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Conservative intensional extension of Tarski's semantics.

1. Introduction

In "Uber Sinn und edeutung," Frege concentrated mostly on the senses of names, holding that all names have a sense (meaning). It is natural to hold that the same considerations apply to any expression that has an extension. But two general terms can have the same extension and different cognitive significance; two predicates can have the same extension and different cognitive significance; two sentences can have the same extension and different cognitive significance. So, general terms, predicates, and sentences all have senses as well as extensions. The same goes for any expression that has an extension or is a candidate for extension.

The significant aspect of an expression's meaning is its extension. We can stipulate that the extension of a sentence is its truth-value, and that the extension of a singular term is its referent. The extension of other expressions can be seen as associated entities that contribute to the truth-value of a sentence in a manner broadly analogous to the way in which the referent of a singular term contributes to the truth-value of a sentence. In many cases, the extension of an expression will be what we intuitively think of as its referent, although this need not hold in all cases. While Frege himself is often interpreted as holding that a sentence's referent is its truth-value, this claim is counterintuitive and widely disputed. We can avoid that issue in the present framework by using the technical term "extension." In this context, the claim that the extension of a sentence is its truth-value is a stipulation.

"Extensional" is most definitely a technical term. Say that the extension of a name is its denotation, the extension of a predicate is the set of things it applies to, and the extension of a sentence is its truth value. A logic is extensional if coextensional expressions can be substituted one for another in any sentence of the logic "salva veritate," that is, without a change in truth value. The intuitive idea behind this principle is that, in an extensional logic, the only logically significant notion of meaning that attaches to an expression is its extension. An intensional logics is exactly one in which substitutivity salva veritate fails for some of the sentences of the logic.

The first conception of intensional entities (or concepts) is built into the possible-worlds treatment of Properties, Relations, and Propositions (PRPs). This conception is commonly attributed to Leibniz and underlies Alonzo Church's alternative formulation of Frege's theory of senses ("A formulation of the logic of sense and denotation" in Henle, Kallen, and Langer, 3-24, and "Outline of a revised formulation of the logic of sense and denotation" in two parts, Nous, VII (1973), 24-33, and VIII, (1974), 135-156). This conception of PRPs is ideally suited for treating the modalities (necessity, possibility, etc.) and to Montague's definition of intension of a given virtual predicate [phi] ([x.sub.1], ..., [x.sub.k]) (a FOL open-sentence with the tuple of free variables ([x.sub.1], ... [x.sub.k])), as a mapping from possible worlds into extensions of this virtual predicate. Among the possible worlds, we distinguish the actual possible world. For example, if we consider a set of predicates, of a given Database, and their extensions in different time-instances, then the actual possible world is identified by the current instance of the time.

The second conception of intensional entities is to be found in Russell's doctrine of logical atomism. In this doctrine, it is required that all complete definitions of intensional entities be finite as well as unique and noncircular: it offers an algebraic way for definition of complex intensional entities from simple (atomic) entities (i.e., algebra of concepts), conception also evident in Leibniz's remarks. In a predicate logics, predicates and open-sentences (with free variables) express classes (properties and relations), and sentences express propositions. Note that classes (intensional entities) are reified, that is, they belong to the same domain as individual objects (particulars). This endows the intensional logics with a great deal of uniformity, making it possible to manipulate classes and individual objects in the same language. In particular, when viewed as an individual object, a class can be a member of another class.

The distinction between intensions and extensions is important (as in lexicography [1]), considering that extensions can be notoriously difficult to handle in an efficient manner. The extensional equality theory of predicates and functions under higher-order semantics (e.g., for two predicates with the same set of attributes, p = q is true iff these symbols are interpreted by the same relation), that is, the strong equational theory of intensions, is not decidable, in general. For example, the second-order predicate calculus and Church's simple theory of types, both under the standard semantics, are not even semi-decidable. Thus, separating intensions from extensions makes it possible to have an equational theory over predicate and function names (intensions) that is separate from the extensional equality of relations and functions.

Relevant recent work about the intension, and its relationship with FOL, has been presented in [2] in the consideration of rigid and nonrigid objects, with respect to the possible worlds, where the rigid objects, like "George Washington," are the same things from possible world to possible world. Nonrigid objects, like "the Secretary-General of United Nations," are varying from circumstance to circumstance and can be modeled semantically by functions from possible worlds to domain of rigid objects, like intensional entities. But in his approach, differently from that one, fitting changes also the syntax of the FOL, by introducing an "extension of" operator, J, in order to distinguish the intensional entity "gross-domestic-product-of-Denmark," and its use in "the gross domestic product of Denmark is currently greater than gross domestic product of Finland." In his approach, if x is an intensional variable, [down arrow] x is extensional, while [down arrow] is not applicable to extensional variables, differently from our where each variable (concept) has both intensional and extension. Moreover, in his approach the problem arises because the action of letting x designate, that is, evaluating [down arrow] x, and the action of passing to an alternative possible world, that is, of interpreting the existential modal operator 0, are not actions that commute. To disambiguate this, one more piece of machinery is needed as well, which substantially and ad-hock changes the syntax and semantics of FOL, introduces the Higher-order Modal logics, and is not a conservative extension of Tarski's semantics.

In most recent work in [3, 4] it is given an intensional version of first-order hybrid logic, which is also a hybridized version of Fitting's intensional FOL, by a kind of generalized models, thus, different from our approaches to conservative extension of Tarski's semantics to intensional FOL.

Another recent relevant work is presented by I-logic in [5], which combines both approach to semantics of intensional objects of Montague and Fitting.

We recall that Intensional Logic Programming is a new form of logic programming based on intensional logic and possible worlds semantics and is a well-defined practice in using the intensional semantics [6]. Intensional logic allows us to use logic programming to specify nonterminating computations and to capture the dynamic aspects of certain problems in a natural and problem-oriented style. The meanings of formulas of an intensional first-order language are given according to intensional interpretations and to elements of a set of possible worlds. Neighborhood semantics is employed as an abstract formulation of the denotations of intensional operators. The model-theoretic and fixpoint semantics of intensional logic programs are developed in terms of least (minimum) intensional Herbrand models. Intensional logic programs with intensional operator definitions are regarded as metatheories.

