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Thought and Measurement

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Judgments and Measurement

Our scope here is termed judgments rather than propositions for the plain orientation of the former towards existence and possible actions. Having judgments in the focus keeps in friendly view Ayn Rand’s resting of logic “on the axiom that existence exists” (1957, 1016). My philosophy of logic is tuned to Rand’s conception of logic as the art of noncontradictory identification, under the metaphysical background that existence is identity.

Rand’s conception of logic is consonant with the conception of formal logic stated by Edward Zalta “the study of the forms and consequences of predication” (2004, 433), consonant as an animal with skeleton is consonant with its skeleton. Propositions in propositional and predicate logic do not constitute the entirety of right judgment, though they are our skeleton of right judgment. To make a true judgment is to hold existence in its identity, which has as one requirement that when such a judgment is entered as a premise in the truth-preserving inferences of deductive logic, no concluded judgment shall by presence of that premise lack the holding of existence in its identity.

Judgments are conceptual identifications, and in the present study, I shall extend my earlier work incorporating modern theory of measurement into the theory of concepts on into theory of predications as identifications, which is to say, on into theory of judgments. Modern first order predicate logic, including logical quantification with identity, (or an equally comprehensive predicate logic) is a presupposition of the modern mathematical/logical analysis of measurement (Suppes 2002, 24–26). That makes it presupposition in my Randian theory of judgment twice over: once as skeleton of right judgment, then again as requirement for every use of measurement theory in this theory of judgment.

Predication with Identity

I have analyzed in terms of modern measurement theory Rand’s formulation of concepts as comparatively similar items reduced to a singular item of thought with particular measure values along common dimensions suspended. That is, I brought modern measurement theory to the measurement-omission analysis of concepts that is presupposed in Rand’s account of the genesis of concepts. I did not accept her view that all concepts acquired in early development are by a process of measurement omission. Rand could be wrong about that point, yet right in a presupposition of that point: all concretes can be placed under some concept or other(s) whose structure includes suspension of measure values along dimensions of the concrete. Because in Rand’s view and mine those dimensions, such as length, curvature, or heat-flow rate, are concrete dimensions belonging to the concrete world, from modern measurement theory in which quantitative character of a dimension(s) dictates appropriate type of measurement, minimal quantitative character, or magnitude character, of all concretes (minimal above their affordance of counting) could be inferred from Rand’s thesis that all concretes are susceptible to at least ordinal measurement (Boydstun 2004).

This implication of Rand’s presuppositions for her analysis of concepts and her theory of their genesis was unknown to her because she mistakenly thought that differences in the types of scales of measurement to which we resort are merely according to our state of knowledge concerning the nature of what is being measured (Rand 1966–67, 39; 1970–71, 189–90, 199–200, 223) and because she did not posses the knowledge of modern measurement theory (including modern geometry) I was privileged to have acquired. I continue to regard as correct the analytic presupposition I exposed for Rand’s theory of concepts and the minimal quantitative character (above denumerability) of all concretes I inferred from it.

It was on account of research in early cognitive development that I had rejected Rand’s proposal that all concepts are formed by a process of measurement omission (Boydstun 1990, 33–34;* for update, see 2012a;* Lourenco and Longo 2011*). However, I have since noticed a reason of logical order for restricting the range of items that can possibly be analyzed as under the pattern of measurement omission in the measure-value sense. The logical and mathematical concepts presupposed by the concept measurement, as that concept is analyzed in modern measurement theory, cannot themselves be analyzed in terms of measurement omission. The concepts of logic such as every, some, and, or, not, and if-then and the concepts of set theory such as mapping and is an element of are not to be explained in terms of measurement omission of the Randian strength.

Recall that what is distinctive and substantial in Rand’s model of right concepts is their unity under suspension of their particular measure values, not merely their suspension of particular identity (Boydstun 2004, 273–74). The latter, substitution-unit standing of the extension of concepts is recognized in all theories of concepts, not only in Rand’s. A measure-value suspension accompanying all suspensions of particular identity in the substitution-unit aspect of concepts is Rand distinctive proposal. I say the logical and mathematical concepts presupposed by the concept measurement do not have that measure-value suspension, only the substitution-unit character. All of our other concepts remain candidates for measurement-value-omission analysis and strengthening, and the thesis that all concretes can be brought under some concept or other(s) explicable in terms of measurement-value-omission remains standing. (Eventually, our power to conceive conceptual classes in terms of measurement omission needs to be situated in the computational brain, in specific parallel distributed processing in the brain; see Rogers and McClelland 2004, 376–80.)

The purely algebraic structures in mathematics, such as groups and algebraic vector spaces, are not classes affording analysis by measurement omission at ordinal scale or above. That kind of magnitude character is not part of their identity. When it comes to mathematical kinds that are at least partly topological, such as in topological vector spaces or in the geometry of Euclid, Randian form of measurement omission is logically implicit in some distinctions of kinds and of individuals within kinds (cf. Heath 1925, 132–33).

I proposed a triple-identity model of predication under Rand’s metaphysics and epistemology in 1991.* That model was incomplete in the types of judgment it addressed, and I shall remedy that incompleteness in a moment. The model so extended is correct so far as it goes, though it yet remains to incorporate Rand’s distinct contribution of measurement omission into the theory of judgment. Launching that incorporation is the main task of this essay.

I drew a distinction between particular identity and specific identity. Concrete existents are always of both types, though we can separate the two kinds of identity in thought. Existence of each concrete is identity, particular and specific. Particular identity answers to that, which, where, and when. Specific identity answers to what (Boydstun 1991, 43–44). To that original distinction, I should add that particular identity also answers to how much.

Particular identity is the it-ness or that-ness of a thing, whether the thing is entity, action, attribute, or relation. Particular identity is the target of reference. It is just the existent(s) itself as against all others and through all the times of its existence. The relation of a thing to itself at another time is a relation of particular identity. The part-whole relation is one of particular identity. Then too, a thing under one apprehension or description stands in a relation of particular identity to itself under another apprehension or description. A thing under its species stands in a relation of particular identity to itself under its genus (ibid.).

Furthermore, a thing may in certain respects stand in a relation of particular identity to another thing. Here we have the identity of membership in a class or set, logical identity and equivalence, mathematical mapping and equivalence, as well as the identity in physical symmetries and functional relations. Here we have the identity of the specific identities of all electrons (of the same spin) with each other; apart from their separate world lines in spacetime or difference in their associated de Broglie waves, they are identical. Particular identity is the identity of correspondence (ibid.).

Specific identity is the what-ness of a thing. Classifications, attributions, and descriptions capture specific identity. Specific identity is the identity of character or nature, of kind, capability, or susceptibility (Boydstun 1991, 44).

Every existent is the totality of its kinds and the totality of its parts. This is an important way in which each existent is both specific and particular identity. We rightly integrate our taxonomies and our partonomies, as in “Its wing is part of the bird, which is a type of vertebrate” (cf. Rand 1957, 1016; 1969–71, 264–67; see further, Boydstun 1990, 30–31).

Though the membership relation of concepts and the measure-value suspension of concepts are based on concrete relations in the world, they are not themselves concrete relations. Yet predicative structure is in its source the general concrete circumstance that existence is identity (Boydstun 1991, 45; Kelley 1996, 7–9).

Consciousness is identification (Rand 1957, 1016). It is grasp of specific and particular identity of existents. In judgment, or propositional consciousness, there are both types of identification. In Rand’s view, every process of consciousness, or identification, is a process of both differentiation and integration (1966–67, 5, 13–14, 19, 21–24, 41–42; 1969–71, 138, 143–44). Particular and specific identities ground the distinctions and the unities, or integrations, attained in judgment.

In my 1991, I analyzed propositions such as “Spot is a dog” and “Spot is brown and white” in the following way. The copula is coordinates three identities. It wires the subject, the proper name Spot, and the predicate, brown and white coloration, to the same existent, namely, the dog who is Spot. This is a relation of particular identity. Secondly, it specifies Spot in respect of his coloration. This is a relation of specific identity. Thirdly, it affirms that this particular identity and this specific identity obtain in reality. This last is a relation of particular identity between the proposition and reality. For the third identity to obtain, for the proposition to be a true judgment, the proposition’s first particular identity and its specific identity must obtain in reality (Boydstun 1991, 44; on copula and predication, see Oderberg 2005, 184–85, 191–92).

Going beyond my 1991 work, what of the proposition “Spot is in the back yard”? In this case, the second identity in the triple coordination is one of particular identity rather than specific identity. The predicative identity is location, a particular identity, though expressing the location required “back yard,” which invokes specific identity. Still, the second identity, the predicative identity, is here a particular identity, and all three identities coordinated in the judgment are particular.

With predication of particular identity, it would be widely recognized that the relation of predicates to subjects can be further articulated by measurements. Spatial relations are widely recognized to be measureable. To say that predication of specific identity, as in “Spot is colored” or “Spot is a dog,” can always be further articulated by measurements is a distinctively Randian thesis implicit in her analysis of the superordinate hierarchical relations of concepts in terms of measurement omission (Rand 1966–67, 21–22, 41–42; Boydstun 2004, 281–82).

What of “Spot exists”? Some of the meaning in that proposition is “Spot is part of existence.” The predicative identity in the latter proposition is particular, not specific, for it is a relation of part to whole. The first identity of “Spot is brown and white” and of “Spot is in the back yard” remains in play in this case as well. The subject Spot and the predicate part of existence are wired to the same existent, and that coordination is a relation of particular identity. However, the third identity in the triple coordination is degenerate with the second in “Spot is part of existence,” because a claim to part of existence is a claim to existence. That is, the predicative identity in this judgment is already the particular identity that is correspondence to reality. Notice that because physical existence is widely regarded as necessarily spatial, dynamical, and so forth, the predicative identity in “Spot is part of existence” would be widely recognized to be amenable to further articulation by measurement.

Another part of the meaning of “Spot exists” is “Spot is a type of existent.” The latter is of my original triple-identity form particular-specific-particular. I have shown that in taxonomies from the individual to the most general categories—existence, entity, attribute, action, or relation—there will not be common magnitude dimension(s) for measurement-value omission spanning the taxonomy (Boydstun 2004, 280–84; Rand 1966–67, 42; 1969–71, 146–47, 274–76). In the present case, that means there is not common magnitude dimension(s) spanning the predicative identity of “Spot is a dog is an animal is an organism is an open thermodynamic system is physical is an entity is an existent.” Or, in short, “Spot is a type of existent.”

It remains, however, that a major portion of the subordinate-superordinate relations in this tall taxonomy can be analyzed in Rand’s distinctive way, that is, in terms of measure-value omission. Moreover, it remains that every individual existent has some measureable dimensions (measureable beyond denumerablity) by which it can be conceived as belonging to each level. Spanning the various levels in a measurement analysis will sooner or later require a shift in kinds of magnitude dimensions, and that is a connection of particular identity between kinds, not specific identity between them, and so, not an instance of what is distinctive in Rand’s measure-analysis between levels, that is, not an instance of measure-value omission.

Rand vacillates at times between units as substitution units and as measure-value units without recognition of the vacillation (e.g., 1966–67, 21–22). Creating a link by substitution units and not recognizing that measure-value suspension was not employed in the link created an illusion inflating the extent to which measure-value omission permeates abstraction. Yet there was a distinct virtue of Rand’s effort at that full permeation. She conceived of magnitude dimensions as key to abstraction, and this is itself a novel and substantial proposal, one I take for true. That wider proposal as well as Rand’s distinctive measure-value omission are incorporated in my amended version of Rand’s definition of concepts: mental integrations of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted or with the particular measureable forms of their common distinguishing characteristic(s) omitted (Boydstun 2004, 285).

