# Formal Axiology

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Formal Axiology is the attempt to lay out the principles of ethics with mathematical rigor.

"*Formal*" refers to the fact that *formal logic* is involved.

"*Axio-*" is a Greek term and means "*Value*".

"*-ology*" also has a Greek origin, and it means "*the study of*."

"**Formal Axiology**" is the application *formal logic* to the *study of Value*.

Formal Axiology begins by positing an axiom of what value is, and then proceeds to deduce conclusions from the axiom.

### Contents

- Background on Axiomatic Systems
- A Simple Concrete Example
- Hartman's Formal Axiology
- Definition of Value in Terms of a Concept's Intension
- The Axiom
- Three Classes of Objects of Value
- Three Classes of Valuing Perspective
- Resulting Conclusions
- Validation of the Conclusions
- Advantages
- Disadvantages
- Concluding Remarks
- References
- External Links

### Background on Axiomatic Systems

In an *axiomatic system*, the axiom describes very general relationships between
mental constructions, or terms. These mental constructions must be further defined,
and tied down to observables, in the world. For example, in geometry, the invisible
point is said to be like a point on a piece of paper, or the invisible line is said
to be similar to a line drawn with a pencil. Once the terms have been clearly defined,
and tied down, one can proceed to perform mental operations (adding, subtracting,
etc.) on the clearly defined mental terms. These mental operations lead to theorems
that were not originally present in the axiom. This allows one to try out many new
combinations without actually having to go out into the world to manipulate objects.
This is the power of an axiomatic system - it saves one much physical labor. The
theorems that are mentally deduced can then be compared to the results of performing
the same operations in the physical world. If the physical world results are isomorphic
with the mental conclusions, one can conclude that the axiomatic system accurately
models the physical world. One can also infer that the axiom accurately states some
very abstract relationship that does actually exist in the world.

Since ethics is such an abstract subject, it may help to precede the following discussion with a more concrete example from the field of geometry.

### A Simple Concrete Example

**Example Axiom:** The volume of a sphere is *dependent upon* its diameter.

**Breaking down the axiom:** The *terms involved* in the axiom are volume,
sphere, and diameter. The relationship that is posited to exist is that volume *depends*
*on* the diameter.

**Analysis of the term diameter:** Assume that there are three categories of
sphere diameters: small (under two inches); medium (under two feet); and large (anything
over two feet).

**Applying operations to the terms in the axiom:** Let us combine two of the
three types of spheres in every possible way.

**Resulting Theorem:** The combination of the large sphere with another large
sphere creates the greatest volume.

**Verification of the model:** By comparing the volume of the combination of
two large spheres to the volumes of all other possible combinations of types of
spheres, we can determine if the model is isomorphic with reality.

###
Hartman's Formal Axiology

Robert S. Hartman
constructed a system of values based on formal axiology. Other systems are possible,
just as non-Euclidean geometries were created after Euclidean geometry. Each system
can be experimentally verified to determine which is more *isomorphic* with
the underlying data it is attempting to explain.

### Definition of Value in Terms of a Concept's Intension

Hartman began by surveying how people used the words "good" and "value". He looked
at writings, newspaper clippings, dictionaries, and so on. After looking at everyday
use of the words, he abstracted the idea that things are good to the extent that
they match up to what we think they should **ideally** be. If something exactly
matches the ideal, it is "perfect." If it closely matches the ideal, it is "good."
Things that have very little in common with their ideal are "bad."

The advantage of taking this empirical starting point of surveying how people use the words "good" and "value" is that it gives one the power to see when those words are being used according to mainstream usage and when they are not.

Hartman defined **Value** in a manner that is parallel to the way that **Number**
is defined in the "Principia Mathematica." Russell and Whitehead defined number
in terms of the extension of a class. The *extension* of a class refers to
the number of members contained in the class. Number is defined as being the class
of all classes with the same extension. For example, the Number Five represents
the class of all classes that have an extension of five members.

