Hi David,

For relations in extension (RIE), the question boils down to sets whose membership is defined in the usual way.  Relations in intension (RII) are something else.  I agree with you that the tradition of linking logic to ontology here is a bad idea, i.e., it's a property if it's exemplified by one thing, otherwise it's an RII.  The matter came to a head for me many moons ago when I was thinking about the Russell Paradox.  If we allow "~(x exemplifies x)" as a substitution instance for an abstraction axiom for properties, his famous argument goes through and we're dead.  The gambit I tried is to define (contra Frege) n-adicity-in-intension based on variable occurrences rather than variables: intuitively, one variable occurrence is a property, otherwise it's an RII.  Russell's sentence expresses a funny kind of relation, not a property, so property abstraction is okay.  I also used this idea to resolve the Third Man Argument's hang-up over self-exemplification -- the point being that Plato should have anticipated Russell's non-self-exemplification "property," so the solution to the Russell Paradox should also fix Plato's TMA.  I need to post on this site references to the papers I published on this stuff years ago -- maybe over the weekend.

Cheers,

Arnold    

As of the back and forth on this thread it appears you were asking a question in relational logic, not metaphysics. For the record, the answer to which you took apparent exception was a metaphysical response that was quizzical for what proved to be the right reasons. Audi’s Dictionary of Philosophy suggests some of the relevant distinctions (under ‘relation’).

In relational logic, I believe we can define polyadicity quite simply as follows: The differential distribution of a property. Whatever receives the property is, by the semantics, forcing a polyadic facticity. Note that in relational logic it is property that is employed to define relations. But the real definition of the plural aspect appears to require the distinction herein offered up for further discussion by your readers.

Note, however, why a metaphysician tends to question such methods of definition. I stressed the word ‘receives’ to suggest that where one element is passive, another is presumed transitive or otherwise active. But is it the “relation” that is active? Or is the “relation” merely a label by which to describe the presence of what is taken to be epistemic? The fallacy of realist attribution looms large and is obviously best avoided, which is why in my first attempt I didn’t “go there”. I am, however, a ‘soft’ realist myself, and I wish not to imply a nominalist persuasion by these comments.

In utilizing the comparative weight issue you are doing the same as asking of the image (in a mirror, for example) with respect to the original. In either instance we have, as doubtless you are aware, the reflexive type, R such that for ‘a’ and ‘b’ aRb, 'a' weighs the same as 'b', 'a' appears the same as 'b', etc. Unfortunately, this does not (properly in my estimation) define either identity or plurality, and is yet another reason to be overly careful in taking relational logic too seriously.

What happens when an adjective comes to take on the character of the noun it modifies, or when a noun takes on adjectival properties of its grammatical modifier? Where are we with relational logic? We are at square one, or nearly so. Indexical v. indexicality, philosophic v. philosophical. What distinguishes each member of the pair from the other if not what I have just described? How can we make relational logic account? I believe there are possible ways, but I am not certain anyone in love with relational logic will be ready to listen anytime soon. As for methods claimed to work, I am distrustful. There is something slightly irrational in these cross-fertilizations, and most forms of logic are practically by definition ill-equipped to handle anything between here and the Andromeda nebula as regards the irrational.

From the vantage of metaphysics, what might happen were these considerations to become relevant to the discernment and definition of plurality in relations. How would the “differential distribution” notion have to be modified, if at all? For example, would we be required to assert specific properties, for example: Pervasiveness? Penetration?  See-through? Governance? The problem, of course, is that elements within cognatic dyads might not easily fall into the usual categories of relation as defined in relational logic. This is where I was coming from the first time around, sorry for the misimpressions conveyed. Anyhow, just a thought. Have fun.

CSH

BIG, BIGGER, BEST: ON ABSOLUTE VS. RELATIVE JUDGMENT

DB: "I would like a criterion of polyadicity. What jumps to mind is that polyadic properties require more than one object for their instantiation."

What I am suggesting is that if you look at objects and properties epistemically rather than ontically -- i.e., in terms of how the brain sorts its sensorimotor inputs, rather than in terms of what objects and properties generate those inputs -- it is not clear whether a criterion of polyadicity is still needed (or perhaps whether it's always needed): What looks relational when described ontically may be holistic (monadic) epistemically.

Another way to put it is that there are ontic objects out there, but the "objects" that the brain actually operates on depend on how the brain has learned to categorize and chunk its inputs. What is polyadic and relational out there, might be monadic and nonrelational for the brain.

And, most important, the brain's "objects" and properties are not immutable: they can be rechunked.

(In my ignorance of formal work on properties, relations and mereology, though, I wonder whether, for example, the distinction between isosceles and scalene triangles is based on a monadic property of triangles or a relational property of sides.)