We have seen that two parts of the meaning of “Spot exists” are “Spot is part of existence” and “Spot is a type of existent.” Exists, existence, and existent are concepts used in the logical predicate. They can also be used as the logical subject as in “Something existent is Spot,” “Something part of existence is Spot,” and “Some type of existent is Spot.” Notice also, for the sake of the syllogistic, that Spot being the only Spot there is, he is all of them. For use in some syllogisms, “Spot is mortal” can be taken to mean “All Spots are mortal.” Then too, in other syllogisms, “Spot is mortal” can be taken to mean “Some Spot is mortal,” where we continue to speak of one particular doggy (Sommers 2005, 13–14).

That the predicative identity, my second identity of the triple, could be a particular identity rather than a specific identity was logically implicit in my earlier, less complete model of predication. I had not realized this implication, but it is there. By the logical principle of conversion, “All Spots are mortal” entails “Some mortal things are Spots.” The predicative identity in the first assertion is specific; in the second, it is particular. I do not think this indicates any conflict of my model of predicative judgment with formal logic, only exposure in logic of the metaphysical harmony of specific and particular identity.

Conversion of “Spot exists” to “Something existent is Spot,” as well as the two other, parallel conversions paragraph before last, raises awareness of the circumstance that the whole of existence and one’s grasp of it whole is a background of the identifications in existential judgments. Identifications of particular and specific existents are made only within one’s own cumulative identification of particular and specific existents throughout a whole that is one’s understanding of all existence. That same background frame of cognition is at work when we make the particular existential judgment “It is raining.” The it has as nested existential backgrounds the existence of one’s surrounding outdoors and its particular and specific identities in the whole in one’s grasp: existence. Negative existential judgments, too, require flex of one’s hold on the totality of existence in all its particular and specific identity (cf. Hermann Lotze and Heinrich Maier in Martin 2006, 131–36; Branden 1963a; 1963b; c. 1967, 72–83; Peikoff 1991, 122–26, 137–38, 163–71).

In moving from exists to is part of existence or to is a type of existent, the is does not of itself mean exists. The is is here the copula of a subject-predicate logical form, the copula itself being open for topic. In these two cases, the topic beyond the subject—the predicate, the identification—is part of existence or is a type of existent, either of which entails assertion of existence. Drawing out of those predications from the simpler assertions such as “Spot exists” or “Something exists” or “It is” is logical inference upon the general schema “Existence is identity,” but these two-word statements do not depend on those implications, rather they are solid assertions as they stand. The is of “It is” is not a copula; it is simply expressing and affirming existence. I think such two-word assertions are in fact the most basic form of judgment, with the subject-copula-predicate form presupposing them. This one point I take from Franz Brentano (Martin 2006, 63–67). It echoes the circumstance that “Existence is identity” is the fundamental circumstance close to but beyond “Existence exists.”

Genetically too, not only logically, those two-word assertions seem to be prior. I had the privilege of observing the progress of my partner’s grandson from early infancy. The first two-word utterance I heard from him was “Guy there.” He made that remark as his mother was getting him bundled up. By his glance, it was clear he was referring to me standing a few feet away, by the door. “There is a guy there” was not yet in his repertoire, and I think “Guy there” by itself suffices for judgment, if not full predicative judgment (cf. Bogdan 2009, 18–19, 70–74, 83–106).

We should note that the distinction between analytic and synthetic judgments pertains only to the subject-copula-predicate form. Contrary the Randian position set out by Leonard Peikoff (1967), I have argued there is a legitimate analytic-synthetic distinction, one in which the distinctive necessity of analytic judgments is taken from reality, given the character of our cognition. The logical priority of “It is” (meaning “It exists”) over “It is such-and-such” (with is acting additionally as copula) commends the rejection of the versions of analyticity rejected in Peikoff’s essay and commends acceptance of my own version (Boydstun 2012b*).

Predication with Measurement

If all concepts over concretes can be formulated to abstractly reflect magnitude structures of those concretes, then magnitude structures enter all judgments in which such concepts enter. Logical copula (is, have), logical operators (not, and, or, not both, if-then), and logical quantifiers (all, some) are not themselves reflectors of magnitude structures. They relate magnitude structures of the magnitude-structured existential concepts they relate.

Entities can be subjects or predicates. Likewise it goes for actions, attributes, and relations. All of these four categories are categories of existents. Concrete relations of entities, actions, and attributes include proximities, containments, and magnitude structures (see further, Suppes 2002, 105–10, 282–300; 2005, 102–4; Choi 2006). Concretely existing relations do not include certain logical and mathematical relations such as class membership, maps, or transitivity as a general relation.

Existence as the totality of concrete existence has, among all the concrete magnitude structures it contains, a minimal magnitude structure beyond the meager structure required for counting. I have argued that by metaphysical presupposition of Rand’s model of concepts as occasions of measure-value omission, there is a minimal concrete structure contained in any concrete existence, and that structure is the one-dimensional linear order that is a uniform topological lattice (Boydstun 2004, 278–80).

Existence includes things with more magnitude structure than that. The three-dimensional purely spatial slices of spacetime have the full structure required for familiar ratio-scale measurement. Spacetime itself is four-dimensional and has less magnitude structure than a four-dimensional Euclidean structure, though more than a merely ordered four-dimensional space, which has only a linear order along its dimensions.

Rand had it that entities and their actions are measurable (beyond counting) only by measure of their attributes (Rand 1966–67, 7). Because in Rand’s view entities not only have identities, but are their identities, entities can be fully measured by the measures of their actions, attributes, and relationships. (I shall qualify this thesis in a moment.) Then again, actions are fully measured by the measures of their attributes and relations, including their join to other attributes and relations of the entities to which the actions belong.

Divide the how-much genre of particular identity into two sorts. There is on the one hand what I shall call the item-measure particularity of the existent. That would be its possibilities for being placed in sequences, for being counted, and for being placed in frequency distributions, discrete or continuous. On the other hand, there is what I shall call the trait-measure particularity of the existent. I am using trait as a short cover for actions, attributes, and dimensioned relations. The relations implicated in item-measure particularity are not relations along dimensions; they are purely numerical relations.

Item-measure coincides with the type of measurement now called absolute in the measurement literature. Trait-measure includes the types the experts call ordinal, hyperordinal, interval, and ratio (Suppes 2002, 63–73, 110–20). Trait-measure includes also all the multidimensional forms of measurement, such as topological vector spaces, which are used in physics, or trigonometry, which has been used in our house in making drapery. Measurement restricted to the sense of trait-measures is what Rand presumes when she writes “entities (and their actions) are measured by their attributes (length, weight, velocity, etc.)” (1966–67, 7). Not only attributes of entities and attributes of their actions, but relations of where and when are also specified by trait-measures.

My caveat concerning the idea that entities can be fully measured by the measures of their actions, attributes, and dimensioned relations is this. I should caution against saying “an entity is in every way its attributes,” for an entity has the capability of entering item-measures logically independent of its attributes and their trait-measures, even though there is no such thing as an attribute-bare particular entity (cf. Rand 1969–71, 276, 278–79; Sciabarra 1995, 146–47; Oderberg 2005a, 196–203).

The first identity in the triple-identity model of existential predication ties a subject in its aspect for item measurability to a predicate in its aspect for trait measurability by tying them both to a self-same existent. That tie of particular identity between subject and predicate includes a tie of item-measure structure to trait-measure structure. Because the tie is by the existent under judgment, this is a facet of Rand’s dicta that logic rests on the axiom that existence exists.

The second identity in an existential predication, the second being a relation of either specific or particular identity, includes specification of the subject by trait-measure structure. Where that second identity is specific, rather than particular, we have a distinctive Randian contribution to a measurement theory of predicative judgments (cf. Kelley 1996, 11), in addition to the general measurement-saturated approach to this theory continuing Rand’s approach to concepts and their setting with definitions.

I have said that trait-measures, like item-measures, are part of particular identity; they answer to how much. Where the second identity in a predication is a specific identity, we have specific identity fused to the particular identity that is trait-measure. Suspension of particular values in those trait measures is what yields species for our discernment of specific identity. That unity of particular identity and specific identity was implicit in Rand’s measurement-omission formula for best concepts true to existence.

The third identity in the triple of predicative existential judgment is the relation of particular identity between the topic existent and the subject-predicate tie of the first identity as well as between the topic existent and the specification in the second identity. Whereas the first two identities coordinated under the copula is unfold it as merely copula, merely connective, the third is concerned with that same is as the is of existence. The third identity of predicative existential judgment is an elaboration upon the basic judgment “It is.” This third identity I have for the coordinations performed by is in subject-is-predicate is another facet of the idea of logic as resting on the fact most basic: existence exists.

My Randian analysis of predicative judgment has introduced into the theory mathematical resources additional to purely logical resources. That means we have availed ourselves of set-theoretic resources, and that means we are equipped to treat relational propositions such as “Lassie is taller than Spot” as subject-predicate judgments by taking the ordered pair (Lassie, Spot) as the logical subject and taking for predicate of the pair “member of the set of ordered pairs such that the former is taller than the latter” (Martin 2006, 80n19). Membership, thence this predicate, is particular identity. That is, the second identity in my triple is here a specification by particular identity. The selection rule for membership in this set, expressed in the predicate, plainly invokes ratio-scale magnitude structure.

Further development of this theory of judgment should include consideration of judgments that are definitions of existential concepts. It may well be that Randian mathematical rendition of the essential characteristic in the definition of an existential concept can strengthen the explanatory power of that characteristic, help prepare our concepts for potential scientific sophistication, and lead us more deeply into the unity of existence.

Appendix on Hypothetical Judgments

In this essay, I analyzed some forms of judgment in terms of their identification structures, that is, in terms of the identities they profess. Three existential forms were analyzed under the principle and fundamental pattern “Existence is identity.” A fourth form, the negative existential was also noted for the salience of its tie to one’s comprehensive cognition of existence in its totality of kinds and parts.

Those four forms of judgment are exemplified by (i) “Spot is brown and white” (or “is a dog” or “is in the back yard”), by (ii) “Lassie is taller than Spot,” by (iii) “Something existent is Spot” (or “Spot exists” or “Something part of existence is Spot” or “Some type of existent is Spot”), and by (iv) “Spot’s coat is not any color but brown or white” or “Spot is not a television” or “There is no native intelligence on Mars” or “There is no God,” where Spot, like Mars or God, is proper name of a unique individual.

Identity analysis of (ii) required set-theoretic organization, supplementing general logic, to transform it into subject-is-predicate form. Its analysis then is as in (i), with its specifying predicative identity (the second in the triple identity invoked in predicative judgments) being particular identity, not specific identity, making its predicative form under (i) as in “Spot is in the back yard.” Only (ii), not (i) or (iii) or (iv), required set-theoretic organization for my identity analysis, though all four forms had their coordinations of measurement structures spelled out in addition to their logical identity structures. The third identity in the triple of predication was for all cases linked to the particular, correspondence identity of the more fundamental form of judgment “It is” or “Existence exists.”

We are in position to add a fifth form of judgment to our analysis. Its form is exemplified by (v) “If Spot is free to roam, he does.” This form can be reduced to a combination of (i) and (iv). Unlike the transformation of (ii) to (i), the transformation required for this reduction and identity analysis does not require set-theoretic organization, only general logic. Our (v) transforms to the following: “Not so: Spot is free to roam, yet does not roam” or “Not both: Spot is free to roam, and Spot does not roam.” The conjunct “Spot is free to roam” is of the form (i), whose identity and measurement analyses therefore apply to it. The conjunct “Spot does not roam” is of the form (iv). I remarked in the essay that negative existentials, as also the form “It is raining” (or “It is some quadruped trotting”), token more saliently the circumstance that all existential judgments ply one’s cumulative understanding of spheres of existence around their subject and predicate. In the case of negative existentials, those spheres are more expressly implicated, and they expand, even up to the sphere that is the whole of existence. In the transformation of hypothetical judgments to judgments negated over conjunction of an affirmative and a negative judgment, we apply negation twice, we deny twice, and the background condition of our existential judgments, the condition that one have a certain understanding of existence whole, is in much show, down to the seventh veil as it were, shy the full disclosure “Existence exists.”

References

Bogdan, R. J. 2009. Predicative Minds – The Social Ontogeny of Propositional Thinking. MIT.

