Concepts, Language and Ontologies
(from the logical point of view)

Marie Duží
VSB-Technical University of Ostrava
17. listopadu 15, Ostrava, 708 33, Czech Republic
marie.duzi@vsb.cz

Motto: Es gibt eine und nur eine vollstaendige Analyse des Satzes.
Wittgenstein, Tractatus, 3.25

Abstract. Traditional, intuitive conceptions of concepts and their shortcomings are summarised first. We pose five worrisome questions, answering which becomes extremely important when we want to build ontologies of particular domains of interest, to ”conceptualise” them and to perform a terminological classification, which should be used for an intelligent search in the huge amount of data on web pages. Since traditional, set-theoretical concept theories do not answer these questions in a satisfactory way, we introduce a new non-traditional theory of concepts. A concept is explicated by the key notion of Transparent intensional logic – logical construction, i.e. structured procedure. We show that between an expression (of any language) and the denoted object beyond the language (denotatum) there is just the meaning of the expression, i.e., the respective concept, which identifies (or fails to identify – in case of empty concepts) the denotatum. Whereas the denotatum is a “flat” set-theoretical entity (intension in case of empirical expressions), the meaning (concept) is structured procedure consisting of parts – constituents. This conception enables to adequately solve the problem of the so-called hyper-intensional contexts of knowledge, beliefs and hypotheses. Thus our theory not only provides a precise explication of intuitively used notions like concept, ontology, meaning, etc., but it also provides an essential extension of traditional theories including Kauppi’s axiomatic theory and Ganter-Wille conception.
Key words. Concept, structured meaning, construction, ontology, conceptual lattice.
1. Introduction
Ontology is a notion borrowed from philosophy where it stands for handling possibilities and conditions of being, so that it is closely related to epistemology, which envisages the possibilities and limits of human perceiving and knowledge. Answering the fundamental ontological question ”What is there?” is in no way as trivial as it may seem, since (quoting from [16]):
a)  our perception of reality by our organs is filtered, so that we cannot be sure that the world so is and behaves as we perceive it
b)  answering has necessarily to use linguistic means, so that we possibly digress from the properly ‘being’ when choosing our concepts and using special words.
And we add: 
c)  There are also many abstract objects the acquisition of which by our ‘mind eyes’ is limited by our mind capabilities.
	In many areas of informatics we meet the task of representing what is known, believed, inferring other pieces of deduced knowledge and hypotheses, and communicating such knowledge. Hence knowledge is inherently endowed and connected with concepts. Men can obtain, save and exploit concepts intuitively, using their mind capabilities.  If, on the other hand, automata take over the search, inference and communication, they need the representation of the respective concepts (and their interrelations). Thus in the recent years we can notice a proliferation of the term ‘ontology’, since some branches of informatics have for the above purpose coined the notion ontology. T. Gruber [15] has defined ontology as ”an explicit formal specification of a shared conceptualisation”. In this sense ontology describes a knowledge domain by means of a standardised terminology, concepts and inference rules. Since there are many knowledge domains – using specific or even competing vocabularies – it is meaningful to use (unlike in philosophy) also plural ”ontologies”. The use of ontologies in informatics is very broad, in particular in those domains that concern saving, combination and derivation of knowledge, i.e. in artificial intelligence, (deductive) databases and (global) information systems (”the web”).
	Building ontologies (and semantic search methods in particular) is, in general, based on a traditional conception of concepts: Concept is generally understood as a ‘universal’, which is expressed by a meaningful expression and is determined by its content (intension) and extent (extension). Further, the extension of a concept is understood as a set of objects that ‘fall under’ the concept; the intension of a concept is the set of attributes (aspects) that characterise these objects. Thus, for instance, ‘municipalities of the Czech republic’ and ‘cities and districts of the Czech Republic’ have the same extension (the set of cities, towns, etc.), but different intensions, which means that these are distinct concepts. Classically it is affirmed that between an intension and extension of a concept there is an inverse relation: Increasing the content of a concept decreases its extent, and vice versa. Thus, for instance ‘university student’ has a greater content than ‘student’, but smaller extent. 
	This might seem to be a satisfactory theory, but accepting it, we meet many problems and questions. First, it was already Bernard Bolzano [2] who realised that it does not answer a fundamental ontological question, namely:
a) What kind of entity is a concept?
It only says that a concept consists of content and extent; but what is the content and what is the extent? If these are not defined independently of a ”concept”, we have a case of a circular ”definition”. If, on the other hand, content and extent are conceived in a naïve way, just “syntactically” as a set of sub-expressions, we meet serious problems. Thus, e.g., the concept of  ‘prime(numbers)’ has a content distinct from that of the concept of ‘numbers that have exactly two factors’, and yet these are evidently identical concepts, what else might the expression ‘prime’ mean than the latter?
	The inverse relation between content and extent of a concept is also not precisely specified: For instance the concept ‘student of a Prague university’ has smaller content but also smaller extent than the concept ‘student of a university in Prague or in Brno’. (B. Bolzano adduced a similar well-known example: ‘Man who understands all European languages’ vs. ‘Man who understands all living European languages’.) Moreover, it is obvious that concepts are not independent. For instance, the concept ‘cities and districts of the Czech republic’ has content distinct from (greater than?) the content of ‘cities and districts in Moravia’, but the extension of the latter is necessarily a subset of the extension of the former. There is also a problem of adjectives, which modify a property to form another (possibly independent of the original) property: ‘Wooden horse’ has a greater content than ‘horse’, but the extent of the former is never a subset of the latter (wooden horse is not a horse!). Is the concept of ‘horse’ contained in the concept of ‘wooden horse’? We can formulate two further questions:
b) What kind of entity is the content and extent of a concept?
c)  What can be deduced from (”follows from”) a concept?
