The cross of invention: a tool to guide creative exploration

Arnold Wentzel [email protected]

The cross of invention: a tool for finding zones of creative potential Draft: October 2014

Arnold Wentzel, University of Johannesburg

+27 11 559 2615, [email protected]

1 The creative principle

Invention is generating new ideas to solve problems. Over the years many approaches have been

developed to invention, but the effective ones are based on a single principle: that useful novelty

emerges from a productive interplay between negation and affirmation. This is best seen from the role

that constraints play in creative thinking.

Problems are usually constituted by a set of constraints (Nickles 1981), and break-through ideas are

made possible when at least one of those constraints is challenged and found to be non-binding or

self-imposed (Ackoff 1978). The act of negating constraints is therefore critical to creativity, for

without it one would just have the production of the same old ideas that allow the non-binding

constraints to remain in place.

The problem with negation is that there are almost an infinite number of ways to negate something.

An analogy might help here: when asked to recount an event, a reasonable person will find that there

is usually only one way to tell the truth, but many ways to distort the truth and even more ways to

make up complete lies. It is not surprising then that deception has been recognized as a creative act

(Hirstein 2005). But that creative potential needs to be controlled in some way, or else it becomes

simply overwhelming – having thousands of ideas is not useful unless one knows how to make sense

of them and ultimately settle on one of them. The explosive creative potential of negation can be

anchored by affirmation, specifically the affirmation of constraints.

Accepting constraints is known to be an important part of the creative process (Costello & Keane

2000) partly because it constrains the possibility space and so make the creative process more

manageable. Constraints also help to maintain the connection between the familiar and the new –

people are familiar with (if not always aware of) the constraints, and if new ideas connect to the

familiar they are easier to accept. If a new idea negates every single constraint, people would find it

impossible to understand and would describe it as ahead of its time or absurd (Davis 1971). But

affirming too many constraints, or affirming the wrong constraints, is what causes a problem in the

first place, so negation is required in turn counteract this rigidity.

On their own negation would be useless and affirmation stifling. Negation opens up thinking, while

affirmation stabilises it, and both are needed. Ever since Alex Osborne invented ‘brainstorming’ for

use in advertising, the necessity of the interplay between the two has been recognised. This paper

develops a more formal and structured way to approach to describe this interplay with the aid of a

number of case studies: a physical survival problem, a business problem, a scientific problem and an

educational problem.

2 The cross of invention

The creative process needs to have a defined beginning and a defined end. These points need not stay

the same throughout the process, but they provide a channel within which exploration can take place.

This exploration is provoked by the interplay of negation and affirmation. In this paper, the starting

point in creative exploration is a question triggered by a problem, the mid-point is a conjunction of an

affirmation and a negation, and the end point is a solution that satisfies the conjunction and solves the

problem (see figure 1).


Figure 1: The channel of exploration

The question should have a clear subject and object since it is from them that the appropriate

negations and affirmations are derived. The affirmations and negations interact in two intersecting

channels that resemble a cross, hence the name “cross of invention” which is shown as figure 2 in its

most abstract form. It will be illustrated below by means of a simple example.

Figure 2: The cross of invention in its most abstract form

The cross can be used without the symbols; they are used more as a shortcut because without them

one would have to repeatedly write out long sentences. The symbol “~” represents negation. So A is

the affirmation and ~A (not-A) is the negation of A. The middle of the cross directs us to find

possibilities by asking what would make either (A∧~B) true or (~A∧B) true. The symbol “∧” stands

for “and” and it conjoins the affirmation on one bar with the negation on the other bar. This is written

as (A∧~B)∨(~A∧B), since the symbol “∨” stands for “or”.

To show how to construct a cross of invention to guide the generation of new ideas I’ll use the

example of an actual escape from death. In 1949 fifteen smokejumpers parachuted into Mann Gulch

to bring a forest fire under control. Soon after that twelve of them were dead after trying to escape the

fire that chased them up a steep hill. Only two of them managed to run to safety.

