s challenge is: “Use cheating as a weapon. How can you use the idea of cheating as a tool to take apart the structures that you work in? What does it say about learning? About power? About how you see teaching?”


You can only cheat if there is a set of rules that determine accepted behaviour. If “cheating” itself is the goal, there is actually no “cheating” in the usual sense, for the cheat is actually accepted behaviour. This creates a certain kind of logical paradox.

Suppose, as a teacher in a classroom, you give a task to students, and this task is: “to make mistakes”. You create a paradox in which (a) if you make a mistake, this is not a mistake, but rather a success – but if it is a success, it is a mistake (because to succeed in the task would be a mistake), and so on; or (b) if you don’t make a mistake, this is a mistake, and then it is a success (which is actually a mistake)… and so on.

This is something like the “Liar’s paradox”, and other kinds of self-referential paradoxes. Something like: “This sentence is false.” If it is false, then it is true, and if it is true, then it is false. “If A, then not-A” and “If not-A, then A”, which puts us in an infinite loop.

A solution to this kind of paradox was proposed by Bertrand Russell and Alfred Whitehead in their Principia Mathematica. Their solution came to be known as the “Theory of Logical Types”, a theory that postulates a hierarchy of types of logical objects, and then assigns every conceivable entity to a type. Objects of one type are always made of objects of a lower type. You can think about sets of numbers, then sets of sets of numbers, sets of sets of sets, and so on.

Applying this idea to rules, we have rules about behaviour, then in the next logical type rules about rules, and so on.

In this framework, a cheat is a break in the rules in a certain logical level. If you establish cheating as a goal, this can then be called a “meta-rule” –  that is, a rule about the rules in the lower logical level. It is not a “cheat” in its own logical level, but an accepted rule in that level, that tells you to break the rules of the lower level.

To understand cheating as a meta-rule means that the cheating refers only to rules in the lower levels of logical types. Cheating in the meta-level is still cheating in the usual sense, but I can also imagine multiple logical levels subverted by meta-rules established in the levels above… I don’t know where this could lead.

To me, it seems that accepting cheating as a meta-rule allows us to subvert rules in lower levels. Some interesting degree of freedom is added to the game…

In practice: if, for example, a teacher allows “cheat sheets” in tests, they are not cheats anymore, because they are allowed. They become tools that students can use to pass tests. They are still called “cheats” only in relation to the old tradition of tests in ordinary classrooms.

Lots of ideas can sprout from this approach. I would say that playing with subversion in this manner seems a very interesting way of dismantling limiting structures of power that condition learning.

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4 Responses to Cheating and Logical Types

  1. Pingback: Dimensions of power, knowledge and rhizomatic thinking | Francesbell's Blog

  2. Mark says:

    Dave’s construction of “cheating” reminded me immediately of something Ken Robinson said in Changing Education Paradigms (YouTube). At min 10:07 he says,

    “Now a lot of things have happened to these kids as they have grown up. But one of the most important things is that by now, they have become educated. They have spent ten years at school being told that there is one answer – it’s at the back…and don’t look, and don’t copy, because that’s cheating. I mean, outside schools, that’s called collaboration.”

    I’ve thought about this a lot since I first heard it, and standing in a classroom most of the day I have plenty of opportunity for that. Robinson’s words struck a chord with me: I do not see myself as an enforcer of anything, and, I’ll admit, while considering this cheating / collaboration thing, have not thought much about ethics or theft or moral philosophy. None of this seems relevant given my role as a teacher …

    Too many teachers don’t actually teach, they police.

    For more on this see Rebecca Moore Howard,

    • arca says:

      Mark, I see myself represented in your comments and in your posts – both as teacher and as student. It is enlightening to talk to others who are dealing with similar issues in their practice. This comment took my mind straight to Pink Floyd… lol Thanks man.

      I couldn’t access the link, though. Only for subscribers… is there a way to cheat around it? : )

  3. Frances Bell says:

    In my post about Cheating and Learning I linked to a rhizome that cheats It is of a genus that grows from corms but the schizostylis grows from a rhizome to fit its habitat. I am wondering if that has relevance for the diversity on this MOOC where people may be adapting the idea of rhizomatic to fit their (mainly) formal education habitat.

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