Still, there are other worry-some questions: The extent (as ”defined” above) of a concept C keeps changing! For instance, the extent of the concept ‘university student’ varies with each newly accepted student or a student who graduates. Hence this concept keeps changing? But it is all the same concept! Yet, it is often said that ”concepts change, evolve, arise, cease, etc.”. Did the concept of ‘ontology’ evolve? Or did the concept of a computer exist before people invented computers? The other question to answer is:
d) Do concepts ”change”?
	If we want to build ontologies of particular domains of interest, to ”conceptualise” them and to perform a terminological classification, which should be used for an intelligent search in the huge amount of data on web pages, these questions are extremely important. For instance, using the Law of inverse proportion of extension E and intension I, a partial ordering on a set of ‘concepts’ is defined: I1, E1  I2, E2  iff I1  I2 (or E2  E1, which is said to be the same as the former). If correctly defined (e.g. on the basis of Galois connections [14]), the poset of ‘concepts’ forms a lattice, and a theory of conceptual lattices has been developed, which is broadly used just for the intelligent ‘semantic’ search. In order that the theory can be used properly, the above partial ordering should be based on the relation of ”what follows” from a concept, which is trivially fulfilled by the traditional conjunctive conception. However, it is just a special case, not exhausting all the possibilities. For instance, if somebody is a university student, than he certainly is a student (concept ‘student’ is contained in the concept ‘university student’), but not only that. We can also deduce that he is tax sheltered, etc. Moreover, there are important concepts which cannot even be defined using conjunction of some attributes, e.g., ‘ancestors of an entity’.
	It might seem, that concept can be identified with a meaningful expression, which is actually the case in traditional conceptions. But how would we then answer the question
e) What does it mean that some expressions are synonymous?
Well, they are expressions with the same meaning, but then there is another question what the meaning is. Denotational semantic theories answer that meaning of an expression is the entity that is ‘talked about’, i.e. the entity denoted by the expression, its denotatum. Leaving aside for a moment the question ‘what the denotatum is’, let us contemplate whether denotational semantics is fine-grained enough. Consider the classical example ‘equilateral triangle’ vs ‘equiangular triangle’. These expressions denote exactly the same set of geometrical figures; does it mean that they are synonymous, have the same meaning? But these are evidently distinct concepts (with different contents), and if they had the same meaning we would not have to study geometry so as to know that they have the same denotation, linguistic competence would do. Or, our students would not have to learn logic, they would know immediately that sentences like ‘It is not true that if A then B’ and ‘A and not B’ are equivalent, i.e. denoting the same proposition (and up to logical symbols they have the same ”content”), knowledge of mother tongue would suffice. How would we perform translation into another language? On the basis of identical denotations? But then it would not matter whether we translated a sentence of the former type or of the latter, which again is obviously not true. Moreover, some meaningful mathematical expressions, like ‘the greatest prime’, do not denote any object; does it mean that they do not have any meaning? But we can reasonably claim and prove that ‘the greatest prime does not exist’; how then could we compose a non-reasonable expression into a reasonable true sentence? What are we talking about? According to Parmenides, we cannot talk about anything that does not exist!
An attempt at a more sophisticated theory of concepts is presented by Ganter & Wille in [14], and Kauppi in [18] introduces perhaps the best work out of axiomatic set-theoretical concept theories. K.D. Schewe presented in [26] a very good survey of these traditional conceptions, including the intuitionistic approach. Summarising briefly, all these set-theoretical conceptions could be in general characterised as follows: a concept is a logical theory, axioms of which form the intension of the concept, and a set of models of the theory is the extension of the concept. These axiomatic conceptions are more adequate in solving the problems as stated above, still do not answer all the questions we pose. In particular, they do not distinguish between identical and equivalent concepts. In the concluding Chapter 5 we summarise particular items in which the theory of structured meaning proposed in this paper form an essential extension of these traditional theories.
	In this paper we answer the above questions, exploiting modern theory of concepts, first proposed by Materna in [20]. This theory conceives concepts as structured, ”hyper-intensional” entities, i.e., as logical constructions. The origins of the theory can be found already at B. Bolzano [2] and in part also at G. Frege [13], and B. Russell [25], but its full development has been made possible by a new paradigm of Transparent Intensional Logic (TIL), the author of which is late Prof. Pavel Tichý [29]; his follower Pavel Materna [20] and others [6], [7], [8], [9] explicated concept by the key notion of TIL, namely logical construction. Chapter 2 is a brief introduction into the philosophy and basic notions of TIL. The exposition of the theory of concepts can be found in Chapter 3, and in Chapter 4 we deal with the problem of analyzing natural language. The concluding Chapter 5 contains an outline of possible applications. The goal of this brief study cannot be a detailed recapitulation and exhibition of the entire theory and its applications, we would rather like to motivate the reader so as to find the theory interesting, worth studying, and to stimulate further discussion on its possible promising use.
2. Transparent Intensional Logic
Due to the sensitivity of a meaning to the logical structure of an expression, the analysis of  ‘what is known, believed’ and other attitudes has become a stumbling block for all the denotational (set-theoretical) semantic theories that take into account just the denoted entity. Our knowledge and inferential capabilities are inherently connected with the sense (meaning) of an expression. Since Frege’s times, many leading logicians strived after logically handling structured meanings, to name at least Russell’s structured propositions, Carnap’s [4] attempts at the formulation of a stronger criterion for the identity of belief, i.e. intensional isomorphism between substituted expressions, Cresswell’s tuples [5], etc., etc. B. Russell’s results were perhaps the closest to a satisfactory solution: The world consists of ”complexes” that are to be understood as ”logical constructions” formed from the immediately given entities of sensation, viz., ”sensibilia”. (Russell 1918, [25]). Still, none of these attempts carries conviction of a full adequacy and correctness (see, e.g., Tichy [29], Materna [20]).
