Conservative Intensional Extension of Tarski's Semantics

Abstract

We considered an extension of the first-order logic (FOL) by Bealer's intensional abstraction operator. Contemporary use of the term “intension” derives from the traditional logical Frege-Russell doctrine that an idea (logic formula) has both an extension and an intension. Although there is divergence in formulation, it is accepted that the “extension” of an idea consists of the subjects to which the idea applies, and the “intension” consists of the attributes implied by the idea. From the Montague's point of view, the meaning of an idea can be considered as particular extensions in different possible worlds. In the case of standard FOL, we obtain a commutative homomorphic diagram, which is valid in each given possible world of an intensional FOL: from a free algebra of the FOL syntax, into its intensional algebra of concepts, and, successively, into an extensional relational algebra (different from Cylindric algebras). Then we show that this composition corresponds to the Tarski's interpretation of the standard extensional FOL in this possible world.

1. Introduction

In “Über 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 𝜙(𝑥₁,…,𝑥_𝑘) (a FOL open-sentence with the tuple of free variables (𝑥₁,…𝑥_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, ↓, 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, ↓𝑥 is extensional, while ↓ 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 ↓𝑥, and the action of passing to an alternative possible world, that is, of interpreting the existential modal operator ⋄, 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^𝐴 the set of all functions from 𝐴 to 𝐵, and by 𝐴^𝑛 an 𝑛-folded cartesian product 𝐴×⋯×𝐴 for 𝑛≥1. By 𝑓,𝑡 we denote empty set ∅ 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 𝒟⁰ is a singleton set {⟨⟩}, so that {𝑓,𝑡}=𝒫(𝒟⁰), 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₋₁⋃𝐷_𝐼, where a subdomain 𝐷₋₁ is made of particulars (extensional entities), and the rest 𝐷_𝐼=𝐷₀⋃𝐷₁⋯⋃𝐷_𝑛⋯ is made of universals (𝐷₀ for propositions (the 0-ary concepts)), and 𝐷_𝑛,𝑛≥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 𝜙,” where 𝜙 is a proposition, the “that 𝜙” is denoted by the ⋖𝜙⋗, where ⋖⋗ is the intensional abstraction operator, which transforms a logic formula into a term. Or, for example, “𝑥 believes that 𝜙” is given by formula 𝑝²_𝑖(𝑥,⋖𝜙⋗) (𝑝²_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 (∧,¬,∃), 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 𝑡₁,…,𝑡_𝑘 are terms, then 𝑝^𝑘_𝑖(𝑡₁,…,𝑡_𝑘) is a formula (𝑝^𝑘_𝑖∈𝑃 is a 𝑘-ary predicate letter);(3)if 𝜙 and 𝜓 are formulae, then (𝜙∧𝜓), ¬𝜙, and (∃𝑥)𝜙 are formulae;(4)if 𝜙(𝐱) is a formula (virtual predicate) with a list of free variables in 𝐱=(𝑥₁,…,𝑥_𝑛) (with ordering from-left-to-right of their appearance in 𝜙), and 𝛼 is its sublist of distinct variables, then ⋖𝜙⋗^𝛽_𝛼 is a term, where 𝛽 is the remaining list of free variables preserving ordering in 𝐱 as well. The externally quantifiable variables are the free variables not in 𝛼. When 𝑛=0,⋖𝜙⋗ is a term that denotes a proposition, for 𝑛≥1 it denotes an 𝑛-ary concept.
An occurrence of a variable 𝑥_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 (∃𝑥_𝑖)𝜙 (or a term of the form ⋖𝜙⋗^𝛽_𝛼 with 𝑥_𝑖∈𝛼). 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 𝑝²₁ for identity is singled out as a distinguished logical predicate, and formulae of the form 𝑝²₁(𝑡₁,𝑡₂) are to be rewritten in the form 𝑡₁≐𝑡₂. We denote by 𝑅₌ the binary relation obtained by standard Tarski’s interpretation of this predicate 𝑝²₁. The logic operators ∀,∨,⇒ are defined in terms of (∧,¬,∃) in the usual way.