In what follows, we denote by [B.sup.A] the set of all functions from A to B, and by A" an n-folded cartesian product A x ... x A for n [greater than or equal to] 1. By f, t we denote empty set 0 and singleton set {<>}, respectively (with the empty tuple <> i.e., the unique tuple of 0-ary relation), which may be thought of as falsity f and truth t, as those used in the relational algebra. For a given domain D, we define that [D.sup.0] is a singleton set {(}}, so that {f, t} = P([D.sup.0]), where P is the powerset operator.

2. Intensional FOL Language with Intensional Abstraction

Intensional entities are such concepts as propositions and properties. The term "intensional" means that they violate the principle of extensionality, the principle that extensional equivalence implies identity. All (or most) of these intensional entities have been classified at one time or another as kinds of Universals [7].

We consider a nonempty domain D = [D.sub.-1] [union] [D.sub.I], where a subdomain [D.sub.-1] is made of particulars (extensional entities), and the rest [D.sub.I] = [D.sub.0] [union] [D.sub.I] ... [union] [D.sub.n] ... is made of universals ( [D.sub.0] for propositions (the 0-ary concepts)), and [D.sub.n], n [greater than or equal to] 1, for n-ary concepts.

The fundamental entities are intensional abstracts or so-called, that-clauses. We assume that they are singular terms; Intensional expressions like "believe," "mean," "assert," "know," are standard two-place predicates that take "that" clauses as arguments. Expressions like "is necessary," "is true," and "is possible" are one-place predicates that take "that"-clauses as arguments. For example, in the intensional sentence "it is necessary that [phi]," where [phi] is a proposition, the "that [phi]" is denoted by the <[phi]>, where [??] is the intensional abstraction operator, which transforms a logic formula into a term. Or, for example, "x believes that [phi]" is given by formula [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([p.sup.2.sub.i] is binary "believe" predicate).

Here we will present an intensional FOL with slightly different intensional abstraction than that originally presented in [8].

Definition 1. The syntax of the first-order logic language with intensional abstraction [??], denoted by L, is as follows:

logic operators ([conjunction], [logical not], [there exists]), predicate letters in P (functional letters is considered as particular case of predicate letters), variables x, y, z, ... in V, abstraction [??], and punctuation symbols (comma, parenthesis). With the following simultaneous inductive definition of term and formula,

(1) all variables and constants (0-ary functional letters in P)are terms;

(2) if [t.sub.1], ..., [t.sub.k] are terms, then [p.sup.k.sub.i] ([t.sub.1], ..., [t.sub.k]) is a formula ([p.sup.k.sub.i] [member of] P is a k-ary predicate letter);

(3) if [phi] and [psi] are formulae, then ([phi] [conjunction] [psi]), [logical not][phi], and ([there exists]x) [phi] are formulae;

(4) if [phi] (x) is a formula (virtual predicate) with a list of free variables in x = ([x.sub.1], ..., [x.sub.n]) (with ordering from-left-to-right of their appearance in 0), and a is its sublist of distinct variables, then [<[phi]>.sup.[beta].sub.[alpha]] is a term, where [beta] is the remaining list of free variables preserving ordering in x as well. The externally quantifiable variables are the free variables not in [alpha]. When n = 0, <[phi]> is a term that denotes a proposition, for n [greater than or equal to] 1 it denotes an nary concept.

An occurrence of a variable [x.sub.i] in a formula (or a term) is bound (free) if and only if it lies (does not lie) within a formula of the form ([there exists][x.sub.i]) [phi] (or a term of the form [<[phi]>.sup.[beta].sub.[alpha]] with [x.sub.i] [member of] [alpha]). A variable is free (bound) in a formula (or term) if and only if it has (does not have) a free occurrence in that formula (or term).

A sentence is a formula having no free variables. The binary predicate letter [p.sup.2.sub.1] for identity is singled out as a distinguished logical predicate, and formulae of the form [p.sup.2.sub.1] ([t.sub.1], [t.sub.2]) are to be rewritten in the form [t.sub.1] [??] [t.sub.2]. We denote by [R.sub.=] the binary relation obtained by standard Tarski's interpretation of this predicate [p.sup.2.sub.1]. The logic operators [for all], [disjunction], [??] are defined in terms of ([conjunction], [logical not], [there exists]) in the usual way.

Remark 2. The k-ary functional symbols, for k [greater than or equal to] 1, in standard (extensional) FOL are considered as (k + 1)-ary predicate symbols p +1:the function f: [D.sup.k] [right arrow] D is considered as a relation obtained from its graph R={([d.sub.1], ..., [d.sub.k], f([d.sub.1], ..., [d.sub.k])) | [d.sub.i] [member of] D}, represented by a predicate symbol [p.sup.k+1].

The universal quantifier is defined by [for all]=[logical not][there exists][logical not]. Disjunction and implication are expressed by [phi] [disjunction] [psi] = [logical not]([logical not][omega] [conjunction] [logical not][psi]) and [phi] [??] [psi] = [logical not][phi] [disjunction] [logical not][psi]. In FOL with the identity [??], the formula [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We denote by [R.sub.=] the Tarski's interpretation of [??].

In what follows, any open-sentence, a formula [phi] with nonempty tuple of free variables ([x.sub.1], ..., [x.sub.m]), will be called a m-ary virtual predicate, denoted also by [phi] ([x.sub.1], ..., [x.sub.m]). This definition contains the precise method of establishing the ordering of variables in this tuple: such a method that will be adopted here is the ordering of appearance, from left to right, of free variables in 0.This method of composing the tuple of free variables is the unique and canonical way of definition of the virtual predicate from a given for mula.