Frege’s contextualist approach inherited by most of current logicians (including Montague [22]) has been criticised by Pavel Tichý who presented a fundamental revision of Frege’s (sense-reference) semantic scheme, which makes it possible to adequately explicate the ”behaviour” of expressions even in hyper-intensional belief/knowledge contexts. Tichý’s solution respects the distinction between the meaning (sense) of an expression and the object denoted by the expression; but it differs from most current conceptions (incl. Montague’s) in at least two points:
a)	it logically handles the hyperintensional structure of the meaning, sense is explicated as a hyperintensional entity (construction)
b)	No contextualism is present; expressions simply denote either an extension or an intension via its sense. Empirical expressions always denote intensions, and where it seems that they denote extensions they only possess de re supposition.
In contrast to a model theory, which starts with a naked syntax and only subsequently proceeds to a semantic interpretation in a model, TIL is ‘transparent’ not only that it is anti-contextualistic but also not formalistic. Notion of a naked formal expression as a pure graphic shape can be arrived at only through abstracting its sense from it. In terms of conceptual priority, TIL starts with sense-endowed expressions, which is to say that the ”formal language” of TIL-constructions constitutes an ”interpreted formalism”. Every factor that is semantically salient is explicitly present in the respective formalism. This is evident, for instance, in the explicit typing of the theory, the types of TIL being exclusively objectual. So what qualifies the formal language of TIL as transparent, inherently interpreted, is that a naked shape can be introduced as an expression only if it is paired off with a construction constructing an object of a particular type.
Definition 1 (Simple types of order 1):
(An objectual) base is a collection of mutually disjoint nonempty sets. 
i)	Every member of the base is a type over base.
ii)	Let a, b1,..., bm  be types over base, then (a b1... bm), i.e. the set of all m-ary (total and partial) functions with an argument (a tuple) áb1,...,bmñ, where bi  (1  i  m) is a member of the type bi, and at most one value of type a, is a type over base.
iii)	Nothing is a type over base unless it so follows from i) - ii).
An object O (that is a member) of a type  will be called an -object, denoted O/.
	An objectual base is a special kind of base, over which (an infinite) hierarchy of functions and constructions can be built up, and our conceptual scheme can be adequately modelled within this system. This base consists of sets of objects of four basic categories: i, o, w, t, where i is a type (set) of individuals. The objectual base together with the interpretation of other elements constitutes an epistemic frame. Interpretation of its other elements is as follows: o is the type (set) of truth-values {T, F}; w is the type (set) of possible worlds, and t is the type (set) of time points (or real numbers). The collection of pre-theoretically given (basic) features (traits), by the usage of which all the other notions are defined, constitutes the intensional base of the given system.
Empirical expressions denote intensions, i.e. functions from possible worlds to chronologies of members of a type . Hence -intensions are functions of type ((at)w), which will be abbreviated by atw. Extensions are objects of other types (-objects), namely they are not functions from possible worlds …
Note: Intention (extension) in the above sense, viz., -object (-object) must not be confused with ‘intention of a concept’ (‘extension of a concept’). We will better use the terms ‘content/intent’ and ‘extent’ for the latter.
Examples of intensions: 
Individual concepts (offices), like ‘the Pope’, ‘the richest man’, are objects of type itw;
properties of individuals, like (being a) ‘student’, are objects of type (oi)tw; 
binary relations-in-intensions between individuals, like ‘kicking’, are objects of type (oii)tw;
propositions, like ‘The Pope is in danger’, are objects of type otw. 
According to Zalta [30], Russellian ‘structured propositions’ play the desired role of complexes that result by ‘plugging’ objects into the gaps of properties and relations. In our opinion, properties, relations, i.e. functions in general, have no ‘gaps’; particular objects simply are members of (the arguments of) these flat functions. But we can accept the possible-world semantics of propositions, while the demand of structured meanings is met by another entity: Between an expression and the denoted flat object there is a structured mode of presentation (construction in our terminology) of the object, i.e. meaning (perhaps the Fregean sense) of the expression. It is a complex, a (declaration of a) procedure that consists in a creation of a function by abstracting over objects and/or in applying the function to its arguments. But particular (physical/abstract) objects cannot be ”plugged” into such a (conceptual) procedure; they must always be presented in an (albeit primitive) way, i.e., their concepts are constituents of the procedure. There are two such primitive modes of presentations that fill in the objects into the construction: variables and trivialisations. The other two kinds of constructions working over these ones are more complex; they are closure (creating a function by abstraction) and composition (applying a function to an argument).
The TIL language of constructions can be viewed as a typed -calculus whose terms are names of (denote) constructions. They might be conceived as Montague’s -terms with fixed (intended) interpretation of components, which denote not the function (or its value), but the way of arriving at it. Due to the perfect correspondence between terms and constructions it is idle to mention the terms, and we transparently talk about the constructions. Thus, e.g., instead of claiming that ‘x [0> x 00]’ denotes the construction x [0> x 00] which constructs the class of positive numbers, we simply say x [0> x 00] is the construction.
