\documentclass[11pt]{article}

% userpackage
\usepackage{geometry,amssymb,color,graphicx}
%,geometry,enumerate,longtable,epstopdf,url,graphicx,epsfig,color}
\usepackage[all]{xy}

\geometry{paperwidth=128mm,paperheight=96mm,textwidth=110mm, textheight=75mm}

%\setlength \paperwidth{128mm}
%\setlength \paperheight{96mm}
%\setlength \textwidth{110mm}
%\setlength \textheight{75mm}



% newcommand for \tab
\newcommand{\tab}{\hspace{5mm}}

% newcommand for list without indentantion
\newcommand{\bslist}{\begin{list}{$\vartriangleright$}{\setlength{\labelwidth}{5mm}\setlength{\leftmargin}{5mm}\setlength{\topsep}{1mm}\setlength{\parsep}{0mm}\setlength{\itemsep}{2mm}}}
\newcommand{\eslist}{\end{list}}

% newcommand for list ox formula environnement
\newcommand{\bflist}{\begin{list}{}{\setlength{\topsep}{2mm}\setlength{\parsep}{0mm}}}
\newcommand{\eflist}{\end{list}}

% newcommand for labels of axioms, theorems and definitions 
\newcommand{\AxLabel}{A}
\newcommand{\DefLabel}{D}
\newcommand{\ThLabel}{T}
\newcommand{\QueLabel}{?}

% counter and newcommand for numbering formulas
\newcounter{cntax}
\newcommand{\myaxiom}[1]{\refstepcounter{cntax}(\AxLabel\thecntax)\label{#1}}
\newcounter{cntdef}
\newcommand{\mydefinition}[1]{\refstepcounter{cntdef}(\DefLabel\thecntdef)\label{#1}}
\newcounter{cntth}
\newcommand{\mytheorem}[1]{\refstepcounter{cntth}(\ThLabel\thecntth)\label{#1}}
\newcounter{cntque}
\newcommand{\myquestion}[1]{\refstepcounter{cntque}(\QueLabel\thecntque)\label{#1}}

% newcommand for kinds
\newcommand {\stateofaffairs} {\ensuremath{State}}
\newcommand {\fact} {\ensuremath{Fact}}
\newcommand {\goal} {\ensuremath{Goal}}
\newcommand {\softgoal} {\ensuremath{SoftGoal}}
\newcommand {\hardgoal} {\ensuremath{HardGoal}}
\newcommand {\achievable} {\ensuremath{Achievable}}
\newcommand {\actor} {\ensuremath{Actor}}
\newcommand {\member} {\ensuremath{Member}}
\newcommand {\group} {\ensuremath{Group}}
\newcommand {\SelfSufficient} {\ensuremath{{\it SelfSufficient}}}
\newcommand {\goaldepender} {\ensuremath{Depender^G}}
\newcommand {\depressed} {\ensuremath{Depressed^G}}

% newcommands for relations/predicates
\newcommand {\Part} {\ensuremath{\mathsf{part}}}
\newcommand {\ppart} {\ensuremath{\mathsf{ppart}}}
\newcommand {\atom} {\ensuremath{\mathsf{atom}}}
\newcommand {\atpart} {\ensuremath{\mathsf{atpart}}}
\newcommand {\obtains} {\ensuremath{\mathsf{obtains}}}
\newcommand {\wantsachieve} {\ensuremath{\mathsf{wants}}}
\newcommand {\itswantsachieve} {\ensuremath{\mathsf{wants_{its}}}}
\newcommand {\othwantsachieve} {\ensuremath{\mathsf{wants_{oth}}}}
\newcommand {\prefersachieve} {\ensuremath{\mathsf{prefers}}}
\newcommand {\intendsachieve} {\ensuremath{\mathsf{intends}}}
\newcommand {\ableachieve} {\ensuremath{\mathsf{able}}}
\newcommand {\socableachieve} {\ensuremath{\mathsf{able_{soc}}}}
\newcommand {\commitsachieve} {\ensuremath{\mathsf{commits}}}
\newcommand {\offersachieve} {\ensuremath{\mathsf{offers}}}

\newcommand {\soaconj} {\ensuremath{\sqcup}}

\newcommand {\depends} {\ensuremath{\mathsf{depends}}}
\newcommand {\strongdepends} {\ensuremath{\mathsf{depends}_{\mathsf{strong}}}}
\newcommand {\socdepends} {\ensuremath{\mathsf{depends}_{\mathsf{soc}}}}
\newcommand {\multidepends} {\ensuremath{\mathsf{depends}_{\mathsf{multi}}}}
\newcommand {\reciprocaldepends} {\ensuremath{\mathsf{depends}_{\mathsf{rec}}}}
\newcommand {\mutualdepends} {\ensuremath{\mathsf{depends}_{\mathsf{mut}}}}
\newcommand {\goaldepresses} {\ensuremath{\mathsf{depresses}}}
\newcommand {\selfsufficient} {\ensuremath{\mathsf{self.sufficient}}}
\newcommand {\selfinsufficient} {\ensuremath{\mathsf{self.insufficient}}}

\newcommand {\delegates} {\ensuremath{\mathsf{delegates}}}
\newcommand {\unilateraldelegates} {\ensuremath{\mathsf{delegates}_{\mathsf{uni}}}}
\newcommand {\acceptancedelegates} {\ensuremath{\mathsf{delegates}_{\mathsf{acc}}}}

\newcommand {\adopts} {\ensuremath{\mathsf{adopts}}}
\newcommand {\unilateraladopts} {\ensuremath{\mathsf{adopts}_{\mathsf{uni}}}}
\newcommand {\acceptanceadopts} {\ensuremath{\mathsf{adopts}_{\mathsf{acc}}}}

\newcommand {\contracts} {\ensuremath{\mathsf{contracts}}}


%\newcommand {\Part} {\ensuremath{\mathsf{P}}}
%\newcommand {\Overlap} {\ensuremath{\mathsf{O}}}
%%\newcommand {\ssum} {\ensuremath{\mathsf{SSum}}}
%\newcommand {\Sum} {\ensuremath{\mathsf{Sum}}}
%\newcommand {\PPart} {\ensuremath{\mathsf{PP}}}

%\newcommand {\inheres} {\ensuremath{\mathsf{i}}}
%\newcommand {\abstracts} {\ensuremath{\mathsf{a}}}
%\newcommand {\position} {\ensuremath{\mathsf{p}}}
%\newcommand {\qlspace} {\ensuremath{\mathsf{space}}}

%\newcommand {\QualityKind} {\ensuremath{\mathcal{QT}}}
%\newcommand {\QualeKind} {\ensuremath{\mathcal{QL}}}
%\newcommand {\SpaceKind} {\ensuremath{\mathcal{S}}}

%
%\newcommand {\dolce} {\sc dolce}


\newcommand {\titline} {
\vspace{-0.5cm}
---------------------------------------------------------------------------------------
\vspace{-0.5cm}}



%%-------------- BEGIN TITLE ----------------
%% title page
%\title{Qualities in Formal Ontology}
%\author{Claudio Masolo and Stefano Borgo \\  \small{Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy} \\ \\ \\ Koblenz, 11.09.2005}
%\date{}
\title{}
\author{}
\date{}

%------------- BEGIN DOCUMENT ----------
\begin{document}

%-------------- BEGIN TITLE ----------------
% title page
%\maketitle
\mbox{}
\vspace{0.2cm}

%-----------------------------------------------------------------
\section*{In Defense of a Trope-Based Ontology for Conceptual Modeling: An example with the foundations of Attributes, Weak Entities and Datatypes}
%-----------------------------------------------------------------

\vspace{0.3cm}
{\it Giancarlo Guizzardi}$^{1,2}$, {\it Claudio Masolo}$^{1}$, and {\it Stefano Borgo}$^{1}$ \\
\begin{tiny}
$^{1}$Laboratory for Applied Ontology, ISTC-CNR, Trento, Italy\\
$^{2}$Computer Science Departement, Federal University of Espirito Santo, Vitoria-ES, Brazil\\ \\
\end{tiny}
ER 2006 -- Tucson, 07.11.2006

\newpage

%%--------------Abstract--------------
%\begin{abstract}
%\noindent Write Abstract.
%\end{abstract}


%-----------------------------------------------------------------
\section*{Goal}
%-----------------------------------------------------------------
\titline

\bslist
\item
Explore different philosophical positions on {\it properties} and {\it predication}. 

