The first work in this program was my essay “Universals and Measurement.”

It was published in The Journal of Ayn Rand Studies in 2004.
It has now become freely available online, in six segments:

Universals and Measurement

              Stephen Boydstun

First

I. Orientation

Second

II. Analysis

     Affordance of Ratio or Interval Measures

Third

II. Analysis (cont.)

     Affordance of Ordinal Measures

     Superordinates and Similarity Classes

     Amended Measure-Definitions of Similarity and Concepts

     Conclusion of Core Task

Fourth

III. Genesis

       Elaboration of Identity

       First Words, First Universals

       Analytic Constraint

Fifth

Notes

Sixth

References

{[(Ordered) Affine] Euclidean}

{[(Ordered) Absolute] Euclidean}

{[(Ordered) Absolute] Hyperbolic}

{[(Projective) Affine] Euclidean}

[(Projective) Elliptic]

I should have just moved on till I could find an area that didn't glaze my eyes, But... I started thinking and became interested.

Given that the best way to view a "thought" may be as a process, an action, does that change things?

I started thinking about it from the view point of grammar (which is about the rules for getting our words to best represent our thoughts).

The simplest of complete thoughts, just like the simplest of sentences in grammar, requires a subject and a predicate asserting an action, property or state of being about the subject.  So we have three primary concepts: "Today," "is," and "day." 

Then there are some modifiers to the concept of "day."   The first, "a," is identifying that we are talking about a single member of a class of entities (a dog, a day, a concept - one of a possible many and not a concrete that has been identified) and the more important modifier (in the context of this sentence/thought), "nice," is the property type of modifier of the class name.  Because it takes "day" as its antecedent, and because "a" says "day" is a member of a class, then we know that the part of the predicate (separate from the verb) means " a member of the class of 'day' with the property known as 'nice.'   We get to imply that "day" may have other categories, and that 'nice' is but one of the conditions that would work in the chosen category.  But we probably care about those implications, instead we have some purpose.

The verb, "is" establishes the relationship between the subject and the rest of the predicate.  With "is" telling us that this is an assertion of a kind of equality - Say for example I had reworded it to make the non-verb portion of the predicate into an appositive ("Today, a nice day, was my first time in Denver).  In that case the "Today" and "a nice day" both function as nouns - the equality is that they are refering to the same existent but with different measurements.  "Today" being the identifier of the particular day of all possible days, and "a nice day" being the condition, needing only the implied context of my time and location of the thought, say, about noon on July 9th, 2008 in my location)

The process then, of this thought, is that of identification of the specific category of 'day' that we wish to make for some purpose not clear in that sentence.  There are two levels of purpose in this thought: How we construct it, and the other being what is the purpose of having the throught?  We have chosen the category which contains, among other conditions, the one called "nice."  Again with the use of appositives I'll illustrate what I mean by saying that our purpose (context) is undetermined in the sample sentence, by creating another sentence where the context gives the purpose: "Today, a nice day, a good day for our picnic, energized me as I drove home."

I like that every subordinate clause, every modifier, every object, every phrase is identifying properties of the subject or properties about the assertion - an act of identifying and joining - and a rich thought can tell us purpose, time, location, emotional context, etc.  Are we quantifying the process of the joining these elements till they reache completion as a thought?

sigh... I look up in my post and see those {[(Ordered) Affine] Euclidean} thingies and figure I may be spewing forth foolishly so I'll stop and maybe someone with an excess of spare time will put me out of my misery by telling what fills the enormous gap between what I've been writing in my grammar-like description and what everyone else is writing about.

Thanks, Ted.

Steve Wolfer, thank you for the thoughtful post.

To find a measure-theoretic account of predication right for my Randian measure-theoretic account of concepts (analysis, not genesis) is part of my ongoing research program.

My triple-identity view of predication, which is now being cast in appropriate measurement terms, can be found in §10 Existence Is Identity (scroll down to p. 43) of “Induction on Identity” (1991). I no longer draw the distinction between particular and specific identity in entirely the same way as I did in that early essay. (The distinction was my own, but it has been also landed upon by philosopher John Campbell.) The basic simple distinction remains and is as I stated it in Note 34 of “Universals and Measurement.”