Definition 2 (Constructions):
i)	Variables are constructions. Variables and constructions involving variables construct objects dependently on a valuation v, they v-construct.
ii)	If X is an entity whatsoever, even a construction, then 0X is a construction called trivialisation. Trivialisation 0X constructs X without any change. 
iii)	If X0 is a construction that v-constructs a function (mapping) F, i.e. an (a b1...bn)-object, and X1,...,Xn are constructions that v-construct b1-,...,bn-objects b1,...,bn, respectively, then [X0 X1 ... Xn] is a construction called composition. If F is defined on the argument áb1,...,bnñ, then the composition [X0 X1...Xn] v-constructs the value of F on áb1,...,bnñ; otherwise it does not construct anything, it is v-improper.
iv)	Let x1,...,xn be pairwise distinct variables that range over types b1,...,bn, and let X be a construction that v-constructs an a-object for some type . Then [lx1...xn X] is a construction called closure (abstraction). It v-constructs the following function F (of the type ( 1…n) ): Let v´ be a valuation that differs from v at most by assigning objects b1,...,bn, (of the respective types) to variables x1,…,xn, respectively. Then the value of the function F on the argument áb1,...,bnñ is the object v´-constructed by X. If X is v´-improper, then F is undefined on the given argument. 
v)	Nothing is a construction unless it follows from i) - iv).
Notes: 
1.	The simplest constructions are variables; they are open constructions that construct objects dependently on valuation (they v-construct). They are no letters, characters, ‘x’, ‘y’, ‘z’, ... are names of variables.
2.	Trivialisation consists in grasping an object and its ”delivering” without any change. If X is an entity, then 0X is a presentation of X without a ”perspective”. The term ‘0X’ might be likened to a constant of a formal language; but unlike such a formal constant, which can be interpreted in many ways so as to denote different entities and thus actually not being a constant but a ”variable construction”, ‘0X’ rigidly denotes construction 0X that constantly constructs the X. A possible objection against such a conception might be: Well, your transparent approach is punctilious, but you lose the expressive power of model theories that enable us to examine common features of properties and relations between objects of all the particular models. Our answer is: Not at all; TIL transparent approach is more precise without losing anything; due to the infinite hierarchy of types we have at our disposal variables ranging over objects at any level, which makes it possible to render particular ”models” by valuations of (higher-level) variables.
3.	A composition corresponds to the traditional operation of application (of a function to an argument). (Only) composition may fail to construct anything, it may be (v-)improper, namely in two cases: First, the component X0  constructs a function F and components X1,...,Xn construct áb1,...,bnñ, but F is not defined at this argument. Second, some of the components X0, X1, ..., Xn fail to construct an object (they are v-improper). 
	(In case X1,...,Xn  do not construct objects of proper types to create an argument of F, the expression ‘[X0 X1 ... Xn]’ is meaningless, it does not denote a construction.)
4.	Closure (l-abstraction) enables us to construct a function, and thus to analyse talking about the whole function (to ”mention” it), not only talking about a particular value at a given argument (to ”use” the function). Closure can never be improper, even if it constructs a (degenerated) function that is undefined at any of its arguments, like, e.g., x[0: x 00].
We commonly use variable w as ranging over , and variable t as ranging over . If X is a construction that constructs an intension of type atw, then instead of [[Xw]t] we write Xwt. 
Quantifiers - general  and existential $ - are functional objects of type (o(oa)). We will write xA, xA instead of [0a lx A], [0$a lx A], respectively. Quantifiers are ”totalising”, i.e., they always return a truth value when being applied to a class (even if the characteristic function of the class were undefined at some arguments), namely [0a lx A] returns True iff [lx A] constructs the whole type  (A v-constructs True for all x ranging over ), otherwise False, [0$a lx A] constructs True iff [lx A] constructs a non-empty subset of  (A v-constructs True for some x), otherwise False. Singulariser a is an object of type (a(oa)),  and instead of [0a lx A] we will use xA (the only x such that A); [0a lx A] constructs the only member of the class constructed by [lx A] iff [lx A] constructs a singleton, otherwise it is an improper construction. We will use a standard infix notation without trivialisation in case of using truth-value functions (, , …), less-than, greater-than and identity functions (, , =, …), but we have to keep in mind that these are just abbreviations that conceal the self-contained intrinsic meaning of the respective ‘logical symbols’. 
	The bridge between an expression and a construction (logical analysis of the expression) is provided by a principle of subject-matter, which says, roughly, that an expression is about all and only those objects, incl. constructions, which receive mention in the expression (see [10]). Constructions are mentioned, e.g., in hyper-intensional contexts of knowledge, belief, etc., where the meaning, i.e. the expressed construction plays a crucial role. Thus a construction/meaning is a ‘full-right’ object to talk about, and has to be of a definite (higher-order) type, which is not possible within the simple hierarchy of types. What follows is a modification of Russell’s ramified hierarchy of types:
Definition 3 (Ramified hierarchy of types)
Let B be an objectual base, i.e. a collection of mutually disjoint non-empty sets.
1. Types of order 1 over B 
	defined according to Definition 1.
2. Constructions of order n 	over B.
(Cni)	Let a be a type of order n over B. If x is a variable that ranges over , then x is a construction of order n over B.
(Cnii)	If X is a member of a type of order n, then 0X is a construction of order n over B. 
(Cniii)	If X0, X1,..., Xm  are constructions of order n over B, then [X0 X1...Xm] is a construction of order n over B.
(Cniv)	If distinct variables x1, ..., xm, as well as X, are constructions of order n over B, then [lx1...xmX] is a construction of order n over B.
Let *n  be the collection of all constructions of order n over B. Types of order n+1 over B are defined as follow:
3. Types of order n+1 over B.
(Tn+1 i)	 *n  and all the types of order n are types of order n+1 over B.
(Tn+1 ii)	If a, b1, ..., bm are types of order n+1, then the set (a b1...bm) of all m-ary (total and partial) functions from b1 x... x bm to a is also a type of order n+1 over B.
(Tn+1 iii)	Nothing is a type of order n+1 over B  unless it so follows from (Tn+1i) or (Tn+1ii).