%\item 
%Take into account:
%\bslist
%\item[$\bullet$]
%{\sl objective} vs. {\sl contextual} properties;

%\item[$\bullet$]
%{\sl change} of properties in time;
%\eslist

\item 
Compare the {\it Bunge-Wand-Weber} (BWW) position and a {\it trope-based} one with respect to:

\bslist
\item[$\bullet$]
{\sl ontological commitment} (kinds of entities and relations);
\item[$\bullet$]
{\sl expressivity} (ontological interpretation of CM constructs);
\item[$\bullet$]
{\sl adequacy} (for some modeling tasks).
\eslist

\eslist
\newpage


%-----------------------------------------------------------------
\section*{Main philosophical positions on properties}
%-----------------------------------------------------------------
\titline\\
{\bf Example}. The particulars $a$ and $b$ have the property ``being red".\\


%\item[\footnotesize{1.}]
%Universalism.

%\item[\footnotesize{2.}]
%Trope theory.

%\item[\footnotesize{3.}]
%Endurants+Tropes+Universals (inherence and instantiation).\\
%(In this case, the resemblances can be defined by means of the instantiation). \\

\mbox{} \hspace{-1cm} 
\begin{tabular}{ccc} 
\footnotesize{{\bf Universalism}} & \footnotesize{{\bf Trope theory}} & \footnotesize{{\bf Universals+Tropes}}\\ 
\xymatrix@C20pt{
a \ar[r]^(.4){I} & Red \\
b \ar[ru]_{I}} &

\xymatrix@C20pt{
a & a_{red} \ar[l]_{i} \ar@{<->}[d]^{\thickapprox}  \ar[r]^(.4){\in} & |red|_{_\thickapprox} \\
b & b_{red} \ar[l]^{i} \ar[ru]_{\in}} &

\xymatrix@C20pt{
a & a_{red} \ar[l]_{i} \ar[r]^{I} & Red \\
b & b_{red} \ar[l]^{i}  \ar[ru]_{I}}\\

%&
%\xymatrix@C20pt{
%a \ar[r]^{\in} & Red \\
%b \ar[ru]_{\in}} 
\end{tabular}

\bslist
\item[]
\item
{\bf Natural classes}. Properties are classes of particulars, {\it natural} classes correspond to ``universals" 

\item
{\bf Resemblance Nominalism}. Properties are classes of resembling particulars (resembling couples, etc.) 
\eslist

\newpage

%%-----------------------------------------------------------------
%\section*{(??) Particularity vs. Bundles}
%%-----------------------------------------------------------------
%\titline\\
%Is {\it particularity} a fundamental ontological category, or are particulars just {\it bundles of properties}, i.e. particulars are ``constructed" from properties (universals or tropes)?

%\bslist
%\item 
%{\sl Ontologically}, this is a very important question because it address directly the problem of what are the basic entities in the world.

%\item
%{\sl Representationally}, the expressive power of bundles theories is equivalent to the expressive power of theories based on particulars and the two alternatives have some advantages and drawbacks (in particular taking into account change and extrinsic/intrinsic properties).
%\eslist

%\noindent (??) in linea con BWW (che si focalizza alle thing), e per essere piu' neutri, non consideriamo necessariamente che i particolari siano riducibilli a proprieta', quindi considerimano sempre due tipi di entita': particolari e proprieta'
%%{\bf Assumption}. In order to simplify the comparison, here we consider particularity as a fundamental category.

%
%%\bslist
%%\item[$\bullet$]
%%{\bf Bundles of tropes}. Beside an {\it exact resemblance} relation, a principle (or some principles) of {\it bundling} is necessary. In this case, particulars can be conceived (as universals) as equivalent classes of tropes (with respect to two different relations).

%%\item[$\bullet$]
%%{\bf Bundles of universals}. Particulars are conceived as ``classes" of universals (also in this case one or more equivalence relations are necessary). If extensionality is accepted, no two different particulars can be ``constituted" by the same universals.

%%\eslist
%\newpage


%%-----------------------------------------------------------------
%\section*{(?) Universals vs. Classes}
%%-----------------------------------------------------------------
%\titline
%\bslist
%\item
%Universals differ from classes because:

%\bslist
%\item[$a.$] 
%universals are {\it wholly} present in their instances (immanentism); they are constituent parts of the instances {\bf while} classes are {\it partially} present in their instances; the instances are the constituents of classes;

%\item[$b.$]
%universals are {\it sparse} and {\it minimal} to capture all the distinction in the world {\bf while} classes are {\it redundant} and {\it abundant}.\\
%\mbox{}[nota che nel caso della teoria dei tropi, le classi che considero sono soltanto classi di eq. di tropi, quindi sono anch'esse sparse ecc.]
%%{\it note}: the disjunction/negation is sometime refused as property;
%\eslist

%\item (??) forse qui e' piu' interessante dire qualche cosa sul fatto che nella teoria dei tropi senza universali, in qualcha maniera si riducono le proprieta' ad una astrazione rispetto ai tropi, quindi in questo senso le classi non sono di piu' degli universali, e sono soltanto un modo di atrarre
%\eslist
%\newpage


%-----------------------------------------------------------------
\section*{Things vs. Particulars}
%-----------------------------------------------------------------
\titline

\bslist
\item BWW (and universalism).\\({\sc Post 1}) The world is made of {\it things} that possess {\it properties}.
 
\item Tropes inhere in (and existentially depend on) {\it things} and possess {\it properties}.

\bslist
\item[--] {\sc Note}. The difference between tropes as {\it members} of classes of resembling entities vs. tropes as {\it instances} of universals is not relevant for the following arguments.

%COMMENTO:
%la distinzione tra trope-theory e universals+tropes dal punto del conceptual modeling non e' fondamentalissima in quando comunque non ci sono problemi di estensionalita' ne' di densita'/sparsita' con le classi di tropi, l'unico problema e' proprio sulla diversa natura ontologica di classi e tropi\\
\eslist

\item Therefore, {\it tropes} are existentially dependent particulars.\\ In this sense they are conceptually similar to {\it weak-entities} and different from {\it things}.

\item Can tropes inhere in tropes? 

\eslist

\newpage

%-----------------------------------------------------------------
\section*{More specific goals}
%-----------------------------------------------------------------
\titline\\
Show that a trope-based theory

\bslist
\item[\small{1.}] can be used to provide an alternative (w.r.t. BWW) ontological interpretation  of some CM fundamental constructs/notions;

\item[\small{2.}]  leads to a more explicit ontological characterization of some of these CM constructs/notions;

\item[\small{3.}] allows for representing additional situations in an ontologically well founded way, e.g. change in time, properties of properties, measurement, etc. 

%COMMENTO
%(tropi+classi: proprieta' come astrazione, tropi+universali: cambiamento nel tempo come sostituzioni di parti)\\
%(CM....)