Remark 2. The 𝑘-ary functional symbols, for 𝑘≥1, in standard (extensional) FOL are considered as (𝑘+1)-ary predicate symbols 𝑝^𝑘+1: the function 𝑓∶𝒟^𝑘→𝒟 is considered as a relation obtained from its graph 𝑅={(𝑑₁,…,𝑑_𝑘,𝑓(𝑑₁,…,𝑑_𝑘))|𝑑_𝑖∈𝒟}, represented by a predicate symbol 𝑝^𝑘+1.
The universal quantifier is defined by ∀=¬∃¬. Disjunction and implication are expressed by 𝜙∨𝜓=¬(¬𝜙∧¬𝜓) and 𝜙⇒𝜓=¬𝜙∨𝜓. In FOL with the identity ≐, the formula (∃₁𝑥)𝜙(𝑥) denotes the formula (∃𝑥)𝜙(𝑥)∧(∀𝑥)(∀𝑦)(𝜙(𝑥)∧𝜙(𝑦)⇒(𝑥≐𝑦)). We denote by 𝑅₌ the Tarski’s interpretation of ≐.
In what follows, any open-sentence, a formula 𝜙 with nonempty tuple of free variables (𝑥₁,…,𝑥_𝑚), will be called a 𝑚-ary virtual predicate, denoted also by 𝜙(𝑥₁,…,𝑥_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 𝜙. This method of composing the tuple of free variables is the unique and canonical way of definition of the virtual predicate from a given formula.
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→𝒟, which is a kind of “conceptualization”, such that an open-sentence (virtual predicate) 𝜙(𝑥₁,…,𝑥_𝑘) with a tuple of all free variables (𝑥₁,…,𝑥_𝑘) is mapped into a 𝑘-ary concept, that is, an intensional entity 𝑢=𝐼(𝜙(𝑥₁,…,𝑥_𝑘))∈𝐷_𝑘, and (closed) sentence 𝜓 into a proposition (i.e., logic concept) 𝑣=𝐼(𝜓)∈𝐷₀ with 𝐼(⊤)=𝑇𝑟𝑢𝑡ℎ∈𝐷₀ for a FOL tautology ⊤. A language constant 𝑐 is mapped into a particular (an extensional entity) 𝑎=𝐼(𝑐)∈𝐷₋₁ if it is a proper name, otherwise in a correspondent concept in 𝒟.
An assignment 𝑔∶𝒱→𝒟 for variables in 𝒱 is applied only to free variables in terms and formulae. Such an assignment 𝑔∈𝒟^𝒱 can be recursively uniquely extended into the assignment 𝑔^∗∶𝒯→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^∗(𝑡)=𝑔(𝑥)∈𝒟 if the term 𝑡 is a variable 𝑥∈𝒱;(2)𝑔^∗(𝑡)=𝐼(𝑐)∈𝒟 if the term 𝑡 is a constant 𝑐∈𝑃;(3)if 𝑡 is an abstracted term ⋖𝜙⋗^𝛽_𝛼, then 𝑔^∗(⋖𝜙⋗^𝛽_𝛼)=𝐼(𝜙[𝛽/𝑔(𝛽)])∈𝐷_𝑘,𝑘=|𝛼| (i.e., the number of variables in 𝛼), where 𝑔(𝛽)=𝑔(𝑦₁,…,𝑦_𝑚)=(𝑔(𝑦₁),…,𝑔(𝑦_𝑚)) and [𝛽/𝑔(𝛽)] is a uniform replacement of each 𝑖th variable in the list 𝛽 with the 𝑖th constant in the list 𝑔(𝛽). Notice that 𝛼 is the list of all free variables in the formula 𝜙[𝛽/𝑔(𝛽)].
We denote by 𝑡/𝑔 (or 𝜙/𝑔) the ground term (or formula) without free variables, obtained by assignment 𝑔 from a term 𝑡 (or a formula 𝜙), and by 𝜙[𝑥/𝑡] the formula obtained by uniformly replacing 𝑥 by a term 𝑡 in 𝜙.
The distinction between intensions and extensions is important especially because we are now able to have and equational theory over intensional entities (as ⋖𝜙⋗), 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∈𝐷₀, ℎ(𝑢)∈{𝑓,𝑡}⊆𝒫(𝐷₋₁) is its extension (true or false value); for each 𝑛-ary concept 𝑢∈𝐷_𝑛, ℎ(𝑢) is a subset of 𝒟^𝑛 (𝑛th Cartesian product of 𝒟); in the case of particulars 𝑢∈𝐷₋₁, ℎ(𝑢)=𝑢.
The sets 𝑓,𝑡 are empty set {} and set {⟨⟩} (with the empty tuple ⟨⟩∈𝐷₋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∈𝐷₀ is the concept (intension) of the tautology.
We define that 𝒟⁰={⟨⟩}, so that {𝑓,𝑡}=𝒫(𝒟⁰), where 𝒫 is the powerset operator. Thus we have (we denote the disjoint union by “+”): ℎ=ℎ₋₁+_𝑖≥0ℎ_𝑖∶_𝑖≥−1𝐷_𝑖⟶𝐷₋₁+_𝑖≥0𝒫𝐷^𝑖, (1) where ℎ₋₁=𝑖𝑑∶𝐷₋₁→𝐷₋₁ is identity mapping, the mapping ℎ₀∶𝐷₀→{𝑓,𝑡} assigns the truth values in {𝑓,𝑡} to all propositions, and the mappings ℎ_𝑖∶𝐷_𝑖→𝒫(𝐷^𝑖), 𝑖≥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 𝜙∈ℒ with a tuple of free variables (𝑥₁,…,𝑥_𝑘), and ℎ∈ℰ, the following conservative conditions for all assignments 𝑔,𝑔′∈𝒟^𝒱 have to be satisfied: (T)ℎ(𝐼(𝜙/𝑔))=𝑡 if and only if (𝑔(𝑥₁),…,𝑔(𝑥_𝑘))∈ℎ(𝐼(𝜙)) and if 𝜙 is a predicate letter 𝑝^𝑘, 𝑘≥2 which represents a (𝑘−1)-ary functional symbol 𝑓^𝑘−1 in standard FOL,(TF)ℎ(𝐼(𝜙/𝑔))=ℎ(𝐼(𝜙/𝑔^′))=𝑡, and ∀_{1≤𝑖≤𝑘−1}(𝑔^′(𝑥_𝑖)=𝑔(𝑥_𝑖)) implies 𝑔^′(𝑥_𝑘+1)=𝑔(𝑥_𝑘+1).

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, ℰ denotes the set of possible extensionalization functions ℎ satisfying the constraint (T). Each ℎ∈ℰ 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≃ℰ.

