# Logic of Typical and Atypical Instances of a Concept—A Mathematical Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- Theorem 1 which represents a fundamental theorem for the model of the LDO. It asserts that the LDO structure can be represented by a Galois lattice;
- The quasi topology structure (QTS) of the fully determinate objects Ext f in LTA.

## 2. The Logic of Determination of Objects (LDO)

#### 2.1. Informal Description

- It supplys a solution for the mismatch between logic categories and linguistic categories (adjectives, intransitive verbs often represented by unary predicates);
- It considers the determination as a logic operator in order to represent linguistic expression as a book, a red book, a book which is on the table;
- It reconsiders the duality of extension–intension via its theory of typicality; the entension and the intension of a concept are no longer in duality.

- A network of concepts;
- A set of objects;
- A type theory.

- An object called typical object τf, which represents the concept f as an object. This object is completely (fully) indeterminate;
- A determination operator σf, constructing an object more determinate than the object to which it is applied;
- The intension of the concept f, Int f, conceived as the class of all concepts that the concept f «includes», that is, a semantic network of concepts structured by the relation «IS-A»;
- The essence of a concept f, Ess f; it is the class of concepts such that they are inherited by all objects falling under the concept f;
- The expanse of the concept f, Exp f, which contains all more or less determinate objects to whom the concept f can be applied;
- A part of the expanse is the extension Ext f of the concept f; it contains all fully (completely, totally) determinate objects such that the concept f applies to.

_{1}rule of Da Costa [12]. We arrive at the following interpretation of the weakening of the principle of contradiction (¬(B ∧ ¬B)) contained by the RA

_{1}rule inside the LDO: an object obtained by a LDO-rule using this form of the weakening of the principle of contradiction (¬(B ∧ ¬B)) is an atypical object. From the point of view of managing negation, we can conclude that LDO is a particular case of a paraconsistent logic. For its power of description and especially for its basic notions (to emphasise the distinction between object and concept and between extention and intension), we can state that LDO is a description logic capturing at least one more cognitive feature: the typicality of objects.

#### 2.2. Formal Description of LDO

- Primitive types are: J individual entity type, H truth value (sentence) type;
- Functional type constructor:
**F**; - Rules.

- Primitive types are types;
- If α and β are types, then
**F**αβ is a type; - All types are obtained by one of the above rules.

- All objects are operands of type J; all propositions are of type H;
- All concepts are operators of type
**F**JH.

_{1}is the operator of negation defined as:

_{1}f) (x) = T, if and only if (f x) = ⊥

_{1}(N

_{1}g)) = g.

_{0}is the negation of a sentence defined as:

_{0}(f (x)) = T, if and only if f (x) = ⊥

- Objects of type J and concepts of type
**F**JH. - Predicates defined on individual objects (concepts of type
**F**JH) and the relators between individuals with respective types**F**J**F**JH,**F**J**F**J**F**JH, etc.); - Proposition of type of H;
- Connectives between propositions are of the type
**F**H**F**HH; - Fregean quantifiers: simple quantifiers with the type
**FF**JHH; restricted quantifiers with the type**FF**JH**FF**JHH; - Operators of negation with the type
**F**HH (classical negation) defined only on propositions.

#### 2.3. Basic Operators of LDO

#### 2.3.1. The Constructor of the “Typical Object”: the Operator τ

**FF**JHJ; it canonically associates to each concept f, an indeterminate object τf, called “typical object”. Its applicative scheme is:

#### 2.3.2. The Operator of Determination: the Operator δ

**F**JJ to each concept f.

**FF**JH

**F**JJ. Its applicative scheme is:

_{1}o δg

_{2}, …, o δg

_{n}. The composition of determinations is associative and supposed to be commutative.

#### 2.3.3. Objects in LDO

#### More or Less Determinate Objects

- τf is a more or less determinate object;
- If δf is a chain of determinations, then y = ((δg
_{1}o δg_{2}, …, o δg_{n}) x) = (δg_{1}(δg_{2}(…, δg_{n}(x), …)) is a more or less determinate object; - Each more or less determinate object is obtained by the above rules.

#### Fully Determinate Objects

- More or less determinate objects;
- Fully determinate objects.