\eslist


\newpage


%-----------------------------------------------------------------
\section*{Attribute Functions \hfill{\small{(1/5)}}}
%-----------------------------------------------------------------
\titline\\
 Alternative representations of attribute functions in UML:

\bslist
\item[]
% FIGURE
\vspace{-0.2cm}
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.65]{applecolor.pdf}
\end{center}
\label{FigAppleColor}
\end{figure}
\vspace{-0.3cm}
\item In UML, a {\it datatype} is a class whose instances are {\it values} not {\it objects}. A value does not have an identity: two occurrences of the same value cannot be differentiated:

\item[] {\sf color}: {\sf Apple} $\rightarrow$ {\it Color} 

\item In BWW, {\sf Apple} is a set of things, {\it Color} is a set of values, and {\sf color} is a property (an attribute). A set $M$ representing the ``observation conditions" (times, contexts, etc.) is added.

\item[] {\sf color}: {\sf Apple} $\times$ $M$  $\rightarrow$ {\it Color}
\eslist

\newpage

%-----------------------------------------------------------------
\section*{Attribute Functions  \hfill{\small{(2/5)}}}
%-----------------------------------------------------------------
\titline
\bslist
\item Intuitively, ``being coloured" is different from  ``being red" or from properties individuating a specific color shade.

\item Each value in {\it Color} individuates a specific property, e.g. ``being scarlet", ``being crimson", etc.

\item  {\it Color} (and {\sf color}) individuates the set of {\it specific properties} (by means of values) that specialize a ``common aspect", a general property, of things, ``being coloured" in this case.

\item In trope theory, specific properties are classes of {\it exactly} resembling tropes.

\item {\bf (?)} How can these notions (specific vs. general properties) be characterized in a trope-based theory?

%inhering in {\sf Person} is a set of objects in which different types of tropes can inhere. In the example we are interesting in the tropes that regards the ``heigh"
% considering {\sf weight} as a property, it is a set of tropes inhering in apples, i.e.:
\eslist
\newpage

%-----------------------------------------------------------------
\section*{Qualia}
%-----------------------------------------------------------------
\titline
\mbox{}\\
\xymatrix@R12pt@C15pt{
& & Colored \\
& Red \ar[ru]^{dD} & & Blue \ar[lu]_{dD}\\
*+[F]{Scarlet} \ar[ru]^{dD} & *+[F]{Crimson} \ar[u]_{dD} & & \dots \ar[u]^{dD} & \dots \ar[lu]_{dD}\\
%\dots \ar[u]^{dD} & \dots \ar[u]_{dD}
}\\
\vspace{0.2cm}
\bslist
\item
$dD(p_1,p_2)$: property $p_1$ is a determinate of $p_2$ (the determinable): 
\bslist
\item[$\bullet$]
having a {\sl determinate} property entails having a {\sl determinable} property;
\item[$\bullet$]
having a {\sl determinable} property entails having (at least) one of the properties that are its {\sl determinates}.
\eslist

\item 
 {\bf Qualia}: determinates that are not determinables, i.e. the more specific properties (that, intuitively, correspond to values).
\eslist

\newpage

%-----------------------------------------------------------------
\section*{Predication of determinables}
%-----------------------------------------------------------------
\titline
\begin{tabular}{cc} 
\mbox{} \\
\footnotesize{{\bf Universalism}} & \footnotesize{{\bf Trope theory}}\\
%\xymatrix{
%a \ar[r]^{I} & F \\
%b \ar[ru]_{I}} &
\xymatrix{
a \ar[r]^{I} & Scarlet \ar[r]^{dD} & Red \\
b \ar[ru]_{I}} &

\xymatrix@C35pt@R15pt{
a & a_{sc} \ar[l]_{i} \ar@{<->}[d]^{\thickapprox}  \ar[r]^(.5){\in} \ar[rd]_(.6){\in} & |Scarlet|_{_\thickapprox} \ar[d]^{dD}\\
b & b_{sc} \ar[l]^{i} \ar[ru]^(.6){\in} \ar@{<->}[d]^{\thickapprox_d} \ar[r]_(.4){\in}  & |Red|_{_{\thickapprox_d}}\\
c & c_{cr} \ar[l]^{i} \ar[ru]_{\in}
}
\end{tabular}
\\ \\
\bslist
%\item
%Qualia, but not all the determinables of qualia, are universals: ``non-objective" properties (or concepts) need to be added.

%\item
%$a$ has the property $F$ iff there exists a {\it quale} $Q$, such that $I(a,Q) \wedge {\sf dD^*}(Q,F)$ (where ${\sf dD^*}$ is the transitive closure of ${\sf dD}$)
\item Universalism. $dD$ is $(a)$ primitive; $(b)$ based on resemblance with degrees {\em between universals}; $(c)$ based on partial identity.
\item
Trope theory. $dD$ is based on the inexact resemblance with degree $d$ {\em between tropes} ($\thickapprox_d$): classes of exactly (inexactly) resembling tropes are qualia (determinables, resp.).
\eslist
\newpage

%-----------------------------------------------------------------
\section*{Qualia kinds}
%-----------------------------------------------------------------
\titline
\mbox{}\\
\xymatrix@R12pt@C15pt{
& & *+[F]{Colored} \\
& Red \ar[ru]^{dD} & & Blue \ar[lu]_{dD}\\
Scarlet \ar[ru]^{dD} & Crimson \ar[u]_{dD} & & \dots \ar[u]^{dD} & \dots \ar[lu]_{dD}\\
\mbox{} \\
}\\ 
\bslist


\item
{\it Incompatibility of qualia}. One (atomic) entity can have only a coloured-qualia (not the case of $Coloured\_OR\_Shaped$).

\item
{\it Comparability of qualia}. Coloured-qualia are at least qualitatively comparable (they are related). No coloured-quale resembles more closely a shaped-quale than a volume-quale. 

%\item 
%{\it Additivity}. ``Magnitudes" of volume-qualia can be added to magnitudes of volume-qualia but not to ones of mass-qualia.
\eslist
\medskip
\noindent
{\bf Qualia kinds} are maximal wrt incompatibility and comparability.

\newpage


%%-----------------------------------------------------------------
%\section*{Qualia types (2)}
%%-----------------------------------------------------------------
%\titline
%\bslist
%\item
%{\sl Quality types} are determinable properties maximal w.r.t {\it resemblence}/{\it addiction} and with incompatible determinates.

%\item
%Quality types have different structures, e.g. {\it volume} is linear, {\it color} is a spindle, {\it shape} is ??, and the determinates are ``points".

%\item[]
%\mbox{}
%\xymatrix@R=6pt@C=35pt{
%a \ar[r]^(.5){I/\in} & Q \ar[r]^{\sf dD^*} & F \ar[r]^{\sf dD^*} & QT  \\
%b \ar[ru]_(.5){I/\in}} 

%\item
%Some universalists consider {\it quality types} as universals while their determinates (that are no qualia) just as {\it concepts}/classes of determinates.

%\item
%{(\bf ??)} {\it Qualities} ($\neq$ individual qualities) are determinates of a quality type (e.g., for {\sl color}: red, yellow, scarlet, dark-scarlet).