The other survivor was the leader Wagner Dodge who found an ingenious way to escape and

motioned his team to join him. He set a fire ahead of them, which created a safety zone that the

pursuing fire could not reach. But everyone in his team thought he was being irrational, and instead

continued trying to outrun the forest fire that was gaining on them. As the rest of his team continued

running, Wagner Dodge dived into the ashes of the fire he set and survived as the pursuing fire passed

around him.

Problematic question

Conjunction of negation and affirmation

Idea that makes conjunction

possible


The Mann Gulch disaster was made famous in management studies by Weick (1993) as a case of

failure of sensemaking in organisations. Klein (2013) saw in Dodge’s thinking the overturning of an

assumption (or self-imposed constraint) which led to his idea. The cross of invention described in this

paper gives more substance to Klein’s discussion and explains a systematic way through which Dodge

might have reached his idea.

Wagner Dodge’s problem expressed as a question was: how do I survive the fire? From this problem

the cross of invention can be constructed as follows:

1. Identify the subject and the object in the question. The subject is Wagner Dodge and the object

is the fire.

2. Either the subject or the object is entered in the vertical bar of the cross, appropriately

affirmed and negated. In this case the subject is used. The subject in the problem remains

unchanged or in a constant state when it stays in the fire so “stay in the fire” becomes the

affirmation (and labelled A). The negation is then simply “don’t stay in the fire” (and labelled

~A).

3. Identify what verb or action describes the interaction between the subject and the object. In this

case it is the act of putting out the fire. This interaction is entered in the horizontal bar of the

cross.

4. Describe the interaction in a way that is expected or in harmony and enter it into the vertical

bar, appropriately affirmed and negated. The smokejumpers were expected to extinguish the

fire so “put out the fire” is the affirmation (and labelled B). The negation is the interaction that

is unexpected or conflicting in some way (“don’t put out the fire” and labelled ~B). At this

point the cross has been constructed, and is shown in figure 3.

Figure 3: Wagner Dodge’s cross of invention at Mann Gulch

5. Use the cross to generate the mid-point which is the conjunction of an affirmation drawn from

one bar of the cross and a negation from the other bar of the cross. It makes us search for

possibilities that would satisfy one of the following conjunctions: A∧~B (“stay in the fire” and

“don’t put out the fire”) or ~A∧B (“don’t stay in the fire” and “put out the fire”) true. These


two conjunctions are placed in the middle of the cross. Rephrased as questions they become:

“how do I stay in the fire and not put out the fire?” or “how do I not stay in the fire and put out

the fire?”

Once the cross is constructed it is time to use the power of negation to generate combinations of

negation and affirmation. One first takes each negation and asks: what are the possible forms this

negation may take. Table 1 shows two possible ways not to stay in the fire and two possible ways not

to put out the fire. These negations generate the creative possibilities.

Table 1: Possible negations

~A: don’t stay in the fire ~B: don’t put out the fire

Neg

ati

on

~A1: run away from the fire

~A2: run into the fire

~B1: do nothing

~B2: make fire

For the creative potential of negation to be utilised, it needs to be constrained by an affirmation. This

is done by conjoining the negations on one bar of the cross to an affirmation on the other bar of the

cross, as given in table 2.

Table 2: Possible conjunctions

Conjunctions 1-2 (~A∧B) Conjunctions 3-4 (A∧~B)

Neg

ati

on

~A1: run away from the fire

~A2: run into the fire

~B1: do nothing

~B2: make fire

Con

join

wit

h…

B: put out the fire A: stay in the fire

From table 2 it is clear how negation expands the possibilities, while affirmation limits them.

Conjoining the negations in every column with their relevant affirmation gives us four possible

conjunctions to explore (~A1∧B, ~A2∧B, A∧~B1, A∧~B2).