Boydstun, S. 1990. Capturing Concepts. Objectivity 1(1):13–41.

——. 1991. Induction on Identity. Objectivity 1(3):1–56.

——. 2004. Universals and Measurement. The Journal of Ayn Rand Studies 5(2):271–305.

——. 2012a. Capturing Quantity. Objectivist Living.

——. 2012b. Objective Analyticity. Objectivist Living.

Branden, N. 1963a. The Stolen Concept. The Objectivist Newsletter 2(1):2, 4.

——. 1963b. Agnosticism. The Objectivist Newsletter 2(4):15.

——. c. 1967. The Basic Principles of Objectivism. In The Vision of Ayn Rand. 2009. Cobden.

Choi, S. 2006. Preverbal Spatial Cognition and Language-Specific Input: Categories of Containment and Support. In Action Meets Word – How Children Learn Verbs. K. Hirsh-Pasek and R. M. Golinkoff, editors. Oxford.

Heath, T. 1925 [1956]. Euclid’s Elements. Volume 1. Dover.

Kelley, D. 1996. Concepts and Propositions. Institute of Objectivist Studies.

Lourenco, S. F., and M. R. Longo 2011. Origins and Development of Generalized Magnitude Representation. In Space, Time, and Number in the Brain. S. Dehaene and E. Brannon, editors.

Martin, W. M. 2009. Theories of Judgment – Psychology, Logic, Phenomenology. Cambridge.

Oderberg, D. S., editor, 2005. The Old New Logic – Essays on the Philosophy of Fred Sommers. MIT.

Oderberg, D. S. 2005a. Predicate Logic and Bare Particulars. In Oderberg 2005.

Peikoff, L. 1967. The Analytic-Synthetic Dichotomy. In Rand 1966–67.

——. 1991. Objectivism: The Philosophy of Ayn Rand. Dutton.

Rand, A. 1957. Atlas Shrugged. Random House.

——. 1966–67. Introduction to Objectivist Epistemology. 2nd ed. Meridian.

——. 1969–71. Transcript from Rand’s Epistemology Seminar. Appendix to Rand 1966–67.

Rogers, T. T., and J. T. McClelland 2004. Semantic Cognition – A Parallel Distributed Processing Approach. MIT.

Sciabarra, C. M. 1995. Ayn Rand: The Russian Radical. Penn State.

Sommers, F. 2005. Intellectual Biography. In Oderberg 2005.

Suppes, P. 2002. Representation and Invariance of Scientific Structures. CSLI.

——. 2005. The Syntax and Semantics of English Prepositional Phrases. In Oderberg 2005.

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Existence may imply identity but the converse is not so. Consider: All unicorns are 4 legged animals with a horn in the middle of the forehead. That is a true characterization of unicorns but it does not imply unicorns exist.

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Quite so, Bob. Discernment of concrete existence requires discernment of identities in or rightly tied to perception.

Beyond pure fictions such as unicorns, we entertain also things for which we have some identity in our conception of them, but the conception does not entirely hold up logically when pressed, and the proposed existent can be shown not to exist. These too, are cases of identity without existents. One example would be the concept “absolute perfect being,” which contradicts the life-context of goodness and the thermodynamic nature of life. Another would be “an odd counting number N such that [(N x N) - 1] is not divisible by 4,” which we can prove, by mathematical induction, does not exist.

Rand’s thesis “Existence is identity” means anything without identity is nothing. Does it mean further that identity is convertible with existence? In a very inclusive sense of convertibility, I would say Yes. An existent could be nothing but its identities provided one really included all of them. An existent, then, is not more than all its identities. Then too, an existent is not less than all of its identities, including all its character as specific relata in all its relations to other existents. For the concrete existent, that would include all its susceptibilities for perception and observation, direct or indirect. For a number, that would include its susceptibility for standing in all of the mathematical relations in which it stands, those we have learned and those we have yet to discover. --Stephen

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Universals and Measurement (2004)

I. Orientation

Rand spoke of universals as abstractions that are concepts (1966–67, 1,13). Quine spoke in the same vein of “conceptual integration—the integrating of particulars into a universal” ([1961] 1980, 70). Those uses of universal engage one standard meaning of the term. Another standard meaning is the potential collection to which a concept refers. This is the collection of class members consisting of of all the instances falling under the concept.[1] In the present study, the latter, metaphysical aspect will be brought into fuller articulation and relief. That vantage will be attained through amplifications of the former, epistemological aspect and its mathematical-metaphysical requirements of Rand's theory of universals as conceptual abstractions.

To begin I situate the topic of the present study within Rand’s larger system of metaphysics and epistemology. My core task for the present study will then emerge fully specified.

Rand’s system relies on three propositions taken as axioms: (E) Existence exists. (I) Existence is identity. (C ) Consciousness is identification.[2] Rand’s set of axioms conveys the fundamental dependence of consciousness on existence. Existence is and is as it is independently of consciousness. But consciousness is dependent on existence and characters of existence (Rand 1957, 1015–16; 1966–67, 29, 55–59; 1969–71, 228, 240–41, 249–50).

As part of the meaning of (I), Rand contended (Im): All concretes, whether physical or mental, have measurable relations to other concretes (1966–1967, 7–8, 29–33, 39; 1969–71, 139–40, 189, 199–200, 277–79).[3] Every concrete thing—whether an entity, attribute, relation, event, motion, locomotion, action, or activity of consciousness—is measurable (Rand 1966–67, 7, 11–17, 25, 29–33; 1969–71, 184–87, 223–25).

As part of the meaning of (C ), Rand contended (Cm): Cognitive systems are measurement systems (1966–67, 11–15, 21–24; 1969–71, 140–41, 223–25). Perceptual systems measure,[4] and the conceptual faculty measures. Concepts can be analyzed, according to Rand’s theory, as a suspension of particular measure values of possible concretes falling under the concept. Items falling under a concept share some same characteristic(s) in variable particular measure or degree. The items in that concept class possess that classing characteristic in some measurable degree, but may possess that characteristic in any degree within a range of measure delimiting the class (Rand 1966–67, 11–12, 25, 31–32).[5] This is Rand’s “measurements-omitted” theory of concepts and concept class.

All concretes can be placed within some concept class(es). All concretes can be placed under concepts. Supposing those concepts are of the Randian form, then all concretes must stand in some magnitude relation(s) such that conceptual rendition of them is possible. What is the minimal magnitude structure (minimal ordered relational structure) that all concretes must have for them to be susceptible to being comprehended conceptually under Rand's measurement-omission formula?

That is to say, what magnitude structure is implied for metaphysics, for all existence, by the measurement-ommission theory of concepts in Rand’s epistemology? My core task in the present study is to find and articulate that minimal mathematical structure. With that structure in hand, we shall have as well the new, fuller articulation of the class character of universals implied by Rand’s theory of concepts.

Such mathematical structure obtaining in all concrete reality is metaphysical structure. It is structure beyond logical structure; constraint on possibility beyond logical constraint. Yet it is structure ranging as widely as logical structure through all the sciences and common experience.

The minimum measurement and suspension powers required of the conceptual faculty by Rand’s theory of concepts calls for neuronal computational implementation. Is such implementation possible, plausible, actual? This is a topic for the future, bounty beyond the present study.

We must keep perfectly distinct our theoretical analysis of concepts and universals on the one hand and our theory of the developmental genesis of concepts on the other. Analytical questions will be treated in the next section, and it is there that I shall discharge the core task for this study.

The logicomathematical analysis of concepts characterizes concepts per se. It characterizes concepts and universals at any stage of our conceptual development, somewhat as time-like geodesics of space-time characterize planetary orbits about the sun throughout their history. The analysis of concepts and universals offered in the next section constrains the theory of conceptual development, as exhibited in Section III.

(Continued)

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II. Analysis

Rand gave three definitions of concept. I shall tie them all together in the next section, but for the present section, we need this one alone: Concepts are volitional mental integrations of “two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted” (Rand 1966–67, 13).[6]

The units spoken of in this definition are items appropriately construed as units by the conceiving mind. They are items construed as units in two senses, as substitution units and as measure values (Rand 1969–71, 184, 186–88). As substitution units, the items in the concept class are regarded as indifferently interchangeable, all of them standing as members of the class and as instances of the concept. Applied to concept units in their substitution sense, measurement omission means release of the particular identities of the class members so they may be treated indifferently for further conceptual cognitive purposes.[7] This is the same indifference at work in the order-indifference principle of counting. The number of items in a collection may be ascertained by counting them in any order. Comprehension of counting and count number requires comprehension of that indifference.

The release of particular identity for making items into concept-class substitution units is a constant and necessary part of Rand’s measurement-omission recipe. But this part is not peculiar to Rand’s scheme. What is novel in Rand’s theory is the idea that in the release of particular identity, the release of which-particular-one, there is also a suspension of particular measure values along a common dimension.

Before entering argumentation for the minimal mathematical structure implied for the metaphysical structure of the world, let us check that we have our proper bearings on objective structure and intrinsic structure. I have ten fingers, eight spaces between those fingers, and two of my fingers are thumbs. That’s how many I have of those items. Period. Those numerosities are out there in the world, ready to be counted, and they are what they are whether I count them or not. In our positional notation for expressing and calculating numbers, we choose the number base, but the different base systems designate the same things, the numbers. In base ten, my (fingers, spaces, thumbs) are (10,8,2); in base eight, (12,10,2); and in base two (1010,1000,10). The three numbers referred to in all these bases are the same three numbers. In Rand’s terminology, the various bases are objective schemes; they are appropriate tools for getting to the intrinsic structure of numbers. But the numbers have intrinsic character—even or odd, whole or fraction, rational or irrational, analytic or transcendental—quite independently of our choices, such as choice of number base.

In asking for the minimal magnitude structure that all concretes must possess if all concretes can be subsumed under concepts for which Rand’s measurement-omission analysis holds, we are seeking intrinsic structure, obtaining under every adequate objective expression of that structure. Now we are ready.

II.A Affordance of Ratio or Interval Measures

II.B Affordance of Ordinal Measures

II.C Superordinates and Similarity Classes

II.D Amended Measure-Definitions of Similarity and Concepts

II.E Conclusion of Core Task

(Continued)

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II.A Affordance of Ratio or Interval Measures

I have said that the units suspended in the formula “two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted” are units in a double sense: substitution units and measure values. We focus now on units in the latter sense. Rand spoke of measurement as “identification of a relationship in numerical terms” (1966–67, 39) and as “identification of a relationship—a quantitative relationship established by means of a standard that serves as a unit” (7; also 33; further 1969–71, 188, 199–200). The measure-value sense of unit is at work here. By the expression “a standard that serves as a unit” and by some of her examples of concepts and their measurement bases, one might suppose that Rand’s theory of concepts entails that all concretes stand under magnitude relations affording some sort of concatenation measurement. That supposition would be incorrect.

Rand illustrates her theory with the concept length. The pertinent magnitudes of items possessing length are magnitudes of spatial extent in one dimension. Another illustration of Rand’s is the concept shape (1966–67, 11–14; 1969–71, 184–87). The pertinent magnitudes of items possessing shape, in 3D space, are pairs of linear, spatial magnitudes such as curvature and torsion (as functions of arc length) for shapes of curves or the two principal curvatures for shapes of surfaces.[8]

Shapes must possess such pairs of magnitudes in some measure but may possess them in any measure. Observe that Rand’s measurement-omission theory does not entail what number of dimensions for the magnitude relations among concretes is appropriate for the concept. Length requires 1D, shape requires 2D. Rand’s theory works for any dimensionality and does not entail what the dimensionality must be, except to say that it must be at least 1D. Observe also that the conception of linearity to be applied here to each dimension is not the more particular linearity familiar from analytic coordinate geometry or from abstract vector spaces. It is merely the linearity of a linear order.[9]

The magnitude structure of the concretes falling under the concept length affords concatenations. Take as unit of length a sixteenth of an inch. Copies of this unit can be placed end-to-end, in principle, to form any greater length, such as foot, mile, or light-year. This standard concatenation of lengths is properly represented mathematically by simple addition. That is a numerical rule of combination appropriate to concatenations of the concrete magnitude structure in the case of length.