Classes have two main properties: extension and intension. In "The Structure of
Value," Hartman defined Value in terms of the intension of a class. The *intension*
of a class refers to the properties of the class. The intension of the class is
often thought of as the **meaning** of the class. There are two important types
of intension.

The **first type of intension** refers to the properties that an object **must**
possess in order to be a member of that class. This first type of intension can
be referred to as the *"minimum criteria"* because it is a list of the minimum
properties that something must have to be included in the class. Historically, this
first type of intension has been referred to as the *essential definition*
of a class.

The **second type of intension** refers to all the properties that an **ideal member**
of the class would have. This second type of intension can be referred to as the
*"maximum criteria"* because it describes every property that the ideal member
of that class must possess.

### The Axiom

The value of an object depends directly on how well the properties of the actual
object match up with the properties of the maximum criteria. To put it simply, the
value of an object depends on how close the object matches its ideal. Translating
this idea into formal language results in the following **axiom**:

**"A thing is good to the degree that it fulfills its concept."**

The formal statement of this axiom may be a little confusing to the person on the street, and a more down to earth rephrasing of the statement may be to say that a thing is good to the extent that it lives up to it's ideal. The power of this axiom lies in the fact that it allows one to deduce ethical conclusions in the same way that Russell and Whitehead deduced the rest of mathematics from the definition of number.

Since the words "fulfilled" and "concept" are used in the axiom, their meanings need to be further analyzed.

*Fulfillment* of a concept occurs when the properties of the actual object
match up to the properties of the ideal for that concept. Goodness results when
a concept is *fulfilled* just as a glass is *fulfilled* by pouring water
into it. Consequently, more value results when larger concepts are fulfilled than
when smaller ones are.

Based on the sizes of sets in set theory, there are three different sizes of *concepts*
(finite, denumerably infinite, and non-denumerably infinite). These three concept
sizes are based on the number of properties contained in the concept, and a simpler
way to think about concept size for this discussion is to just think of there being
"small", "medium", and "large" concepts.

Now that the axiom has been laid down, the next step is to analyze the **valuing situation
regarding its two most important dimensions**:

1) **objects of value**, and

2) **valuing perspectives**.

(These two dimensions of "value space" can be thought of as being analogous to the three dimensions involved in "geometrical space.")

### Three Classes of Objects of Value

By applying the measuring tool of concept size to the first dimension (**objects of
value**), one discovers three distinct classes. When proceeding from largest
concept size to smallest, the three distinct classes of objects of value are:

**a) People** - large-size concept because a person is a thing plus a thinking
being. This thinking being can think any of an unlimited number of thoughts;

**b) Things** - medium-size concept because they cannot think, but they are extended
in space and time and can be looked at from many different perspectives and subdivided
in many ways; and

**c) Ideas** - small-size concept because, for example, it only takes three properties
to define a triangle.

### Three Classes of Valuing Perspective

Similarly, if the measuring tool of concept size is applied to the second dimension
(**valuing perspectives**), one finds three different perspectives:

**a) Intrinsic** - large-size concept because one is not limiting the possibilities
by trying to use something to obtain a particular goal;

**b) Extrinsic or practical** - medium-size concept because one has limited the
possibilities by trying to get to a specific goal; and

**c) systemic, or labeling** - small-size concept because one is just trying
to determine if something has a few essential properties that are needed to make
it a member of a class.

### Resulting Conclusions

By performing combinational logic on the three classes of objects and on the three classes of valuing perspective, one is able to determine which combinations create the most value. For example, combining the intrinsic valuing perspective with a person produces a great deal of value because both concepts involved in the combination are "large" ones. The conclusion that results from this combinational logic is that loving a person is the most valuable, and killing, or hating, a person is the least valuable. Although that conclusion may not be all that astounding, it is still a purely logical deduction.