%%VEDI SE PUO' SERVIRE
%%\item
%%{(\bf ??)} --forse non serve--discorso della dipendenza tra le qualita', dipendenza tra i determinati di un certo tipo

%\eslist
%\newpage


%-----------------------------------------------------------------
\section*{Attribute Functions  \hfill{\small{(3/5)}}}
%-----------------------------------------------------------------
\titline\\
Let us make use of the distinctions just introduced to interpret the attribute functions.\\

\noindent We introduce:
\bslist
\item[\small{1.}] the set of things {\sf Apple};

\item[\small{2.}] the qualia kind (a set of inexactly resembling tropes) {\sf Colored};   

\item[\small{3.}] the (second order) axiom

\item[] ${\sf Apple}(x) \rightarrow \exists t,Q(i(t,x) \wedge Q(t) \wedge dD(Q,{\sf Colored}) \wedge \neg \exists Q'(dD(Q',Q)))$

\item[] {\sc Note}. In BWW given a $m \in M$, the value of an attribute needs to be defined:

\item[]$(x \in {\sf Apple} \wedge m \in M) \rightarrow \exists v \in Color ({\sf color}(x,m) = v)$ 

\eslist

\newpage

%-----------------------------------------------------------------
\section*{Attribute Functions  \hfill{\small{(4/5)}}}
%-----------------------------------------------------------------
\titline\\
To avoid second order quantification, we reify qualia kinds and their determinates:
\bslist
\item {\sf colored} is the reification of the attribute (qualia kind) {\sf Colored};

\item $q$ is the reification of property $Q$ that is a determinate of a qualia kind, in particular qualia are identified by:

\item[] ${\sf Qualia}(q)$\, {\bf iff}\, $\neg \exists q'(dD(q',q))$

%\item properties linked to attributes (quality kinds, qualia, and other ``intermediate" properties) in the domain of quantification;

\item a {\it classification} relation ($::$) between tropes and properties is introduced (a generalization of membership and instantiation).

%\item the $dD$ relation, a {\it subsumption} relation between properties;

\item The previous axiom can be rewritten as:

\item[] ${\sf Apple}(x) \rightarrow \exists t,q(i(t,x) \wedge {\sf Qualia}(q) \wedge t::q \wedge dD(q,{\sf colored}))$
\eslist
\newpage

%-----------------------------------------------------------------
\section*{Attribute Functions  \hfill{\small{(5/5)}}}
%-----------------------------------------------------------------
\titline
\bslist
\item A function {\sf color} from {\it things} to {\it qualia} can be defined as:

\item[] ${\sf color}(x) =  q$\,  {\bf iff}\, $\exists t(i(t,x) \wedge t::q \wedge dD(q,{\sf colored}))$

\item[] assuming the incompatibility of qualia of the same quality kind:
\bslist
\item[$\bullet$] $(i(t,x) \wedge t::{\sf colored}) \rightarrow \neg \exists t'(t \neq t' \wedge i(t',x) \wedge t'::{\sf colored})$
\item[$\bullet$] $t::{\sf colored} \rightarrow \exists q({\sf Qualia}(q) \wedge t::q \wedge dD(q,{\sf colored}))$ 
\eslist
\eslist

\vspace{0.5cm}
\noindent Two basic differences with respect to the {\sf color} function in BWW:
\bslist
\item the additional argument ($M$) is missing;
\item {\sf color} yields now ``qualia" instead of ``values"\\ (we will go back to this point).
\eslist

%\item to each reified qualia kind (e.g. {\sf colored}) a predicate (e.g. {\sf Colored}) on objects can be defined\\
%(assuming that $(t::q \wedge dD(q,q')) \rightarrow t::q')$ ):\mbox{}
%\vspace{3pt}

%${\sf Colored}(t) \triangleq t::{\sf colored}$

%\item for each quality kind a predicate on objects can be defined:
%\mbox{}
%\vspace{2pt}

%${\sf High}(o) \triangleq \exists t(i(t,o) \wedge {\sf Height}(t))$

%\item function {\sf height} from objects to qualia can be defined as:
%\mbox{}
%\vspace{2pt}

%${\sf height}(o) =  q$ iff $\exists t(i(t,o) \wedge t::q \wedge dD(q,{\sf qheight})  \wedge \neg \exists q'(dD(q',q))) $
%\eslist
\newpage


%-----------------------------------------------------------------
\section*{Time and change in time}
%-----------------------------------------------------------------
\titline
\bslist
\item One of the reason of the argument $M$ in BWW is the encoding of the change of properties of things through time.

\item Like other particulars, tropes can have a temporal extension. 

\item Let us suppose that the function {\sf time} yields the temporal extensions of particulars, then, we can introduce a temporal argument in the previous {\sf color} function:
\mbox{}
\vspace{5pt}

${\sf color}(x,m) =  q$\, {\bf iff}\, $\exists t(i(t,x)\, \wedge\, {\sf time}(t)=m\, \wedge\, t::q\, \wedge\, dD(q,{\sf colored}))$

\bslist
\item[$\Rightarrow$] change in time as {\it substitution} of tropes;

\item[$\Rightarrow$] explicit recording of the ``color history" of an object.
\eslist

\item[--] {\sc Note}. The same can be done for relationships.
\eslist

\newpage

%-----------------------------------------------------------------
\section*{Attributes of Attributes  \hfill{\small{(1/2)}}}
%-----------------------------------------------------------------
\titline

\bslist
\item In the previous example, the function {\sf time} can be seen as an attribute of tropes that yields temporal qualia.

\item Consequently, we admit tropes that inhere in tropes.

\item Very useful in the case of complex tropes like {\it symptoms}, e.g. John's headache and influenza are tropes inhering in John and they are different from the ones inhering in another patients.

\item Different {\it symptoms} can:

\bslist
\item[$\bullet$] occur at different times;
\item[$\bullet$] have specific temporal/causation relations;
\eslist
\eslist
\newpage 

%-----------------------------------------------------------------
\section*{Attributes of Attributes  \hfill{\small{(2/2)}}}
%-----------------------------------------------------------------
\titline\\
Another interesting representational problem regards roles, e.g.:
\bslist
\item[$\bullet$]
if the instances of {\sf Customer} are persons (or organizations) and {\sf code} is an attribute of {\sf Customer}, therefore to each person it is possible to associate only one customer code.

\item[] {\bf But}, at the same time, the same person can be customer of different stores, therefore he can have a multitude of different codes, one for each store.

\item A possible solution consists in introducing {\sf code} as an attribute of a class of (relational) tropes that  inhere in persons and stores.
\eslist

%\item The idea is to introduce between relations between tropes, or properties of tropes, i.e. tropes of tropes. 

%\item Do we consider properties that have properties as instances?

%\item What about relations between properties?\\
%e.g. ISA [in trope theory, it is possible to use inexact resemblance, I don't know if universalism accepts to reduce ISA to ``extensional inclusion of instances"--all the instances of u1 are instances of u2 too, or ``intensional inclusion of attributes" (see in BWW)] or laws--here the problem is much more difficult

%\item How the philosophical approaches solve this problem? 

%\item In principle, if properties are defined just by ``having instances", then nothing forbids that their instances are properties, i.e. the following (1) and (2) are not inconsistent: (1) there are instances of  properties [or, properties (can) have instances]; (2) properties can have properties

%\item (2) puo' essere giustificato da un p.to di vista di linguaggio di rappresentazione, ma non troppo a livello filosofico forse [vedi anche Armstrong]


\newpage



%-----------------------------------------------------------------
\section*{Qualia vs. Values}
%-----------------------------------------------------------------
\titline
\bslist
\item What is the ontological nature of values in BWW?

\bslist
\item[\small{1.}] Can the same value  be used for different attributes? For example, can ``1m" be used for {\it height} and {\it length}? 

\item[\small{2.}] Do ``1m" and ``100cm" refer to two different values?
\eslist

\item Qualia are specific properties, therefore  ``being 1m high" and ``being 1m long"  are just two different properties.

\item The same qualia can be ``measured" in different ways: ``being 1m high" and ``being 100cm high" refer to the same property but to different {\it measurement systems}.

\item ``m" and ``cm" can refer to different granularities or measurement's precisions.
\eslist
\newpage




%-----------------------------------------------------------------
\section*{Spaces of properties \hfill{\small{(1/3)}}}
%-----------------------------------------------------------------
\titline
%\mbox{}\\
%\xymatrix@R12pt@C15pt{
%& & *+[F]{Coloured} \\
%& Red \ar[ru]^{dD} & & Blue \ar[lu]_{dD}\\
%*+[F]{Scarlet} \ar[ru]^{dD} & *+[F]{Crimson} \ar[u]_{dD} & & \dots \ar[u]^{dD} & \dots \ar[lu]_{dD}\\
%}\\ 

%{\bf Idea}: representing {\it relative similarities} between entities, i.e. similarities dependent on cultures, available informations, measurement instruments and methods, etc.
% 
\bslist
\item
Objects sharing a {\it quale} are {\it exactly similar} (w.r.t. some given aspect).