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

Definition 4. A two-step intensional semantics.
Let ℜ=⋃_𝑘∈ℕ𝒫(𝒟^𝑘)=∑_𝑘∈ℕ𝒫(𝐷^𝑘) be the set of all 𝑘-ary relations, where 𝑘∈ℕ={0,1,2,…}. Notice that {𝑓,𝑡}=𝒫(𝒟⁰)∈ℜ, that is, the truth values are extensions in ℜ.
The intensional semantics of the logic language with the set of formulae ℒ can be represented by the mapping ℒ−−→^𝐼𝒟⟹_𝑤∈𝒲ℜ, (2) where →−^𝐼 is a fixed intensional interpretation 𝐼∶ℒ→𝒟 and ⇒_𝑤∈𝒲 is the set of all extensionalization functions ℎ=𝑖𝑠(𝑤)∶𝒟→ℜ in ℰ, where 𝑖𝑠∶𝒲→ℰ is the mapping from the set of possible worlds to the set of extensionalization functions.
We define the mapping 𝐼_𝑛∶ℒ_𝑜𝑝→ℜ^𝒲, where ℒ_𝑜𝑝 is the subset of formulae with free variables (virtual predicates), such that for any virtual predicate 𝜙(𝑥₁,…,𝑥_𝑘)∈ℒ_𝑜𝑝 the mapping 𝐼_𝑛(𝜙(𝑥₁,…,𝑥_𝑘))∶𝒲→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∈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 𝑝¹₁(𝑥), “𝑥 has been bought”, and 𝑝¹₂(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(𝑝¹₁(𝑥)) and 𝑣=𝐼(𝑝¹₂(𝑥)) in ∈𝐷₁. 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))∈𝐷₂, such that (∀𝑤∈𝒲)(𝑖𝑠(𝑤)(𝐼𝑑)=𝑅₌), where ≐(𝑥,𝑦) (i.e., 𝑝²₁(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 ≐∈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_⟨⟩∶ℜ→ℜ, such that for any relation 𝑅∈ℜ, 𝑓_⟨⟩(𝑅)={⟨⟩} if 𝑅≠∅; ∅ otherwise. Let us define the following set of algebraic operators for relations in ℜ.(1) Binary operator ⋈_𝑆∶ℜ×ℜ→ℜ, such that for any two relations 𝑅₁,𝑅₂∈ℜ, the 𝑅₁⋈_𝑆𝑅₂ 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 𝑅₁ while the second argument is the column index of the joined column of the relation 𝑅₂); otherwise it is equal to the cartesian product 𝑅₁×𝑅₂. For example, the logic formula 𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚)∧𝜓(𝑥_𝑙,𝑦_𝑖,𝑥_𝑗,𝑦_𝑗) will be traduced by the algebraic expression 𝑅₁⋈_𝑆𝑅₂ where 𝑅₁∈𝒫(𝒟⁵),𝑅₂∈𝒫(𝒟⁴) are the extensions for a given Tarski’s interpretation of the virtual predicate 𝜙,𝜓 relatively, so that 𝑆={(4,1),(2,3)} and the resulting relation will have the following ordering of attributes: (𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚,𝑦_𝑖,𝑦_𝑗). (2) Unary operator ∼∶ℜ→ℜ, such that for any 𝑘-ary (with 𝑘≥0) relation 𝑅∈𝒫(𝒟^𝑘)⊂ℜ, we have that ∼(𝑅)=𝒟^𝑘⧵𝑅∈𝒟^𝑘, where “⧵” is the substraction of relations. For example, the logic formula ¬𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚) will be traduced by the algebraic expression 𝒟⁵⧵𝑅 where 𝑅 is the extensions for a given Tarski’s interpretation of the virtual predicate 𝜙. (3) Unary operator 𝜋_−𝑚∶ℜ→ℜ, such that for any 𝑘-ary (with 𝑘≥0) relation 𝑅∈𝒫(𝒟^𝑘)⊂ℜ, we have that 𝜋_−𝑚(𝑅) is equal to the relation obtained by elimination of the 𝑚th column of the relation 𝑅 if 1≤𝑚≤𝑘 and 𝑘≥2; equal to 𝑓_⟨⟩(𝑅) if 𝑚=𝑘=1; otherwise it is equal to 𝑅.  For example, the logic formula (∃𝑥_𝑘)𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚) will be traduced by the algebraic expression 𝜋₋₃(𝑅) where 𝑅 is the extensions for a given Tarski’s interpretation of the virtual predicate 𝜙 and the resulting relation will have the following ordering of attributes: (𝑥_𝑖,𝑥_𝑗,𝑥_𝑙,𝑥_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 𝒜_int = (𝒟,𝑓,𝑡,𝐼𝑑,𝑇𝑟𝑢𝑡ℎ,{con𝑗_𝑆}_{𝑆∈𝒫(ℕ}²₎,neg,{exist𝑠_𝑛}_𝑛∈ℕ), with binary operations con𝑗_𝑆∶𝐷_𝐼×𝐷_𝐼→𝐷_𝐼, unary operation neg∶𝐷_𝐼→𝐷_𝐼, unary operations exist𝑠_𝑛∶𝐷_𝐼→𝐷_𝐼, such that for any extensionalization function ℎ∈ℰ, and 𝑢∈𝐷_𝑘,𝑣∈𝐷_𝑗, 𝑘,𝑗≥0,(1)ℎ(𝐼𝑑)=𝑅₌ and ℎ(𝑇𝑟𝑢𝑡ℎ)={⟨⟩}.(2)ℎ(con𝑗_𝑆(𝑢,𝑣))=ℎ(𝑢)⋈_𝑆ℎ(𝑣), where ⋈_𝑆 is the natural join operation defined above and con𝑗_𝑆(𝑢,𝑣)∈𝐷_𝑚 where 𝑚=𝑘+𝑗−|𝑆| if for every pair (𝑖₁,𝑖₂)∈𝑆 it holds that 1≤𝑖₁≤𝑘, 1≤𝑖₂≤𝑗 (otherwise con𝑗_𝑆(𝑢,𝑣)∈𝐷_𝑘+𝑗).(3)ℎ(neg(𝑢))=∼(ℎ(𝑢))=𝒟^𝑘⧵(ℎ(𝑢)), where ∼ is the operation defined above and neg(𝑢)∈𝐷_𝑘.(4)ℎ(exist𝑠_𝑛(𝑢))=𝜋_−𝑛(ℎ(𝑢)), where 𝜋_−𝑛 is the operation defined above and exist𝑠_𝑛(𝑢)∈𝐷_𝑘−1 if 1≤𝑛≤𝑘 (otherwise exist𝑠_𝑛 is the identity function).
Notice that for 𝑢∈𝐷₀, ℎ(neg(𝑢))=∼(ℎ(𝑢))=𝒟⁰⧵(ℎ(𝑢))={⟨⟩}⧵(ℎ(𝑢))∈{𝑓,𝑡}.
We define a derived operation union∶(𝒫(𝐷_𝑖)⧵∅)→𝐷_𝑖, 𝑖≥0, such that, for any 𝐵={𝑢₁,…,𝑢_𝑛}∈𝒫(𝐷_𝑖) we have that union({𝑢₁,…,𝑢_𝑛})=_def𝑢₁ if 𝑛=1; neg(con𝑗_𝑆(neg(𝑢₁),con𝑗_𝑆(…,neg(𝑢_𝑛))…), where 𝑆={(𝑙,𝑙)|1≤𝑙≤𝑖}, otherwise. Than we obtain that for 𝑛≥2,ℎ(union(𝐵))=ℎnegcon𝑗_𝑆neg𝑢₁,con𝑗_𝑆…,neg𝑢_𝑛…=𝒟^𝑖⧵𝒟^𝑖⧵ℎ𝑢₁⋈_𝑆⋯⋈_𝑆𝒟^𝑖⧵ℎ𝑢_𝑛=𝒟^𝑖⧵𝒟^𝑖⧵ℎ𝑢₁⋯𝒟^𝑖⧵ℎ𝑢_𝑛=ℎ𝑢_𝑗∣1≤𝑗≤𝑛={ℎ(𝑢)∣𝑢∈𝐵}. (3)
Intensional interpretation 𝐼∶ℒ→𝒟 satisfies the following homomorphic extension.(1)The logic formula 𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚)∧𝜓(𝑥_𝑙,𝑦_𝑖,𝑥_𝑗,𝑦_𝑗) will be intensionally interpreted by the concept 𝑢₁∈𝐷₇, obtained by the algebraic expression con𝑗_𝑆(𝑢,𝑣) where 𝑢=𝐼(𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚))∈𝐷₅,𝑣=𝐼(𝜓(𝑥_𝑙,𝑦_𝑖,𝑥_𝑗,𝑦_𝑗))∈𝐷₄ are the concepts of the virtual predicates 𝜙,𝜓, relatively, and 𝑆={(4,1),(2,3)}. Consequently, we have that for any two formulae 𝜙,𝜓∈ℒ and a particular operator con𝑗_𝑆 uniquely determined by tuples of free variables in these two formulae, 𝐼(𝜙∧𝜓)=con𝑗_𝑆(𝐼(𝜙),𝐼(𝜓)). (2)The logic formula ¬𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚) will be intensionally interpreted by the concept 𝑢₁∈𝐷₅, obtained by the algebraic expression neg(𝑢) where 𝑢=𝐼(𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚))∈𝐷₅ is the concept of the virtual predicate 𝜙. Consequently, we have that for any formula 𝜙∈ℒ, 𝐼(¬𝜙)=neg(𝐼(𝜙)). (3)The logic formula (∃𝑥_𝑘)𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚) will be intensionally interpreted by the concept 𝑢₁∈𝐷₄, obtained by the algebraic expression exist𝑠₃(𝑢) where 𝑢=𝐼(𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚))∈𝐷₅ is the concept of the virtual predicate 𝜙. Consequently, we have that for any formula 𝜙∈ℒ and a particular operator exist𝑠_𝑛 uniquely determined by the position of the existentially quantified variable in the tuple of free variables in 𝜙 (otherwise 𝑛=0 if this quantified variable is not a free variable in 𝜙), 𝐼((∃𝑥)𝜙)=exist𝑠_𝑛(𝐼(𝜙)).
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 a way that a given singular abstracted term ⋖𝜙⋗^𝛽_𝛼 will denote an appropriate property, relation, or proposition, depending on the value of 𝑚=|𝛼|. 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 ⋖𝜙⋗^𝛽_𝛼, we define that 𝐼⋖𝜙⋗^𝛽_𝛼𝐼𝜙𝛽=union𝑔𝛽∣𝑔∈𝒟^𝛽, (4) where 𝛽 denotes the set of elements in the list 𝛽, and the assignments in 𝒟^𝛽 are limited only to the variables in 𝛽.