#### 2.3.4. Concepts and Objects

#### Classes of Concepts Associated with a Concept f

#### Classes of Objects Associated with a Concept f

#### 2.3.5. Theory of Typicality in LDO

#### Typical Object of f

- Either each determination concept is in the intension of f and its negation is not in the intension of f;
- Or if there is a determination concept such that itself and its negation are in the intension of f, then this determination concept belongs to the characteristic intension of x;
- Roughly speaking, a typical object of f is an object that inherits all the concepts of the intension of f, Int f.

#### Atypical Object of f

- Either there is a determination concept that it is not in the intension of f, but its negation belongs to this intension;
- Or if all determination concepts are in the intension of f, then x has an atypical “ascendant” as object.

#### 2.3.6. The Logic of Determination of Objects (LDO) as a Deductive System

## 3. A Formal Description of the Logic of Determination of Objects (LDO) as a Preordered Set Model

_{1}, o

_{2}∈ $\mathcal{O}$, o

_{1}---> o

_{2}iff the object o

_{2}is more determinate than the object o

_{1}

^{n}f can be interpreted as the number of properties in depth subsumed by the concept f.

#### The Galois Connexion and the Galois Lattice of LDO

**Definition**

**1**

**[14].**

_{P}) and (Q ≤

_{Q}) is a couple of functions m

_{1}and m

_{2}such that:

_{1}: (P ≤

_{P}) → (Q ≤

_{Q}), m

_{2}: (Q ≤

_{Q}) → (P ≤

_{P}) with

For all p ∈ P, p ≤

_{P}m

_{2}(m

_{1}(p)) and for all q ∈ Q, q ≤

_{P}m

_{1}(m

_{2}(q)).

**Definition**

**2**

**[13].**

**Remark**

**1.**

- A single ideal corresponding to all typical objects denoted by I
_{Ext}_{τ}(in Figure 2, in magenta); - Several filters containing concepts from the intension Int f and, in the counterpart NInt f (see Figure 2) negations of concepts in Int f. Such a filter is denoted by F
_{Int-NInt}; - Several ideals containing some typical and some atypical objects. Such an ideal is denoted by I
_{typ-atyp}.

**Definition**

**3.**

_{1}: ($\mathcal{P}$(Fil), ⊆) → ($\mathcal{P}$(Id), ⊆) and) by:

_{1}(F

_{Int-NInt}) = Extτ if F

_{Int-NInt;}= F

_{Int}

m

_{1}(F

_{Int-NInt}) = I

_{typ-atyp}otherwise

_{2}: ($\mathcal{P}$(Id), ⊆) → ($\mathcal{P}$(Fil), ⊆) by:

_{2}(I

_{typ-atyp}) = F

_{Int}if I

_{typ-atyp}= Extτ

m

_{2}(I

_{typ-atyp}) = F

_{Int-NInt}otherwise

**Theorem**

**1.**

_{1}, m

_{2}) is a Galois connection on the double network (($\mathcal{P}$(Fil) ⊆), ($\mathcal{P}$(Id) ⊆))).

**Corollary**

**1.**

## 4. The Logic of Typical and Atypical Instances (LTA)

#### Informal Description of LTA Versus LDO

- In LTA the vertices of $\mathcal{F}$ are properties, a concept is represented by entire lattice $\mathcal{F}$. That is because it is based by the difference between property and concept.
- In LTA in the part of objects $\mathcal{O}$, there are at least two categories of objects unless typical objects and atypical objects:
- Strong exceptions (the “homeless”, in Example);
- Weak exceptions (the “irregular inhabitant” in Example).

- The LDO describes the structure of a concept f;

## 5. LTA as a Quasi Topology Structure

#### 5.1. Quasi Topology Structure (QTS) Definition

**Definition**

**4**

**[5].**

_{1}and O

_{2}of O, and two closed sets F

_{1}and F

_{2}such that:

_{2}⊂ O

_{1}⊆ E ⊆ F

_{1}⊂ F

_{2}

_{1}is the biggest open set contained in E, that is O

_{1}= Int (E),

_{1}is the smallest closed set containing E, that is F

_{1}= Cl(E),

_{2}is the biggest open set strictly contained in O

_{1},

_{2}is the smallest closed set strictly containing F

_{1}.

