Wednesday, August 14, 2013

Ludwig von Mises on Fuzzy Set-Theoretics

 Ludwig von Mises never heard of fuzzy sets, but I believe he would have been fond of what has been going on with fuzzy-sets as used currently in the social sciences.  Outside of the quotation below I going to list a few reasons why Mises would have excepted an approach such as this:

  1. Mises understood the different between asymmetrical and symmetrical relations.  This can be found in his division between case probability and class probability, i.e. frequentism.  Here, I assume that Mises would have the same division for truth claims as he would for probability theory.  (However, I would argue that Mises' case probability is not a probability theory but rather a truth claim involving thought experiments and deductive analysis).  Fuzzy set-theoretics can measure the degrees of membership of asymmetrical relationships.  Statistical notions such as the correlation coefficient measure symmetrical or class distributional relationships utilizing the arithmetical mean of the central tendency theorem. 
  2. Mises recognized the problem of incomplete observations.  He doesn't say much about the topic but he does say that observations are fragmentary (see quotation below).  In terms of fuzzy set-theoretics we can suppose that there are conditions A, B, C, D and that each combination of truth values of true and false possible for each of the conditions.  In addition, suppose that in a particular observation there were not observations of the combination TTFF of A, B, C, D and then the correlation coefficient is calculated.  The correlation coefficient would not contain all the possible cases. 
  3. Suppose we have some event that includes the condition A, B, C, D and there are observations of all truth value combinations.  However, there are 371 observations of TTTF but only 134 observations of TTFF and smaller observations of all other combinations of truth values.  A correlation coefficient would be a bias measure for an asymmetrical measure.   
There are changes whose causes are, at least for the present time, unknown to us.  Sometimes we succeed in acquiring a partial knowledge so that we are able to say: in 70 per cent of all cases A results in B, in the remaining cases in C, or even in D, E, F, and so on. In order to substitute for this fragmentary information more precise information it would be necessary to break up A into its elements.  As long as this is not achieved, we must acquiesce in a statistical law.  But this does not affect the praxeological meaning of causality.  Total or practical ignorance in some areas does not demolish the category of causality.                                                                                Ludwig von Mises, Human Action, pg. 22
Inductive methodologies (, i.e. set-theoretic and statistical analysis) have to take account that economic events are asymmetrical.  Nonetheless, there are symmetrical trends that can be observed in areas such as investment finance.  For example, the correlation between the dollar and the bond yield, in which case there is a chart here that indicates that the United States has lost the safe haven effect of foreign investment around 2012.  The concern here would be for what actually was observed with the historical frequencies retained.  On the other hand, asymmetrical events are concerned with necessity and sufficiency.  Set-theoretics can produce truth tables and measures of necessity and sufficiency (consistency and coverage).  I've produced a chart two months ago that shows that among countries in the Heritage Foundation's Economic Freedom Index 2013, there was strong set-theoretic relationship between the Rule of Law (the minimum of property rights and freedom from corruption) and the GDP per Capita (percentile).  The was a consistency of 0.914838 (91.48% sufficiency) and a coverage of 0.704714 (70.47% necessity).  The correlation coefficient was .745736 and the standard deviation was 0.265695.  The high sufficiency is normal in social sciences, since obviously there can be other causes of higher GDPs, such as the existence of a large amount of natural resources.  Here is the chart I created, which unfortunately uses the frequentist notion of percentile as opposed to the calibrated method used in the books by Charles Ragin. ...I image if I change my equation to account for the sum of Rule of Law and natural resources, that it would increase the measure of coverage.  I added the correlation coefficient and the standard deviation to show the skewed sensitivity of correlational reasoning (I suggest reading book on fuzzy set theory).  Nonetheless, the low standard deviation indicates a strong relationship.  The chart below shows that the Rule of Law, which is the sum of property rights and freedom from corruption, is a highly sufficient and necessary cause of higher GDP per Capita, a measure of prosperity.

Note that set-theoretic charts look differently than statistical charts.  Set-theoretic should tend towards having observations in the upper-left hand corner but close to the diagonal center.