An intensional interpretation of this intensional FOL is a mapping between the set L of formulae of the logic language and intensional entities in D, I: L [right arrow] D, which is a kind of "conceptualization", such that an open-sentence (virtual predicate) [phi] ([x.sub.1], ..., [x.sub.k]) with a tuple of all free variables ([x.sub.1], ..., [x.sub.k]) is mapped into a k-ary concept, that is, an intensional entity u = I([phi]([x.sub.1], ..., [x.sub.k])) [member of] [D.sub.k], and (closed) sentence y into a proposition (i.e., logic concept) [upsilon] = I ([psi]) [member of] [D.sub.0] with I([??]) = Truth [member of] [D.sub.0] for a FOL tautology [??]. A language constant c is mapped into a particular (an extensional entity) a = I(c) [member of] [D.sub.-1] if it is a proper name, otherwise in a correspondent concept in D.

An assignment g: V [right arrow] D for variables in V is applied only to free variables in terms and formulae. Such an assignment g [member of] [D.sup.V] can be recursively uniquely extended into the assignment [g.sup.*] : T [right arrow] D, where T denotes the set of all terms (here I is an intensional interpretation of this FOL, as explained in what follows), by

(1) [g.sup.*] (t) = g(x) [member of] D if the term t is a variable x [member of] V;

(2) [g.sup.*] (t) = 1(c) [member of] D if the term t is a constant c [member of] P;

(3) if t is an abstracted term [<[phi]>.sup.[beta].sub.[alpha]], then [g.sup.*] ([<[phi]>.sup.[beta].sub.[alpha]]) = I ([phi][[beta]/g([beta])]) [member of] [D.sub.k], k = [absolute value of ([alpha])] (i.e., the number of variables in [alpha]), where g([beta]) = g([y.sub.1], ..., [y.sub.m]) = (g([y.sub.1])), ..., g([y.sub.m])) and [[beta]/g([beta])] is a uniform replacement of each tth variable in the list [beta] with the ith constant in the list g([beta]). Notice that a is the list of all free variables in the formula [phi][[beta]/g([beta])].

We denote by t/g (or [phi]/g) the ground term (or formula) without free variables, obtained by assignment g from a term t (or a formula [phi]), and by [phi] [x/t] the formula obtained by uniformly replacing x by a term t in [phi].

The distinction between intensions and extensions is important especially because we are now able to have and equational theory over intensional entities (as <[phi]>), that is, predicate and function "names," which is separate from the extensional equality of relations and functions. An extensionalization function h assigns to the intensional elements of D an appropriate extension as follows: for each proposition u [member of] [D.sub.0], h(u) [member of] {f, t} [subset or equal to] P([D.sub.- 1]) is its extension (true or false value); for each n-ary concept u [member of] [D.sub.n], h(u) is a subset of [D.sup.n] (nth Cartesian product of D); in the case of particulars u [member of] [D.sub.-1], h(u) = u.

The sets f, t are empty set {} and set {<>} (with the empty tuple (} [member of] [D.sub.-1], i.e., the unique tuple of 0-ary relation) which may be thought of as falsity and truth, as those used in the Codd's relational-database algebra [9], respectively, while Truth [member of] [D.sub.0] is the concept (intension) of the tautology.

We define that [D.sup.0] = {<>}, so that {f, t} = P([D.sup.0]), where P is the powerset operator. Thus we have (we denote the disjoint union by "+"):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1)

where [h.sub.-1] = id:[D.sub.-1] [right arrow] [D.sub.-1] is identity mapping, the mapping [h.sub.0] : [D.sub.0] [right arrow] {f, t} assigns the truth values in {f, t} to all propositions, and the mappings [h.sub.i] : [D.sub.i] [right arrow] P([D.sub.i]), i [greater than or equal to] 1, assign an extension to all concepts. Thus, the intensions can be seen as names of abstract or concrete entities, while the extensions correspond to various rules that these entities play in different worlds.

Remark 3 (Tarski's constraints). This intensional semantics has to preserve standard Tarski's semantics of the FOL. That is, for any formula [phi] [member of] L with a tuple of free variables ([x.sub.1], ..., [x.sub.k]), and h [member of] [??], the following conservative conditions for all assignments g, g' [member of] [D.sup.V] have to be satisfied:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, intensional FOL has a simple Tarski first-order semantics, with a decidable unification problem, but we need also the actual world mapping which maps any intensional entity to its actual world extension. In what follows, we will identify a possible world by a particular mapping which assigns, in such a possible world, the extensions to intensional entities. This is a direct bridge between an intensional FOL and a possible worlds representation [10-15], where the intension (meaning) of a proposition is a function, from a set of possible W worlds into the set of truth values. Consequently, W denotes the set of possible extensionalization functions h satisfying the constraint (T). Each h [member of] [??] may be seen as a possible world (analogously to Montague's intensional semantics for natural language [12, 14]), as it has been demonstrated in [16, 17]and given by the bijection is: W [equivalent] [??].

Now we are able to define formally this intensional semantics [15].

Definition 4. A two-step intensional semantics.

Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the set of all k-ary relations, where k [member of] N = {0, 1, 2, ...}. Notice that {f, t} = P([D.sup.0]) [member of] R, that is, the truth values are extensions in [Real part].

The intensional semantics of the logic language with the set of formulae L can be represented by the mapping

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2)

where [??] is a fixed intensional interpretation I: L [right arrow] D and [[??].sub.w[member of]W] is the set of all extensionalization functions h = is(w) : D [right arrow] [Real part] in [??], where is : W [right arrow] [??] is the mapping from the set of possible worlds to the set of extensionalization functions.

We define the mapping [I.sub.n] : [L.sub.op] [right arrow] [R.sup.W], where Lop is the subset of formulae with free variables (virtual predicates), such that for any virtual predicate [phi] ([x.sub.1], ..., [x.sub.k]) [member of] [L.sub.op] the mapping [I.sub.n] ([phi] ([x.sub.1], ..., [x.sub.k])) : W [right arrow] R is the Montague's meaning (i.e., intension) of this virtual predicate [10-14], that is, the mapping which returns with the extension of this (virtual) predicate in each possible world w [member of] W.

We adopted this two-step intensional semantics, instead of well-known Montague's semantics (which lies in the construction of a compositional and recursive semantics that covers both intension and extension), because of a number of weakness of the second semantics:

Example 5. Let us consider the following two past participles: "bought" and "sold" (with unary predicates [p.sup.1.sub.1] (x),"x has been bought", and [p.sup.1.sub.2] (x), "x has been sold"). These two different concepts in the Montague's semantics would have not only the same extension but also their intension, from the fact that their extensions are identical in every possible world.