Remark 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^∗∶𝒯→𝒟. The answer is positive, because the abstraction symbol ⋖_⋗^𝛽_𝛼 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 𝐼(⋖_⋗^𝛽_𝛼)∶ℒ→𝒟, so that for any 𝜙∈ℒ we have that 𝐼(⋖𝜙⋗^𝛽_𝛼) is equal to the application of this function to the value 𝜙, that is, to 𝐼(⋖_⋗^𝛽_𝛼)(𝜙). 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)=𝜙∈ℒ, can be given by 𝑔^∗⋖_⋗^𝛽_𝛼(𝑋)=𝐼⋖_⋗^𝛽_−𝛼𝑔(𝑋)=𝐼⋖_⋗^𝛽_𝛼𝜙=⟨⟩∈𝐷₋₁,if𝛼⋃𝐼𝜙𝛽𝛽isnotequaltothesetoffreevariablesin𝜙;=union𝑔^′𝛽∣𝑔^′∈𝒟^𝛽∈𝐷_|𝛼|,otherwise. (5)

Notice than if 𝛽=∅ is the empty list, then 𝐼(⋖𝜙⋗^𝛽_𝛼)=𝐼(𝜙). Consequently, the denotation of ⋖𝜙⋗ is equal to the meaning of a proposition 𝜙, that is, 𝐼(⋖𝜙⋗)=𝐼(𝜙)∈𝐷₀. In the case when 𝜙 is an atom 𝑝^𝑚_𝑖(𝑥₁,…,𝑥_𝑚), then 𝐼(⋖𝑝^𝑚_𝑖(𝑥₁,…,𝑥_𝑚)⋗_{𝑥1,…,𝑥𝑚}) = 𝐼(𝑝^𝑚_𝑖(𝑥₁,…,𝑥_𝑚))∈𝐷_𝑚, while 𝐼(⋖𝑝^𝑚_𝑖(𝑥₁,…,𝑥_𝑚)⋗^{𝑥1,…,𝑥𝑚}) = union({𝐼(𝑝^𝑚_𝑖(𝑔(𝑥₁),…,𝑔(𝑥_𝑚)))|𝑔∈𝒟^{{𝑥1,…,𝑥𝑚}}})∈𝐷₀, with ℎ(𝐼(⋖𝑝^𝑚_𝑖(𝑥₁,…,𝑥_𝑚)⋗^{𝑥1,…,𝑥𝑚})) = ℎ(𝐼((∃𝑥₁)⋯(∃𝑥_𝑚) 𝑝^𝑚_𝑖(𝑥₁,…,𝑥_𝑚))) ∈ {𝑓,𝑡}.