The magnitude structure of the concretes falling under the concept length also affords ratios [these being] independent of our choice of elementary unit. The ratio of the span of my left hand, thumb-to-pinky, to my height is simply the number it is, regardless of whether we make those two measurements using sixteenths of an inch as elementary unit or millimeters as elementary unit.

Mass is another concept whose concept-class magnitude structure affords simple-addition concatenations and affords ratios of its values that are independent of choice of elementary unit. Because of the latter feature, conversion of pounds to kilograms requires only multiplication by a constant. Such measurement scales are called ratio scales.[10] The mathematical combinations reflecting the concatenations need not be simple addition. This category of scales is somewhat more inclusive than that. It would include the scale for the concept grade (grades of roads, say). Grades can be concatenated, although the proper mathematical reflection of this concatenation is not simple addition.[11]

Finest objectivity requires measurement scales appropriate to the magnitude structures to which they are applied. What does appropriate mean in this context? It means that all of the mathematical structure of the measurement scale is needed to capture the concept-class magnitude structure of concretes under consideration. It means as well that all the magnitude structure pertinent to the concept class is describable in terms of the mathematical structure of the measurement scale.[12]

What is the magnitude structure of concretes that is appropriately reflected by ratio-scale characterization? It is a magnitude structure whose automorphisms are translations.[13] Translations are transformations of value-points (i.e., points, which may be assigned numerical values) of the magnitude structure (the ordered relational structure of the concept-class concretes) that shift them all by the same amount, altering no intervals between them.

Rand’s measurement-omission analysis of concepts and concept classes applies perfectly well to cases in which the measurement scale appropriate to the pertinent magnitude structure of concretes is ratio scale. But Rand’s theory does not entail that all concretes afford ratio-scale measures. For Rand’s theory does not necessitate that the scale type from which measurements be omitted be ratio scale. Her analysis also works perfectly well for scales having less structure. The magnitude structure entailed for all concretes by Rand’s theory is less than the considerable structure that ratio scales reflect.

An analogous conclusion obtains for multidimensional magnitude structures of concept classes. Rand’s theory does not entail that all 2D or 3D magnitude structures have, such as Euclidean geometry has, not only affine structure, but absolute structure.[14] That is, Rand’s theory does not entail that multidimensional magnitude structures of concept classes afford a metric (a measure of the interval between two value-points) definable from a scalar product (a measure of perpendicularity of value-lines).[15]

Physical temperature, sensory qualities, and utility rankings are examples of concretes whose magnitude structures afford what are now called interval measures, but evidently do not afford ratio measures.[16] The measurement scale appropriate to the magnitude structures of these three is the interval scale. The magnitude structure underlying the concept class temperature affords only an interval scale of measure. Such magnitude structures do not afford concatenations, unlike the natures of length or mass, but they do afford ordering of differences of degree, and they afford composition of adjacent difference-intervals.[17]

Such magnitude structures do not afford ratios of degrees that are independent of choice of unit, but they afford ratios of difference-intervals that are independent of choice of unit and choice of zero-point.[18] Ratio scales have one free parameter, requiring we select the unit, such as yard or meter. These scales are said to be 1-point unique. Interval scales have two free parameters, requiring we select the unit, such as ˚F or ˚C, and requiring we select the zero-point, such as the freezing point of an equally portioned mixture of salt and ice or the freezing point of pure ice. These scales are said to be 2-point unique.[19]

The magnitude structure of concretes affording interval-scale characterization is one whose automorphisms are fixed-point collineations, preeminently stretches.[20] Stretches are transformations of the value-points of a magnitude structure such that one point remains fixed and the intervals from that point to all others are altered by a single ratio.

Rand’s measurement-omission analysis of concepts and concept classes applies perfectly well to cases in which the measurement scale appropriate to the pertinent magnitude structure of concretes is interval scale. The temperature attribute of a solid or fluid must exist in some measure, but may exist in any measure.[21] But Rand’s theory does not entail that all concretes afford interval-scale measures. For Rand’s theory does not necessitate that the scale type from which measurements be omitted be interval scale. Her analysis also works perfectly well for a kind of scale having less structure. The magnitude structure entailed for all concretes by Rand’s theory is still less than the considerable structure that interval scales reflect.

An analogous conclusion obtains for multidimensional magnitude structures of concept classes. Rand’s theory does not entail that all 2D or 3D magnitude structures have, such as Euclidean or Minkowkian geometry have, not only order structure, but affine structure.[22] That is, Rand’s theory does not entail that multidimensional magnitude structures of concept classes afford a metric definable from a norm (a measure on vector structure).[23]

(Continued)

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II.B Affordance of Ordinal Measures

Recall again Rand’s characterization of measurement: identification of “a quantitative relationship established by means of a standard that serves as a unit” (1966–67, 7). The phrase “a standard that serves as a unit” suggests that Rand’s conception of measurement for her measurement-omission analysis of concepts was ratio-scale or interval-scale measurement. These two types possess interval units that can serve as interval standards. They possess interval units that can be meaningfully summed to make measurements. The quantitative relationship established in measurements equipped with interval units entails summation of elementary units. The summation might be simple addition or a more elaborate mathematical combination, and the basis for the summation in concrete reality might be susceptibility to concatenation (for ratio scales) or to composition of ordered difference-intervals (for interval scales).

The measure values required for Rand’s theory need not be interval units. As Rand realized, merely ordinal measurement suffices for her measurement-omission scheme (33). I say that the magnitude structure captured by ordinal measurement is the minimal structure implied for metaphysics if, as I supposed at the outset, all concretes fall under one or more concepts for which Rand’s measurement-omission analysis holds. What is the magnitude structure captured by ordinal measurements?

All magnitude structures captured by ratio- or interval-scale measurements contain a linear order relation. A magnitude structure consisting only of such a linear order relation is a structure for which merely ordinal measurement is appropriate. An example is the hardness of a solid. I mean specifically the scratch-hardness, which is measurable using the Mohs hardness scale. Calcite scratches gypsum, but not vice versa; quartz scratches calcite, but not vice versa; therefore, yes, quartz scratches gypsum, but not vice versa. Degrees of hardness have an order that is anti-symmetric and transitive.

Mohs scale assigns the numbers (2, 3, 8) to the degrees of hardness for (gypsum, calcite, quartz). All that is intended by the scale is to be true to the order of the degrees of hardness. That Mohs has chosen these three numbers to be integers is of no significance. They could as well be the rational triple (14.7, 55.3, 56.9) or the irrational triple (√2, π, 1.1π). Unlike the numbers on interval scales, the ratios of difference-intervals between the numbers on these scales are not meant to be of any significance. The hardness degrees (2, 3, 8) are not intended to imply that the hardness of calcite is closer to the hardness of gypsum than it is to the hardness of quartz. For all we know, and for all our ordinal measurements signify, there simply may be no fact of the matter whether the scratch-hardness of calcite is closer to that of gypsum than to that of quartz.

The magnitude structure of hardness (scratch-hardness, not dent-hardness) evidently does not warrant summations or equal subdivisions of some sort of interval unit of hardness. This particular hardness concept is founded analytically on merely ordinal measure. To fall under this concept hardness, an occasion need only present the quality at some measure value on the merely ordinal scale, and that may be any measure value on that scale. Affordance of ordinal measurement is all that Rand’s measurement-omission recipe entails for the magnitude structure of all concretes. Her theory does not entail that every attribute of concretes—hardness, for example—must in principle afford ratio- or interval-scale measurements. Her theory does not imply that, were only our knowledge improved enough, it would be possible to make ratio- or interval-scale measurements of scratch-hardness.[24]

The magnitude structure affording merely ordinal measurement is a linear order whose automorphisms are the order-automorphisms of (same-order subsets of) the real numbers in their natural order. Such a magnitude structure affords characterization by a lattice (a type of partially ordered set) formed of sets and subsets of possible Dedekind-cuts of its linear order. This linear order might be scattered or dense; ordinal measurement is possible in either case.[25]

The magnitude structure affording merely ordinal-scale measurement affords metrics. Each of the three scales adduced above to capture degrees of hardness bears a metric defined by the absolute values of those scales’ numerical differences. A magnitude structure affording a (separable) metric belongs to the topological category known as a (separable) uniformity. Topologies that are uniformities in this sense are Hausdorff topologies, but they need not be compact nor (topologically) connected.[26] The topological character of the magnitude structure entailed for all concretes by Rand’s measurement-omission theory of concepts is the character of a uniformity.

The magnitude structure entailed by Rand’s theory has the algebraic character of a lattice, which has more structure than a partially ordered set (or a directed set) and less than a group (or a semi-group). In terms of the mathematical categories, Rand’s magnitude structure for metaphysics is a hybrid of two: the algebraic category of a lattice and the topological category of a uniformity. Rand’s structure belongs to the hybrid we should designate as a uniform topological lattice.

Concerning multidimensional magnitude structures of concept classes, I concluded in the preceding subsection that Rand’s theory entails neither affine nor absolute structure. What is entailed: concept classes with a 2D or 3D magnitude structure will have the structure of at least an ordered, distance geometry.[27] Significantly, it is implied that planes and spaces concretely realizable will have at least that much structure.

(Continued)

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II.C Superordinates and Similarity Classes

Hardness, fatigue cycle limit, critical buckling stress, shear and bulk moduli, and tensile strength all fall under the superordinate concept strength of a solid. The conceptual common denominator (Rand 1966–67, 15, 22–25; 1969, 143–45) of these various strengths of solids is that they are all forms of resistance to degradations under stresses. What is the common measure of this resistance the different species of strength have in common? What is the common measure of strength that all specific forms of the strength of solids have in common? The magnitude structure of hardness affords only ordinal-scale measurement. The magnitude structure of tensile strength affords ratio-scale measurement. Only the ordinal aspect of the tensile-strength measure could be common with the hardness measure. Is the ordinal aspect of each specific form of strength a same, single, common measure? No, the ordinal measure of hardness is not the same as the ordinal measure of tensile strength. Resistance to being scratched is not the same as resistance to being pulled apart under tensile stress.

The way in which an ordinal measure of hardness and an ordinal measure of tensile strength can form a common ordinal measure for the general concept strength of a solid is only as substitution units, not as distinct measure values along some common ordinal measurement scale. The some-any locution can be applied to substitution units (e.g., Rand 1966–67, 25). We sensibly say that strength of a solid in general must have some type of ordinal strength measure but may have any such type. That sort of use of some-any pertains to units as substitution units: there must be some specific form of strength to instantiate the general concept strength of a solid, but it may be any of the specific forms.

The substitution-unit standing of concepts under their superordinate concepts is a constant and necessary part of Rand’s measurement-omission recipe as applied to the superordinate-subordinate relationship. But this part is not peculiar to Rand’s scheme for that relationship. Here is what is novel in Rand’s measurement-omission theory for superordinate constitution, as I have dissected it: Whichever concept is considered as instance of the superordinate concept, not only will that subordinate concept and its instances stand as substitution instance of the superordinate, each instance of the subordinate will have some particular measure value along a specific dimension. And that particular value is suspended for the subordinate concept, thence for the superordinate concept.

Analytically, identity precedes similarity.[28] For purposes of her theory of concepts and concept classes, Rand defined similarity as “the relationship between two or more existents which possess the same characteristic(s), but in different measure or degree” (13). I concur. Occasions of scratch-hardness are similar to each other because they are all occasions of scratch-hardness, exhibiting that hardness in various measurable degrees. This much accords with Rand’s definition and use of similarity in the theory of concepts.

Occasions of scratch-hardness are also more like each other than they are like occasions of tensile strength. This is a perfectly idle invocation of comparative similarity (comparative likeness). The work that comparative similarity pretends to be doing here can be accomplished fully by simple identity (sameness) without any help from similarity: scratch-hardness is itself and not something else, such as tensile strength.

