# Definitions

Theorymaker native speakers often find it convenient within their Theories to include calculations on other Variables. So they can apply a Rule to one or more Variables but this Rule is not a prediction of the Level of some other measurable Variable; rather it is just a Rule. Theorymaker native speakers sometimes call it a “bare Rule”. This Rule still, in a sense, creates a Variable - a “bare Rule Variable”, one that is defined solely in terms of the parent Variable(s).

So there is no text before the “!Rule” mark in the written Theorymaker; if there was, it would name a measurable Variable, one which could agree with the prediction or not. In the diagram, there is just a blank line at the top of the Variable.

So if we have a set of Variables for the number of health centres in each region, we can also define a Variable “total number of centres”.

!Rule Total number of health centres opened

Total number of health centres opened in North Region

Total number of health centres opened in South Region 

If the parent Variables in a simple Theory are joined to the child Variable by definition, we just make sure there is no text before the : !Rule.

Classically, the Variables in a Theory must be logically independent of one another (see USAID & EU guidelines xx). So, classically, a Theory can’t include, say, both a Variable number of children in the school and another Variable number of girls in the school.

But in practice, it is sometimes very convenient to include these kinds of Variables too (chapter xx). We have to be careful when doing so. Otherwise, for example, someone will come and ask how we are going to measure some defined Variable; but of course we can’t and don’t need to, because if we know the Levels of the Variable(s) which define them, we already know (or can calculate definitively) the Levels of the defined Variable.

(Now there is a complicated calculus when we have compound Theories.)

Need to talk about how Definitions work through chained Definitions and in particular in the case where some of the links are causal, some not.

Theorymaker native speakers often call a Variable which has been defined just for the purpose of synthesising information from one or more sources a simple definition. They call any part of a composite Theory composed only of interconnected simple definitions a composite definition.

!Rule (add) Total number of children vaccinated

!Rule (add) Number of children vaccinated in North region

Number of children vaccinated in North-East region

Number of children vaccinated in North-West region

!Rule (add) Number of children vaccinated in South region

Number of children vaccinated in South-East region

Number of children vaccinated in South-West region

!Rule (take average) Average number of children vaccinated per region

Number of children vaccinated in North-East region

Number of children vaccinated in North-West region

Number of children vaccinated in South-East region

Number of children vaccinated in South-West region

Any block of text means the same as the same block of text with the addition of any Definition, providing the defining Variables are already mentioned in the block.

!Rule Defined Variable

Defining Variable 1

Defining Variable 2 

That’s a bit of a stretch? Take two Theories, the second of which is the same as the first with the addition of some Definitions. Sure, they might have the same truth conditions. But the one with extra Definitions certainly adds something.

While Theorymaker allows us to give a defined Variable a title (like “total number of children involved”), it is important to understand that the entire meaning of a defined Variable is just its Rule. If by some mistake the title deviates from what the Rule means, just ignore it.

If we do include any defined Variables in a Theory of Change, it is very important to clearly highlight them.

## Definitions with arithmetical Variables

Most often in M&E, the Definition is arithmetical, as in these examples.

!Rule (add) Total number of project beneficiaries; Note=double counting possible!

Total number of beneficiaries receiving psychosocial support

Total number of beneficiaries receiving livelihoods training

Total number of beneficiaries receiving school support  

## Definitions with lo-hi Variables

However, in M&E as in social science, we often encounter Variables which are defined as some kind of total or union or summary of other Variables, but as the defining Variables do not have numerical Levels it is not possible to formulate an arithmetical definition as in the above examples.

!Rule (add?) Child's overall well-being

Child's social well-being

Child's emotional well-being  

So in this case if we were going to measure “Child’s overall well-being” we would do so by measuring the other two concepts and somehow combining the score. But the definition, the basic idea, happens prior to any actual numerical operationalisation. In any case, just as with ordinary Rules, we can ask for more details about this Definition - in this example, by convention the influences are roughly monotonic and combine independently.

So if we were going to measure the defined Variable, we might use a questionnaire consisting of two sub-chapters, one for each defining Variable, and the score for the defined Variable might be the sum or average of the scores for the defining Variables.

## Definitions with thresholds

In social science Theories we often see a different type of Definition, in which the Defined Variable is only satisfied if its defining Variables reach certain thresholds, as in this example:

!Rule (each type of well-being must be above a certain threshold) Child has adequate well-being ((no,yes))

Child's social well-being ((lo-hi))

Child's emotional well-being ((lo-hi)) 

Both of these examples are completely different from a Theory in which Child well-being is a separate, independently measurable Variable which is influenced by two other Variables which happen to also have names involving “well-being”:

Child's overall well-being

Child's social well-being

Child's emotional well-being

In this case, there can and should be a separate way of measuring or assessing Child well-being.

