D.1. The Primacy of Deductive Logic: Economic Analysis is about deducing the implications about the human behaviour from the Given Data.
E.1. From 1.1.1 and 1.2.1, it follows that Economics only studies the implications of given means and given ends for our behaviour, and excludes any theoretical or empirical explanation of choice of ends, or presence of means. All of these fall out of bounds of Economics. Economics, following 1.1.1 and 1.2.1, takes the data about means and ends as given, and proceeds to study the “implications” of these for our behaviour. Since, the implications can be studied only by the method of deduction, it follows that due to the definition of the subject, Economics remains only a mathematical, and deductive, branch of knowledge.
E.2. A logical derivation of aforesaid is as follows, based on original text summarized (Robbins, L. (2007). An essay on the nature and significance of economic science.) and infusing some additional knowledge of social science. The purpose is to make explicit the aforesaid.
i. We have only two logical methods; Either Deductive, or Inductive.
ii. Economics takes the information about means and ends as given, and studies the “implications” of these given data. (From 1.1.1, and 1.2.1)
iii. The implications of something are found by method of deduction. (By definition, see below)
iv. By “implication”, we mean to deduct additional concepts without modifying our given data.
Deduction identifies relations between concepts, variables etc in a more direct manner. For an instance we are given: X > Y, A < Z, Y = Z+1, and we are interested only in a relation between X and A, so we deduce X > A which deducts all other concepts but shows relevant information explicitly. This is deductive logic, and this makes implications of given data explicit for our use.
v. All observational studies only yield probabilities.
The method of inductive reasoning, observational studies or experimentation, only points to a probability in the data generated by statistically controlled or lab controlled setting. Dealing with the data, all an observational study does is to report variances and covariances between multiple variables, and a non-zero covariance by definition implies a non-uniform probability distribution. If all values are the same i.e. no probability distribution exists, variance will be zero, covariance will be zero, and thus coefficients in regression will be an error (See below), and experimental results will report probabilities of 1.0. This is never the case. Thus, observational studies only report, implicitly or explicitly, probabilities, often from non-uniform distributions.
If there is a uniform distribution, for our purpose of empirical study etc, it is same as if we have one number, and no probabilities. Besides, uniform distribution is a rarity among rarities. Moreover, since the formula of bivariate (simplest) regression coefficient is Covariance (XY)/Variance(Y), a zero in all cases gives an error in result. Finally, since experimental setting uses real life data, it is least likely to have probability of 1.0 in results. Because the most frequent, almost universal, are the distributions which not uniform.
vi. A probability is only a statistic, and not an implication.
vii. Therefore, the method of inductive reasoning does not allow for the identification of implications.
viii. Thus, the method of Economics is necessarily only deductive logic .
E.3. This is a logical necessity of the definitions in i and ii, and the proceeding propositions about inductive versus deductive logic.
E.4. Since probabilities are subsets on a set of real numbers from 0 to 1, Set Theory, which is deductive logic, is employed to deduce probabilities which are not directly given, but implied.
E.5. Jevons was a logician and an economist who wrote extensively on both topics eventually developing the first ever mathematical work in economics. Mathematics is deductive logic.