For example, ℎ𝐼⋖𝑝¹_𝑖𝑥₁∧¬𝑝¹_𝑖𝑥₁⋗^𝑥1=ℎ𝐼∃𝑥₁⋖𝑝¹_𝑖𝑥₁∧¬𝑝¹_𝑖𝑥₁⋗^𝑥1=𝑓. (6)

The interpretation of a more complex abstract ⋖𝜙⋗^𝛽_𝛼 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(𝑝¹_𝑖(𝑥)∧𝑝¹_𝑘(𝑥))=con𝑗_{(1,1)}(𝐼(𝑝¹_𝑖(𝑥)),𝐼(𝑝¹_𝑘(𝑥))), 𝐼(¬𝜙)=neg(𝐼(𝜙)), 𝐼(∃𝑥_𝑖)𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑖,𝑥_𝑘) = exist𝑠₃(𝐼(𝜙)).

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 ⋖𝜙⋗^𝛽_𝛼 with |𝛼|≥1, we have that ℎ𝐼⋖𝜙⋗^𝛽_𝛼=𝜋_−𝛽ℎ𝐼𝜙, (7) where 𝜋_{−(𝑦1,…,𝑦𝑘)}=𝜋_−𝑦1∘⋯∘𝜋_−𝑦1, ∘ is the sequential composition of functions, and 𝜋_−∅ is an identity.

Proof . Let 𝐱 be a tuple of all free variables in 𝜙, so that 𝐱=𝛼⋃𝛽, 𝛼=(𝑥₁,…,𝑥_𝑘), then we have that ℎ(𝐼(⋖𝜙⋗^𝛽_𝛼)) = ℎ(union({𝐼(𝜙[𝛽/𝑔(𝛽)])∣𝑔∈𝒟^𝛽})), from Definition 7  = ⋃{ℎ(𝐼(𝜙[𝛽/𝑔(𝛽)]))∣𝑔∈𝒟^𝛽} = ⋃{{(𝑔₁(𝑥₁),…,𝑔₁(𝑥_𝑘))∣𝑔₁∈𝒟^𝛼andℎ(𝐼(𝜙[𝛽/𝑔(𝛽)][𝛼/𝑔₁(𝛼)]))=𝑡}∣𝑔∈𝒟^𝛽} = {𝑔₁(𝛼)∣𝑔₁∈𝒟^𝛼⋃𝛽  and  ℎ(𝐼(𝜙/𝑔₁))=𝑡}=𝜋_−𝛽({𝑔₁(𝐱)∣𝑔₁∈𝒟^𝐱  and  ℎ(𝐼(𝜙/𝑔₁))=𝑡}) = 𝜋_−𝛽({𝑔₁(𝐱)∣𝑔₁∈𝒟^𝐱  and  𝑔₁(𝐱)∈ℎ(𝐼(𝜙))}), by (𝐓) = 𝜋_−𝛽(ℎ(𝐼(𝜙))).

We can correlate ℰ 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_int=(𝒲,𝒟,𝑉), where 𝒲={𝑖𝑠⁻¹(ℎ)|ℎ∈ℰ}, a mapping 𝑉∶𝒲×𝑃→⋃_𝑛<𝜔{𝑡,𝑓}^𝒟𝑛, with 𝑃 a set of predicate symbols of the language, such that for any world 𝑤=𝑖𝑠⁻¹(ℎ)∈𝒲,𝑝^𝑛_𝑖∈𝑃, and (𝑢₁,…,𝑢_𝑛)∈𝒟^𝑛 it holds that 𝑉(𝑤,𝑝^𝑛_𝑖)(𝑢₁,…,𝑢_𝑛)=ℎ(𝐼(𝑝^𝑛_𝑖(𝑢₁,…,𝑢_𝑛))). The satisfaction relation ⊨_𝑤,𝑔 for a given 𝑤∈𝒲 and assignment 𝑔∈𝒟^𝒱 is defined as follows:(1)ℳ⊨_𝑤,𝑔𝑝^𝑘_𝑖(𝑥₁,…,𝑥_𝑘) if and only if 𝑉(𝑤,𝑝^𝑘_𝑖)(𝑔(𝑥₁),…,𝑔(𝑥_𝑘))=𝑡,(2)ℳ⊨_𝑤,𝑔𝜑∧𝜙 if and only if ℳ⊨_𝑤,𝑔𝜑 and ℳ⊨_𝑤,𝑔𝜙, (3)ℳ⊨_𝑤,𝑔¬𝜑 if and only if not ℳ⊨_𝑤,𝑔𝜑,(4)ℳ⊨_𝑤,𝑔(∃𝑥)𝜙 if and only if (4.1)ℳ⊨_𝑤,𝑔𝜙, if 𝑥 is not a free variable in 𝜙;(4.2)exists 𝑢∈𝒟 such that ℳ⊨_𝑤,𝑔𝜙[𝑥/𝑢], if 𝑥 is a free variable in 𝜙.
It is easy to show that the satisfaction relation ⊨ for this Kripke semantics in a world 𝑤=𝑖𝑠⁻¹(ℎ) is defined by ℳ⊨_𝑤,𝑔𝜙 if and only if ℎ(𝐼(𝜙/𝑔))=𝑡.
We can enrich this intensional FOL by another modal operators, as, for example, the “necessity” universal logic operator □ with accessibility relation ℛ=𝒲×𝒲, obtaining an S5 Kripke structure ℳ_int=(𝒲,ℛ,𝒟,𝑉). In this case, we are able to define the following equivalences between the abstracted terms without free variables ⋖𝜙⋗^𝛽1_𝛼/𝑔 and ⋖𝜓⋗^𝛽2_𝛼/𝑔, where all free variables (not in 𝛼) are instantiated by 𝑔∈𝒟^𝒱 (here 𝐴≡𝐵 denotes the formula (𝐴⇒𝐵)∧(𝐵⇒𝐴)).(i)(Strong) Intensional equivalence (or equality) “ ≍” is defined by ⋖𝜙⋗^𝛽1_𝛼/𝑔≍⋖𝜓⋗^𝛽2_𝛼/𝑔 if and only if □(𝜙[𝛽₁/𝑔(𝛽₁)]≡𝜓[𝛽₂/𝑔(𝛽₂)]), with ℳ⊨_𝑤,𝑔′□𝜑 if and only if for all 𝑤^′∈𝒲, (𝑤,𝑤^′)∈ℛ implies ℳ⊨_{𝑤′,𝑔′}𝜑. From Example 5, we have that ⋖𝑝¹₁(𝑥)⋗_𝑥≍⋖𝑝¹₂(𝑥)⋗_𝑥, that is, “𝑥 has been bought” and “𝑥 has been sold” are intensionally equivalent, but they have not the same meaning (the concept 𝐼(𝑝¹₁(𝑥))∈𝐷₁ is different from 𝐼(𝑝¹₂(𝑥))∈𝐷₁). (ii)Weak intensional equivalence “ ≈” is defined by ⋖𝜙⋗^𝛽1_𝛼/𝑔≈⋖𝜓⋗^𝛽2_𝛼/𝑔 if and only if ◊𝜙[𝛽₁/𝑔(𝛽₁)]≡◊𝜓[𝛽₂/𝑔(𝛽₂)]. The symbol ◊=¬□¬ 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 𝒜_int for the “necessity” modal logic operator □.