It seems to me that this kind of distinction, which is glass clear in Theorymaker, is absolutely central to constructing and understanding Theories of Change; and I’d go as far as to say that the majority of actual logframes and Theories of Change don’t make this distinction clearly - either they include both Definitions and ordinary Rules in the same Theory without noting the difference, and/or they present Variables for which it is not even clear if they are supposed to be influenced or defined by the Variables pointing to them.

## Reality of “defined Variables”

The important thing to realise is that the defined Variable in the Mechanism referred to by a Theory is just as real as any other Variable. In the example above, the total number of health centres opened is just as real as the regional Variables. It is the link between the three, not any of the Variables themselves, which is special.

So though sometimes even Theorymaker native speakers talk about a “defined Variable”, what they really mean is “a Variable at the leaf end of a Definition”.

In a very loose way you could say, the defining Variables and the defined Variable share one helping of reality between them.

Ancient Theorymaker Proverb: A defined hammer still hurts when it hits you.

This is just the same as with influence Rules - the happiness of the child on her birthday is partly determined by the quantity and quality of the presents, but the happiness is real just as the presents are.

We have to be very careful with these kinds of Rules, because it would be a mistake to try to measure all of them independently - in logframe-speak, we shouldn’t try to set indicators for all of them.

## Strange duality of Theories, Mechanisms and Definitions: Interchangeability of perspectives

We can always flip our perspective between normative (seeing a set of Definitions) and descriptive (seeing an actual fallible Mechanism which might well deviate from those principles). So we can understand ‘2+2’ as a Rule which can underlie either a Definition or a corresponding Mechanism (e.g. a person using an abacus according to the normal abacus rules) which is interpreted as reproducing that Definition but which can always fail.

The Definition follows the Mechanism like the shadow follows the jumping child.

Wittgenstein quote about the ideal machine (Wittgenstein 1978)

What is very important for us: the whole bewildering, unlimited array of kinds of Rule available to simple Theories are available for Definitions too.

## Incomplete Definitions

We already looked at incomplete Rules in Theories xx. We can apply the same idea of (in)completeness to Definitions as well.

Example of an incomplete Definition:

!Rule:incomplete (add) Level of corruption in a country

Level of corruption in education in a country

Level of corruption in commerce in a country

The two Variables seem to be connected by some kind of definitional relationship but it is incomplete; there is more to corruption than corruption in education and commerce. You might say, who would be so stupid as to write down such an incomplete definition but a) sometimes this is just the situation we are faced with particularly at “upper levels of the logframe”, see xx; and b) we couldn’t in fact easily list “all” the kinds of corruption needed to complete the definition.

## Tricks with Definitions: collapsing and factorising, etc

Here are some more examples of Definitions.

Previously we saw xx that under certain circumstances we can collapse several Variables into one with a pre-set Rule, namely “one Level of the defined Variable for every combination of the defining Variables”.

So this

Person's age ((younger, older))

Person's sex ((male, female))

can (roughly) be replaced with this:

Person's socio-demographic group ((girl, boy, woman, man))

But we can also leave the “replaced” Variables as part of the Theory, defining a new Variable.

!Rule (combine) Person's socio-demographic group ((girl, boy, woman, man))

Person's age ((younger, older))

Person's sex ((male, female))

And some of the Variables might be just noise Variables, Pearl’s “U” Variables.

This gets interesting when the set of many Variables is fuzzily defined. For example, when we talk about “the impact” of a project or programme: the whole range of different possible (imaginable / unimaginable) outcomes on a whole (enumerable / not enumerable) range of different Variables.

### Combining several Variables into one with fewer Levels

Much more frequent and useful is the case when we define new Variables by collapsing several Variable into a single Variable with less information.

Example: An evaluation Terms of Reference might specify how project performance, recorded in a whole array of numerical scores, is to be rated on, say, a 1-5 scale, with 5 meaning “exceptional performance, about the best which could ever be achieved with the given inputs” and 1 meaning “overall project impact neutral or even negative”. Here the ToR defines a new Variable, overall 1-5 judgement. Now, how is this collapsing of information to be done? By an algorithm or expert judgement? See xx.

### Aggregating “for” Variables

See later.

Example: given twenty Variables representing the success of the twenty project offices in a country, we can define the Variable best performing office which has one Level representing each of the twenty offices. Or, if the success Variables are numerical, you can define a Variable average office performance.

## And that is it!

These basic ingredients - Variables and Rules - are all we need to build all the other features for any major kind of Theory including Theories of Change.

Theories of Change often include some additional features, many of which are mentioned below, to make Theories more succinct and easier to understand. But philosophers, nerds and geeks will be impressed that all of these extra features can be defined in terms of the basic Theorymaker ingredients - Variables linked together by Rules.

### References

Wittgenstein, L. 1978. “Remarks on the Foundations of Mathematics.” http://philpapers.org/rec/WITROT-2.