The shapes of balls are similar to each other because they have principal-curvature measures at various values within certain ranges. Likewise for the shapes of cups (to keep the illustration simple, consider a Chinese teacup, not a cup with a handle). Moreover, ball shapes are more like one another than they are like cup shapes because ball values of principal curvatures are closer to each other than they are to cup values of principal curvatures. Here the invocation of comparative similarity is not idle. To say that ball shapes are more like one another than they are like cup shapes is to say something beyond what is claimed in saying: Shapes that balls have are themselves and not something else, such as shapes that cups have.

The strengths of a solid are of various kinds that are not simply of various values along some common dimension(s). The shapes of a solid are of various kinds, and unlike kinds of strengths, these kinds are of various values along some common dimension(s).

Rand’s conception of similarity as sameness of some characteristic, but difference in measure, can be put squarely to work in analyzing comparative similarities of shapes of solids with each other. Then this conception of similarity is a genuine worker, too, in the analysis of the concept shape of a solid, superordinate for the concepts ball-shape and cup-shape. This employment of Rand’s conception of similarity in the analysis of comparative similarity, thence in the analysis of superordinates, is just as Rand would have it (14). But such an employment of Rand’s conception of similarity as sameness of some characteristic, but difference in measure, is incorrect in application to the comparative similarities of the various strengths of solids, thence to their superordinate concept strength of a solid.

What will be the proper analysis as we move on up the superordinates? Strengths of a solid are more like strengths of a solid than they are like shapes of a solid. Let us suppose, as Rand supposed, that the reason we can say that strengths of a solid are more like each other than they are like shapes of a solid is because there is some common dimension, the dimension of the conceptual common denominator, between strengths and shapes of a solid. Property of a solid fits the bill for conceptual common denominator. Strengths and shapes of a solid are both properties of a solid. What is the measurable dimension of the concept property of a solid that is common to both strength and shape of a solid? Like the common dimension for strength, it is a dimension consisting of nothing more than various substitution dimensions. The measurable dimension of property of a solid will be the hardness dimension or the tensile-strength dimension or the principal-curvature dimensions or . . . There is no single, common measure of property of a solid that all specific properties of solids have in common. Rand supposed in error that there were, for she supposed it always the case that there is some same, common measurable dimension supporting the conceptual common denominator for any superordinate concept (23).[29] That supposition is here rejected, and measurement-omission analysis of superordinate concepts is here corrected in this respect.

Suppose for a moment, though it be false, that there were some common measurable dimension of property of a solid that was singular, not common merely by substitutions. Then in saying that strengths of a solid are more like each other than they are like shapes of a solid, we could reasonably contend that the values of strengths are closer to each other on the hypothetical common property-of-a-solid dimension than they are to the values of shapes on that common dimension (14).[30] Then the magnitude structure of the common dimension for property of a solid could not be one that affords only ordinal measures. On such measures, there is no telling whether a value between two others is closer to the one than to the other. (Then in an order of values ABCD, one has no measure-basis for clustering B or C with A or D: B might cluster with A, and C might cluster with D; or B and C might both cluster with A; or . . .) The magnitude structure of the common dimension for property of a solid would have to afford additional measurement structure. It would need to afford ratio- or interval-scale measurements. But it is not at all plausible that a measurable dimension common to each instance of property of a solid should have not only ordinal-scale structure, but ratio- or interval-scale structure, when hardness, for instance, has only ordinal structure.

(Continued)

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II.D Amended Measure-Definitions of Similarity and Concepts

With possible exception for the most general concepts such as property, Rand supposed that concept classes are always similarity classes (1969–71, 275–76). This is immediately apparent from comparison of her definition of similarity with her definition of concepts. In the present study, I likewise make Rand’s supposition.

Now I have said that a solid’s resistance to being scratched is not the same as its resistance to being pulled apart under tensile stress. Nonetheless, these two sorts of strength of a solid are similar. Occasions of hardness are similar to occasions of tensile strength because the same characteristic, limit of resistance to some sort of stress, is possessed by both in different measurable forms. These measurable forms could be merely ordinal, yet in this way be a basis of similarity. Moreover, hardness and tensile strength are more similar to each other than to shape because hardness and tensile strength are two different measurable forms of the same characteristic that is different from the measurable characteristic (pair of principal curvatures) shared by shapes in different degrees.

So I should amend Rand’s definition of similarity as follows: Similarity is the relationship between two or more existents possessing the same characteristic(s), but in different measurable degree or in different measurable form.

The corresponding definition of concepts would be: Concepts are mental integrations of two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted or with the particular measurable forms of their common distinguishing characteristic(s) omitted.

Every concrete falls under both sorts of concept. Both sorts of conceptual description have application to every concrete. Occasions of hardness fall under the hardness concept by sameness of characteristic in various measures. Those very same occasions of hardness fall under the strength concept by sameness of characteristic varying in measurable form. Occasions of cup-shape fall under the cup-shape concept by sameness of (pairs of) characteristics in various measures. Those very same occasions of cup-shape fall under the spatial property concept by sameness of a characteristic, spatial extension, that has various measurable forms.

(Continued)

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II.E Conclusion of Core Task

My amendments to Rand’s definitions of concept and concept class (similarity class) do not implicate metaphysical structure beyond what is already implied by Rand’s definitions. Where I have spoken of various measurable forms of a characteristic, all of those forms are the same measurable dimensions that are also at work in concept classes based on variation of measure values along a dimension.

What is the magnitude relation under which all concretes must stand such that conceptual rendition of them is possible? They must stand in the relation of a uniform topological lattice, at least one-dimensional. This is the magnitude structure implied for metaphysics, for all existence, by the theory of concepts in Rand’s epistemology. The same magnitude structure is implied by that theory with my friendly amendment.

What is the mathematical character of universals, of the collection of potential concept-class members, implicit in Rand’s theory of concepts? In Rand’s theory, universals are recurrences, repeatable ways that things are or might be. Properties, such as having shape or having hardness, are examples of such ways. That universals are recurrences is a traditional and current mainstay in the theory of universals.[31] In Rand’s theory, however, universals are not only recurrences, they are recurrences susceptible to placement on a linear order or they are superordinate-subordinate organizations of recurrences susceptible to placement on such linear orders.

Universals as (abstractions that are) concepts are concept classes with their linear measure values omitted. If the concept is a superordinate, then the linear measurable form might also be omitted, that is, be allowed to vary across acceptable forms. Universals as collections of potential concept-class members are recurrences on a linear order with their measurement values in place.[32] For either sense of the term universals, they are an objective relation between an identifying subject and particulars spanned by those universals (Rand 1966–67, 7, 29–30, 53-54; 1965, 18; 1957, 1041).

(Continued)

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III. Genesis

Rand takes concepts to be mental products of a mental process “that integrates and organizes the evidence provided by man’s senses” ([1970] 1982, 90). She gives three definitions of concepts:

(1) Concepts are mental integrations of “two or more perceptual concretes, which are isolated by a process of abstraction and united by means of a specific definition” ([1961] 1964, 20);

(2) More generally in terms of the data processed, concepts are mental integrations of “two or more units which are isolated according to a specific characteristic(s) and united by a specific definition” (1966–67, 10);

(3) Finally and most deeply, concepts are mental integrations of “two or more units possessing the same distinguishing characteristic(s), with their particular measurements omitted” (12).

The “two or more perceptual concretes” spoken of in definition (1) are the elementary type of “two or more units” spoken of in (2) and (3). Rand proposes, in a general way, a developmental intellectual ascent from apprehending the world only in terms of perceptual concretes and actions they afford to apprehending that same world in terms of units in classes. That ascent is a refinement and sophistication in our apprehensions of existents: an ascent from apprehending existents as entities to apprehending them as identities to apprehending them as units (6–7; 1969–71, 180–81).

Rand’s measurement-omission analysis of concepts could be correct even if her account of their genesis were incorrect. In particular, her analysis could be correct even if her proposed developmental intellectual ascent were incorrect. I contend that her general proposed ascent is correct. I shall give a thumbnail sketch of the developments I think should be seen as tracing an entity-identity-unit ascent in the apprehension of existents.

III.A Elaboration of Identity

III.B First Words, First Universals

III.C Analytic Constraint

(Continued)

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III.A Elaboration of Identity

For the first day or two after birth, existents for us are plausibly only entities. Such would be the occasions of Mother’s face or voice.[33] Very soon existents become for us not mere entities, but identities, particular and specific.[34]

At 20 days, there is expectation of the reappearance of a visual object gradually occluded by a moving screen; rudimentary particular identity of visual objects in general (Bremner 1994). At 4 weeks, there is some oral tactile-to-visual transfer of object features, without opportunity for associative learning; rudimentary specific identity (Meltzoff 1993). At 5 weeks, there is recognition memory of color and form; growth of specific identity. At 8 weeks, there is onset of attention toward internal features of patterns and onset of smooth visual tracking; also, hand tactile-to-visual transfer of object features; growth of particular and specific identity. At 10 weeks, there is expectation that one visual solid object cannot move through another (Bremner 1994). By 3 months, visual tracking is becoming anticipatory (Johnson 1990); there is visual fill-in of invisible parts of objects (Bremner 1994); visual objects are being identified as separate using various static-separation and motion traits (Spelke and Van de Walle 1993); there is categorical perception of objects and events (Quinn 1987). At 4 months, haptic apprehensions of shapes can be transferred to the visual mode (Streri and Spelke 1988); visual solid objects are expected to endure and retain size when occluded for a brief period (Bremner 1994); objects are expected to fall if not supported (Needham and Baillargeon 1993).

The infant’s world of entities-identities will continue to elaborate. Units are not yet. At 6 months, the infant will have some sensitivity to numerosity; will be able to detect numerical correspondences between disparate collections of items, even correspondences between visible objects and audible events; and will be able to detect the equivalence or nonequivalence of numerical magnitudes of collections (Starkey, Spelke, and Gelman 1990). At 7 months, still without words, the infant distinguishes global categories (e.g., animals v. vehicles) which will later become superordinates of so-called basic-level categories (e.g., dog v. car) yet to be formed (Mandler and Bauer 1988; cf. Rand 1969–71, 213–15). By 12 months, the infant reliably interprets adult pointing, looking from hand to target (Butterworth and Grover 1988).

(Continued)

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III.B First Words, First Universals

At around 12 months, the infant puts first words, single-word utterances, into her play. Words at this stage are used only in play, not for communication, which is still accomplished with cries, gestures, and gazes (Bremner 1994, 249–51; Nelson 1996, 105, 112). An infant in my family, just past his first birthday, uses the word ba. He says it quietly to himself whenever he sees or is handed a spherical ball; he does not say his word when the ball is a football. We should not suppose too hastily, I should note, that his word ba refers simply to the spherical ball with which he is engaged. At this first-words stage, his utterance may designate the object as component of his whole activities that go with those objects (activities like training the adults to fetch) (Bremner 1994, 251–52; Nelson 1996, 97, 109–10, 115, 227–29; Bloom 2000, 35–39).

By 14 months, the toddler points to indicate items (Butterworth and Grover 1988). By 16 months, she spontaneously groups objects of a single category (Bremner 1994, 173). In another month or two, comes the naming explosion, naming of objects especially (Nelson 1996, 111–15; Macnamara 1986, 144–45; Bloom 2000, 91–100). (That there really is such a dramatic burst in the rate of word acquisition at this time is disputed; Bloom 2000, 39–43.)

By that time, at 17 or 18 months, the toddler is using single words to refer (Macnamara 1986, 56–57). These words (50 to 100 words) include demonstratives such as that, common nouns such as ball, and proper names such as Star, say, to refer to a particular ball. The use of common nouns and proper names in single-word reference indicates certain competencies of identification, certain representational comprehensions of identities specific and particular. The representational comprehensions of specific and particular identity that are evidently coming into operation at this stage are class-membership relation, individuation within a class, and particular identity over time.