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 ⋖𝜙⋗^𝛽1_𝛼/𝑔≍⋖𝜓⋗^𝛽1_𝛼/𝑔, then ℎ(𝐼(⋖𝐴⋗^𝛽1_𝛼/𝑔))=ℎ(𝐼(⋖𝜓⋗^𝛽2_𝛼/𝑔)) for all extensionalization functions ℎ∈ℰ (i.e., possible worlds 𝑖𝑠⁻¹(ℎ)∈𝒲).

It is easy to verify that the intensional equality means that in every possible world 𝑤∈𝒲 the intensional entities 𝑢₁ and 𝑢₂ have the same extensions.

Let the logic modal formula □𝜙[𝛽₁/𝑔(𝛽₁)], where the assignment 𝑔 is applied only to free variables in 𝛽₁ of a formula 𝜙 not in the list of variables in 𝛼=(𝑥₁,…,𝑥_𝑛), 𝑛≥1, represents an 𝑛-ary intensional concept such that 𝐼(□𝜙[𝛽₁/𝑔(𝛽₁)])∈𝐷_𝑛 and 𝐼(𝜙[𝛽₁/𝑔(𝛽₁)])=𝐼(⋖𝜙⋗^𝛽1_𝛼/𝑔)∈𝐷_𝑛. Then the extension of this 𝑛-ary concept is equal to (here the mapping necess∶𝐷_𝑖→𝐷_𝑖 for each 𝑖≥0 is a new operation of the intensional algebra 𝒜_int in Definition 6) ℎ𝐼𝛽□𝜙¹𝑔𝛽₁𝐼𝜙𝛽=ℎnecess¹𝑔𝛽₁=𝑔^′𝑥₁,…,𝑔^′𝑥_𝑛∣ℳ⊨_𝑤,𝑔′𝛽□𝜙¹𝑔𝛽₁,𝑔^′∈𝒟^𝒱=𝑔^′𝑥₁,…,𝑔^′𝑥_𝑛∣𝑔^′∈𝒟^𝒱,∀𝑤₁𝑤,𝑤₁∈ℛimpliesℳ⊨_{𝑤1,𝑔′}𝜙𝛽₁𝑔𝛽₁=_ℎ1∈ℰℎ₁𝐼𝜙𝛽₁𝑔𝛽₁, (8)

while ℎ𝐼𝛽◊𝜙¹𝑔𝛽₁𝐼𝛽=ℎ¬□¬𝜙¹𝑔𝛽₁𝐼𝛽=ℎnegnecess¬𝜙¹𝑔𝛽₁=𝒟^𝑛𝐼𝛽⧵ℎnecess¬𝜙¹𝑔𝛽₁=𝒟^𝑛⧵_ℎ1∈ℰℎ₁𝐼𝛽¬𝜙¹𝑔𝛽₁=𝒟^𝑛⧵_ℎ1∈ℰℎ₁𝐼𝜙𝛽𝑛𝑒𝑔¹𝑔𝛽₁=𝒟^𝑛⧵_ℎ1∈ℰ𝒟^𝑛⧵ℎ₁𝐼𝜙𝛽₁𝑔𝛽₁=_ℎ1∈ℰℎ₁𝐼𝜙𝛽₁𝑔𝛽₁. (9)

Consequently, the concepts □𝜙[𝛽₁/𝑔(𝛽₁)] and ◊𝜙[𝛽₁/𝑔(𝛽₁)] 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, ⋖𝜙⋗^𝛽1_𝛼/𝑔≍⋖𝜓⋗^𝛽2_𝛼/𝑔, if and only if ℎ(𝐼(𝜙[𝛽₁/𝑔(𝛽₁)]))=ℎ(𝐼(𝜓[𝛽₂/𝑔(𝛽₂)])) for every ℎ.