We do not conjoin A and B because it is this conjunction that describes the status quo. While we

could conjoin ~A and ~B it is less useful because without affirmation to anchor the creative process

all the negations together will very likely give rise to too many possibilities (especially as the number

of negation possibilities rise).

The end point of the exploration can be determined by finding an idea that makes one of the four

conjunctions possible and answers the question. The conjunction that appeared to have generated the

crucial insight was A∧~B specifically A∧~B2. The question that directs thinking here is whether it is

possible to stay in the fire by making fire. The answer was that by making fire Dodge created a safe

zone which the pursuing fire could not reach. Or in formal logical terms: (A∧~Bn)→(~A∨B) which

means that the conjunction ultimately led to not staying in the dangerous fire or putting out the fire

before it could reach him.


3 Additional case studies

As further illustrations of the cross of invention, three more case studies follow. The first two explain

past instances of good ideas: the first one is discussed in Slywotzky and Weber’s (2012) useful book

on tapping latent demand, and the second one was analysed by Klein (2013) in his interesting book on

how we generate insights. While most creative thinking techniques seem to work in hindsight, I

wanted to see if it was possible to apply the cross in a situation where I did not yet know the solution

– which is what the third case is about.

3.1 Getting Zipcar adopted

In 1999, Robin Chase started Zipcar in the Boston area of the USA. It was meant to be a convenient

and accessible car-sharing service that made it unnecessary to own your own vehicle. Members could

access cars kept at prearranged parking locations throughout the area, and reserve the closest one

through a Web-based rental system. Robin Chase explained that her goal was “to make access to cars

as easy as getting cash from an ATM machine” (Slywotzky & Weber 2012:21).

But the service was not getting adopted fast enough, and investors were getting impatient. In 2003

Scott Griffith was called in to solve the problem. He identified the problem as a chicken-or-egg

situation. To increase adoption rates, the cars had to be within close walking distance of potential

customers, but this is expensive because it required acquiring more cars than the company could

afford at the time. The only way to generate the income with which to buy more cars was to increase

adoption of the service, but adoption was not happening because cars were not close enough to

potential customers. Furthermore investors needed a return on their investment, which could be

generated by expanding the service, but to expand the service would have resulted in cars being

spread too far apart for customers to access conveniently. So, in short, lack of adoption was limiting

expansion of the service, and lack of expansion was limiting adoption of the service.

The problem facing Griffith was: how do we expand Zipcar to more areas? From this problem the

cross of invention can be constructed as follows:

1. Identify the subject and the object in the question. The subject is Zipcar and the object is the

areas to which the service can be expanded.

2. Either the subject or the object will be entered in the vertical bar of the cross. In this case the

subject is used. Since the subject is intended to expand, the affirmation which leaves the subject

in its intended form is “expand Zipcar to more areas” (and labelled A). This is then negated

(“don’t expand Zipcar to more areas” and labelled ~A).


case it is the act of adopting the service. This interaction is entered in the horizontal bar of the

cross.

4. Describe the interaction in a way that is expected (“get more people to adopt Zipcar” and label

it B) and also in a way that is unexpected or conflicting in some way (“don’t get more people to

adopt Zipcar” and label it ~B).

5. At this point the cross has been constructed, and is shown in figure 4. This cross can now be

used to generate the mid-point which is the conjunction of an affirmation drawn from one bar

of the cross and a negation from the other bar of the cross. It makes us search for possibilities

that would satisfy one of the following conjunctions: A∧~B (“expand Zipcar to more areas”

and “don’t get more people to adopt Zipcar”) or ~A∧B (“don’t expand Zipcar to more areas”

and “get more people to adopt Zipcar”) true.


Each negation (~A and ~B) now needs to be explored for the different ways of negation, as shown in

table 3. This is what generates the creative possibilities since it shows two possibilities consistent with

each negation (more are possible).