Within the two-step formalism, we can avoid this problem by assigning two different concepts (meanings) u = I([p.sup.1.sub.1] (x)) and v = I([p.sup.1.sub.2] (x)) in [member of] [D.sub.1]. Note that we have the same problem in the Montague's semantics for two sentences with different meanings, which bear the same truth value across all possible worlds: in Montague's semantics, they will be forced to the same meaning.

Another relevant question with respect to this two-step interpretations of an intensional semantics is how in it the extensional identity relation [??] (binary predicate of the identity) of the FOL is managed. Here this extensional identity relation is mapped into the binary concept Id = I(= (x, y)) [member of] [D.sub.2], suchthat ([for all]W [member of] W)(is(w)(Id) = [R.sub.=]), where = (x, y) (i.e., [p.sup.2.sub.1] (x, y)) denotes an atom of the FOL of the binary predicate for identity in FOL, usually written by FOL formula x [??] y.

Note that here we prefer to distinguish this formal symbol [??] [member of] P of the built-in identity binary predicate letter in the FOL, from the standard mathematical symbol "=" used in all mathematical definitions in this paper.

In what follows, we will use the function [f.sub.()] : [Real part] [right arrow] [Real part], such that for any relation R [member of] [Real part], [f.sub.{)] (R) = {<>} if R [not equal to] 0; 0 otherwise. Let us define the following set of algebraic operators for relations in [Real part].

(1) Binary operator [[??].sub.s] : [Real part] x [Real part] [right arrow] , such that for any two relations [R.sub.1], [R.sub.2] [member of] [Real part], the [R.sub.1] [[??].sub.s] [R.sub.2] is equal to the relation obtained by natural join of these two relations if S is a nonempty set of pairs of joined columns of respective relations (where the first argument is the column index of the relation [R.sub.1] while the second argument is the column index of the joined column of the relation [R.sub.2]); other wise it is equal to the cartesian product [R.sub.1] x [R.sub.2].

For example, the logic formula [phi] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m]) [conjunction] [psi] ([x.sub.i], [y.sub.i], [x.sub.j], [y.sub.j]) will be traduced by the algebraic expression [R.sub.1] [[??].sub.s] [R.sub.2] where [R.sub.1] [member of] P([D.sup.5]), [R.sub.2] [member of] P([D.sup.4]) are the extensions for a given Tarski's interpretation of the virtual predicate [phi], [psi] relatively, so that S = {(4, 1), (2, 3)} and the resulting relation will have the following ordering of attributes: ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m], [y.sub.i], [y.sub.j]).

(2) Unary operator ~: [Real part] [right arrow] [Real part], such that for any k-ary (with k [greater than or equal to] 0) relation R [member of] P([D.sup.k]) [subset] R, we have that ~ (R) = [D.sup.k]\R [member of] [D.sup.k], where "\" is the substraction of relations. For example, the logic formula -[phi] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m]) willbe traduced by the algebraic expression [D.sup.5]\R where R is the extensions for a given Tarski's interpretation of the virtual predicate [phi].

(3) Unary operator [[pi].sub.-m] : [Real part] [right arrow] R, such that for any k-ary (with K [greater than or equal to] 0)relation R [member of] P([D.sup.k]) [subset] R, we have that [[pi].sub.-m] (R) is equal to the relation obtained by elimination of the mth column of the relation R if 1 [less than or equal to] m [less than or equal to] k and k [greater than or equal to] 2; equal to [f.sub.<>] (R) if m = k=1; otherwise it is equal to R.

For example, the logic formula ([there exists][x.sub.k]) [phi] ([x.sub.i], [x.sub.p], [x.sub.k], [x.sub.i], [x.sub.m]) will be traduced by the algebraic expression [[pi].sub.-3] (R) where R is the extensions for a given Tarski's interpretation of the virtual predicate 0 and the resulting relation will have the following ordering of attributes: ([x.sub.i], [x.sub.j], [x.sub.l], [x.sub.m]).

Notice that the ordering of attributes of resulting relations corresponds to the method used for generating the ordering of variables in the tuples of free variables adopted for virtual predicates.

Analogously to Boolean algebras, which are extensional models of propositional logic, we introduce now an intensional algebra for this intensional FOL, as follows.

Definition 6. Intensional algebra for the intensional FOL in Definition 1 is a structure [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with binary operations [conj.sub.s] : [D.sub.j] x[D.sub.I] [right arrow] [D.sub.i], unary operation neg : [D.sub.I] [right arrow] [D.sub.I], unary operations [exists.sub.n] : [D.sub.I] [right arrow] [D.sub.I], such that for any extensionalization function he W, and u [member of] [D.sub.k], [upsilon] [member of] [D.sub.j], k, j [greater than or equal to] 0,

(1) h(Id) = [R.sub.=] and h(Truth) = {(}}.

(2) h(conjs(u, v)) = h(u) [[??].sub.s] h(v), where [[??].sub.s] is the natural join operation defined above and [conj.sub.s] (u, v) [member of] [D.sub.m] where m = k+j - [absolute value of (S)] if for every pair ([i.sub.1], [i.sub.2]) [member of] S it holds that 1 [less than or equal to] [i.sub.1] [less than or equal to] k, 1 [less than or equal to] [i.sub.2] [less than or equal to] j (otherwise con [j.sub.s] (w, v) [member of] [D.sub.k+j]).

(3) h(neg(w)) =~(h(u)) = [D.sup.k]\(h(u)), where ~ is the operation defined above and neg(w) [member of] [D.sub.k].

(4) h([exists.sub.n] (u)) = [[pi].sub.-n] (h(u)), where [[pi].sub.-n] is the operation defined above and [exist.sub.sn] (u) [member of] [D.sub.k-1] if 1 [less than or equal to] n [less than or equal to] k (otherwise existsn is the identity function).

Notice that for u [member of] [D.sub.0], h(neg(u)) = ~(h(u)) = [D.sup.0]\ (h(u)) = {<>}\(h(u)) s[member of][micro]{f, t}.