Analogously, two concepts are weakly equivalent, that is, ⋖𝜙⋗^𝛽1_𝛼/𝑔≈⋖𝜓⋗^𝛽2_𝛼/𝑔 if and only if ℎ(𝐼(◊𝜙[𝛽₁/𝑔(𝛽₁)]))=ℎ(𝐼(◊𝜓[𝛽₂/𝑔(𝛽₂)])).

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) 𝐼_𝑇 consists in a nonempty domain 𝒟 and a mapping that assigns to any predicate letter 𝑝^𝑘_𝑖∈𝑃 a relation 𝑅=𝐼_𝑇(𝑝^𝑘_𝑖)⊆𝒟^𝑘, to any functional letter 𝑓^𝑘_𝑖∈𝐹 a function 𝐼_𝑇(𝑓^𝑘_𝑖)∶𝒟^𝑘→𝒟, or, equivalently, its graph relation 𝑅=𝐼_𝑇(𝑓^𝑘_𝑖)⊆𝒟^𝑘+1 where the 𝑘+1th column is the resulting function’s value, and to each individual constant 𝑐∈𝐹 one given element 𝐼_𝑇(𝑐)∈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_𝑇∈𝒲. The correspondent extensionalization function is ℎ=𝑖𝑠(𝑤)=𝑖𝑠(𝐼_𝑇). We define the satisfaction of a logic formulae in ℒ for a given assignment 𝑔∶𝒱→𝒟 inductively, as follows. If a formula 𝜙 is an atomic formula 𝑝^𝑘_𝑖(𝑡₁,…,𝑡_𝑘), then this assignment 𝑔 satisfies 𝜙 if and only if (𝑔^∗(𝑡₁),…,𝑔^∗(𝑡_𝑘))∈𝐼_𝑇(𝑝^𝑘_𝑖); 𝑔 satisfies ¬𝜙 if and only if it does not satisfy 𝜙; 𝑔 satisfies 𝜙∧𝜓 iff 𝑔 satisfies 𝜙 and 𝑔 satisfies 𝜓; 𝑔 satisfies (∃𝑥_𝑖)𝜙 if and only if there exists an assignment 𝑔′∈𝒟^𝒱 that may differ from 𝑔 only for the variable 𝑥_𝑖∈𝒱, and 𝑔^′ satisfies 𝜙. A formula 𝜙 is true for a given interpretation 𝐼_𝑇 if and only if 𝜙 is satisfied by every assignment 𝑔∈𝒟^𝒱. A formula 𝜙 is valid (i.e., tautology) if and only if 𝜙 is true for every Tarksi’s interpretation 𝐼_𝑇∈ℑ_𝑇. An interpretation 𝐼_𝑇 is a model of a set of formulae Γ if and only if every formula 𝜙∈Γ is true in this interpretation. We denote by FOL(Γ) the FOL with a set of assumptions Γ, and by ℑ_𝑇(Γ) the subset of Tarski’s interpretations that are models of Γ, with ℑ_𝑇(∅)=ℑ_𝑇. A formula 𝜙 is said to be a logical consequence of Γ, denoted by Γ⊩𝜙, if and only if 𝜙 is true in all interpretations in ℑ_𝑇(Γ). Thus, ⊩𝜙 if and only if 𝜙 is a tautology. The basic set of axioms of the FOL are that of the propositional logic with two additional axioms: (A1) (∀𝑥)(𝜙⇒𝜓)⇒(𝜙⇒(∀𝑥)𝜓) (𝑥 does not occur in 𝜙 and it is not bound in 𝜓), and (A2) (∀𝑥)𝜙⇒𝜙[𝑥/𝑡], (neither 𝑥 nor any variable in 𝑡 occurs bound in 𝜙). For the FOL with identity, we need the proper axiom (A3) 𝑥₁≐𝑥₂⇒(𝑥₁≐𝑥₃⇒𝑥₂≐𝑥₃). The inference rules are Modus Ponens and generalization (G) “if 𝜙 is a theorem and 𝑥 is not bound in 𝜙, then (∀𝑥)𝜙 is a theorem.”

The standard FOL is considered as an extensional logic because two open sentences with the same tuple of variables 𝜙(𝑥₁,…,𝑥_𝑚) and 𝜓(𝑥₁,…,𝑥_𝑚) are equal if and only if they have the same extension in a given interpretation 𝐼_𝑇, that is, if and only if 𝐼^∗_𝑇(𝜙(𝑥₁,…,𝑥_𝑚))=𝐼^∗_𝑇(𝜓(𝑥₁,…,𝑥_𝑚)), where 𝐼^∗_𝑇 is the unique extension of 𝐼_𝑇 to all formulae, as follows.(1)For a (closed) sentence 𝜙/𝑔, we have that 𝐼^∗_𝑇(𝜙/𝑔)=𝑡 if and only if 𝑔 satisfies 𝜙, as recursively defined above.(2)For an open-sentence 𝜙 with the tuple of free variables (𝑥₁,…,𝑥_𝑚), we have that 𝐼^∗_𝑇(𝜙(𝑥₁,…,𝑥_𝑚))=_def{(𝑔(𝑥₁),…,𝑔(𝑥_𝑚))∣𝑔∈𝒟^𝒱 and 𝐼^∗_𝑇(𝜙/𝑔)=𝑡}.

It is easy to verify that for a formula 𝜙 with the tuple of free variables (𝑥₁,…,𝑥_𝑚), 𝐼^∗_𝑇(𝜙(𝑥₁,…,𝑥_𝑚)/𝑔)=𝑡 if and only if (𝑔(𝑥₁),…,𝑔(𝑥_𝑚))∈𝐼^∗_𝑇(𝜙(𝑥₁,…,𝑥_𝑚)).

This extensional equality of two virtual predicates can be generalized to the extensional equivalence when both predicates 𝜙,𝜓 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 𝜙≡𝜓.

