*The Structure of Value: Foundations of Scientific Axiology* is Robert S. Hartman’s seminal work on Formal Axiology. Robert S. Hartman was born in Germany in 1910. Seeing the Nazis organize evil, he fled Nazi Germany for his opposition to Hitler. He devoted the rest of his life to organize good. This led him to a life-long quest to answer the question, ”What is good?” and how to apply the answer to help preserve and enhance the value of human life. Here’s an overview of the book together with some conclusions.

**Part One: The Structure of Science**

**Philosophy vs. Science**

Hartman starts his book with an examination of the transition from *value philosophy* to *value science*.^{1} For him, the difference between philosophy and science is a methodological one.

- The method of philosophy is
*analysis*. The concepts are relatively*unstructure*d, in a definite logical way. Analysis not only has relative lack of structure, due to its abstractive nature, but it also has relative lack of relevance to actuality.^{2} - The scientific method is one of
*synthesis*. The concepts in science are precisely*structured*. Axiomatic synthesis gives rise to systems which mirror the total variety of the corresponding actuality.^{3}

**Extensional vs. Intensional Logic**

Surprisingly, the concept *concept* has never been fully treated in the history of logic.^{4} The development from natural philosophy to natural science is based on *extensional* logic. This is the part of logic which has been determined with precision, for example mathematics. The other part of logic which has been neglected is *intensional logic.^{5}*

**Value vs. Number**

A logic of intensions investigates and structures the interrelationships between intensions, up to and including the totality of all intentions. This gives us *meaning*, rather than just an inventory of the world.^{6} The definition of *Value* is the intensional analogue of the logical definition of *Number*. Inversely, the definition of *Number* is the extensional analogue of the definition of *Value*.^{7}

**Formal Axiology vs. Mathematics**

*Formal axiology* is with respect to intension what *mathematics* is with respect to extension. And what *mathematics* is to natural philosophy, *formal axiology* is to moral philosophy.^{8} Extensional logics is applied to *mathematics*, and *mathematics* to the natural sciences. Intensional logic is applied to *formal axiology*, and *formal axiology* to the moral sciences.^{9}

It is relatively easy to follow an analytic argument, but it is difficult to follow a synthetic one. To do so, it is necessary to think both formally and systematically. It is, furthermore, a true art to find a correspondence between reality and a formal system. This can only can be learned by practice.^{10}

**Analytic vs. Synthetic Concepts**

The difference between *analytic* and *synthetic concepts* defines, as we have seen, the difference between philosophy and science.^{11} The intension of the *analytic concept* contains within itself other concepts equally abstracted.^{12} A *synthetic* concept, on the other hand, is very different. It consists of terms related to terms. The model of a synthetic intension is a network rather than a nest of Chinese boxes.^{13} The difference between *term* and *concept* is that the *term* has neither intension nor extension. The term is a *constructed variable*. All its significance derives from its interrelationship with other terms.^{14}

**Part Two: The Foundations of Value Science**

**Axiological Value**

If value theory is to become a science, then *Value* must be determined by an *axiom* which identifies it with some notion or application of logic.^{15} *The Axiom of Formal Axiology* is the *definition of Good*:

**” A thing is good if it fulfills the intension of its concept.”**

^{16}

This axiom defines axiological Value in general.^{17} Axiological interpretation is subjective, axiological formalization is objective.^{18}

**Exact Value Measurement**

The application of combinatorial calculus makes exact measurement of value possible. There are three possible kinds of sets, *finite*, *denumberably infinite*, and *nondenumberably infinite*.^{19}

*Finite sets*define*formal concepts*. The things corresponding to them are constructions of the human mind and are called*systemic values*. Such things either fulfill their concept or they are no such things.^{20}*Denumberably infinite sets*define*abstract concepts*. These properties are denumerable, for they must be abstracted one by one. Fulfillment by a thing of an abstract concept constitutes*extrinsic value*.^{21}*Nondenumerably infinite sets*define*singular concepts*. Things corresponding to such concepts are unique. Uniqueness is the intensional counterpart to extensional singularity. The fullfilment by a thing of a singular concept constitutes*intrinsic value*.^{22}

**Systemic, Extrinsic, and Intrinsic Value**

Systemic value, extrinsic value, and intrinsic value are the three value dimensions. They constitute a hierarchy of value. Intrinsic value is more valuable than extrinsic value, and extrinsic more than systemic value. The hierarchy of value is a valuation of value.^{23}

**Part Three: The Structure of Value**

**Systematic vs. Empirical Import**

Both the axiom and the system following from it have systematic and empirical import. The systematic import of the system is its logical structure. The empirical import is its capacity of accounting for the value realm, its applicability.^{24}

**Intensional Structures
**Formal axiology is based upon the logical structure of intension. Various kinds of intensional structure are arrived at by applying the rules of set theory:

*finite*,

*denumerably infinite*, and

*nondenumerably infinite*. These structures determine systemic, extrinsic, and intrinsic value. The intension, in formal axiology, is an

*axiometric structure*.