Skillful reference for the utterance ball indicates that the beginning speaker has some working principles for deciding whether a given item qualifies as being in the category ball. Then such a speaker has some operational sense of class-membership relation (61–62, 72–74, 124–28, 148–49, 152–56). Ball is a count noun. Although the beginning speaker does not yet possess the principles of counting, not even implicitly, she has some working principles of individuation within a class, some principles for holding in mind individual balls as distinct from one another (128–30).[35] Moreover, ball refers to any individual ball as a distinct individual over time (59–60, 130–36, 141–42, 152). Finally, the name Star is attached to a particular one of those individual balls over time (55–62, 71–83).[36]

I suggest that even at the single-words stage of language development, the toddler has entered the conceptual level of consciousness in Rand’s sense of that level. The utterance ball refers, and marks a concept, already at this stage.

One problem for that conjecture is the following. Rand required that the items falling under a concept be united with a specific definition. [See (1) and (2) above and Rand 1966–67, 48–50; 1969–71, 177–81.] But at the single-words stage of development, the toddler cannot yet form two-word expressions. That competence will not be attained for another six months or so, at around 24 months of age.[37] Not yet having two-word expressions, she cannot yet form a sentence, cannot yet use words in assertive sentences. Without propositions one is without defining propositions, hence, without definitions. Then at the single-words stage of development, the items falling under a “concept” cannot be united by a specific definition. Then it would seem one does not yet possess a concept in Rand’s sense. I think that conclusion would be an overstatement.

For an older child or an adult, of course, “a concept identifying perceptual concretes stands for some implicit propositions” (Rand 1966–67, 48, 21). For a single-words toddler, no propositions can be adduced. Actions can be adduced. A ball is something that can be handled and thrown down. It bounces and rolls. These things are clearly known of balls even by the one-year-old whose first and only word is ba. The concept ball is likely held in mind in the form of image and action schemata as well as by the term ball (13, 20, 43; 1969, 167–70).[38]

There is something else, something profoundly conceptual, at hand in linguistic competence at least by the time of the naming explosion. John Macnamara concludes that having a word such as ball at this stage means having a logical principle of application. That is a surrogate for definition at this single-words stage. A principle of application is the working principle, spoken of above, for determining whether an item is or is not a ball (Macnamara 1986, 124–28).[39] A principle of application determines class membership. That is the basic function a definition accomplishes for more advanced language users (Rand 1966–67, 40; 1969–71, 231–32).

To have an operational grasp of the class-membership relation is to have a tacit grasp of the notion of unit in the sense of a substitution unit, which is the unit for counting. That does not mean that one has yet grasped the elementary principles of counting (nor that one can put the notion of a substitution unit to work in counting). At 18 months, one has evidently gotten some working hold on the notion of a substitution unit, the notion of a simple member-of-a-class, without yet having the principles of counting. But at this single-words stage, one has taken the first step into the dual realms of the conceptual and the mathematical. With a tacit grasp of the notion of unit in the substitutional sense, “man reaches the conceptual level of cognition, which consists of two interrelated fields: the Conceptual and the Mathematical” (Rand 1966–67, 7). Rand was correct in thinking that “man’s mathematical and conceptual abilities develop simultaneously,” even though she was incorrect in thinking that “a child learns to count when he is learning his first words” (9; 1969–71, 200).[40]

Having ball, one is getting hold of “ball, any one.” That is the membership relation and its requisite principle of application. Having ball, one is also getting hold of “some things of a class, the balls” (Rand 1966–67, 17–18). That is the individuation-within-class relation and principle (Macnamara 1986, 128–30). Then already at the stage of first concepts, one has beginning working principles of universal quantification (any) and existential quantification (some).[41]

(Continued)

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III.C Anyalytic Constraint

As we have seen, in Rand’s view, in the analysis of any concept there can be found a double application for some and any: with respect to substitution units and with respect to measure values. To form our concepts, however, Rand supposes that we do not need to grasp, expressly nor tacitly, the notion of units as measure values. We discern similarities. Where there is similarity, there can be found various measure values along a common dimension, in Rand’s view, but we need not know anything about such measure bases.

When we pick up a ball, our sensory systems measure it in several ways. When we perceive a similarity between two items, according to Rand’s account, we are perceiving some same characteristic(s) they both possess in different measure or degree (1966–67, 13–14; 1969–71, 139–40, 143). They both possess that characteristic in some measure or degree. Items of their class possess that characteristic in some degree, but may possess it in any degree within a range of measure delimiting the class (Rand 1966–67, 11–12, 25, 31–32).

On which characteristic(s) does the similarity class, thence the concept class, rest? Like Ockham, Rand observed that items in a similarity class are more similar to (and less different from) one another than they are to things not in the class. A ball is more similar in various ways to other balls than it is to sticks, hands, and so forth. As we know, Rand analyzed similarity in terms of measurable dimensions, in terms of measures of dimensional characteristics. The characteristic(s) on which the similarity class and its concept ball rests, analytically and genetically, in Rand’s theory, is whichever measurable characteristic(s) makes a ball measurably closer to other balls than to sticks, hands, and so forth (1966–67, 13–14, 21–23, 41–42; 1969–71, 144–47, 217, 274–76).[42]

I have addressed the defect and remedy of this measure-theoretic analysis of similarity classes and concepts in the preceding section. It remains to address the genetic aspect, which I cast as: in forming a similarity class and its concept, one is relying on (tacitly using) whichever measurable characteristic(s) makes items in that class and under that concept measurably closer to one another than to opponent items.

Rand thought, rightly I should say, that formation of any concept whatever requires differentiating two or more existents from other existents. She thought also that such differentiation requires comparative degrees of difference, measurable as such on a dimension(s) common to existents in the class and existents outside the class (Rand 1966–67, 13). What if Rand were right in this second doctrine? What if, in order to form any concept whatever, there had to be a dimension common to the concept class and its opponents and this had to be a dimension along which comparative closeness measurement is possible? What would that imply for metaphysics? It would imply that every concrete can be placed in concept classes whose linear measures are not only ordinal-scale, but interval- or ratio-scale as well.[43]

I avoid that extravagant implication as follows: I retain Rand’s assumption that formation of any concept requires differentiating two or more existents from other existents and her assumption that all concept classes are similarity classes and her measure-definitions, as amended above, of concepts and similarity. I reject the assumption that differentiation between existents included in and existents excluded from a concept class require comparative degrees of difference (beyond the comparative-difference-degree pretender that merely says a thing is less different from itself than it is different from things not itself).

Such differentiation may sometimes be based at least partly on fairly blunt sameness and difference. Spherical balls are the same with one another in that they roll regularly, and in this they are baldly different from floors. A dimension along which items in a concept class have various measure values need not be a dimension common with items in an opponent concept class.

Differentiation of existents included in or excluded from a concept class may enlist nontrivial comparative degrees of difference (or likeness). I see three forms of these. In one the comparative degrees are along dimensions common to both included and excluded existents, and those dimensions afford either ratio- or interval-scale measures. Along the dimensions of shape, a spherical ball can be distinguished from a football in that way. The sets of pairs of principal curvatures (ratio scaling) over the surfaces of spherical balls are less different from each other, from one ball’s set of pairs to another ball’s set of pairs, than they are from the sets of pairs of principal curvatures over the surfaces of footballs.

In light of my amendment to the measure-definition of similarity, we should allow also for a second nontrivial variety of comparative similarity. I observed earlier that hardness and tensile strength are two different measurable forms of a same characteristic (resistance to degradation under some sort of stress) that is different from the measurable characteristic (pairs of principal curvatures, which are spatial extension properties) shared by shapes. This second manner of decomposing a comparative similarity permits concepts based on comparative degrees of similarity without requiring that linear measures of the concept dimensions be anything beyond ordinal measures.

A third decomposition of nontrivial comparative similarity does not rely on shared and unshared dimensions of the relata. It relies simply on numbers of shared and unshared features.[44] Perhaps any concept based on this sort of comparative similarity can be recaptured in a more sophisticated way by ascertaining measurable dimensions on which to base the concept (Boydstun 1990, 31–33). I expect that is so. Notice, however, that the metaphysical implication drawn in the present study (uniform topological lattice structure) need not suppose that all concepts can be analyzed in terms of Rand’s measurement-omission formula; only that all concretes can be placed under one or more concepts analyzable in those terms.

I have exhibited a way in which a measurement analysis of concepts can constrain theorizing about the genesis of concepts. I do not want to create the impression, however, that theory of the genesis of concepts based on observations and empirical testing cannot rightly constrain one’s analysis of concepts. The analytical principles stating that all concretes can be placed in concept classes having a measurement structure and that these structures are of such-and-such characters are conjectures open to restriction through counterexamples. These conjectures of analysis are subject to reform or replacement in the face of contrary analytical and empirical results, somewhat as the General Relativity principle that freely falling bodies follow time-like geodesics of space-time is subject to reform or replacement.[45]

One of the avenues for empirical confrontation of our analytical conjectures concerning concepts and concept classes is research on conceptual development. The core task I have undertaken in the present study has been a certain extension of Rand’s metaphysics arising from her analysis of concepts (not her theory of their formation). I have not undertaken here a survey of the various ways in which empirical research on conceptual development may challenge Rand’s analysis of concepts. But there is one form of challenge that is invalid, and I want to draw attention to this fallacy, which has required a long struggle for me to overcome. That is a fallacy I insinuated in Boydstun 1990, 33–34. It says that because preschoolers do not possess—not even tacitly—mathematical understanding sufficient to be forming their concepts using a principle of measurement-omission, their concepts do not bear analysis in terms of measurement-omission. That is the fallacy of confusing genesis with analysis.[46]

Acknowledgements

I wish to thank Robert Campbell (referee) and Chris Sciabarra (editor) for helpful comments, suggestions, and questions. I also thank my partner's grandson Julian for sharing his progress from age 1 to 2 the year this essay was written.

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Notes

1. Cf. Armstrong 1978a, 25–26.

2. Rand takes propositions (E), (I), and (C ) to express primary facts and to be fundamental compositions upon three concepts she takes as axiomatic: existence, identity, and consciousness. She takes all concepts to bear implicit propositions that elucidate the concepts (Rand 1966, 48; 1969, 177–81, 228). Propositions (E), (I), and (C ) are immediate elucidations of Rand’s axiomatic concepts (1957, 1015–16). Rand does not present (I) and (C ) as axioms, only as most important elucidations of her three axiomatic concepts; for her order of presentation, she follows what she takes to be the order of cognitive development (1966, 3, 55–56, 59). My order of presentation brings the propositions (E), (I), and (C ) to the fore, and this, I hope, is analytically illuminating.

3. In the case of the concrete that is the universe itself, which is all of existence, the measurable relations are to parts of itself. For example, the total mass-energy of the universe is a measure having relation to each of its constituents having mass-energy.

Rand took (Im) to be axiomatic in that she took it to be entailed by her axiom (I). A thing not measurable in any way “would bear no relationship of any kind to the rest of the universe, it would not affect nor be affected by anything else in any manner whatever, . . . in short, it would not exist” (1966, 39). Rand is supposing that anything bearing some relationship to the rest of the universe bears some measurable relationship to the rest of the universe. I think that this supposition, which is tantamount to (Im) (all concretes have measurable relations to other concretes), is a postulate additional to the axiomatic postulate (I) (existence is identity). I do not regard the postulate (Im) to be axiomatic; unlike the axiom (I), the postulate (Im) can be denied without self-contradiction and is therefore open to possible restriction by counterexamples. Like Rand I take (Im) to be an unrestrictedly true postulate.

4. This idea is widespread. Antecedents are to be found in Schopenhauer, Goethe, Helmholtz, James, Bergson, and Dewey. For current applications of the idea that perceptual systems measure, see Krantz, Luce, Suppes, and Tversky 1989, 131–53; Churchland and Sejnowski 1992, 183–233.

5. Pale anticipations of this idea of Rand’s may be found in James [1890] 1950, 270; Johnson 1921, 173–92; and Heath [1925] 1956, 132–33; Prior 1949; Searle 1959. For relations to Aquinas and Hume, see Boydstun 1990, 24–27.