 
For the journey from logical quantification in predicate logic to numerically definite quantification (which is still purely logical quantification) to sets to arithmetic (which adds to logic and is the arena of absolute measurement [see Note 13 of U&M], not to be confused with absolute geometry), I recommend Quine’s Methods of Logic. For the journey from logic and set theory to measurement theory, one-dimensional and multidimensional, I recommend Foundations of Measurement by Krantz, Luce, Suppes, and Tversky.
 

On translation of sentences into logical formulas, one good text with exercises is R. L. Simpson’s Essentials of Symbolic Logic. Some advanced contemporary controversies are treated in Logical Form and Language, G. Preyer and G. Peter, editors.

I should mention for a general audience that there are no mathematical formulas that cannot be translated into a natural language such as English. That is something that needs frequent mention. Similarly, the epigrammatic strings I improvised above concerning geometry are nothing but abbreviations of English statements. One true rendering of the string {[(Ordered)Affine]Euclidean} would be “Axioms that imply ordered geometry can be joined with certain further axioms to imply affine geometry, and those combined axioms can be joined with certain further axioms to imply Euclidean geometry.” Another true rendering would be “A Euclidean plane (or space) is composed of affine structure and specific additional structure, and the affine structure is composed of order structure and specific additional structure.”

(Edited by Stephen Boydstun on 7/10, 3:04pm)

Steve Wolfer,

The essay to which you refer, and the theory of predication begun in it, was written in 1991. I did not have the “with-measurement” program until many years later. I have since begun casting that theory in measurement terms, but that is in progress and nothing of it is yet published.

 
You have my usage of particular and specific reversed, but your substantive points make sense all the same. I’ll need to be thinking about them as I develop the theory of predication further. Thanks.

Glenn and Bill,

I hope to be able to discuss with you soon the character of scientific laws of nature. Meanwhile, I thought you might like to know of this work, a paper delivered at

History of Philosophy of Science 2008

“On Laws of Nature in Aristotle”

  Tiberiu Popa (Butler University)

ABSTRACT

Aristotle’s handling of what we would call laws of nature is rarely tackled directly and by contemporary scholars. In this paper I draw on relevant passages in Parts of Animals II, Generation of Animals V and especially Meteorology IV in order to give prominence to the defining aspects of Aristotle’s approach to the laws of nature.

 

The notion of natural law in Aristotle’s works may not be articulated in theoretical terms as neatly as it will be in later authors, but its importance in his “chemistry” and in some sections of his biological corpus is quite remarkable. Conditional accounts often take the following form: if a uniform body consists of certain ingredients (present in it dunamei) in a particular proportion and if the right external conditions obtain (e.g. if sufficient – dry or moist – heat is applied to it), then a certain property will emerge, or (if already existent) it will be manifested. Conversely, Aristotle will also use conditionals to show that, if a body exhibits a certain behavior, under specific conditions (e.g. when affected by heat or cold in such and such a way), then it is bound to have this or that composition.

 

A conditional analysis of properties and processes does not have to be, for Aristotle in any case, an elegant strategy for doing away with dispositions. Conditional analyses can be used in principle to weaken the status of dispositions; in assuming that ‘if factors X1, X2… obtain, then result Y will be produced’, one can bypass the ascription of dispositions to a certain thing. Such a conditional account can be taken simply to (causally) link a set of categorical factors to an actual event. Aristotle, I believe, shuns this temptation.

 

In this context one might wonder how the laws of nature can conceivably hold in the realm of “for the most part.” Granted that phenomena in the sublunary sphere occur with less regularity than in the outer spheres and among the celestial bodies, necessity and “for the most part” are not exactly mutually exclusive concepts: if/when the right conditions are in place, a particular effect will be produced of necessity. It is just that those conditions for the emergence and then for the manifestation of a certain disposition are not present with unfailing regularity and sheer predictability.

 

My study of the place of laws of nature in Aristotle’s applied science aims to contribute to a fuller understanding of his effort to find order, based on causal connections, in what might otherwise look like a variegated slew of phenomena. Aristotle’s treatment of laws of nature is closer to what we might call dispositionalism than to actualism, to use a deliberate anachronism. In other words, Aristotle did not take laws of nature to be just summary descriptions of strictly actual events (an idea that the Megarians would have found perhaps palatable, just as the positivists would have found it so in more recent times); rather, he used law-like formulations to ascribe dispositions to things (especially organic and inorganic uniform materials) in the sublunary sphere.