Since the terminology in the area of logical analysis of natural language is vague and ambiguous, we exploit the results of Tichy’s followers, as they have been presented, e.g., in  [20], [21]. Just a brief summary: An expression expresses its sense (=meaning, concept) that identifies (non-fregean) denotation. Hence an expression denotes (talks about) its denotation via its meaning-sense. The sense of an expression is in principle a (structured) ‘procedure’ – a closed construction (the concept specified by the expression). A construction identifies (constructs) the denoted object that is (in case of a ”successful” constructing, i.e. construction not being improper) an intension or extension (a first-order, set-theoretical object), or even a higher-order object (involving a lower-order construction). Empirical expressions always denote an intension, and in this case we also speak about a referent or reference of an expression, which is the value of the denoted intension in a given world/time. The relation between the first-order object (intension) and that what is in most semantic theories considered to be a reference of an expression (for instance an individual in space and time, a set of individuals, etc.) does not have a semantic character; it is influenced by an empirical factor – state of affairs, and thus it is not directly a subject of a semantic-theory investigation. Hence a co-reference of expressions is from the TIL viewpoint a contingent, empirical matter. Expressions can be equivalent when they denote one and the same object, but do not have (even in this case) to have the same sense, i.e. do not have to be synonymous. 
Definition 4
(From the TIL viewpoint) logical analysis of an (empirical) expression E consists in specifying a construction expressed by E, which (v-)constructs the object denoted by E.
Note: Correct analysis of an expression E has to respect the principle of subject matter. It means that the very perfect analysis of E (the concept expressed by E) has to be composed of only and just the constructions of all the entities E talks about. 
Example: We will analyse the sentence 
The assets of the richest (i.e. ”the most rich”) man are greater than  $10 000 000. 
a) Type-theoretical analysis of subexpressions:
Assets  Ass / ()		Empirical function that assigns a number to an individual
Most  Most / (())		Function that picks up an individual from a set of individuals
Rich  Rich / (()())	Adjective modifier creates a new property from a  property
Man  Man / ()	Property of individuals
Greater than  > / (()) Relation between numbers; (10 000 000 / )
b) Synthesis (into the construction)
Rich man denotes a property of individuals / (), which is constructed by:
wt [0Richwt 0Man] – concept of a rich man
The richest man denotes an individual office / , which is constructed by:
wt [0Most [wt [0Richwt 0Man]]wt ] – concept of the richest man
Assets of the richest man denotes a magnitude / , which is constructed by:
wt [0Asswt [wt [0Most [wt [0Richwt 0Man]]wt ]]wt ] – 
                                        concept of the assets of the richest man
The whole sentence claims: 
The value of this magnitude (in a world/time) is greater than 10 000 000: 
       wt [wt [0Asswt [wt [0Most [wt [0Richwt 0Man]]wt ]]wt ]]wt > 10 000 000]
This is the very analysis of our sentence, which can be ”hominized” by performing equivalent -reductions:
	wt [ [ 0Asswt [ 0Most [0Richwt 0Man]] ] > 10 000 000 ]
c) Type-theoretical checking:
	wt [ [ 0Asswt [ 0Most [0Richwt 0Man]] ] > 10 000 000 ]
			             ()()  ()
			     ()          ()
		                        
			                                     ()        
	    				
Note: Such a correct analysis is a necessary condition of performing adequate inferences. Now, using inferential rules of the system (Gentzen’s natural deduction in our case) we can deduce that, e.g.:
The richest man exists 
Some rich man possesses more than $10 000 000
Some man possesses more than $10 000 000, etc.
But our sentence does not talk about any particular individual; in particular, it does not talk about Bill Gates (the office of the richest man serves here only as a ‘pointer’ to an unspecified individual). If we perform an empirical investigation (for instance the search in the database), and find out that Bill Gates happens to play the role of the richest man, then only by using this additional premise we can deduce that Bill gates possesses more than $10 000 000.
3. Theory of concepts.
Now you can ask: Did we answer the fundamental ontological question a) What is a concept? Not yet. We just provided a preliminary characterization: It is the meaning of an expression, i.e. the logical construction specifying the way, in which particular entities the expression talks about are composed, so as to identify the resulting denotation of the expression.
	For instance, the meaning of the expression ‘primes’ is a concept of primes, i.e. the construction: x ( [0Nat x]  [0Card y ([0Nat y]  [0Div x y])] = 02 ), which constructs the class of natural numbers that have exactly two factors. Meaning of the expression ‘the greatest natural number’ is the concept: x ( [0Nat x]  y ([0Nat y]  [x  y]) ), which does not identify anything, it is an empty concept.
	Concepts, logical constructions, are abstract objects (like numbers, algorithms, procedures – but not their recording or performing); therefore it is not reasonable to ask whether they exist, or where and when they are. They are beyond time and space [3]. Hence concepts are entities beyond the language, we do not create them, and we just discover some useful ”needed” procedures and coin them by expressions. Conceivng concepts as constructions might answer the question a), but we still have to perform a slight correction of our conception. First, concepts are only closed constructions (without free variables). But constructions are in a way too fine-grained entities to specify (conceptual) procedures. Thus constructions that are almost (‘quasi-‘) identical up to using bound variables, like x[x > 00], y[y > 00], z[z > 00], etc., are, from the conceptual point of view, almost indiscernible. Hence we say that concept is the first (in lexicographic ordering) of them, the other point at the concept. Moreover, concept is not that construction which is ”unreasonably -expanded”. For instance, (primitive) concept of adding is 0+, but not xy [0+ x y] (which just points at the former). (For details see [17].) The following definition provides an answer to the question a):
Definition 5. Concept is a closed construction C such that C is the first one from the ordered set of quasi-identical constructions that point at C.
	Neglecting, for the sake of simplicity, the slight distinction between quasi-identical constructions, we answer another question b):
Definition 6 (content and extent of a concept). 
