Explanatory Conjugates; Formal Implicit Relations; Ideal Frequencies

… V. Lenzen in his Nature of Physical Theory emphasizes the genetic process that begins from experiential contents of force, heat, extension, duration, etc., to move through a process of redefinition towards terms implicitly defined by empirically established principles and laws.  .. Lindsay and Margenau in their Foundations of Physics, … may be said to exhibit a preference for terms implicitly defined by equations.  [CWL 3, 81-82/105]

Macroeconomics is an explanatory science; as science it employs scientific method.

… empirical method involves four distinct elements, namely

    • the observation of data,
    • insight into data,
    • the formulation of the insight or set of insights, and
    • the verification of the formulation. [CWL 3, 78-79/102]

Macroeconomic theory is constituted by the explanatory relations of interdependent velocitous phenomena to one another.  The relations are correlations, thus the terms are correlatives.  The terms in the relations are linked or “yoked” inextricably to one another so as to implicitly and mutually define one another;  and each term of the general explanation is denoted as an “explanatory conjugate.” Cum-iugum!

… verification is of formulations, and formulations state

  • the relations of things to our senses, and
  • the relations of things to one another.

Pure or explanatory conjugates, … are correlatives defined implicitly by empirically established correlations, functions, laws, theories, systems. [CWL 3, 79-80/101-103 ]

If the scientist obtains his hypothesis in the double movement from above downwards (from possibly applicable mathematical forms such as differential equations) and from below upwards  (from data to curve fitting) – i.e. the scissors action – he reaches a formula.  That formula is of itself a hypothesis.  But he does not just announce, ‘I have a hypothesis.’  He makes all possible deductions from that hypothesis, either from it alone or from it in combination with other things.  From the deductions he proceeds to a process of checking.  Does what follows from the hypothesis occur de facto? The fuller that deduction is and the greater the number of checks he makes, the greater the likelihood that he will turn up some facts that his hypothesis does not satisfy.  He then moves to a new insight and a new hypothesis. [CWL 10, 142]

Science employs scientific method.  To accomplish his goal the scientist must be a methodologist.

… No doubt Keynes was an economist first and a methodologist second but he was none the less very articulate about his theorizing……..Lonergan, for his part, is perhaps a methodologist first and an economist second, but, as we shall see, he was able to push his economic reflections further than Keynes because he had a firmer grasp of the essentials of an effective theory.   [Gibbons, 1987]

Lonergan’s critique (shows that) by using the technique of implicit definition, the emphasis shifts from (first) trying to define the relevant variables to (first) searching heuristically for the maximum extent of interconnections and interdependence; and that the variables discovered in this way might not resemble very much the objects which, in the first instance, one was thinking about.  [Gibbons, 1987]

the terms are defined by the relations in which they stand, that is, by a process of implicit definition. [Gibbons 1987, 313]  (Click here)

Keeping in mind D. Hilbert’s method of implicit definition and focusing on one of Lonergan’s implicit equations, we note that the everyday commonsense descriptive terms take on new meaning as general and abstract technical and explanatory terms.  Lonergan’s scientific terms are implicitly defined by the relations in which they stand with one another.

P’Q’ = p’a’Q’ + p”a”Q”   (CWL 15, 157-78)

We may read from left to right, right to left, or back and forth between right and left.  From left to right, expenditures-receipts, P’Q’, define  and determine concomitant macroeconomic costs, p’a’Q’ and p”a”Q”, as they are defined (See CWL 15, 156-58)  From right to left, basic and surplus costs-outlays constitute the incomes which define and determine what is concomitantly expended for basic products.  Travelling back and forth between left and right, the equals sign mandates the reciprocal constraining influence on one another of pretio-quantial expenditures-receipts and pretio-quantital costs-outlays constituting basic incomes.

Lonergan cites examples from mechanics, thermodynamics, and electromagnetics in which “meaning” is determined implicitly by relationship:

… masses might be defined as the correlatives implicit in Newton’s law of inverse squares, (F = Gm1m2/d2).  Then there would be a pattern of relationships constituted by the verified equation; the pattern of relationships would fix the meaning of the pair of coefficients, m1, m2; and the meaning so determined would be the meaning of the name, mass.  In like manner, heat might be defined implicitly by the first law of thermodynamics, [1], and the electric and magnetic field intensities, E and H, might be regarded as vector quantities (whose meaning is) defined (implicitly) by Maxwell’s equations for the electromagnetic field. [CWL 3,  80/103]  [2]

  • Del X E = (-1/c)H’
  • Del X H = (+1/c)E’
  • Del dot H = 0
  • Del dot E = 0

General and abstract explanatory conjugates are implicitly defined by correlations, functions, laws, theories, and systems, rather than by the dictionaries of Webster, Oxford, or American Heritage.