We define a derived operation union :(P(D[s.sub.i)]\0) [right arrow] [D.sub.i], i [greater than or equal to] 0, such that, for any B = {[u.sub.1], ..., [u.sub.n]} [member of] P([D.sub.i]) we have that union({"1 , ..., un})=de{u1 if n=1;neg(conjs (neg([u.sub.1]), [conj.sub.s] (..., neg([u.sub.n])) ...), where S = {(1, 1)] 1 [less than or equal to] I [less than or equal to] i}, otherwise. Than we obtain that for n [greater than or equal to] 2,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

Intensional interpretation I: L [right arrow] D satisfies the following homomorphic extension.

(1) The logic formula [phi] ([x.sub.i] [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m]) [conjunction] [psi] ([x.sub.l], [y.sub.i], [x.sub.j], [y.sub.j]) will be intensionally interpreted by the concept [u.sub.1] [member of] [D.sub.7], obtained by the algebraic expression [conj.sub.s] (u, [upsilon]) where u = I([phi] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m])) [member of] [D.sub.5], v = I([phi]([x.sub.i], [y.sub.i], [x.sub.j], [y.sub.j])) [member of] [D.sub.4] are the concepts of the virtual predicates [phi], [psi], relatively, and S = {(4, 1), (2, 3)}.Consequently, we have that for any two formulae [phi], [psi] [member of] L and a particular operator conjs uniquely determined by tuples of free variables in these two formulae, I([phi] [conjunction] [psi]) = [conj.sub.S] (I([phi]), I([psi])).

(2) The logic formula [logical not] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m]) will be intensionally interpreted by the concept [u.sub.1] [member of] [D.sub.5], obtained by the algebraic expression neg(w) where u = I([phi]([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m])) [member of] [D.sub.5] is the concept of the virtual predicate [phi]. Consequently, we have that for any formula [phi] [member of] L, I([logical not][phi]) = neg(I([phi])).

(3) The logic formula ([there exists][x.sub.k]) [phi] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m]) will be intensionally interpreted by the concept [u.sub.1] [member of] [D.sub.4], obtained by the algebraic expression [exists.sub.3] (u) where u = I(([phi] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m])) [member of] [D.sub.5] is the concept of the virtual predicate [phi]. Consequently, we have that for any formula [phi] [member of] L and a particular operator [exists.sub.n] uniquely determined by the position of the existentially quantified variable in the tuple of free variables in [phi] (otherwise n=0 if this quantified variable is not a free variable in 0), I(([there exists]x)([phi]) = [exists.sub.n] (I(([phi])).

Once one has found a method for specifying the interpretations of singular terms of L (take in consideration the particularity of abstracted terms), the Tarski-style definitions of truth and validity for L may be given in the customary way. What is proposed specifically is a method for characterizing the intensional interpretations of singular terms of L in such away that a given singular abstracted term [<[phi]>.sup.[beta].sub.[alpha]] will denote an appropriate property, relation, or proposition, depending on the value of m = [absolute value of ([alpha])]. Thus, the mapping of intensional abstracts (terms) into D will be defined differently from that given in the version of Bealer [18], as follows.

Definition 7. An intensional interpretation I can be extended to abstracted terms as follows: for any abstracted term [<[phi]>.sup.[beta].sub.[alpha]], we define that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4)

where [bar.[beta]] denotes the set of elements in the list [beta], and the assignments in [D.sup.[bar.[beta]]] are limited only to the variables in [bar.[beta]].

Rema k 8. Here we can make the question if there is a sense to extend the interpretation also to (abstracted) terms, because in Tarski's interpretation of FOL we do not have any interpretation for terms, but only the assignments for terms, as we defined previously by the mapping [g.sup.*] : T [right arrow] D. The answer is positive, because the abstraction symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be considered as a kind of the unary built-in functional symbol of intensional FOL, so that we can apply the Tarski's interpretation to this functional symbol into the fixed mapping [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], so that for any [phi] [member of] L we have that I([<[phi]>.sup.[beta].sub.[alpha]]) is equal to the application of this function to the value [phi], that is, to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In such an approach, we would introduce also the typed variable X for the formulae in L so that the Tarski's assignment for this functional symbol with variable X, with g(X) = [phi] [member of] L, can be given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Notice than if [beta] = 0 is the empty list, then I([<[phi]>.sup.[beta].sub.[alpha]]) = I([phi]). Consequently, the denotation of <[phi]> is equal to the meaning of a proposition [phi], that is, I(<[phi]>) = I([phi]) [member of] [D.sub.0]. In the case when [phi] is an atom [p.sup.m.sub.i] ([x.sub.1], ..., [x.sub.m]), then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] while [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For example,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

The interpretation of a more complex abstract [<[phi]>.sup.[beta].sub.[alpha]] is defined in terms of the interpretations of the relevant syntactically simpler expressions, because the interpretation of more complex formulae is defined in terms of the interpretation of the relevant syntactically simpler formulae, based on the intensional algebra above. For example, I ([p.sup.1.sub.i] (x) [conjunction] [p.sup.i.sub.k] (x)) = [conj.sub.{(lt1)}] (I([p.sub.1.sub.i] (x)), I([p.sup.1.sub.k] (x))), I([logical not][phi]) = neg([phi]), I([there exists][x.sub.i]) [phi] ([x.sub.i], [x.sub.j], [x.sub.l], [x.sub.k]) = [exist.sub.3] (I(([phi])).

Consequently, based on the intensional algebra in Definition 6 and on intensional interpretations of abstracted terms in Definition 7, it holds that the interpretation of any formula in L (and any abstracted term) will be reduced to an algebraic expression over interpretations of primitive atoms in L. This obtained expression is finite for any finite formula (or abstracted term) and represents the meaning of such finite formula (or abstracted term).

The extension of an abstracted term satisfy the following property.

Proposition 9. For any abstracted term [<[phi]>.sup.[beta].sub.[alpha]] with [absolute value of ([alpha])] [greater than or equal to] 1, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the sequential composition of functions, and [[pi].sub.-0] is an identity.