A concept C1 is (intensionally) contained in a concept C2, iff C1 is a sub-construction of C2.
Content (intension) of a concept C is the set of concepts that are contained in C.
Extent (extension) of a concept C is the object E, which is constructed by C.
An empirical concept is such a concept CE, the extent of which is an -intension I (/ ).
The extent of an empirical concept CE in a world/time w,t is the value of its extent in w,t (Iwt).
An empty concept is a concept the extent of which is an empty set.
A strictly empty concept is a concept that does not have an extent (does not construct anything, the construction is improper).
A simple  concept (primitive with respect to a given conceptual system) of an object X is 0X (it cannot be strictly empty).
	Relation of intensional containment, as defined above, is the relation of partial ordering on the set of concepts; it is evidently reflexive, anti-symmetric and transitive. Still, in general, it cannot be used for a correct definition of a semantic conceptual lattice, because it does not have to be in accordance with the relation of ‘deducing from a concept’, which will be defined in the next chapter. Just an enumeration of the set of contained concepts does not suffice. We also have to specify the way in which these contained concepts are composed together to form a complex (see [9]). Otherwise we might obtain such absurd consequences, as have been presented, e.g., in [23].
4. Language and concepts.
As has been stated in Chapter 2, analysis of an expression consists in specifying a construction expressed by the expression that identifies its denotation. But each expression can be usually analysed in more correct ways, less or more fine-grained. A correct analysis follows the principle of subject matter, i.e., only constructions of the objects mentioned by the expression are composed together. This principle has been first formulated by Frege [12]: 
Ueberhaupt ist es unmöglich, von einem Gegenstande zu sprechen, ohne ihn irgendwie zu bezeichnen oder zu benennen.
Such a correct analysis enables us to perform adequate inferences. To be able to perform all the correct inferences, our deduction has to be based on the most fine-grained correct analysis, which takes into account all the objects mentioned or used by the expression. The proof of the claim that to each meaningful expression E the unique best analysis C exists has been adduced in [10]. This analysis C is the concept expressed by expression E, the meaning of E.
Example: Expression ‘assets of the richest man’ might be analysed by 0ARM, where ARM /  is the denoted magnitude. Obviously, such an ”analysis” would not enable us to perform any reasonable inferences. But meaning of the expression is the concept
wt [0Asswt [wt [0Most [wt [0Richwt 0Man]]wt ]]wt ],
-reduced equivalent analysis wt [0Asswt [0Most [0Richwt 0Man] ] ] is easier to read, it points at the concept. The extent of this concept is the magnitude ARM / , the extent in w,t is a number / , i.e. the value of the assets in w,t (the world/time is usually ‘mirrored by’ a knowledge base).
The concept contains (besides the simple concepts 0Ass, 0Most, 0Rich, 0Man) compound empirical concepts:
wt [0Richwt 0Man] – concept of a rich man
wt [0Most [wt [0Richwt 0Man]]wt ] – concept of the richest man
The extent of the former is a property of individuals RM / (), the extent in w,t is a set of rich men. The extent of the latter is an individual office MRM / , the extent in w,t is the individual who plays the role of the richest man (in w, t).
But the assignment ‘expression  concept (=meaning)’ is given by a linguistic convention, it is an empirical relation. Thus an answer to another introductory question d) 
Do concepts change? is no; just the above assignment of concepts to expressions can change, meaning of an expression changes, we even invent new expressions to name some ‘newly discovered’ concepts, and some old expressions cease to be used. Hence a (living) language develops, and moreover, each domain of interest uses actually its own ”jargon”. (We have to take into account a certain fixed, say current, standard, official language of a domain.)
Now we can answer the remaining introductory questions c) and e). We first define synonymy, homonymy and equivalence of expressions, and then explicate a vernacular 
”what follows from a concept”. 
Definition 7.
An expression E is homonymous, if E has more meanings, i.e., E expresses distinct concepts.
Expressions E1, E2 are synonymous, if they have the same meaning, i.e., E1 and E2 express one and the same concept C.
Expressions E1, E2 are equivalent, if they denote one and the same object O, i.e., concept C1 – meaning of E1, and concept C2 – meaning of E2 have the same extent, they construct O.
Empirical expressions E1 and E2 are coincident (co-referential), if they currently refer to the same object (in the actual world/time), i.e., Concepts C1 and C2 happen to have the same extent in the actual, current world/time.
Examples:
•	Coincident (but not equivalent) expressions: Morning Star, Evening Star.
They denote distinct offices / 
•	Equivalent (but not synonymous) expressions:
E1 – Charles likes only football fans
E2 – Charles does not like anybody who is not a football fan.
E3 – There is nobody such that Charles likes him and he is not a football fan.
E1, E2 and E3 are equivalent, they denote the same proposition P / , constructed by:

E1’ – wt x ([0Likewt 0Ch x]  [FootFanwt x])
E2’ – wt x ([FootFanwt x]  [0Likewt 0Ch x])
E3’ – wt x ([0Likewt 0Ch x]  [FootFanwt x])
Definition 8.
Each complex nonempty concept C is an ontological definition of its extent, i.e., concept C defines the object O constructed by C.
Example: Ontological definition of (the class of) prime numbers (/) is:
		x ( [0Nat x]  [0Card y ([0Nat y]  [0Div x y])] = 02 ) 
This is not a common way of using the term ‘definition’; there is no definiendum and definiens! By a ‘definition’ we usually understand the following schema:
Expression E1 (definiendum) =df  expression E2 (definiens).