\item 
In general, objects sharing a {\it determinable} are {\it inexactly similar}, i.e. they resemble each other with a {\it degree}.

%\item 
%Entities sharing a {\it determinable} of a qualia kind are  {\it inexactly similar} but {\it comparable}.
\eslist

\noindent
{\bf But} {\em in applications}, we find a variety of degrees of resemblance 
\bslist
\item
they are {\it empirically} determined by the chosen experiments and depend on species, culture, available information, measurement instruments and methods, etc.
\item
they furnish (roughly speaking) spaces of properties with quite different structures. 
\eslist 
\newpage



%%%-----------------------------------------------------------------
%%\section*{Similarities: the epistemological and empirical levels}
%%%-----------------------------------------------------------------
%%\titline
%%\bslist
%%\item
%%In the {\sl philosophical} approaches considered, two objects are {\it similar}, in some way, iff they share a universal ({\sl universalism}) or they have resemblant tropes ({\sl trope theory}). In this case, {\bf exact similarity is objective}, {\sl mind} and {\sl language} independent and it provides the {\sl finest} possible level of analysis.

%%\item
%%In {\sl cognitive science}, and in {\sl science} in general, (judgments of) {\it similarity} is a central notion too, but it is {\it empirically} built on experiments and {\bf similarity is relative} (actually, many similarities are considered): it may depend on species, on cultures, on the available information, on the measurement instruments/methods, etc.

%%\eslist
%%\newpage

%-----------------------------------------------------------------
\section*{Spaces of properties \hfill{\small{(2/3)}}}
%-----------------------------------------------------------------
\titline
\bslist
\item
Resemblance with {\it degree} simply introduces a partial order among properties.
\bslist
\item[$\bullet$] 
Spaces have more structure: they add further relations like those determining a topological or geometrical space.
\eslist
\item
Each qualia kind is associated to (can be structured in) one or more spaces which depend on culture, instruments of investigation, etc.

\item
Spaces exist {\sl in time}: they are created, adopted, and destroyed by (communities of) intentional agents.
\eslist
\newpage


%-----------------------------------------------------------------
\section*{Spaces of properties \hfill{\small{(3/3)}}}
%-----------------------------------------------------------------
\titline\\
Taking exact similarity and qualia to be {\it objective}, they are {\it contextually} organized in spaces.

\bslist
%\item
%Spaces are associated only to quality types (a link to $QT_i$ is used, but classes of qualia/spaces are alternatives) and their structures correspond to different judgments of similarity. 
\item 
Qualia are linked to possibly different {\it properties} in spaces.

\item[]
\mbox{} 
\xymatrix@R=-2pt@C=25pt{
a \ar[r]^(.3) {::}  & Scarlet  \ar[r]^(.5) {dD} \ar[rdd]_(.5){dD}  & Dark\_Red \ar[r]^(.5){dD} & S^1_1 \ar[rd]\\
& & \vdots & \vdots & Colored\\
& & Red \ar[r]^(.5){dD} & S^1_n \ar[ru]}
%\\
% & Round  \ar[r]^(.5) {dD} \ar[rdd]_(.5){dD}  & R^2_1 \ar[r]^(.5){dD} & S^2_1 \ar[rd]\\
%& & \vdots & \vdots & Shaped\\
%& & R^2_n \ar[r]^(.5){dD} & S^2_n \ar[ru]

\item Structuring relations can be added into specific spaces, e.g. 
${\sf Connected}(Brown,Red)$.

\item Different granularities can be assumed in different spaces, e.g. $Dark\_Red$ is not considered in space $S^1_n$.

\item Different measurement systems can be introduced in one space.

\eslist

\newpage



%-----------------------------------------------------------------
\section*{Multidimensional spaces \hfill{\small{(1/2)}}}
%-----------------------------------------------------------------
\titline
\bslist
\item Simple spaces can be composed in more complex spaces by means of {\it existential dependences} among tropes and qualia, e.g. the color space (trope) can be seen as composed by three spaces (tropes): hue, saturation, and brightness.

\item Constraints ({\it laws}) on qualia in the same simple space or multidimensional spaces, e.g. the linearity of weights, or the splinter shape of the color space, can be introduces as constraints on relations between qualia.
\eslist
\newpage

%-----------------------------------------------------------------
\section*{Multidimensional spaces \hfill{\small{(2/2)}}}
%-----------------------------------------------------------------
\titline\\
Alternative spaces can be considered also for complex attributes like color:
\bslist
\item We can map the same color-quale $q = {\sf color}(x)$ to different {\it regions} (in different spaces).

\item Each region of space can be the result of the composition of other regions belonging to simpler spaces, for example the hue, saturation, and brightness spaces.

\item The qualia kind is associated just to one {\it space kind}, i.e. all the color qualia are mapped to regions in color-spaces. 
\eslist
\newpage

%%%-----------------------------------------------------------------
%%\section*{Trope theory and qualia}
%%%-----------------------------------------------------------------
%%\titline
%%\bslist

%%\item
%%The classes of {\it exactly} resembling tropes correspond to qualia.

%%\item
%%Introducing a resemblance {\it with degree} between tropes it is possible to define classes of {\it inexactly} resembling tropes and a resemblance relation between these classes.

%%\item
%%The classes of inexactly resembling tropes corresponds to determinables properties/concepts.

%%\item[]
%%\mbox{}
%%\xymatrix@C35pt@R15pt{
%%a & F_a \ar[l]_{i} \ar@{<->}[d]^{\thickapprox}  \ar[r]^(.5){\in} \ar[rd]_(.6){\in} & |F|_{_\thickapprox} \ar[d]^{\sf dD^*}\\
%%b & F_b \ar[l]^{i} \ar[ru]^(.6){\in} \ar@{<->}[d]^{\thickapprox_d} \ar[r]_(.4){\in}  & |F|_{_{\thickapprox_d}}\\
%%c & G_c \ar[l]^{i} \ar[ru]_{\in}
%%}
%%\eslist

%%
%%\newpage

%
%%-----------------------------------------------------------------
%\section*{Qualia kinds}
%%-----------------------------------------------------------------
%\titline
%\mbox{}\\
%\xymatrix@R12pt@C15pt{
%& & *+[F]{Coloured} \\
%& Red \ar[ru]^{dD} & & Blue \ar[lu]_{dD}\\
%Scarlet \ar[ru]^{dD} & Crimson \ar[u]_{dD} & & \dots \ar[u]^{dD} & \dots \ar[lu]_{dD}\\
%\mbox{} \\
%}\\ 
%\bslist

%
%\item
%{\it Incompatibility of qualia}. One (atomic) entity can have only a coloured-qualia (not the case of $Coloured\_OR\_Shaped$).

%\item
%{\it Comparability of qualia}. Coloured-qualia are at least qualitatively comparable (they are related). No coloured-quale resembles more closely a shaped-quale than a volume-quale. 

%%\item 
%%{\it Additivity}. ``Magnitudes" of volume-qualia can be added to magnitudes of volume-qualia but not to ones of mass-qualia.
%\eslist
%\medskip
%\noindent
%{\bf Qualia kinds} are maximal wrt incompatibility and comparability.

%\newpage

%
%%%-----------------------------------------------------------------
%%\section*{Qualia types (2)}
%%%-----------------------------------------------------------------
%%\titline
%%\bslist
%%\item
%%{\sl Quality types} are determinable properties maximal w.r.t {\it resemblence}/{\it addiction} and with incompatible determinates.