Formal axiology based on this axiom is not limited to this one conclusion. There are many other theorems that can be deduced from the axiom. For example, another conclusion is that in the world of things, each thing "ought to" become more like its ideal. By analogy, in the world of morality, each person ought to seek to live up to being their ideal self. What is the ideal self? The answer to this question can also be deduced from the axiom. The most fulfilling life is one in which the self is a loving person (i.e., one who intrinsically values people) - the fulfillment of that type of life can be compared to oceans of water being combined rather than only one tiny cup of water being combined with another.

### Validation of the Conclusions

Much scientific research has been conducted to attempt to validate the conclusions of this system. See the external links below to see a list of four of these efforts. However, a subjective analysis of one's own experience may be the easiest way to get existential validation.

Simply ask these questions:

1. "What moments of my life have been the most important to me?" and

2. "Did these moments involve my love for another person?"

### Advantages

1. The main **advantage** of this system is that it provides a unified outlook
on the reason why we call various things good.

2. Another benefit is that it allows one to avoid G.E. Moore's naturalistic fallacy.
In this system, the axiom plainly states that goodness is a **relationship**
- not a **thing**. Goodness is the relationship between the actual and the ideal.

3. One last advantage of this system is that it attempts to provide a way out of
hatred. This system shows that it is actually to the valuer's **benefit** to
be loving **since that creates greater fulfillment.**

### Disadvantages

1. The primary **disadvantage** of the system is that it involves formal logic,
which is difficult for most people to understand.

2. Another disadvantage of the system is that it has not received widespread acceptance.

3. A final criticism is that the system is not precise enough - it is good a deducing
generalities, but not so good at deducing specific conclusions.

###
**Concluding Remarks**

When comparing formal axiology to other theories of value, the difference between the two is great. In Formal axiology, conclusions flow from the axiom with the same authority that geometrical proofs flow from the basic axioms of geometry. One can still doubt the conclusion in formal axiology by arguing against the basic axiom. However, it is difficult to argue with the process by which a conclusion is deduced from an axiom unless one wants to throw out logic and all its benefits out as well.

### References

Rem B. Edwards, ed., *Formal Axiology and Its Critics.* Softback ISBN: :90-5183-879-2
Hardback ISBN: 90-5183-910-3.

Frank G. Forrest, *Valuemetrics: The Science of Personal and Professional Ethics.*
ISBN: 90-5183-683-X.

Robert S. Hartman, *Knowledge of Good: Critique of Axiological Reason.* ISBN:
90-420-1220-X.

Robert S. Hartman,*The Structure of Value.*

Robert S. Hartman, *Can Field Theory be Applied to Ethics?*

G. E. Moore, *Principia Ethica*, ISBN: 0879754982.

Leon Pomeroy, *New Science of Axiological Ethics*, ISBN: 90-420-1826-7.

Whitehead, Alfred North, and Bertrand Russell, *Principia Mathematica*, 3 vols,
Cambridge University Press, 1910, 1912, and 1913. Second edition, 1925 (Vol. 1),
1927 (Vols 2, 3). Abridged as Principia Mathematica to *56,

###
External Links

[[1]]
- Axiology: What is Good?

[[2]]
- Research concepts supplies many excellent books on Formal Axiology.

[[3]]
- Axiometrics International applies axiometrics to help Human Resources find the
best applicants.

[[4]]
- K.T. Connor Ph.D. is a specialist in AxioMetrics.

[[5]] - Inner Talent applies
axiology to improve personal growth.

[[6]] - The Six Advisors apply
axiology to consulting practice

[[7]]
- D.K. Neely and Associates provide assessment tools based on axiology that help
find, develop, and retain the best talent.

Validation Studies:

[[8]]
- lists published studies that have been conducted to validate the conclusions of
formal axiology.

[[9]]
- click on Validation Studies in upper-right hand corner.

[[10]]
- John Austin, a school psychologist, discusses validating conclusions of formal
axiology.

[[11]]
- Leon Pomeroy's recent book describes in detail many validation studies that have
been conducted.