6. How might the concept existence satisfy this definition of concepts? Might the concept of existence as all existents (Rand 1969–71, 241) be rendered as all instances of existents, with all measure values of those existents omitted? (1966–67, 56). See also Armstrong 1997, 194–95.

7. A concept class is at least the sense of class at work for a kind, for mere membership in a kind (Macnamara 1986, 50–53, 152–56). For Rand’s theory of concepts, however, it seems that concept classes might always also be properly regarded as sets. For in Rand’s theory, all concept classes must be measurable. They must afford some appropriate numerical representation, and any such representation can also be expressed in terms of sets.

There are reasons to doubt whether concept classes always satisfy even the extensionality postulate of Zermelo-Fraenkel set theory, the postulate that two classes collecting the same items are the same class (Macnamara 1986, 152; Bigelow 1988, 102). Concept classes not satisfying that postulate could not qualify as either so-called proper classes nor as sets. Even if those doubts can be put to rest (Bigelow 1988, 101–9), there would remain further doubts about whether absolutely all concept classes satisfy the separation axiom of ZF set theory. Some concepts, such as the concept all items (all things that are either a potential or actual existent or a mere posit), are so comprehensive that they do not themselves stand as substitution units in some superordinate concept. Then concept classes need not always be extensionality-satisfying classes that are also sets. In particular a concept class need not always be itself a member of a larger class. Such concepts are extremely rare; almost always an extensionality-satisfying concept class will qualify as a set. I assume, with trepidation, that concept classes appropriate for Rand’s theory of concepts are not only classes in the sense of a kind, but also are rightly construed as classes that satisfy extensionality and, with rare exception, are rightly construed as classes that are sets (cf. Armstrong 1997, 185–95).

The following are proper classes, extensional classes that are not sets: the class of all items (the universe of class discourse), the class of all sets, the class of all ordinals (all order types of total orders having least members), and the class of all cardinals (all least ordinals for sets such that there is a mapping from least order type to set that is one-to-one and onto). On proper and nonproper classes, see Machover 1996, 10–16; von Neumann [1925] 1967, 393–94, 403; Quine [1961] 1980, 90–101, 112–17; 1982, 94, 130–31, 302; and Boolos 1998, 35–36, 42–47, 73–87, 223–24, 238–41.

8. Kline 1972, 554–64, 882–86; Churchland and Sejnowski 1992, 183. My measurement analysis of the concept shape supplements Rand’s. Likewise, it could supplement Armstrong’s (see Armstrong 1997, 55–56).

9. Cf. Rand 1966, 14; Peikoff 1991, 84. On the intended elementary sense of linear order, see Rosenstein 1982, 3.

10. On ratio scales, see Krantz, Luce, Suppes, and Tversky 1971, 3–5, 9–10, 44–46, 71–87, 518; 1990, 10, 108–13.

11. Simple addition: 153.0 yards joined to 153.0 yards is 306.0 yards. Grade addition: 153.0 yards per mile joined to 153.0 yards per mile is 310.3 yards per mile. Note also that there are valid and specific nonstandard ways of concatenating lengths, and these are faithfully represented mathematically by specific nonsimple additions (Krantz, Luce, Suppes, and Tversky 1971, 87–88, 99–102; 1990, 18–56).

12. Cf. Swoyer 1987, 256–58. Here I take a norm accepted in mathematical physics and adapt it for our broader context (Geroch 1985, 86, 81–84, 119). Physical gets replaced by concrete for our metaphysics. Notice, making that replacement, that to obtain the relation of mathematics to metaphysics, we may look to the relation of mathematics to physics (ibid., 1, 17, 111–13, 183–87, 223, 283–90, 324–40; Geroch 1996).

13. Krantz, Luce, Suppes, and Tversky 1990, 112–25; Martin 1982, 14–17. Structures are characterized by their automorphisms, the set of structure-preserving morphisms of that structure into itself. (Consider the set of rotations and reflections, confined to the plane, that transform a square into itself: 90° rotation about the square’s center, reflection through a diagonal line, and so forth.) The identity morphism is among the set of automorphisms for any structure. The set of automorphisms for a totally disorganized structure (a would-be structure, we might say) has only that one member, the identity morphism. The identity automorphism by itself affords counting, which is a form of measurement known as absolute measurement (Suppes 2002, 110–18). That barest structure is less than the minimal structure required for concept classes under Rand’s measurement-omission analysis of concepts.

14. Contrast Rand’s system in this respect with the systems of Descartes and Kant. Rand’s theory does not entail that there is any 2D or 3D magnitude structure of concretes having the structure of Euclidean geometry. In particular Rand’s theory does not imply that physical space is Euclidean.

Let me also note at least some of what is meant by absolute and affine in the present context. Euclidean geometry contains both absolute geometry and an affine geometry. Absolute geometry consists of those propositions of Euclidean geometry that can be obtained from Euclid’s first four postulates alone, neither affirming nor denying the fifth postulate, which is the parallel postulate. These propositions hold not only in Euclidean geometry but in hyperbolic geometry. Absolute structure permits the comparison of lengths along lines whether or not they are parallel to each other.

Affine geometry consists of those propositions that can be obtained from Euclid’s first two postulates (to draw a straight line from any point to any point and to produce a finite straight line continuously in a straight line) together with the fifth postulate (in one version: for any point P off a line L, there exists a unique line through P that is parallel to L). Affine structure permits the comparison of lengths only along lines that are parallel to each other. See Krantz, Luce, Suppes, and Tversky 1989, 109–11; Coxeter 1980; Martin [1975] 1998.

15. Krantz, Luce, Suppes, and Tversky 1989, 31–35.

16. It might be thought that temperature was found to afford ratio scaling once absolute zero was conceived and the “absolute thermodynamic temperature scale” was constructed. That is incorrect. The interval units of the absolute thermodynamic temperature scale (˚K) are the same as the interval units of the Celsius scale (˚C). Like the Celsius and Fahrenheit scales, construction of the absolute thermodynamic temperature scale requires not only that an interval unit be chosen, but that a fundamental fixed point be chosen and assigned a value. The fixed point selected for the absolute thermodynamic temperature scale is the triple point of water (unique temperature and pressure at which water, ice, and vapor coexist). Absolute zero is then defined to be 273.16 ˚K below the triple point exactly.

What if, contrary to my supposition, temperature were found to be a physical quantity that affords ratio measures? That would not change the outcome of my core task in this study. I am to delineate and put aside the richer types of magnitude structures affording measurement until we arrive at the minimal structure required for Rand’s measurement-omission recipe. The physical examples presented need be, for our purpose, only hypothetical illustrations of types of magnitude structures.

On applications of interval-scale measurement in psychophysics, see Krantz, Luce, Suppes, and Tversky 1971, 139, 519–20; 1989, 177–78, 184–85; also, Michell 1999, 20–21, 74–76, 81–87, 147–52, 172–77, 189–90, 198–200, 205–8. On applications of interval-scale measurement in utility theory, see Krantz, Luce, Suppes, and Tversky 1971, 17–21, 139–42; also, Nozick 1985. That there are magnitude structures affording interval-scale measurement in the realms of utility and psychophysics does not mean that every magnitude structure in those realms affords such measures; some may afford merely ordinal measurement. Rand likely supposed that only ordinal measurement is appropriate in utility theory (1966–67, 32–34), under tutelage from Austrian-school economists (cf. Rothbard 1962, 15–28, 222–31, 276–79).

17. On interval scales, see Krantz, Luce, Suppes, and Tversky 1971, 10, 17–21, 136–48, 170–73, 515–20; 1990, 10, 108–13. Throughout this paper, I use simply concatenation in place of the usual technical expression positive concatenation. That the concatenations are positive means that the resulting, concatenated magnitude is greater than either of the magnitudes entering into the concatenation. So, I say simply that magnitude structures of concretes such as temperature (or chemical potential) do not afford concatenations, rather than say, as would be usual technically, that such structures afford concatenations qualified as intensive in contrast to positive.

18. A body or fluid at 43˚C is at 109.4˚F. If at 45˚C, then at 113.0˚F. If at 56˚C, then at 132.8˚F. The Celsius difference-interval ratio (45 – 43)/(56 – 45) equals the Fahrenheit difference-interval ratio (113.0 – 109.4)/(132.8 – 113.0). The simple ratios of degrees such as 43/45 and 109.4/113.0 are not equal, unlike the character of ratio scales. We should be aware too of an important respect in which magnitude structures affording interval scales are like magnitude structures affording ratio scales. For either type of structure and their scale types, it is the case that whether two intervals in the structure are equal is independent of which measurement scale in the scale type is used. The interval between 43˚C and 45˚C equals the interval between 47˚C and 49˚C. That equality remains when those values are converted to ˚F, though the value of each equal interval changes from 2˚C to 3.6˚F.

19. Ratio scales stand to each other as a metal bar under uniform thermal expansions. A single number characterizes a particular state of expansion, a particular ratio scale. Interval scales stand to each other as an elastic band pinned at some point, then stretched to some degree from that pinned point, for various such pinnings and stretches. Two numbers characterize a particular pinning and stretch, a particular interval scale. On characterization of scale types by degrees of uniqueness and homogeneity, see Krantz, Luce, Suppes, and Tversky 1990, 112–25, 142–50; Suck 2000; Cameron 1989.

20. Krantz, Luce, Suppes, and Tversky 1990, 112–22; Martin 1982, 136–44.

21. Temperature attributes are relational attributes, specifically, difference attributes. When we sense the warmth or coolness of a body by touching it, we are sensing the rate of heat flow into or out of our own body at the contact surface. Rate of heat flow reflects size of temperature difference between the two bodies in contact.

22. Krantz, Luce, Suppes, and Tversky 1989, 107–8; Coxeter 1980; Martin [1975] 1998.

23. Krantz, Luce, Suppes, and Tversky 1989, 42–46. The concept color (resolved as hue, saturation, and brightness) is a 3D magnitude structure that is affine, but not also absolute (Krantz, Luce, Suppes, and Tversky 1971, 515–20; 1989, 40, 243–50, 279–85). The concept space-time (flat space-time) is a 4D magnitude structure that is affine, but not also absolute (though it has absolute substructures).

24. Rand did not herself reach a stable understanding of these entailments. See Rand 1966–67, 31; 1969, 189–90, where she expresses her supposition that our resort to measurements less rich than ratio-scale measurement is a resort to measurements that are less “exact” and reflects our relative ignorance of the thing we are measuring.

25. On linear orders, see Rosenstein 1982. On ordinal measurement, see Krantz, Luce, Suppes, and Tversky 1971, 2–3, 11, 14–15, 38–43; 1989, 83–89; and Droste 1987a; 1987b.

26. The absolute value function here is not taken over the real numbers in their character as a vector space. Then the absolute value function in our merely ordinal context is not a norm (Bartle 1976, 54–55). Our metric is not being derived from a norm; we do not magically convert our merely ordinal scale to an interval one by taking absolute values of numerical differences.

On topological, uniform, and metric spaces, see James 1999 and Geroch 1985. That the topology of a magnitude structure affording ordinal-, interval-, or ratio-scale measurement be a Hausdorff topology seems fitting. In such a topology, any two distinct points have some nonintersecting neighborhoods, and this would seem to be a natural condition for any sort of measurement at all.

27. On ordered geometry, see Krantz, Luce, Suppes, and Tversky 1989, 104–7, and Coxeter 1980. I say a distance geometry rather than a metric geometry because the distance function need be only positive and symmetric. The triangle inequality, an additional requirement for a metric, need not be satisfied (Martin [1975] 1998, 68–69; Coxeter 1980, 175–81; Blumenthal 1970, 16; consider also, Krantz, Luce, Suppes, and Tversky 1989, 186–87, 205–8).