(One correction: the kind of subitizing and "pop-out" that I was referring to was a result of overlearning, rather than an alternative to it. The slower, serial, analytic (and usually more conscious and deliberate) processing is what occurs before and during the learning and overlearning, whereas the automatized, holistic processing happens after the overlearning.)

-- SH

SUBITIZING, INNATE AND LEARNED

MM: "isn't there evidence that some subitizing procedures are innate?"

CATEGORY INTERCONFUSABILITY AND THE CONTEXT-DEPENDENCE OF CATEGORIES

DB: "...a given scene can be "seen" in different ways.. one way, you see one big object bearing a monadic property... the other way, you see smaller objects related to one another... [R]elative and non-relative judgements... can be correct at the same time. I don't think it follows from this that there is no fact of the matter about which properties are relational or not... we can distinguish between contextual and relative judgements..."

If I could put it in slightly more specific language: Yes, a given input can be categorized in different ways (and, because of categorical perception effects, this also means it can be "seen" in different ways), depending on context. 

But there is a very specific reason why the context-dependence of all categorization (with the possible exception of what is based exclusively on a stipulated formal definition) entails that, in a sense, all category judgments are relative rather than absolute: To be able to categorize correctly you have to be able to sort the category members correctly from the nonmembers based on instances encountered individually, alone, one at a time. So far this is absolute, not relative. But on what basis do you categorize individual instances absolutely as members or nonmembers of the category, reliably and correctly? On the basis of whatever distinguishing features (called "invariants") will reliably resolve any confusion between members and nonmembers.

Now a nontrivial category will have (1) an infinite number of members and nonmembers (otherwise categorization would just be extensional: the rote memorization of all the finite members and nonmembers) and (2) the category's invariants will be underdetermined, i.e., hard to find, and possibly multiple and non-unique. How does the brain find the invariants of a nontrivial category that provide a sufficient basis for sorting instances correctly? 

A general mechanism is not yet known, but it is clear that (unless the category-detector is inborn, which simply raises the same question about evolution) it has to be found on the basis of trial-and-error learning through the sampling of members and nonmembers, guided by corrective feedback (in the form of adaptive and maladaptive consequences, in the case of evolved category-detectors), in order to somehow discover and extract or abstract the invariants that are sufficient for correct future sorting. Once you have found a reliable set of invariants, you have a category-detector, which is a kind of filter, applicable to new individual instances, and capable of correctly sorting them as members or nonmembers.

The categorization is still absolute, in that it is based on individual instances, encountered alone, not on pairwise or multiple comparisons. But it is relative in that it depends on the "context of alternatives" that have been sampled to date -- as well as on the empirical interconfusability among present and future instances that needs to be successfully resolved by the category-detector in order to generate error-free categorization. 

Because categorization here consists of a sensorimotor judgment by a categorizer, there is no guarantee that the category invariants will always work. Since we are not doing ontology here (even though this is a "metaphysics" thread), all we can say is that there is whatever there is in the world, and categorizers have whatever capacity they have to categorize some instances of those things correctly (in terms of the feedback from the adapative/maladaptive consequences of categorizing correctly and incorrectly), based on what sensorimotor (or symbolic) "input" instances they have sampled and what confusion-resolving invariants their category-detectors have successfully extracted therefrom. 

It is never certain that the context of alternatives sampled to date, on the basis of which the invariants were extracted, was a sufficiently representative and exhaustive sample to have provided a reliable basis for correctly categorizing all actual future instances, let alone all possible ones: +STRIPES and -LONG-NECK might be enough to sort zebras from giraffes for a lifetime on our planet, but they would no longer be enough if there were striped giraffes or long-necked zebras. (Some of this might be familiar to philosophers in the form of "grue" predicates and "twin-earth" and "category/schmategory" examples.)

So in that sense all category judgments and the category invariants on which they are based are relative to the context-of-alternatives sampled, rather than absolute: "How's yir woif?" "Compayured to whot?" This is just as true of the category BIRD as it is of the category BIG.

Harnad, S. (1987) Category Induction and Representation, Chapter 18 of: Harnad, S. (ed.) (1987) Categorical Perception: The Groundwork of Cognition. New York: Cambridge University Press. 

Harnad, S. (2005) To Cognize is to Categorize: Cognition is Categorization, in Lefebvre, C. and Cohen, H., Eds. Handbook of Categorization. Elsevier.  

Harnad, S. (2007) From Knowing How To Knowing That: Acquiring Categories By Word of Mouth. Presented at Kaziemierz Naturalized Epistemology Workshop (KNEW), Kaziemierz, Poland, 2 September 2007. 

Harnad, S; Blondin-Massé; A, St-Louis, B; Chicoisne, G; Gargouri, Y;&Picard, O.  (2008) Symbol Grounding, Turing Testing and Robot Talking.  RoadMap Workshop on Action and Language Integration, Rome on 22-24 September 2008.  

-- SH