Method 33

Use defeasible reasoning.  This is reasoning based upon what "usually happens", what is "usually done", or what "usually works" in similar‡ circumstances.

Class of Method

The 4th and Most Important of The Keys to Problem-Solving.

Use When

When you create a plan, or you are creating a general approach or process to solve a problem without the details of the process included or filled in.  When you do steps of other processes that are difficult to do using logic, or when you need to search for truth you do not yet know.

How

1.  Use your intuition to consider what seems to be the best choice or is the most accurate of one of the following of a through d:

(Your choice of these options depends upon what you are using defeasible reasoning for.)

a.  Interpretation of a statement.  This is done by interpreting a statement by what usually or most likely is being communicated given similar past statements made in similar circumstances.  For example, you might take the absolute logical interpretation of a statement as your most likely candidate to testing that interpretation’s validity, and then if that interpretation wasn’t correct, to then try the next most likely true interpretation and so on.

b.  A strategy or general approach to some problem type.  This is done by considering what usually or most likely would be the best general approach or strategy given your interpretation of similar approaches or strategies in previous similar circumstances.  You take the seemingly most promising approach, and if that approach was problematic, to then try the next most promising strategy or approach, etc.  For hard problems, you may need to return to previous strategies or approaches you considered earlier to reconsider their use.

c.  Deciding the most likely truths.  This is done by considering what usually or most likely are truths given similar past truths that existed in similar circumstances.

d.  Any other defeasible reasoning needs where you reason by what usually occurs in similar circumstances.

2.  Take what your intuition tells you, and begin to skeptically check what it gives you by experimenting and testing the information if possible.   For example, you may test various special cases of a proposed truth, or test a general plan for solving a problem by specifying simpler forms of your problem and the implications of the general plan on those simpler forms.

3.  When the information obtained from defeasible reasoning has been tested carefully and is likely useful, then use the information given from the last step towards solving your problem.  Otherwise, consider restarting this method and find the next most promising approach (or potential truth, etc).

Note:  In the steps above, you may use this defeasible reasoning to create or discover minor ideas (or also potential truths), and then construct the major ideas (or also potential truths) by doing defeasible reasoning on the supposed minor ideas (or potential truths) you considered.

Notes

Doing defeasible reasoning is hard-wired into our brains and it is usually quite automatic.  You may need to ponder on your defeasible reasoning for much time and sleep on a part of your problem to allow your mind to "figure it out" in less familiar situations.

This method is essential to creating a plan, and to guiding experimentation.

This method is important for finding supposed truths that you can use.

Defeasible reasoning is reasoning by using:

-What usually works

-What usually happens in certain similar circumstances

-What usually causes a certain type of result

There are three types of problem situations you will encounter depending on your resources and experience with using defeasible reasoning to solve problems in order of difficulty.  (See Schoenfeld [51]) 

1.  A mathematical situation or pattern that you directly recognize and its associated solution.  A “condition-action” pair according to your conditioning from experience.

2. A mathematical situation that you can classify as a type of problem, or a problem that you classified into various types according to your experience with problems.  With that type of problem you may know a general approach that tends to work well with that type of problem such as heuristics to use, information to consider, etc.

3. A complex but somewhat familiar or similar situation to other mathematical situations you are familiar with.  When solving other problems that have similarities to your new problem, you likely subconsciously classified and categorized those experiences into types.  Then you likely tried to categorize new experiences into classes (schema) that are already in your mind according to similarity before analyzing the problem in detail.  You use bodies of information about categories of experience that are potentially useful for the schema you felt to classify your problem as.  However you must always question and monitor the approach your schema dictates in case the approaches are faulty or lead to little progress.

Developing this skill (defeasible-reasoning) requires experience solving problems, observing many problems get solved, and having familiarity with truths from a variety of circumstances similar to situations of interest.  It requires a lot of mathematical study to gain a better intuition.  This is "crystalline intelligence".  Whereas abstraction, defining, language synchronization, model creation, and especially openness of mind to new models and ideas is "fluid intelligence".  Crystalline intelligence can often inhibit fluid intelligence, because experience often suggests against the recommendations of fluid intelligence.  Be careful to not let experience inhibit fluid intelligence.  You can accomplish this by always questioning the validity of your intuition (having skepticism always accompany defeasible reasoning).  This helps you to avoid locking yourself inside of a “box”, and stagnating your thinking abilities.

‡Two entities, relationships, or etc. being "similar" means that many defining parts of the entities, relationships, etc. are equivalent or that the defining parts of the entities have equivalencies in aspects of the definitions that apply to defining the entities, relationships, or etc.

Method 34

Think outside the box [45].  Relax the pre-conceptions you have about the problem.

Class of Method

A Type of Skepticism to Defeasible Reasoning.

Use When

When thinking inside the box didn't work to help find a solution to your problem.

How

1.  List all of your pre-conceptions on how to solve the problem and assumptions about the information in the problem.

2.  Remove a subset of the assumptions and pre-conceptions on the list that seem most reasonable—meaning that removal of a pre-conception or assumption would not significantly complicate the problem-solving process or the possibilities for solving the problem.  

However all other problem-solving options you have up to this point of solving your problem may be more complicated than complications that would be caused by removing a pre-conception or assumption.  If so, then remove the pre-conceptions or assumptions that would best balance complicating the problem as little as possible and allowing for sufficient possibilities for options to create good plans to solve the problem.

3.  Taking into account your success or failure of applying this method, choose the next heuristic (method) you will use based upon good defeasible reasoning or by using the general method given on page 100.

Notes

Thinking inside the box first is an important control principle for problem-solving efficiency because usually your preconceptions about a problem are most likely correct.

Try reinventing a completely new way of dealing with the problem by removing your previous beliefs of restrictions you had on how the problem should be solved.  Don’t be boxed in by standard thinking, or what methods you are used to seeing.

Example

See example problem 4 of chapter 6 (Examples of Problem Solving).

In chess often sacrificing a piece seems like a bad idea and out of the norm, but could win a game.

The development of the theory of relativity is a good example of discovery that came from thinking outside the box.  Henri Poincaré thought outside the box by considering the idea that in the universe, there is no absolute reference frame.  From that he set up a theory that Einstein built upon to derive special relativity.  Also the discovery of wave particle theory came from thinking outside the box by questioning the then currently accepted models.  Many great discoveries in physics come from questioning the accepted theories of the time.