Table 3: Possible negations

~A: don’t expand Zipcar to more areas ~B: don’t get more people to adopt Zipcar

Neg

ati

on

~A1: remain in the same areas

~A2: contract to fewer areas

~B1: same number of people to use Zipcar

~B2: fewer people to adopt Zipcar

But for the creative potential of negation to be utilised, it needs to be constrained by an affirmation.

This is done by conjoining the negations on one bar of the cross to an affirmation on the other bar of

the cross, as given in table 4.

Table 4: Possible conjunctions

Conjunctions 1-2 Conjunctions 3-4

Neg

ati

on

~A1: remain in the same areas

~A2: contract to fewer areas

~B1: same number of people to use Zipcar

~B2: fewer people to adopt Zipcar

Con

join

wit

h…

B: get more people to adopt Zipcar A: expand Zipcar to more areas

Conjoining the negations in every column with their relevant affirmation offers four possible

conjunctions to explore (~A1∧B, ~A2∧B, A∧~B1, A∧~B2). These are placed in the middle of the

cross, and we ask how it is possible for one or more of them to be true.

Figure 4: Griffith’s cross of invention at Zipcar

The conjunction that appeared to have

generated the crucial insight and answered the

problematic question was ~A∧B, specifically

~A2∧B. The question that directs thinking here

is whether it is possible to reduce the number

of areas where the service is offered and

increase adoption.

The answer is that as the service is (initially)

contracted to fewer areas, the density of the

service increases. In other words, there will be

more cars in any single area, which places cars

closer to potential customers in that area. This

increases adoption in that area, which in turn

generates the income that allows for expansion

to more areas. Or in formal logical terms:

(~A2∧B)→A which means that the conjunction


ultimately led to expansion to more areas. This idea could also have resulted from ~A2∧~B1. (is it

possible that the same number of people would adopt Zipcar if we contracted to fewer areas?).

3.2 Einstein and the constant speed of light

Einstein is famous for his theory of relativity which he discovered through his thought experiment of

riding on a beam of light. The puzzle he confronted was why light moves away from an observer at

the constant speed of light (the c in e=mc2) regardless of the speed at which the observer is travelling.

It is a puzzle because one would expect to gain on a light particle as one moved faster, yet the light

particle moves away at a constant speed.

From this puzzle, a cross of invention can be constructed from the question: why does light move at a

constant speed relative to the observer no matter how fast the observer moves?

1. Identify the subject and the object in the question. The subject is the observer and the object is

the speed of light.

2. Either the subject or the object is entered in the vertical bar of the cross. In this case it is most

fruitful to place the puzzling object (the speed of light) here. It is then affirmed (that light

moves at a constant speed and labelled A) and negated by being changed (light does not move

at a constant speed and labelled ~A).


case it is the act of travelling and moving away.

4. Describe the interaction in a way that is in line with what is known (that light moves away at a

constant speed from the observer – and label it B) and also in a way that conflicts with what is

known (that light does not move away from the observer at a constant speed and label it ~B).


used to generate the mid-point which is the conjunction of an affirmation drawn from one

channel of the cross and a negation from the other channel of the cross. It could conceivably

have made Einstein search for possibilities that would satisfy one of the following

conjunctions: A∧~B (“light moves at a constant speed” and “light does not move away from

subject at a constant speed”) or ~A∧B (“light does not move at a constant speed” and “light

moves away from subject at a constant speed”) true.


Figure 5: Cross of invention for Einstein

Now that the cross is constructed it should be used to direct the creative exploration. The conjunction

that appeared to have generated the crucial insight was A∧~B possibly because a constant speed of

light (A) was probably in Einstein’s mind the least open to be challenged. That meant that possible

negations of B had to be generated, and this is shown as conjunctions 4-6 in table 5.