From the logical point of view this is a linguistic definition. It associates the expression E1 (which is usually, but not necessarily, a simple expression that has not been used in the language till now) with the meaning of the expression E2. In other words, linguistic definition assigns a new meaning to E1, namely the ontological definition of the object denoted by E2, i.e., it makes the two expressions synonymous on the basis of linguistic convention. Thus simple expressions often do not express primitive simple concepts (trivialisation of a denoted object), but complex concepts, see also [21].
Example: Primes =df Numbers that have exactly two factors
	      Cat =df Domestic carnivorous animal, a feline, …
To answer the last question c), which means to explicate vernacular ”what follows from a concept”, we need to define the so-called requisites of a property (/of an office) and properties typical for a property (/for an office):
Definition 9. Let P, Q, G be any properties and U an office. We define:
[Reqpr P Q]  =  wt x [[Qwt x]  [Pwt x]]			(P is a requisite of Q)
[Reqof P U]  =  wt [[0Ewt U]  x [[Uwt = x]  [Pwt x]]]	
(P is a requisite of  U, where E is the property (of an office) of existence)
[TPpr P Q G] =  wt x [[Gwt x]  [[Qwt x]  [Pwt x]]]	
(P is typical for Q, unless G)
[TPof P U G]  =  wt [[0Ewt U]  x [[Gwt x]  [[Uwt = x]  [Pwt x]]] ]
			(P is typical for U, unless G)
Note: In artificial intelligence the condition G in the definition of a typical property is called the guard of a rule. The existential E / ( ) is a property of an office, returning True (in w,t) when the office is occupied (in this w,t). Generally, existential presupposition concerning the ‘office rules’ can be also conceived as a guard of the rule. Such an existential guard is indispensable in case of offices, because if the office U is vacant in some world/time w,t, then the composition Uwt, and thus the whole composition [[Uwt = x]  [Pwt x]], are v-improper, see Definition 2. 
	For instance, sentence The King of France is a ruler of France is ambiguous. It can be read as a true sentence claiming that being a ruler of France is a requisite of the King of France (which does not imply the existence of the King), or as a sentence claiming that the (current occupant of the office of the) King of France is a ruler of France, which does not have any truth value, because its existential presupposition is not met. If it were true or false, then it would imply that the King of France exists (see [28]).
	
Examples of typical properties are, e.g.:
A typical property of a bird is flying, unless it is a penguin or an ostrich.
A typical property of a swan is being white, unless it has been born in Australia or New Zealand.
	Now we can answer question c):
Definition 10. 
Let C be an empirical concept, the extent of which is an intension I, and let the ”population” of I in a world/time be Iwt. Then it follows from the concept C that:
Any member of the population Iwt has any property R such that R is a requisite of I, and has any property T such that T is typical for I, if the guard is satisfied.
Hence we can say that it follows from the concept of cat that my ”Minka” is a feline, or, it follows from the concept of spider that any spider has eight limbs, etc.
	Finally, we define a semantic partial ordering on the set of (equivalence classes of) concepts that can induce a conceptual lattice. 
Definition 11. 
Let C1 and C2 be empirical concepts such that C1 constructs a requisite R of the extent I constructed by C2. Then C1 is weaker than or equivalent to C2, denoted C1  C2. 
Claim: 
Let properties EC1, EC2 be extents of concepts C1, C2, respectively, such that C1  C2. Then necessarily, i.e., in all world/times w,t, EC2wt  EC1wt. 
Proof: Follows from the definition of requisites.
As a special case, if the number of all the requisites Ri of an intension I is finite, a concept C can construct I by means of ”conjuncting” Ri:
wt x ([R1wt x]  …  [Rnwt x]).
Then, obviously, for any Cj  such that 
Cj = wt x ([Ri1wt x]  …  [Rikwt x]), where {i1,…,ik}  {1,…,n} it holds that Cj  C and in any world/time the extent of C is a subset of the extent of Cj (ECwt  ECjwt), i.e., the law of inverse proportion is valid. 
We have explicated and obtained the classical conjunctive conception of Galois ordering, which is just a special case of our general definition.
	Another ”weaker semantic” ordering on the set of concepts can be defined on the basis of typical properties: Let C1 and C2 be empirical concepts such that C1 constructs a property T typical for the extent I constructed by C2. Then C1 G C2; 
Now the inverse relation between the extents in world/times holds only on the condition that the respective guards are satisfied (which can be formulated as some non-intrinsic integrity constraints, and their validity will be checked by the program): 
Claim: 
Let intensions EC1, EC2 be extents of concepts C1, C2, respectively, such that C1 G C2. Then a guarded rule is valid on the assumption that the respective guard G is satisfied (in world/time w,t): EC2wt  EC1wt (in case EC2 is an office and the office is not vacant, then EC2wt  EC1wt ).
Proof: Follows from the definition of requisites and typical properties.
5. Conclusion.
We presented logical apparatus together with the philosophical background of a new modern theory of concepts, and proved that this theory not only provides an exact explication of frequently used but vaguely ”defined” traditional terms like ‘concept’, ‘ontologies’, ‘conceptual lattice’, ‘semantic search’, etc., but that this theory is an essential extension of the traditional conception. Thus the simplest case, when the definition of the denoted ‘sought’ object is given in the form of a conjunction of basic traits (requisites in our terminology, or perhaps Bolzano’s ”Merkmale”) is just a special case of our general theory. Since such a conjunctive definition (though frequently used) is not the only possible one, and it is often the case that the object which is sought cannot be even defined in this way, our general theory may lead to a substantial qualitative extension of the methods of building particular domain ontologies, and provide a powerful tool supporting semantic search in the huge amount of data spread in web files. Last but not least, the TIL method of finding the analysis of an expression serves as a common unique tool for knowledge representation and acquisition. Knowledge can be not only described and specified in this language, but also particular queries on both explicit and implicit (deduced) knowledge can be specified (see [11]).