Now, at the root of classical method there are two heuristic principles.  The first is that similars are understood similarly, that a difference of understanding presupposes a significant difference of data.  The second is that similarities, relevant to explanation, lie not in the relations of things to our senses but in their relations to one another.  Next, when these heuristic principles are applied, there  result classifications by sensible similarity, then correlations, and finally the verification of correlations and of systems of correlations.  But verified correlations necessarily involve the verification of terms implicitly defined by the correlations; and they do not involve more than such implicitly defined terms as related, for what is verified accurately is not this or that particular proposition but the general and abstract proposition on which ranges of ranges of particular propositions converge. … there is a fundamental heuristic structure that leads to the determination of conjugates, that is, of terms defined implicitly by their empirically and explanatory relations.  Such terms as related are known by understanding, and so they are forms. Let us name them conjugate forms. (CWL 3, 435/460-61)

Repeating for emphasis:

Pure or explanatory conjugates, … are correlatives defined implicitly by empirically established correlations, functions, laws, theories, systems. [CWL 3, 79-80/101-03]

… pure (explanatory) conjugates satisfy (empirical science’s) Canon of Parsimony.  For the equations (stating the relations among the explanatory conjugates) are or can be established empirically.  And by definition pure conjugates mean no more than necessarily is implicit in the meaning of such verified equations. [CWL 3, 80/103]

(The Canon of Parsimony) is at once obvious and difficult. … it forbids the empirical scientist to affirm what, as an empirical scientist, he does not know. … empirical method involves four distinct elements, namely …

    • the observation of data,
    • insight into data,
    • the formulation of the insight or set of insights, and
    • the verification of the formulation. (CWL 3, 78-79/102)

Econometricians in academia, the Bureau of Economic Analysis, and the Federal Reserve must verify the hypothesis of Functional Macroeconomic Dynamics (also known as Macroeconomic Field theory).

Now the empirical investigator cannot be said to know what is not verified and he cannot be said to be able to know the unverifiable.  Because, then, verification is essential to his method, the canon of parsimony in its most elementary form excludes from scientific affirmation all statements that are unverified and, still more so, all that are unverifiable.  [CWL 3, 78-79/]

the pure (explanatory) conjugate has its verification, not in contents of experience nor in their actual or potential correlatives, but only in combinations of such contents and correlatives.  I see, for instance, a series of extensions and alongside each I see a yard-stick; from the series of combinations I obtain a series of measurements; from the correlation of the two series, together with a leap of insight, I am led to posit as probably realized some continuous function; pure conjugates are the minimal correlatives implicit in such functions; and their verification finds its ground, not in experience as such, but only in the combination of combinations, etc., etc., etc., of experiences. [CWL 3, 80/103-04  ]

Note that Lonergan’s primitive terms “basic” and “surplus” are precisely and analytically distinguished and defined by their point-to-point and point-to-line correspondence with elements passing out of under-process into the standard of living.

Also note that in the general case prediction is impossible.

However, besides classical laws, there are also statistical laws; and since the latter as well as the former are verifiable, it would seem that, besides pure and experiential conjugates, one must also recognize events. … The law of nature, then, is one thing.  The event of its illustration is another.  And such events are subject to laws of a different type named statistical. [CWL 3, 82/]

… formulations that concern events are formulations that answer the question for intelligence, How often?  They concern events, for the frequency they assign is a frequency of events.  Finally the frequency assigned by a statistical law is verifiable; for the assigned frequency is an ideal frequency; it is distinct from the actual frequencies that can diverge from it in non-systematic fashion; and it can be verified by appealing to those actual frequencies. [CWL 3, 83/  ]


[1] The first law says that the change in the internal energy of a system  is equal to the sum of the heat gained or lost by the system and the work done  by or on the system.

[2] Lonergan was aware that Newton’s mechanics was a theory of efficient causes and contained the residue of the experience of force and muscular effort in contrast to Einstein’s field equations.

☐  CWL 15, Appendix p. 180

Part Two … belongs almost entirely in what I call the Einsteinian context of Part Three, in contrast to the Newtonian achievement of Part One, (but) it is still somewhat transitional in system and expression.  So, for example, to take the central character in the drama, pure surplus income is there named systematic profits. CWL 21, 325