Table 5: Possible forms of negation


Neg

ati

on

~A1: light decelerates

~A2: light accelerates

~A3: light moves at a variable speed

~B1: the observer decelerates relative to

light

~B2: the observer accelerates relative to

light

~B3: the observer’s speed varies relative to

light

Con

join

wit

h…

B: light moves away from subject at a

constant speed A: light moves at a constant speed

The conjunction A∧~B1 appears (in hindsight) to be the most interesting one to explore for an

observer trying to catch up with a light particle – that is the idea that for light to move at a constant

speed at all times, the speeding observer in pursuit is probably slowing down relative to light. Speed is

calculated from changes in distance and time, so if the observer decelerates relative to light even as he

is speeding up something strange has to be happening to the space and time around him. Either space

must be expanding around the pursuing observer as he is trying to cover the same distance in less

time, or time must be slowing down for him as he is trying to cover a greater distance in the same

time.

But light continues to move away from the observer even if the observer slows down, so the

conjunction A∧~B2 is useful for the case of an observer that slows down relative to light. So, the


conjunction A∧~B3 is the most general case that captures the insights of both A∧~B1 and A∧~B2. The

conclusions Einstein drew from this were that space and time are firstly not constant (as imagined by

classical physics), and secondly that they are interdependent, and best referred to as space-time.

3.3 Solving a professor’s problem

To show how to construct a cross of invention to guide the generation of new ideas in a situation

where I did not yet have any insights, I consulted with a professor faced with a problem: she is trying

to maintain the academic standard in her courses, but this requires that she gives the students regular

written assignments that take a very long time to assess properly. She wanted new ideas beyond the

obvious ones of self-assessment and peer assessment. Her question became: how can I reduce the

work involved in assessment without dropping the standard of my course?

From this question the cross can be constructed as follows:

1. Identify the subject and the object in the question. The subject is the professor who is marking

the assignments and the object is the academic standard of the course.

2. Either the subject or the object will be entered in the vertical bar of the cross. In this case the

object is affirmed by being left unchanged (“maintain the standard” and labelled A) and negated

(“don’t maintain the standard” and labelled ~A).


case it is the act of marking that makes the professor interact with the standard of the course.

4. Describe the interaction in a way that is expected (“mark the same number of assignments” and

labelled B) and also in a way that is unexpected or conflicting in some way (“don’t mark the

same number of assignments” and labelled ~B). This is entered on the horizontal bar.

Figure 6: Cross of invention for the professor


used to generate the mid-point which is the conjunction of an affirmation drawn from one

channel of the cross and a negation from the other channel of the cross. It makes us search for


possibilities that would satisfy one of the following conjunctions: A∧~B (“maintain standards”

and “do not mark the same number of assignments”) or ~A∧B (“do not maintain standards” and

“mark the same number of assignments”) true. These two conjunctions are placed in the middle

of the cross.

To expand the possibilities in these two conjunctions we generated possible negations. In a situation

where the solution is not known, the most useful negations are not obvious, so many will be generated

as is evident from table 6.

Table 6: Possible forms of negation


Neg

ati

on

~A1: lower the academic standard

~A2: increase the academic standard

~A3: have no academic standard

~A4: change the standard to something else

~B1: mark fewer assignments

~B2: mark more assignments

~B3: do not mark

~B4: mark something different

~B5: mark in a different way

~B6: someone else marks

Con

join

wit

h…

B: mark the same number of assignments A: maintain academic standards

From table 6 it is clear how negation expands the possibilities, while affirmation limits them.

Conjoining the negations in every column with their relevant affirmation gave us ten possible

conjunctions to explore (~A1∧B, …, ~A4∧B, ~B1∧A, ~B2∧A, …, ~B6∧A).

In this case the professor initially found the end point A∨~B most attractive which means she explored

possibilities to make any one of conjunctions 1-4 true, but she knew not be closed to the possibilities

of conjunctions 5-10 or even possibilities that conjoin negations of A and B (though this would add

another 24 conjunctions to explore).