This extensional equivalence between two relations 𝑅₁,𝑅₂∈ℜ with the same arity will be denoted by 𝑅₁≅𝑅₂, while the extensional identity will be denoted in the standard way by 𝑅₁=𝑅₂.

Let 𝒜_FOL=(ℒ,≐,⊤,∧,¬,∃) 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 ¬⊤), 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 (∃𝑥). In what follows, we will make an analog extensional algebraization over ℜ but by interpretation of the logic conjunction ∧ 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 𝒜_ℜ=ℜ,𝑅₌,{⟨⟩},⋈_{𝑆𝑆∈𝒫(ℕ2)},∼,𝜋_{−𝑛𝑛∈ℕ}, (10) where {⟨⟩}∈ℜ is the algebraic value correspondent to the logic truth and 𝑅₌ is the binary relation for extensionally equal elements. We will use “=” for the extensional identity for relations in ℜ.
Then, for any Tarski’s interpretation 𝐼_𝑇 its unique extension to all formulae 𝐼^∗_𝑇∶ℒ→ℜ is also the homomorphism 𝐼^∗_𝑇∶𝒜_𝐹𝑂𝐿→𝒜_ℜ from the free syntax FOL algebra into this extensional relational algebra.

Proof. Directly from definition of the semantics of the operators in 𝒜_ℜ defined in precedence, let us take the case of conjunction of logic formulae of the definition above where 𝜑(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚,𝑦_𝑖,𝑦_𝑗) (its tuple of variables is obtained by the method defined in the FOL introduction) is the virtual predicate of the logic formula 𝜙(𝑥_𝑖,𝑥_𝑗,𝑥_𝑘,𝑥_𝑙,𝑥_𝑚) ∧ 𝜓(𝑥_𝑙,𝑦_𝑖,𝑥_𝑗,𝑦_𝑗): 𝐼^∗_𝑇(𝜙∧𝜓) = 𝐼^∗_𝑇(𝜑) = {(𝑔(𝑥_𝑖),𝑔(𝑥_𝑗),𝑔(𝑥_𝑘),𝑔(𝑥_𝑙),𝑔(𝑥_𝑚), 𝑔(𝑦_𝑖),𝑔(𝑦_𝑗)) ∣ 𝐼^∗_𝑇(𝜑/𝑔)=𝑡} = {(𝑔(𝑥_𝑖),𝑔(𝑥_𝑗),𝑔(𝑥_𝑘),𝑔(𝑥_𝑙),𝑔(𝑥_𝑚),𝑔(𝑦_𝑖),𝑔(𝑦_𝑗)) ∣ 𝐼^∗_𝑇(𝜙/𝑔∧𝜓/𝑔)=𝑡} = {(𝑔(𝑥_𝑖),𝑔(𝑥_𝑗),𝑔(𝑥_𝑘),𝑔(𝑥_𝑙),𝑔(𝑥_𝑚),𝑔(𝑦_𝑖),𝑔(𝑦_𝑗)) ∣ 𝐼^∗_𝑇(𝜙/𝑔) = 𝑡 and 𝐼^∗_𝑇(𝜙/𝑔)=𝑡} = {(𝑔(𝑥_𝑖),𝑔(𝑥_𝑗),𝑔(𝑥_𝑘),𝑔(𝑥_𝑙),𝑔(𝑥_𝑚),𝑔(𝑦_𝑖),𝑔(𝑦_𝑗)) ∣ (𝑔(𝑥_𝑖),𝑔(𝑥_𝑗),𝑔(𝑥_𝑘),𝑔(𝑥_𝑙),𝑔(𝑥_𝑚)) ∈ 𝐼^∗_𝑇(𝜙) and (𝑔(𝑥_𝑙),𝑔(𝑦_𝑖),𝑔(𝑥_𝑗),𝑔(𝑦_𝑗)) ∈ 𝐼^∗_𝑇(𝜙)} = 𝐼^∗_𝑇(𝜙)⋈_{{(4,1),(2,3)}}𝐼^∗_𝑇(𝜓).
Thus, it is enough to show that 𝐼^∗_𝑇(⊤)={⟨⟩} is also valid, and 𝐼^∗_𝑇(¬⊤)=∅. The first property comes from the fact that ⊤ is a tautology, thus satisfied by every assignment 𝑔, that is, it is true, that is, 𝐼^∗_𝑇(⊤)=𝑡 (and 𝑡 is equal to the empty tuple {⟨⟩}). The second property comes from the fact that 𝐼^∗_𝑇(¬⊤)=∼(𝐼^∗_𝑇(⊤))=∼({⟨⟩})=𝒟⁰⧵{⟨⟩}={⟨⟩}⧵{⟨⟩}=∅. That is, the tautology and the contradiction have the true and false logic value, respectively, in ℜ.
We have also that 𝐼^∗_𝑇(≐(𝑥,𝑦))=𝐼_𝑇(≐)=𝑅₌ for every interpretation 𝐼_𝑇 because ≐ is the built-in binary predicate, that is, with the same extension in every Tarski’s interpretation.
Consequently, the mapping 𝐼^∗_𝑇∶(ℒ,≐,⊤,∧,¬,∃)→𝒜_ℜ 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 𝐼_𝑇 of the FOL, the following diagram of homomorphisms commutes. (11) where ℎ=𝑖𝑠(𝑤) and 𝑤=𝐼_𝑇∈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_int), and where sense determines denotation for any given extensionalization function ℎ (correspondent to a given Traski’s interpretation 𝐼_𝑇). 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_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 𝐼, and a set of possible denotations (extensions) each determined by a given Tarski’s interpretation of the FOL as follows from Definition 4: ℒ⟶^𝐼𝒟⟹_{ℎ=𝑖𝑠(𝐼𝑇)&𝐼𝑇∈𝒲=ℑ𝑇(Γ)}ℜ. (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_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.