A mathematically determinate form from which measure values may be suspended for the concept shape of a curve (in 3D) is a set of curvature and torsion values, one pair of values for each point of the curve. Consider a 2D graph in which curvature values are plotted along one axis and torsion values are plotted along the other axis. Plotting the particular pairs of values for a particular curve in concrete 3D space will form a particular curve in the plane of our 2D graph. Relations among points in this plane satisfy the axioms of a 2D ordered, distance geometry (as well as axioms for richer 2D geometries).

The concept class shape of a curve satisfies my principle, sprung from Rand’s measurement-omission theory of concepts, that all concept classes having a multidimensional magnitude structure have the structure of at least an ordered, distance geometry. Many of our concepts are obviously multidimensional. Consider a general-purpose definition of the concept animal (metazoa): a multicellular living being capable of nervous sensation and muscular locomotion. Surely the mathematically determinate form of the concept class animal is multidimensional (cf. Rand 1966−67, 16, 24−25, 42). My principle alleges that that multidimensional structure will have the structure of at least an ordered, distance geometry.

28. Cf. Armstrong 1978a, 44–50; 1978b, 95–123; 1997, 17–18, 22–23, 47–57; Jetton 1998, 41–42.

29. But consider Rand’s exchange with Leonard Peikoff (Prof. E) in Rand 1969–71, 275–76.

30. See also Rand’s exchange with Allan Gotthelf (Prof. B ) in Rand 1969–71, 139–40, as well as Peikoff 1991, 85, and Gotthelf 2000, 59. Further, see Kelley and Krueger 1984, 52–61; Kelley 1984, 336–45; Jetton 1998, 63–72.

31. See Armstrong 1978a, 11–12, 77–87, 108–16; 1997, 14–15, 28–31, 49; Bigelow 1988, 4, 18–27, 40–41, 56–57, 121–78.

32. Cf. Armstrong 1997, 185–95.

33. Rand concluded from research literature as of 1966 that the sensory experience of the infant was apparently entirely “an undifferentiated chaos” and did not contain any percepts (1966–67, 5, 6). Subsequent research has dispelled that old vision of cognition in neonates. See Bremner 1994; Meltzoff 1993; Clifton 1992; Kellman 1995.

34. The distinction of particular and specific identity is mine and is as follows. Particular identity answers to that, which, where, or when. Specific identity answers to what. Every existent consists of both a particular and a specific identity (Boydstun 1991, 43–46, and 1995, 110).

35. The sense of implicit here is extracted from the relevant cognitive-development research literature (viz., Gelman and Meck 1983, 344). The child is said to have implicit knowledge of the counting principles if she engages in behavior that is systematically governed by those principles, even though she cannot state them. (See Note 40 for the principles.) Gelman and Meck liken this implicitness of the counting principles at this stage of cognitive development to the way in which we are able to conform to certain rules of syntax when speaking correctly without being able to state those rules. That much seems right, but there is a further distinction I want to make. The child’s implicit counting principles are being learned (and taught) as an integral part of learning to properly count aggregations explicitly, expressly. In contrast, we can (or anyway, my preliterate Choctaw ancestors centuries past could) live out our lives, speaking fine in our mother tongue, following right rules of syntax, yet without being able to state those rules; indeed, without even knowing any of the terminology of syntax. Our learning of tacit rules of syntax is not for the sake of becoming able to follow them explicitly, only tacitly.

In the present developmental discussion, I shall reserve the term implicit to indicate that an operative rule is not only tacit, but has become operative as an integral part of becoming explicitly operative. The tacit logical principles, whose acquisition according to Macnamara is traced in the text, are not implicit in my present sense.

There is, of course, another sense of implicit that I am also happy to use. That is the logicomathematical sense, which was pertinent to our analysis section. It is in that sense that we say a certain theorem is implicit in a set of axioms; Hertz’ wave equation for propagation of electromagnetic radiation is implicit in Maxwell’s field equations; an inverse-cube central force law is implicit in a spiral orbit; dimension reductions are implicit in Kolmogorov superposition-based neural networks; certain measure relations are implicit in any similarity discerned in perception; or certain measure relations are implicit in a concept class. Cf. Rand 1969–71, 159–62; Campbell 2002, 294–96, 300–10; Boydstun 1996, 201–2.

36. Drawn out into our adult expression, here is the logic tacitly put to work by the toddler at this stage: There is a unique kind (class) of which Star is a member, and any object is a ball if and only if it is a member of that kind. For any particular ball, there is a unique member of the kind ball, and as long as that member exists, it is identical (totally same) with that particular ball (Macnamara 1986, 137–39). I should say that such working interpretive principles render one’s perceptual knowledge conceptual. One has conceptual knowledge even at the single-words stage of language development.

My example of proper naming of a special ball Star is contrived for convenience of illustrating the tacit logical resource. Toddlers at this stage are likely to restrict proper names to particular (real or make-believe) animate entities possessing mentality (Bloom 2000, 130–31).

37. By 24 months the child is using two-word utterances such as “Mommy sit!” and “guy there” and “I know [how to do it]” (Bremner 1994, 252–53; Nelson 1996, 112, 124–25). Up to about this time, when grammar begins to develop, “words learned remain tied to their world models and do not form systems of their own” (Nelson 1996, 128). In terms of Deacon’s iconic, indexical, and symbolic levels of representation (1997, 70–83), I should say that concepts at the single-words stage are indexical representations, and these concepts will become symbolic representations with the onset of grammar. Rand’s conceptual level of representation cuts across Deacon’s indexical and symbolic levels.

All three levels of representational cognition—even the iconic level (e.g., drawing a stick man)—are active, deliberate, and constructive. I take the membership relation, which is essential for concepts, classes, and sets, to require this sort of active generation, from our first concept to our last. In this way, the membership relation is unlike perceptual relations of similarity, proximity, or containment (cf. Rand [1961] 1964, 20; Maddy 1997, 90–94, 108–9, 152n30, 172–76, 185–88).

38. See further Boydstun 1990, 16–18; Minsky 1974, 111–17; Johnson 1987, 23–30, 102–4; Iverson and Thelen 1999; Nelson 1996, 16–17.

39. Cf. Kelley and Krueger 1984, 47, 52. In saying that this tacit logical principle of application is a surrogate for a concept’s definition, I mean to say only that the tacit principle accomplishes the main function that an explicit definition accomplishes. I do not mean to say that the tacit principle is additionally an implicit definition in the developmental sense of implicit (as in Note 35). Macnamara’s tacit logical principle of application is needed just as much for concepts of things in terms of merely characteristic features as it is for concepts of things in terms of defining features (cf. Bloom 2000, 18–19).

During the first few years of speech, we evidently tend to conceive of things in terms of characteristic features. After about age 5, there is a developmental shift to conceiving of things in terms of defining features. The course of this shift, which occurs at different times in different domains of knowledge, has been partially charted by Frank Keil 1989; see Boydstun 1990, 34–37. The shift need never occur for all our concepts. [in a preliterate culture (my Choctaw ancestors again), is the shift so extensive as in our culture? See Olson 1994.] Acquiring a tacit logical principle of application is not for the sake of becoming able to conceive of things in terms of defining features.

40. The child has gone far beyond learning first words (roughly months 12 to 18) by the time she is learning to count. By 30 months, the basic linguistic system has become established and is fairly stable (Nelson 1996, 106). Not until around 36 months or beyond does the child have an implicit grasp of the elementary principles of counting: assign one-label-for-one-item, keep stable the order of number labels recited, assign final recited number as the number of items in the counted collection, realize that any sort of items can be counted, and realize that the order in which the items are counted is irrelevant (Gelman and Meck 1983; Butterworth 1999, 109–16).

At 22 months, a child in my family could “say his numbers.” This competence is not essentially different than being able to “say his ABC’s” (Bloom 2000, 215). Rand may have mistaken the onset of recitation of count-word sequences with onset of ability to count.

41. Cf. Macnamara 1986, 143; Burgess 1998, 10–11; Boolos 1984, 72.

42. For Ockham on comparative similarity, see Maurer 1994, 387, 389. For more on comparative difference and comparative similarity in theory of concept formation, especially in Rand’s theory, see Kelley and Krueger 1984, 52–61, and Kelley 1984, 336–45. See also Jetton 1998, 63–72, and Livingston 1998, 15–21.

43. Cf. Armstrong 1997, 64–65, for a related extravagance, which he boldly embraces. The extravagant implication I pose is avoided by me in one way; for another way, consider Jetton 1991.

44. Quine 1969, 117–23; Krantz, Luce, Suppes, and Tversky 1989, 207–22; Nosofsky 1992, 38–40.

45. The General Relativity principle that freely falling bodies follow time-like geodesics of space-time is subject to analytical challenges (Torretti [1983] 1996, 176–81) and to empirical tests, such as whether Earth and Moon have different accelerations towards the sun (Ciufolini and Wheeler 1995, 14, 88, 113–15). Contrast those methods of evaluating conjectures in natural science with the methods of evaluating various candidate axioms for a formal discipline such as set theory (Maddy 1997). We should expect the forms of evaluation appropriate to measurement conjectures for a theory of concepts and concept classes to lie between forms appropriate to natural science and forms appropriate to the formal disciplines of mathematics, set theory, and logic.

46. This essay was studied at the 2003 Advanced Seminar of The Objectivist Center. The significance of the present work was clearly appreciated. The session indicated that reference to an accessible general overview of modern mathematics would be helpful. I heartily recommend MacLane 1986.

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Universals and Measurement (2004)

. . I. Orientation

. . II.A Analysis – Affordance of Ratio- or Interval-Measures
. . II.B Analysis – Affordance of Ordinal Measures
. . II.C Analysis – Superordinates and Similarity Classes

. . II.D Analysis – Amended Measure-Definitions of Similarity and Concepts

. . II.E Analysis – Conclusion of Core Task

. . III. Genesis

. . III.A Genesis – Elaboration of Identity

. . III.B Genesis – First Words, First Universals

. . III.C Genesis – Analytic Constraint

. . . Notes

. . . References

Judgments and Measurement (2013)

Predication with Identity

Predication with Measurement

Appendix on Hypothetical Judgments

(5/13/15)

~~~~~~~~~~~~~~~~~~~~

I don’t know how this happened, but my endnote 5 as shown in this display of my 2004 essay was incomplete. The full note as published in JARS included, after Johnson 1921, also Prior 1949 and Searle 1959.

Johnson, W.E. [1921] 1964. Logic (Part 1). New York: Dover.

Prior, Arthur. 1949. Determinables, Determinates, and Determinants. Mind 58, no. 229:1–20, 179–94.

Searle, John. 1959. Determinables and the Notion of Resemblance. The Aristotelian Society suppl. vol. 33:141–58.

The paper by Prior has been the most important for my purposes subsequent the publication of “Universals and Measurement.” There is an introductory article Determinates vs. Determinables by David Sanford in Stanford Encyclopedia of Philosophy. I was pleased to see challenged therein Searle’s thesis that differentia in a genus-species hierarchy are logically independent of their genus. Sanford writes:


There are both historical and logical difficulties with this view.

The genus-species relation is an ancient philosophical topic. No crisp, clear definition [of genus-species] can be consistent with everything that has been said before. Searle's confident exposition, however, contradicts some standard views. A logic text in wide use for many decades gives the following as a rule of definition:

“The better the definition, the more completely will the differentia be something that can only be conceived as a modification of the genus: and the less appropriately therefore will it be called a mere attribute of the subject defined.” (Joseph, 1916, p. 112)

Aristotle mentions differentia that entail the genus, as in ‘Walking animal’ (Topics, IV. 6) and ‘Footed animal’ (Metaphysics, Z. 12). In his Commentary on Z. 12, Bostock says that the first differentia should entail the genus (Aristotle, 1994, pp. 176–184).

Rand’s treatment of what she called “distinguishing characteristics” hews to this Aristotelian mold. And well it should, for it is of a piece with her analysis of concepts and their genus-species hierarchies in terms of measurement-omission. Far from segregating genus-species hierarchies from determinable-determinate hierarchies, such as Searle had it, Rand explains and reduces the former to the later, to a very specific form of the latter.

~~~~~~~~~~~~~~~~

~some history~

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