%%\item
%%Quality types have different structures, e.g. {\it volume} is linear, {\it color} is a spindle, {\it shape} is ??, and the determinates are ``points".

%%\item[]
%%\mbox{}
%%\xymatrix@R=6pt@C=35pt{
%%a \ar[r]^(.5){I/\in} & Q \ar[r]^{\sf dD^*} & F \ar[r]^{\sf dD^*} & QT  \\
%%b \ar[ru]_(.5){I/\in}} 

%%\item
%%Some universalists consider {\it quality types} as universals while their determinates (that are no qualia) just as {\it concepts}/classes of determinates.

%%\item
%%{(\bf ??)} {\it Qualities} ($\neq$ individual qualities) are determinates of a quality type (e.g., for {\sl color}: red, yellow, scarlet, dark-scarlet).

%%%VEDI SE PUO' SERVIRE
%%%\item
%%%{(\bf ??)} --forse non serve--discorso della dipendenza tra le qualita', dipendenza tra i determinati di un certo tipo

%%\eslist
%%\newpage


%
%%-----------------------------------------------------------------
%\section*{Spaces without qualia}
%%-----------------------------------------------------------------
%\titline\\
%In this case, both exact and inexact resemblance (similarity) are taken to be contextual. These spaces organize objects and not qualia (compare to resemblance nominalism).

%\bslist
%%\item
%%{\sf pos} relation can be interpreted as: $(i)$ $I$, i.e. there are different qualia for each space; or ($ii$) $\in$, i.e. different similarities btw object ``generate" different classes (resemblance nominalism?).
%\item[]
%\mbox{}
%\xymatrix@R=2pt@C=40pt{
%a  \ar[r]^(.5) {\sf pos} \ar[rdd]^(.5){\sf pos} \ar[rddd]_(.85) {\sf pos} \ar[rddddd]_(.5){\sf pos} & Dark\_Red \ar[r]^(.5){P} & S^1_1 \ar[rd]\\ & \vdots & \vdots & Coloured\\
%& Red \ar[r]^(.5){P} & S^1_n \ar[ru]\\
%& R^2_1 \ar[r]^(.5){P} & S^2_1 \ar[rd]\\
%& \vdots & \vdots & Shaped\\
%& R^2_n \ar[r]^(.5){P} & S^2_n \ar[ru]}

%\item
%Note: analogous to G\"{a}rdenfors' {\sl conceptual spaces}.
%\eslist

%\newpage

%
%%%-----------------------------------------------------------------
%%\section*{Ontological similarity}
%%%-----------------------------------------------------------------
%%\titline
%%\bslist
%%\item
%%Universalism and trope theory both consider a ``jugement" of {\it similarity} between (particular aspects of) objects.

%%\item
%%This ``jugement" is based on having some ``common property": two objects are {\it similar}, in some way, iff
%%\bslist
%%\item[$\bullet$]
%%they share a universal ({\sl universalism});
%%\item[$\bullet$]
%%they have resemblant tropes ({\sl trope theory}). 
%%\eslist

%%\item 
%%The {\it similarity} is {\sl objective}, {\sl mind} independent, {\sl language} independent, it is exclusively based on the {\sl ontological nature of objects}, and it provides the {\sl finest} possible analysis 
%%on aspects of objects.

%%\eslist

%%%-----------------------------------------------------------------
%%\section*{Empirical/epistemological level}
%%%-----------------------------------------------------------------
%%\titline
%%\bslist
%%\item
%%In {\sl cognitive science}, similarity is a central notion: \\
%%``[J]udgments of similarity (...) are central for a large number of cognitive processes. (...) such judgments reveal the dimensions of our perceptions and their structures."

%%\item 
%%In this case, {\it similarity} is empirically built on experiments and it is {\it relative}: it may depend on species, cultures, etc. 

%%\item
%%In {\sl science}, the analysis always is conducted at an empirical (or theoretical) level and it depends on the available information, the measurement instruments/methods, etc. 

%%\item
%%It is possible to have different granularities of analysis or qualitative ones and it may interesting to have a way to compare these different analysis.
%%\eslist

%%\newpage

%
%%%-----------------------------------------------------------------
%%\section*{Spaces}
%%%-----------------------------------------------------------------
%%\titline
%%\bslist
%%\item 
%%To each aspect of an object, it is possible to associate different {\it spaces}. The similarity between one aspect of objects can be evaluate in different ways according to different spaces.

%%\item
%%Spaces are {\sl contextual}: they depend on culture, instruments of investigation, etc.

%%\item 
%%Spaces are created, adopted, and destroyed in time by (communities of) intentional agents, they have a temporal extension.

%%\item 
%%Spaces have an internal structure (mereological, topological, geometrical, metrical etc.) that makes it possible to differentiate several quantitative and qualitative degrees of similarity.

%%\eslist
%%\newpage

%
%%%-----------------------------------------------------------------
%%\section*{Introducing spaces: basic ideas}
%%%-----------------------------------------------------------------
%%\titline

%%\bslist
%%\item 
%%Substituting universals (universalism) or classes (trope theory) by spaces.

%%\item
%%Linking together all the spaces relative to the same aspect of the object (it is also possible to define a {\it refinement} relation between spaces). 

%%\item
%%Positioning aspects of the objects in (regions of) the spaces and compare different positions by means of the structure of the space.  

%%\eslist
%%\newpage

%
%%-----------------------------------------------------------------
%\section*{Reconstructing qualia}
%%-----------------------------------------------------------------
%\titline
%\mbox{ }\\
%\bslist
%\item
%If we assume that for each quality kind there is a space $S^*$ with maximal granularity (as defined by, say, a {\em refinement relation}),\\
%then the atomic regions of $S^*$ can be taken to be the ``qualia''.
%\eslist
%%
%%\bslist
%%\item
%%Just select, {\sl from the exterior} and for each quality type, a particular space as the ontological/``real" one. 

%%\item 
%%For each quality space, select the space that is the more finest (maximal granularity) space with respect to a {\it refinement} relation. This presuppose: unique space with the maximal granularity; all spaces are comparable with respect to the refinement. 

%%\item[]
%%\mbox{}
%%\xymatrix@R=10pt@C=35pt{
%%a \ar[r]^(.5) {I/\in} & Q  \ar[r]^(.5) {\sf pos} \ar[rdd]_(.5){\sf pos}  & R^1_1 \ar[r]^(.5){P} & *+[F]{S^1_1} \ar[rd] \ar[d]^{\sf ref}\\
%%& & \vdots & \vdots \ar[d]^{\sf ref} & QT_1\\
%%& & R^1_n \ar[r]^(.5){P} & S^1_n \ar[ru]}
%%\eslist

%\newpage

%%
%%%-----------------------------------------------------------------
%%\section*{Being more ontological: qualia}
%%%-----------------------------------------------------------------
%%\titline\\
%%{\it Qualia} represent {\it partial identities} between objects, and spaces represent different ways of organizing qualia. They do not inhere in specific objects, they are abstract from time, and they do not have any structure (only in variation approach).\\

%%\xymatrix@R=7pt@C=45pt{
%%a & C_a \ar[l]_{i} & q_x \ar[l]_{{\sf abs}_{t_1}} \ar[r]^(.5){{\sf pos}} \ar[rddd]^(.6){{\sf pos}} & R_x^1 \ar[r]^(.5){P} & {S_1}  \\
%%& & q_y \ar[lu]^(.4){{\sf abs}_{t_2}} \ar[r]^(.5){\sf pos}  \ar[rddd]_(.5){{\sf pos}} & R_y^1 \ar[ru]_(.5){P}\\
%%& & & \vdots & \vdots\\
%%& & & R_x^n \ar[r]^(.5){P} & S_n \\
%%& & & R_y^n  \ar[ru]_(.5){P}}   
%%\newpage

%%
%%%-----------------------------------------------------------------
%%\section*{Qualia/spaces and determinates/determinables [DA VERIFICARE]}
%%%-----------------------------------------------------------------
%%\titline

%%\bslist
%%\item
%%Qualia are very similar to {\it determinates}, i.e. universals that can not be further reified. In this sense they are similar to classes of resembling tropes: they are abstract from a specific entity, but they represent the ``finer granularity".