_{2}is said to be the strict interior of E; the set O

_{1}is the large interior of E. The set F

_{2}is said to be the large closure of E and the set F

_{1}the strict closure of E.

_{1}− O

_{2},

_{2}− O

_{1},

_{1}– O

_{1}.

**Remark**

**2.**

#### 5.2. The QTS of the LTA

_{f}, corresponding to the concept ^f = <f, Ess f, Int f, NInt f>, the other F

_{g}, corresponding to concept ^g = < g, Ess g, Int g, NInt g >.

_{f}and O

_{g}, namely Ext f ∪ Ext g. We define a QTS on the fully determinate objects related to the concept ^f.

_{f}.

- The objects verifying all the properties from Ess f and Int f. These objects are typical objects. They form the set O
_{2}; - The objects verifying all the properties from Ess f and at least one property of NInt f. These objects are atypical objects. They form the set O′
_{2}; - The objects verifying some properties from Ess f and some properties of Int. These objects are weak exceptions. They form the set F
_{1}; - The objects verifying only the property f as property. These objects are strong exceptions. They form the set F
_{2}.

**Definition**

**5.**

- typ f (o
_{1},o_{2}) defined by:for all o_{1}, o_{2}, typ (o_{1},o_{2}) iff o_{1},o_{2}are typical objects.It is an equivalent relation. All typical objects form an equivalent class; - atyp
^{k}f (o_{1},o_{2}) defined by:for all o_{1}, o_{2}, typ (o_{1},o_{2}) iff o_{1}, o_{2}are atypical objects with k their degree of atypicality.There are n relations atyp_{k}, f for k = 1, …, n, where n is the degree of atypicality. The degree of typicality of an object o is the number of properties of NInt verifying by this object. All the atypical objects having the same degree of atypicality are considered to be equivalent; - wex
_{i}f (weak exception) defined by:For all o_{1}, o_{2}, wex_{i}f (o_{1}, o_{2}) iff o_{1}, o_{2}are both objects falling under the same number of properties of Ess f.There are m relations wex_{i,}f for i = 1, …. m if Ess f has m properties. All the objects falling under the same number of properties of Ess f are considered to be equivalent; - strongex f (strong exception) defined by:strongex f (o) iff o in an object verifying only the property f.The relation strongex f is an unary (function prototype length 1);

**Remark**

**3.**

**Theorem**

**2.**

_{k=1,…n}(atyp

_{k}f)* and by (wex f)* = ∪

_{i=1,…m}wex

_{i}f. A subset, set E of the approximation space (Ext f ∪ Ext g, typ f ∪ (atyp f)*, (wex f) *∪ strongex f) has the following QTS structure:

_{2}= Ext

_{typ}f

_{1}= (Ext

_{typ}f ∪ (Ext

_{atyp}f)*)

_{1}= Ext

_{wex}f

_{2}= Ext

_{strongex}f

_{2}is the lower approximation of E as regarding to the relation typ f, O

_{1}is the lower approximation of E as regarding to the relation typ f ∪ (atyp f)*, F

_{1}is the upper approximation of E as regarding to the relation (wex f)* and F

_{2}is the upper approximation of E as regarding to the relation strongex f.

_{2}⊂ O

_{1}⊆ E ⊆ F

_{1}⊂ F

_{2}.

**Remark**

**4.**

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

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Desclés, J.-P.; Pascu, A.C.
Logic of Typical and Atypical Instances of a Concept—A Mathematical Model. *Axioms* **2019**, *8*, 104.
https://doi.org/10.3390/axioms8030104

**AMA Style**

Desclés J-P, Pascu AC.
Logic of Typical and Atypical Instances of a Concept—A Mathematical Model. *Axioms*. 2019; 8(3):104.
https://doi.org/10.3390/axioms8030104

**Chicago/Turabian Style**

Desclés, Jean-Pierre, and Anca Christine Pascu.
2019. "Logic of Typical and Atypical Instances of a Concept—A Mathematical Model" *Axioms* 8, no. 3: 104.
https://doi.org/10.3390/axioms8030104