Conjunctions 1-4 made her ask how it is possible to:

~A1∧B: lower the academic standard and mark the same number of assignments

~A2∧B: increase the academic standard and mark the same number of assignments

~A3∧B: have no academic standard and mark the same number of assignments

~A4∧B: change the standard to something else and mark the same number of assignments

This led to interesting discussions about what an “academic standard” is, but no concrete ideas at that

point. Despite initially thinking that conjunctions 1-4 would be most fruitful, it was conjunctions 5-10

that generated the most innovative ideas, specifically:

A∧~B2: maintain academic standards and mark more assignments. A useful idea that emerged

from this line of thinking is that by having much shorter assignments it would not only be

possible to assess more, but also to mark faster. This also allows the exploitation of the so-

called “testing effect” (Roediger & Karpicke 2006).

A∧~B4: maintain academic standards and mark something different. This generated an idea that

is interesting, but needs some refinement. The idea is that a random number generator is used to

give a mark to each student for a specific assignment (the range would for example be between

40% and 60% and may gradually decline over the course of the semester). If a student does not

challenge the mark, the mark stays and their assignment is not marked (though it may be


supplemented with random checking). If a student challenges the mark, they have to provide a

new mark and a detailed defense of the mark. The professor then marks the defense of the new

mark and the professor’s mark is the mark awarded. The act of having to provide a defense taps

the power of self-assessment and enhances the learning.

A∧~B6: maintain academic standards and someone else marks. The professor obtained this idea

from someone else and it involves posting some assignments on social media, and then giving a

mark based on the feedback from others on the posting (e.g. counting the “likes”).

There is one special case where it may be useful to conjoin negations – specifically negations that

describe the exact opposite of what is desired. In this case it may be ~A1 (lower the academic

standard) and ~B2 (mark more assignments). By asking how it is possible to have ~A1∧B2 one

explores how it might happen that the current trajectory (of increasing marking) could have

unintended negative consequences. Useful ideas may then result from creating a reversal of that

situation.

For example, the professor realized that marking more assignments increased her own workload (so

she could no longer give adequate feedback) and increased the workload of students (so they

produced lower quality work because they had less time to spend on assignments). In both cases, the

standard would ultimately suffer. This then led her to explore ways to provide quality feedback more

efficiently and questioning whether it was really important to assess as many topics as she did. This

led her to create ‘feedback codes’ where common mistakes were coded, detailed explanations

attached to each code, and the codes written on the assignments at the relevant places.

Some interesting ideas emerged from the ten conjunctions. It did not guarantee new ideas as much as

it channeled thinking toward areas where useful new ideas could be found.

4 Conclusion

The cross of invention exploits well-known creative principles: the critical role of constraints, the

interplay of expanding and constraining possibilities and the importance of disrupting the thinking

that keeps a problem in place. Creative possibilities are expanded by negating some constraints and

anchoring these possibilities to the constraints not negated. By conjoining negations with constraints,

questions follow that disrupt old thinking and provoke new thinking. The ideas that result from the

exploration lead to ideas that may be considered new, but not so new as to be considered absurd.

5 References Ackoff, R.L. 1978. The art of problem solving. New York: John Wiley & Sons.

Costello, F.J. & Keane, M.T. 2000. Efficient creativity: constraint-guided conceptual combination. Cognitive Science,

24(2):299-349.

Davis, M.S. 1971. That's interesting! Philosophy of the Social Sciences, 1:309-344.

Hirstein, W. 2005. Brain fiction. Cambridge: MIT Press.

Klein, G. 2013. Seeing what others don’t. New York: Public Affairs.

Nickles, T. 1981. What is a problem that we may solve it? Synthese, 47:85-118

Roediger, H.L. & Karpicke, J.D. 2006. Test-enhanced learning: taking memory tests improves long-term retention.

Psychological Science, 17(3):249-255.

Slywotzky, A.J. & Weber, K. 2012. Demand. London: Headline Publishing Group.

Weick, K.E. 1993. The collapse of sensemaking in organizations: the Mann Gulch disaster. Administrative Science

Quarterly, 38:628–652.

The cross of invention: a tool to guide creative exploration

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