	You may wonder, which of the logical features provided by TIL are not covered by traditional systems. What follows is a brief summary specifying the essence of TIL extension as compared with particular traditional logics. Since Ganter-Wille and Kauppian theories are based on a classical predicate logic, this survey provides a comparison of a set-theoretical classical theory of concept vs. TIL-like procedural theory of concepts as well.
a)	Extensional systems of predicate logics (of any order) do not make it possible to distinguish between analytical and empirical expressions, i.e., empirical expressions the reference of which changes dependently on states of affairs (i.e. dependent on modal and/or temporal parameters) are not properly analysed. 
Possible recovery: Using some system of modal and/or temporal logic, or (preferably) intensional logic, e.g. Montagues’s system or TIL.
However, most of these systems (unlike TIL) do not distinguish between temporal and modal parameters.
b)	First order predicate logic does not make it possible to analyse mentioning of functions, propositions, properties and relations, which is to say, the distinction between using an expression (and its respective meaning - construction) in the de dicto vs. de re supposition is not rendered. 
Possible recovery: Using some system of higher order logic.
c)	Denotational approach to semantics (the meaning of an expression is conceived as a denoted object) does not take into account the mode of the presentation of the denoted object, and synonymous expressions are not distinguished from equivalent expressions. Therefore, the proper analysis of the so-called hyper-intensional contexts (of knowledge, beliefs, hypotheses) is a stumbling block for all the denotational semantics, including Montague’s systems. Since Frege’s times, logicians strive for logically treating structured meanings (to mention at least Russell’s structured propositions, Carnap’s intensional isomorphism – critised by Church and Tichý, Cresswell’s structured meanings, i.e. tuples – critised by Tichý and Jespersen, etc., etc.). 
Possible recovery: Using procedural declarative semantics that enables us not only to use concepts – constructions, but also to mention them.
d)	Formalistic approach to semantics does not make it possible to handle a rather fine-grained distinction between a representation of a construction and the construction itself.   
Possible recovery: Using a transparent (formal but anti-formalistic) approach.
Note: Items ad c) and d) are closely interrelated. The need to mention particular constructions can be (to some extend) met even in a formalistic logic using Gödel’s system of numbering formulas. However, a loyal knowledge representation should take into account attitudes to constructions.
e)	Classical systems of predicate logics do not make it possible to handle partial functions. This feature is particularly restrictive when analysing definite descriptions (like the greatest prime, the King of France, etc.) when we have to deal with value gaps (‘holes in reality’). The problems of (non-)existence and sentences with existential presuppositions become a stumbling block of classical logics.
Possible recovery: Using a system that can properly handle partial functions and ‘empty concepts’. There are many of them; the question is which of these is a proper one (from a mathematical point of view, computer scientist’s point of view, analytical-philosophy point of view, etc.). From the viewpoint of analysing natural language, perhaps the most promising are the systems that use the method of ‘partiality being propagated up’, which is adopted by TIL as well. 
We can see that from classical 1st-order predicate logic we would need to proceed to more and more expressive systems, which less or more still lack some expressiveness. An ideal apparatus should cover all the features mentioned above so to say ”under one hat” and without the necessity to use a meta-language. Such an ideal logic that would enable us to cover all the peculiarities of natural language is of course just a dream. Logic can hardly involve, e.g., pragmatic features of a language, particular emotional attitudes, poetic metaphors, context dependent anaphors and situation dependent indexicals, etc. Yet, under some simplification, considering a standard form of current language, we can claim that TIL is the system that meets all the needs we have stated above. In other words, TIL is an ideal system for logical analysis of a language. When performing logical analysis of a language, we also abstract of the ‘evolutionary features’ of the language and of the way in which the concept acquisition is performed.  In cognitive disciplines these features are important, and a form of Quine’s approach (“don’t ask for meaning, ask for using”) might be justified. 
Still, one of the problems that have to be solved is a practical applicability of the method in the web environment comprising a huge amount of heterogeneous documents. Adequate logical analysis of a natural language is only a necessary condition to meet the problem. But we also have to deal with reducing the ‘dimension of the problem’, which is usually not a subject of a logical theory. Anyway, a promising idea of using lattice theory promoting building ontologies in the area of ‘Information retrieval’ has been presented already in the work of Arents and Bogaerts [1]. In the Department of Computer Science of our university the Amphora Research Group (ARG) was founded in 2001, the intention of which is to form an organized research unit specialized in topics of Information Retrieval (IR) and other related disciplines. A software tool has been built up here, which exploits a generalized theory of ‘conceptual lattices’ [14], SVD (singular value decomposition) and other methods [27], [19], to explore web pages so as to ”quarry”, extract particular interesting domains together with their (limited) vocabularies.
	The next step to be done now is a linguistic one: it consists in a disambiguation of the obtained vocabulary, and creation of the so-called ‘intelligent thesaurus’, viz., a semantic dictionary in which each important term is provided with the ontological definition of the denoted object, i.e., the concept (logical construction) expressed by the expression is assigned to it. Using inference rules of the logical system (Gentzen’s natural deduction in case of TIL, or logical programming tools of resolution and unification), we can define a semantic partial ordering on the obtained set of concepts and create the respective conceptual lattice, which will make it possible to conduct a really semantic and at the same time effective ‘web-search’.
	Another promising area of future theoretical research is using soft-computing methods of reasoning and inferring implicit knowledge. TIL, being a higher-order logic, is not and cannot be, of course, complete. But Henkin’s completeness can be obtained in a ‘TIL-like (limited) theory’, and the first theoretical results obtained so far in the area of  ‘designing fuzzy TIL’ are presented by V. Novák in [24].

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This work has been supported by the grant project of GAČR No. 401/03/1403 – Principles of Logical Analysis

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