%%\item
%%Regions in spaces can be viewed as abstraction on determinates, in this sense they are similar to {\it derterminables}. 

%%\eslist
%%\newpage

%%%-----------------------------------------------------------------
%%\section*{Are qualia necessary? (1)}
%%%-----------------------------------------------------------------
%%\titline\\
%%{\sl Conceptual Spaces} organize directly objects not qualia.
%%\bslist
%%\item
%%{\sf pos} relation can be interpreted as: $(i)$ $I$, i.e. there are different qualia for each space; or ($ii$) $\in$, i.e. different similarities btw object ``generate" different classes (resemblance nominalism?).
%%\item[]
%%\mbox{}
%%\xymatrix@R=2pt@C=50pt{
%%a  \ar[r]^(.5) {\sf pos} \ar[rdd]^(.5){\sf pos} \ar[rddd]_(.85) {\sf pos} \ar[rddddd]_(.5){\sf pos} & R^1_1 \ar[r]^(.5){P} & S^1_1 \ar[rd]\\ & \vdots & \vdots & QT_1\\
%%& R^1_n \ar[r]^(.5){P} & S^1_n \ar[ru]\\
%%& R^2_1 \ar[r]^(.5){P} & S^2_1 \ar[rd]\\
%%& \vdots & \vdots & QT_2\\
%%& R^2_n \ar[r]^(.5){P} & S^2_n \ar[ru]}
%%\eslist
%%\newpage

%%
%%%-----------------------------------------------------------------
%%\section*{Are qualia necessary? (2)}
%%%-----------------------------------------------------------------
%%\titline

%%\bslist
%%\item
%%From the expressive point of view, the approach without qualia is equivalent to the one with qualia only if we presuppose, for each quality type, the existence of a maximal refined space.

%%\item
%%This solution is adequate for representing the {\it Conceptual Spaces}.
%%\eslist
%%\newpage

%
%%%-----------------------------------------------------------------
%%\section*{Being less ontological: objects and spaces}
%%%-----------------------------------------------------------------
%%\titline\\
%%In {\sl Conceptual Spaces}, objects are directly organized in spaces: the different similarity relations apply to objects.\\

%%\xymatrix@R=8pt@C=50pt{
%%a  \ar[r]^(.4){{\sf pos}_{t_1}} \ar[rd]^(.7){{\sf pos}_{t_2}}  \ar[rddd]^(.55){{\sf pos}_{t_1}} \ar[rdddd]_(.7){{\sf pos}_{t_2}} & R_x^1 \ar[r]^(.5){P} & {S_1} \\
%%& R_y^1 \ar[ru]_(.5){P}\\
%%& \vdots & \vdots\\
%%& R_x^n \ar[r]^(.5){P} & S_n\\
%%& R_y^n  \ar[ru]_(.5){P}}

%%\noindent
%%A relation between spaces that refer to the ``same" quality type of the objects is needed: colors, weights, heights, etc.

%%\newpage

%%-----------------------------------------------------------------
%\section*{Individual qualities}
%%-----------------------------------------------------------------
%\titline
%\bslist
%\item
%The idea is to consider the particular {\it aspect} of an object linked to a specific qualia kind:

%\bslist
%\item[] The weight of John.\\ The color of the rose.
%\eslist

%\item 
%We call these entities {\it individual} qualities because, like tropes, they inhere in a specific object (the weight of John is different from the weight of Sam).
%\eslist
%\newpage

%
%%-----------------------------------------------------------------
%\section*{Individual qualities and change}
%%-----------------------------------------------------------------
%\titline\\
%The rose $a$ may ``change" its color.\\
%This change is represented (using qualia) in the following picture:
%%\bslist
%%\item
%%{\sl Substitution}. Individual qualities are sums of tropes.

%%\item
%%{\sl Variation}. The introduction of a temporary instantiation (or a modal operator) between the individual quality and the qualia.
%%\eslist
%\vspace{4mm}

%\mbox{} \hspace{-0.5cm}
%\xymatrix@R=10pt@C=20pt{
%a & a_{col} \ar[l]_{i} \ar[r]^{{\sf abs}_{t_1}} \ar[rd]_{{\sf abs}_{t_2}} & Scarlet  \ar[r]^(.4){{\sf pos}} \ar[rddd]^(.6){{\sf pos}} & Red \ar[r]^(.5){P} & S^1_1 \ar[rdd]\\
%& & Crimson  \ar[r]^{\sf pos}  \ar[rddd]_(.7){{\sf pos}} & Red \ar[ru]_(.5){P}\\
%& & & \vdots & \vdots & Coloured \\
%& & & Dark\_Red \ar[r]^(.5){P} & S^1_n \ar[ru]\\
%& & & Crimson  \ar[ru]_(.5){P}}
%\newpage

%
%%-----------------------------------------------------------------
%\section*{The International System of Units}
%%-----------------------------------------------------------------
%\titline
%\mbox{ }\\
%\bslist
%\item
%This approach (but without qualia) is adopted by the SI, where:

%\bslist
%%\item[]
%\item[$\bullet$]
%quantities in the {\it particular} sense correspond to individual qualities;

%\item[$\bullet$] 
%quantities in the {\it general} sense correspond to qualia kinds.
%\eslist 

%\eslist
%\newpage

%
%%-----------------------------------------------------------------
%\section*{Reconstructing individual qualities}
%%-----------------------------------------------------------------
%\titline

%\bslist
%\item
%With qualia, it is possible to reconstruct qualities of kind $i$ as couples $\langle e,Q_i^{|e|} \rangle$, were $e$ is a specific object, and $Q_i^{|e|}$ is the set of all the qualia of kind $i$ that are linked to $e$.

%\item
%Without qualia, the reconstruction becomes possible if we assume a maximal refined space. 

%\item
%In any case the construction is complex and is based on set-theoretical notions (or mereological in alternative)

%\item
%Clearly individual qualities offer a closer interpretation of sentences like ``The weight of John", but it is not clear if they add something to the formal language from the expressive point of view.
%\eslist

%\newpage

%%-----------------------------------------------------------------
%\section*{Conclusions}
%%-----------------------------------------------------------------
%\titline\\
%We have discussed the formalization of unary properties making a selection among existing philosophical approaches and relating them to application requirements.

%\medskip
%\noindent
%This is an ongoing work which, we believe, already shows some interesting results. In particular,

%\bslist
%\item
%The considered approaches have very different ontological commitments.

%\item
%Assuming spaces with maximal granularities and set theory (or mereology) all these approaches are equi-expressive.

%\item
%The approach with objects and spaces only is adequate for conceptual spaces, while the one with objects, individual qualities, and spaces for the International System of Units. 

%\eslist
%\newpage

%
%%-----------------------------------------------------------------
%\section*{Why individual qualities?}
%%-----------------------------------------------------------------
%\titline\\
%Two possible interpretations of ``The weight of John is good".\\
%(let's suppose here that only one space is considered)

%\bslist
%\item
%The weight-quale associate to John belongs to the ``good" region.