Proof. Let x be a tuple of all free variables in [phi], so that [bar.x] = [bar.[alpha]] [union] [bar.[beta]], [alpha] = ([x.sub.1], ..., [x.sub.k]), then we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], from Definition 7 = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We can correlate with a possible-world semantics. Such a correspondence is a natural identification of intensional logics with modal Kripke-based logics.

Definition 10 (model). A model for intensional FOL with fixed intensional interpretation I, which expresses the two-+step intensional semantics in Definition 4, is the Kripke structure [M.sub.int] = (W, D, V), where W = {[is.sup.-1] (h) | h [member of] W}, a mapping [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with P a set of predicate symbols of the language, such that for any world w = [is.sup.-1] (h) [member of] W, [p.sup.n.sub.i] [member of] P, and ([u.sub.1], ..., [u.sub.n]) [member of] [D.sup.n] it holds that V(w, [p.sup.n.sub.])([u.sub.1], ..., [u.sub.n]) = h(I([p.sup.n.sub.i] ([u.sup.1], ..., [u.sub.n]))). The satisfaction relation [[??].sub.wg] for a given w [member of] W and assignment g [member of] [D.sup.V] is defined as follows:

(1) [M[??].sub.w,g] [p.sup.k.sub.i] ([x.sub.1], ..., [x.sub.k]) if and only if V(w, [p.sup.k.sub.i]) (g([x.sub.1]), ..., g([x.sub.k])) = t,

(2) [M[??].sub.w,g] [phi] [conjunction] [phi] if and only if [M[??].sub.w,g] [phi] and [M[??].sub.w,g] [phi],

(3) [M[??].sub.w,g] [logical not][phi] if and only if not [M[??].sub.w,g][phi],

(4) [M[??].sub.w,g] ([there exists]x) [phi] if and only if

(4.1) [M[??].sub.w,g] [phi], if x is not a free variable in [phi];

(4.2) exists u [member of] D such that [M[??].sub.w,g] [phi] [x/u], if x is a free variable in [phi].

It is easy to show that the satisfaction relation N for this Kripke semantics in a world w = [is.sup.-1] (h) is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if h(I([phi]/g)) = t.

We can enrich this intensional FOL by another modal operators, as, for example, the "necessity" universal logic operator a with accessibility relation R = W x W, obtaining an S5 Kripke structure [M.sub.int] = (W, R, D, V). In this case, we are able to define the following equivalences between the abstracted terms without free variables [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where all free variables (not in a)are instantiated by g [member of] [D.sup.V] (here A = B denotes the formula (A [??] B) [conjunction] (B [??] A)).

(i) (Strong) Intensional equivalence (or equality) "[??]" is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if [??] ([phi] [[[beta].sub.1]/g([[beta].sub.1])] = [psi] ([[beta].sub.2]/g [[beta].sub.2])]), with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if for all w [member of] W, (w, w') [member of] R implies [less than or equal to]. From Example 5, we have that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], that is, "x has been bought" and "has been sold" are intensionally equivalent, but they have not the same meaning (the concept I([p.sup.1.sub.1] (x)) [member of] [D.sub.1] is different from I([p.sup.1.sub.2] (x)) [member of] [D.sub.1]).

(ii) Weak intensional equivalence "[??]" is defined by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The symbol [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the correspondent existential modal operator. This weak equivalence is used for P2P database integration in a number of papers [16, 19-24].

Note that if we want to use the intensional equality in our language, then we need the correspondent operator in intensional algebra Aint for the "necessity" modal logic operator a.

This semantics is equivalent to the algebraic semantics for L in [8] for the case of the conception where intensional entities are considered to be equal if and only if they are necessarily equivalent. Intensional equality is much stronger that the standard extensional equality in the actual world, just because it requires the extensional equality in all possible worlds; in fact, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all extensionalization functions h [member of] [??] (i.e., possible worlds [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]).

It is easy to verify that the intensional equality means that in every possible world w [member of] [??] the intensional entities [u.sub.1] and [u.sub.2] have the same extensions.

Let the logic modal formula [??][phi][[[beta].sub.1]/g([[beta].sub.1])], where the assignment is applied only to free variables in [[beta].sub.1] of a formula not in the list of variables in [alpha] = ([x.sub.1], ..., [x.sub.n]), n [greater than or equal to] 1, representsan n-ary intensional concept such that I([??][phi][[[beta].sub.1]/g([[beta].sub.1])]) [member of] [D.sub.n] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then the extension of this n-ary concept is equal to (here the mapping necess : [D.sub.i] [right arrow] [D.sub.i] for each i [greater than or equal to] 0 is a new operation of the intensional algebra [A.sub.int] in Definition 6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

while

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

Consequently, the concepts [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the built-in (or rigid) concept as well, whose extensions do not depend on possible worlds.

Thus, two concepts are intensionally equal, that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for every h.

Analogously, two concepts are weakly equivalent, that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

3. Application to the Intensional FOL without Abstraction Operator

In the case of the intensional FOL defined in Definition 1, without Bealer's intensional abstraction operator <>, we obtain the syntax of the standard FOL but with intensional semantics as presented in [15].

Such a FOL has a well-known Tarski's interpretation, defined as follows.

An interpretation (Tarski) IT consists in a nonempty domain D and a mapping that assigns to any predicate letter [p.sup.k.sub.i] [member of] P a relation R = [I.sub.T] ([p.sup.k.sub.i]) [subset or equal to] [D.sup.k], to any functional letter [f.sup.k.sub.i] [member of] F a function [I.sub.T] ([f.sub.k]): [D.sup.k] [right arrow] D, or, equivalently, its graph relation R = [I.sub.T] ([f.sup.k.sub.i]) [subset or equal to] [D.sup.k+1] where the th column is the resulting function's value, and to each individual constant ceF one given element [I.sub.T] (c) [member of] D.

Consequently, from the intensional point of view, an interpretation of Tarski is a possible world in the Montague's intensional semantics, that is, w = [I.sub.T] [member of] W. The correspondent extensionalization function is h = is(w) = is([I.sub.T]).

We define the satisfaction of a logic formulae in L for a given assignment g: V [right arrow] D inductively, as follows.