^{25}

**Fact and Value**

Formal axiology arrives at *value* being the reality of which *fact* is the measure. *Fact* measures *value*, but that is all.^{26} Any value dimension is fact to the succeeding dimension and value to the preceding dimension.^{27} The relationship between systemic, extrinsic and intrinsic value corresponds to a process of continuous enrichment with leaps from one value dimension to the next.^{28}

**World of Fact vs. World of Value**

As *natural science* creates a *world of fact*, so *axiological science* creates a *world of value*.^{29} The goodness of a thing is not the norm for the thing’s factuality, but for the thing’s value possibilities.^{30}

**Measure Value**

While the systematic import arises from the *axiometric nature* of intension, its empirical import arises from its *axiometric nature*, its capacity to measure value. The *value structure* is the structure of the *value form*. The v*alue measure* is the measure of the *value phenomenon*.^{31}

**Intensional vs. Extensional Structures**

*Intensional structures* are *axiometric* in the same fundamental sense that *extensional structures* are *physiometric*. The former measure value phenomena in the same sense that the latter measure physical phenomena. There is measurement in all three value dimensions. They have great differences among themselves.^{32}

**Dynamic Hierarchy of Values**

The hierarchy of values is dynamic. The experience of the value dimensions follow each other in any order. The application of the combinatorial laws to the value dimensions constitutes the *calculus of value*. Calculus of value is applying *exponentiation* to the value dimensions.^{33}

**Calculus of Value**

The calculus of value arises by combining the three value dimensions S (*systemic*), E (*extrinsic*), and I (*intrinsic*), and their arithmetical values. The combinations of these three value dimensions can either be *compositions* or *transpositions*.

**Value Compositions vs. Value Transpositions**

A *composition* of values is a *positive valuation* of one mode of value by another, while a *transposition* is a *negative valuation*. The most valuable value, that is, the value that fulfills the *Value* concept most fully, is intrinsic value. It is the positive value of a value.^{34}

**Secondary Value Combinations**

There are nine *compositions* and nine *transpositions* of the three value dimensions.^{35} Here are the possible *value combinations* in the order of their *axiological rank*:

I^{I}, E^{I}, S^{I}, I^{E}, I^{S}, E^{E}, S^{E}, E^{S}, S^{S}, S^{-S}, E^{-S}, S^{-E}, E^{-E}, I^{-S}, I^{-E}, S^{-I}, E^{-I}, I^{-I}

The formula I^{I} is , for example, *intrinsic valuation of intrinsic value*, such as valuing a baby. The formula I^{-E} is, on the other hand, *extrinsic disvaluation of intrinsic value*, such as to regard people as functions. Regarding people as functions has, by the way, the same axiological value as making the worst of a good situation. As is obvious, the value combinations can be combined in turn. Thus arise *tertiary*, *quaternary*, etc., *compositions* and *transpositions* of value.^{36}

**Perversion of Value**

*Disvalue posing as value* is a *perversion of value*. It is worse than straightforward disvaluation.^{37} An example is learning children to value not valuing themselves. The value formula (I^{-S})^{-S})^{S} covers, for example, any situation where a systemic disvaluation of a systemic disvaluation of an intrinsic value is systemically valued.^{38} Like a mathematical formula, a value formula is capable of infinite interpretation.^{39}

**Conclusions**

Robert S. Hartman defines science as the application of a *logical frame of reference* to a *subject matter*. Hartman’s specific approach to his *value science* makes use of combinatory mathematics. This makes an exact enumeration of the different value dimensions possible. Even the most complicated axiological arguments and situations can be analyzed by means of this calculus.^{40}

The book itself is an excellent example of *axiomatic synthesis*. I found it interesting to see *how* Hartman constructs the foundations of his value science. He obviously knows philosophy, science, and mathematics very well! The book is well-structured and clearly written, but is also *very* demanding to read!

Hartman’s own hope is that the application of axiology to actual situations will lead to higher levels of insights into the world of value.^{41} For example, that axiology will help to expose the real evils of our civilization.^{42} Hartman’s book is a remarkable achievement and his insights are profound!

**Notes**

^{1} Robert S. Hartman, *The Structure of Value: Foundations of Scientific Axiology* (Wopf & Stock, 2011, first published 1967), p. 14.

^{2} *Ibid.*, p. 46.

^{3} *Ibid.*.

^{4} *Ibid.*, p. 48.

^{5} *Ibid.*, p. 49.

^{6} *Ibid.*.

^{7} *Ibid.*, p. 52.

^{8} *Ibid.*, p. 53

^{9} *Ibid.*, p. 60.

^{10} *Ibid.*, p. 65.

^{11} *Ibid.*, p. 64.

^{12} *Ibid.*, p. 83.

^{13} *Ibid.*, p. 84.

^{14} *Ibid.*, p. 85.

^{15} *Ibid.*, p. 102.

^{16} *Ibid.*, p. 103.

^{17} *Ibid.*, p. 104.

^{18} *Ibid.*, p. 110.

^{19} *Ibid.*, p. 112.

^{20} *Ibid.*, p. 112.

^{21} *Ibid.*, p. 113.

^{22} *Ibid.*.

^{23} *Ibid.*, p. 114.

^{24} *Ibid.*, p. 154.

^{25} *Ibid.*, p. 193.

^{26} *Ibid.*, p. 220.

^{27} *Ibid.*, p. 221.

^{28} *Ibid.*, p. 223.

^{29} *Ibid.*, p. 225.

^{30} *Ibid.*, p. 226.

^{31} *Ibid.*, p. 249.

^{32} *Ibid.*, p. 250.

^{33} *Ibid.*, p. 265.

^{34} *Ibid.*, p. 268.

^{35} *Ibid.*, pp. 272–274.

^{36} *Ibid.*, p. 276.

^{37} *Ibid.*.

^{38} *Ibid.*, p. 277.

^{39} *Ibid.*.

^{40} *Ibid.*, p. 280.

^{41} *Ibid.*, p. 311.

^{42} *Ibid.*, p. 276.

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