%\item
%The weight-quality of John 

%\eslist

%{\bf FINE-FINE}
%\newpage

%

%%-----------------------------------------------------------------
%\section*{Change of properties in time}
%%-----------------------------------------------------------------
%\titline\\
%L'idea base e' di collegare un tropo direttamente al quality type invece che al qualia e quindi il cambiamento rispetto a quel quality type e' una relazione temporalizzata tra la qualita' individuale e la regione che occupa.
%\\
%(legame diretto con la fisica)
%\\
%\\
%Analyze sentences like:
%\bslist
%\item[]
%$a$ is $F$ at $t_1$ and $a$ is $G$ at $t_2$.

%\item[]
%\eslist

%\noindent
%(for simplicity we add the category of {\it times}, also if there are different possible ways to avoid this kind of entities)
%\newpage

%
%%-----------------------------------------------------------------
%\section*{Universalism on change}
%%-----------------------------------------------------------------
%\titline\\
%Different solutions:

%\bslist
%\item 
%The introduction of a temporary instantiation relation: $a$ is an instance of the universal $F$ at $t_1$ and of $G$ at $t_2$. In this case $I$ is a ternary relation, {\it instantiation-at} among particulars, universals, and times.

%\item 
%The introduction of a temporal ``modal" operator, i.e. $I$ remains a binary relation on which some modal operator applies.

%\item
%The commitment on a four-dimensionalist ontology of particulars: $a@t_1$ is $F$, and $a@t_2$, different from $a@t_1$, is $G$. 

%\item[]
%\eslist

%\noindent
%We do not commit on one solution; we simply write $I_{t_1}$.

%\newpage

%
%%-----------------------------------------------------------------
%\section*{Trope theory on change}
%%-----------------------------------------------------------------
%\titline

%\bslist
%\item
%The ``standard" solution consists in the introduction of two tropes, the $a$'s being $F$, and the $a$'s being $G$, respectively existing at $t_1$ and $t_2$.
%\item[]
%Change is reduced to a {\it substitution} of tropes. 

%\item {\bf (??)} additional solutions, maybe too complex to explain

%%Two tropes are necessary, $a$'s being red ($red_a$) and $a$'s being blue ($blue_a$), and this two tropes have a different temporal extension: $red_a$ exists during $t_1$ and $blue_a$ exists during $t_2$.\\
%%{\bf UN'ALTRA possibilita' e' avere $color_a$ e le relazioni di equivalenza temporalizzate, cioe' $color_a \thickapprox_{red,t_1} color_b$ and $color_a \thickapprox_{blue,t_2} color_c$} (forse si tratta semplicemente di dire cho possiamo considerare color come la somma di tutti i tropes di quel tipo, una specie di visione 4D)

%\eslist
%\newpage

%
%%-----------------------------------------------------------------
%\section*{Universals+Tropes on change}
%%-----------------------------------------------------------------
%\titline\\
%{\sl Example}. $a$ is Red ($R$) at $t_1$ and Blue ($B$) at $t_2$.

%\bslist
%\item[\footnotesize{(a)}]
%{\sl Substitution}. Two tropes (as in the case of only tropes) that are instances of  $R$ and $B$ existing, respectively, at $t_1$ and $t_2$. 

%%IMPORTANTE
%%(Campbell: if the object $a$ does not retain its identity).

%\item[\footnotesize{(b)}]
%{\sl Variation}. Only one ``more general" trope, the color ($C$) of $a$, existing during $t_1$ and $t_2$, and a relation of instantiation $I_t$. In this case we have a generalizations of the concept of quality admitting {\it degrees} introduced by Campbell. 

%%IMPORTANTE
%%(Campbell: tropes relative to qualities that have {\it degrees}, in the case of color we do not have really a linear degree, an ordering, but we recognize that red and blue are both colors).

%\item[]
%\begin{tabular}{cc} 
%\xymatrix{
%a & R_a \ar[l]_{i} \ar[r]^{I/\in} & R \\
%   & B_a \ar[lu]^{i} \ar[r]^{I/\in} & B}
%&
%\xymatrix{
%a & C_a \ar[l]_{i} \ar[r]^{I_{t_1}}  \ar[rd]_{I_{t_2}}& R \\
%   & & B}\\
%\footnotesize{(a)} & \footnotesize{(b)}\\ 
%\end{tabular}

%%\item
%%Considering the temporary properties we have 4 possibilities.

%\eslist
%\newpage

%
%%-----------------------------------------------------------------
%\section*{Come back to the example (1)}
%%-----------------------------------------------------------------
%\titline\\
%{\sl Example}. $a$ is Red ($R$) at $t_1$ and Blue ($B$) at $t_2$.

%\bslist
%\item With respect the solution with {\sl substitution}:
%\item[]
%\hspace{-0.4cm}
%\mbox{}
%\begin{tabular}[b]{cc}
%\xymatrix@R=10pt@C=35pt{
%a & R_a \ar[l]_{i} \ar[r]^{I/\in} & R \\
%   & B_a \ar[lu]^{i} \ar[r]^{I/\in} & B}
%&   
%\xymatrix@R=10pt@C=35pt{
%a & R_a \ar[l]_{i} \ar[r]^(.4){{\sf pos}} \ar[rddd]^(.55){{\sf pos}} & R_x^1 \ar[r]^(.5){P} & S_1\\
%& B_a \ar[lu]^{i} \ar[r]^{\sf pos}  \ar[rddd]_(.7){{\sf pos}} & R_y^1 \ar[ru]_(.5){P}\\
%& & \vdots & \vdots\\
%& & R_x^n \ar[r]^(.5){P} & S_n\\
%& & R_y^n  \ar[ru]_(.5){P}}   
%\end{tabular}
%\eslist
%\newpage

%%-----------------------------------------------------------------
%\section*{Come back to the example (2)}
%%-----------------------------------------------------------------
%\titline\\
%{\sl Example}. $a$ is Red ($R$) at $t_1$ and Blue ($B$) at $t_2$.

%\bslist
%\item With respect the solution with {\sl variation}:
%\item[]
%\hspace{-0.4cm}
%\mbox{}
%\begin{tabular}[b]{cc}
%\xymatrix@R=10pt@C=35pt{
%a & C_a \ar[l]_{i} \ar[r]^{I_{t_1}}  \ar[rd]_{I_{t_2}}& R \\
%   & & B}
%&   
%\xymatrix@R=10pt@C=35pt{
%a & C_a \ar[l]_{i} \ar[r]^(.4){{\sf pos}_{t_1}} \ar[rd]^(.7){{\sf pos}_{t_2}}  \ar[rddd]^(.55){{\sf pos}_{t_1}} \ar[rdddd]_(.7){{\sf pos}_{t_2}} & R_x^1 \ar[r]^(.5){P} & S_1\\
%& & R_y^1 \ar[ru]_(.5){P}\\
%& & \vdots & \vdots\\
%& & R_x^n \ar[r]^(.5){P} & S_n\\
%& & R_y^n  \ar[ru]_(.5){P}}   
%\end{tabular}

%\eslist
%\newpage

%%-----------------------------------------------------------------
%\section*{The International System of Units}
%%-----------------------------------------------------------------
%\titline
%\bslist
%\item
%usa una specie di variation, le $C_a$ che variano nel tempo sono rappresentate da funzioni e sono comunque legate ad un sigolo individuo

%\item
%fare in discorso sulla misura, cioe' dare l'intuizione sulla nostra idea delle unita' di misura
%\eslist
%\newpage

%
%%-----------------------------------------------------------------
%\section*{Title}
%%-----------------------------------------------------------------
%\titline
%\bslist
%\item 
%Gli spazi di qualita' permettono una rappresentazione qualitativa dell'istanziazione, e soprattutto permettono di poter parlare delle relazioni che sussistono tra concetti/universali (il rosso e' piu' vicino all'arancio che al blu, ecc.)
%\item

%\item

%\eslist
%\newpage



\end{document} 