If a formula [phi] is an atomic formula [p.sup.k.sub.i] ([t.sub.1], ..., [t.sub.k]), then this assignment g satisfies [phi] if and only if ([g.sup.*] ([t.sub.1]), ..., [g.sup.*] ([t.sub.k])) [member of] [I.sub.T] ([p.sup.k.sub.i]); g satisfies [logical not][phi] if and only if it does not satisfy [phi]; g satisfies [phi] [conjunction] iff g satisfies [phi] and g satisfies [phi]; g satisfies ([there exists][x.sub.i]) [phi] if and only if there exists an assignment g' [member of] [D.sup.V] that may differ from g only for the variable [x.sub.i] [member of] V, and g' satisfies [phi].

A formula [phi] is true for a given interpretation [I.sub.T] if and only if [phi] is satisfied by every assignment g [member of] [D.sup.V]. A formula [phi] is valid (i.e., tautology) if and only if 0 is true for every Tarksi's interpretation [I.sub.T] [member of] [Imaginary part]T. An interpretation [I.sub.T] is a model of a set of formulae [GAMMA] if and only if every formula [phi] [member of] [GAMMA] is true in this interpretation. We denote by FOL([GAMMA]) the FOL with a set of assumptions [GAMMA], and by [[Imaginary part].sub.T] ([GAMMA]) the subset of Tarski's interpretations that are models of r, with [[Imaginary part].sub.T] (0) = [[Imaginary part].sub.T]. A formula [phi] is said to be a logical consequence of [GAMMA], denoted by [[GAMMA].sub.[parallel]-0], if and only if [phi] is true in all interpretations in [[Imaginary part].sub.T] ([GAMMA]). Thus, [parallel]-[phi] if and only if [phi] is a tautology.

The basic set of axioms of the FOL are that of the propositional logic with two additional axioms: (A1) ([for all]x)([phi] [??] y) = ([phi] [??] ([for all]x)[psi]) (x does not occur in [phi] and it is not bound in [psi]), and (A2) ([for all]x) [phi] [??] [phi] [x/t], (neither x nor any variable in t occurs bound in [phi]). For the FOL with identity, we need the proper axiom (A3) [x.sub.1] [??] [x.sub.2] [??] ([x.sub.1] [??] [x.sub.3] [??] [x.sub.2] [??] [x.sub.3]).

The inference rules are Modus Ponens and generalization (G) "if [phi] is a theorem and x is not bound in [phi], then ([for all]x)[phi] is a theorem."

The standard FOL is considered as an extensional logic because two open sentences with the same tuple of variables [phi]([x.sub.1], ..., [x.sub.m]) and [psi] ([x.sub.1], ..., [x.sub.m]) are equal if and only if they have the same extension in a given interpretation 7T, that is, if and only if [I.sup.*.sub.] ([phi]([x.sub.1], ..., [x.sub.m])) = [I.sup.*.sub.T] ([psi]([x.sub.1], ..., [x.sub.m])), where [I.sup.*.sub.T] is the unique extension of [I.sub.T] to all formulae, as follows.

(1) For a (closed) sentence [phi]/g, wehavethat [I.sup.*.sub.T] ([phi]/g) = t if and only if g satisfies [phi], as recursively defined above.

(2) For an open-sentence [phi] with the tuple of free variables ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to verify that for a formula [phi] with the tuple of free variables ([x.sub.1], ..., [x.sub.m]), [I.sup.*.sub.T] (([phi]([x.sub.1], ..., [x.sub.m])/g) = t if and only if (g([x.sub.1]), ..., g([x.sub.m])) [member of] [I.sup.*.sub.T] ([phi]([x.sub.1], ..., [x.sub.m])).

This extensional equality of two virtual predicates can be generalized to the extensional equivalence when both predicates [phi], [psi] have the same set of free variables but their ordering in the tuples of free variables is not identical: such two virtual predicates are equivalent if the extension of the first is equal to the proper permutation of columns of the extension of the second virtual predicate. It is easy to verify that such an extensional equivalence corresponds to the logical equivalence denoted by [phi] [equivalent to] [psi].

This extensional equivalence between two relations [R.sub.1], [R.sub.2] [member of] [Real part] with the same arity will be denoted by [R.sub.1] [congruent to] [R.sub.2], while the extensional identity will be denoted in the standard way by [R.sub.1] = [R.sub.2].

Let [A.sub.FOL] = (L, [??], [perpendicular to], [conjunction], [logical not], [there exists]) be a free syntax algebra for "first-order logic with identity [??]," with the set L of first-order logic formulae, with [??] denoting the tautology formula (the contradiction formula is denoted by [logical not][??]), with the set of variables in V and the domain of values in D. It is well known that we are able to make the extensional algebraization of the FOL by using the cylindric algebras [25] that are the extension of Boolean algebras with a set of binary operators for the FOL identity relations and a set of unary algebraic operators ("projections") for each case of FOL quantification ([there exists]x). In what follows, we will make an analog extensional algebraization over [Real part] but by interpretation of the logic conjunction A by a set of natural join operators over relations introduced by Codd's relational algebra [9]and [26] as a kind of a predicate calculus whose interpretations are tied to the database.

Corollary 11 (extensional FOL semantics [15]). Let us define the extensional relational algebra for the FOL by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where {< >} [member of] [Real part] is the algebraic value correspondent to the logic truth and [R.sub.=] is the binary relation for extensionally equal elements. We will use "=" for the extensional identity for relations in [Real part].

Then, for any Tarski's interpretation IT its unique extension to all formulae [I.sub.*.sub.T] : L [right arrow] [Real part] is also the homomorphism [I.sup.*.sub.T]: [A.sub.FOL] [right arrow] [A.sub.[Real part]] from the free syntax FOL algebra into this extensional relational algebra.

Proof. Directly from definition of the semantics of the operators in [A.sub.[Real] part] defined in precedence, let us take the case of conjunction of logic formulae of the definition above where [phi] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m], [y.sub.i], [y.sub.j]) (its tuple of variables is obtained by the method defined in the FOL introduction) is the virtual predicate of the logic formula [phi] ([x.sub.i], [x.sub.j], [x.sub.k], [x.sub.l], [x.sub.m]) [conjunction] [psi] ([x.sub.l], [y.sub.i], [x.sub.j], [y.sub.j]): [I.sup.*.sub.T] ([phi] [conjunction] [psi]) = [I.sup.*.sub.T] ([phi]) = {(g([x.sub.i]), g([x.sub.j]), g([x.sub.k]), g([x.sub.l]), g([x.sub.m]), g([y.sub.i]), g([y.sub.j])) | [I.sup.*.sub.T] ([phi]/g) = t} = {(g([x.sub.i]), g([x.sub.j]), g([x.sub.k]), g([x.sub.l]), g([x.sub.m]), g([y.sub.i]), g([y.sub.j])) | [I.sup.*] ([phi]/g [conjunction] [psi]/g) = t} = {(g([x.sub.i]), g([x.sub.j]), g([x.sub.k]), g([x.sub.l]), g([x.sub.m]), g([y.sub.i]), g([y.sub.j])) | [I.sup.*.sub.T] ([phi]/g) = t and [I.sup.*.sub.T] ([phi]/g) = t} = {(g ([x.sub.i]), g ([x.sub.j]), g([x.sub.k]), g([x.sub.l]), g([x.sub.m]), g([y.sub.i]), g([y.sub.j])) | (g([x.sub.i]), g([x.sub.j]), g([x.sub.k]), g([x.sub.l]), g([x.sub.m])) [member of] [I.sup.*.sub.T] ([phi]) and (g([x.sub.l]), g([y.sub.i]), g([x.sub.j]), g([y.sub.j])) [member of] [I.sup.*.sub.T] ([phi])} = [I.sup.*.sub.T] ([phi]) [[??].sub.{(4,1),(2,3)}] [I.sup.*.sub.T] ([psi]).

Thus, it is enough to show that [I.sup.*.sub.T] ([??]) = {(}} is also valid, and [I.sup.*.sub.T] ([logical not]T) = 0. The first property comes from the fact that T is a tautology, thus satisfied by every assignment g, that is, it is true, that is, [I.sup.*.sub.T] ([??]) = t (and t is equal to the empty tuple {<>}). The second property comes from the fact that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Thatis, thetautology andthe contradictionhavethe true and false logic value, respectively, in [Real part].

We have also that [I.sup.*.sub.T] ([??] (x, y)) = [I.sub.T] (=) = [R.sub.=] for every interpretation IT because = is the built-in binary predicate, that is, with the same extension in every Tarski's interpretation.

Consequently, the mapping [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [A.sub.[Real] part] is a homomorphism that represents the extensional Tarskian semantics of the FOL.

Consequently, we obtain the following Intensional/extensional FOL semantics [15].

For any Tarski's interpretation IT of the FOL, the following diagram of homomorphisms commutes.

(11)

where h = is(w) and w = [I.sub.T] [member of] W is the explicit possible world (extensional Tarski's interpretation).

This homomorphic diagram formally expresses the fusion of Frege's and Russell's semantics [27-29] of meaning and denotation of the FOL language and renders mathematically correct the definition of what we call an "intuitive notion of intensionality," in terms of which a language is intensional if denotation is distinguished from sense: that is, if both denotation and sense are ascribed to its expressions. This notion is simply adopted from Frege's contribution (without its infinite sense-hierarchy, avoided by Russell's approach where there is only one meaning relation, one fundamental relation between words and things, here represented by one fixed intensional interpretation I), where the sense contains mode of presentation (here described algebraically as an algebra of concepts (intensions) [A.sub.int]), and where sense determines denotation for any given extensionalization function h (correspondent to a given Traski's interpretation IT). More about the relationships between Frege's and Russell's theories of meaning may be found in the Chapter 7,"Extensionality and Meaning", in [18].

As noted by Gottlob Frege and Rudolf Carnap (he uses terms Intension/extension in the place of Frege's terms sense/denotation [30]), the two logic formulae with the same denotation (i.e., the same extension for a given Tarski's interpretation [I.sub.T]) need not have the same sense (intension), thus such codenotational expressions are not substitutable in general.

In fact there is exactly one sense (meaning) of a given logic formula in L, defined by the uniquely fixed intensional interpretation 7, and a set of possible denotations (extensions) each determined by a given Tarski's interpretation of the FOL as follows from Definition 4:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

Often "intension" has been used exclusively in connection with possible worlds semantics; however, here we use (as many others; as Bealer for example) "intension" in a more wide sense, that is, as an algebraic expression in the intensional algebra of meanings (concepts) [A.sub.int], which represents the structural composition of more complex concepts (meanings) from the given set of atomic meanings. Consequently, not only the denotation (extension) is compositional, but also the meaning (intension) is compositional.

4. Conclusion

Semantics is a theory concerning the fundamental relations between words and things. In Tarskian semantics of the FOL, one defines what it takes for a sentence in a language to be truely relative to a model. This puts one in a position to define what it takes for a sentence in a language to be valid. Tarskian semantics often proves quite useful in logic. Despite this, Tarskian semantics neglects meaning, as if truth in language were autonomous. Because of that the Tarskian theory of truth becomes inessential to the semantics for more expressive logics, or more "natural" languages.

Both Montague's and Bealer's approaches were useful for this investigation of the intensional FOL with intensional abstraction operator, but the first is not adequate and explains why we adopted two-step intensional semantics (intensional interpretation with the set of extensionalization functions).

At the end of this work, we defined an extensional algebra for the FOL (different from standard cylindric algebras) and the commutative homomorphic diagram that expresses the generalization of the Tarskian theory of truth for the FOL into the Frege/Russell's theory of meaning.

http://dx.doi.org/10.1155/2013/920157

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Zoran Majkic

International Society for Research in Science and Technology, P. O. Box 2464,

Tallahassee, FL 32316-2464, USA

Correspondence should be addressed to Zoran Majkic; majk.1234@yahoo.com

Received 30 May 2012; Revised 12 October 2012; Accepted 23 October 2012

Academic Editor: Konstantinos Lefkimmiatis
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Title Annotation:Research Article
Author:Majkic, Zoran
Publication:Advances in Artificial Intelligence
Article Type:Report
Date:Jan 1, 2013
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