Functional Language Logic

Abstract

The formalism of Functional Language Logic (FLL) is presented, which is an extension of a logical formalism already introduced to represent sentences in natural languages. In the FLL framework, a sentence is represented by aggregating primitive predicates corresponding to words of a fixed language (English in the given examples). The FLL formalism constitutes a bridge between mathematical logic (high-order predicate logic) and the classical logical analysis of discourse, rooted in the Western linguistic tradition. Namely, FLL representations reformulate on a rigorous logical basis many fundamental classical concepts (complementation, modification, determination, specification, …), becoming, at the same time, a natural way of introducing mathematical logic through natural language representations, where the logic of linguistic phenomena is analyzed independently from the single syntactical and semantical choices of particular languages. In FLL, twenty logical operators express the mechanisms of logical aggregation underlying meaning constructions. The relevance of FLL in chatbot interaction is considered, and a problem concerning the relationship between embedding vectors in LLM (Large Language Model) transformers and FLL representations is posed.

1. Introduction

The logic of natural language is an established field of investigation going back to Aristotile’s logic, middle-age Scholastic philosophy, and Leibniz’s investigation at the beginning of mathematical logic [1]. In his book about the mathematical analysis of logic [2], George Boole emphasizes the logical basis of natural language. In 1879, Gottlob Frege [3] defined first-order predicate logic as a complete conceptual framework. Frege’s language includes predicates of any number of arguments, individual constants and variables, propositional connectives, and quantifiers (universal and existential).

In Principia Mathematica [4], Bertrand Russell and Alfred Withehead introduced the theory of logical types as a remedy to the logical paradoxes discovered within the foundation of mathematics.

In 1928, Alonzo Church introduced the lambda notation, and in 1940 the lambda-typed calculus [5]. However, the notion of function, defined by a mathematical formula, goes back to Leonard Euler [6] and Gottlob Frege [3], who realized the functional nature of predicates. Mathematical function resulted in a powerful foundational concept, in mathematical logic, in computability, up to the new frontiers of artificial intelligence [5,7,8,9,10,11,12,13,14,15,16,17,18]. Hans Reichenbach developed a logical analysis of the conversation language in a chapter of his book on mathematical logic [19].

In 1970, Richard Montague wrote the paper “English as a formal language“, where typed lambda calculus and high-order logic are combined to represent ordinary discourse, and several papers on the same topic followed [20,21,22,23,24].

The elimination of variables is a problem intensively investigated in mathematical logic by many authors, such as Moses Schönfinkel, Harshel Curry, Robert Feys, Alfred Tarski, and Leon Henkin [25,26]. In natural language, neither apparent nor free variables are used; therefore, a logical analysis of language has to cope with this phenomenon for a full comprehension of its internal mechanisms. We will show that this aspect is strictly related to the monadic nature of predicates and the possibility of having high-order predicates.

In [27], a formalism of logical semantics for natural languages was introduced within the High-order Monadic Logic (HML), which is essentially a typed lambda calculus based on unary functions with a new logical operator of “Predicate Abstraction”, making logical representations of sentences completely adherent to the usual linguistic constructions. In the same paper, an experiment is reported on teaching the given logical formalism to ChatCPT3.5. This shows interesting perspectives on interacting with chatbots, which reveal a surprising ability to use such logical formalism.

In this paper, the approach of [27] is developed, by defining the more complete and motivated formalism of Functional Language Logic (FLL), strictly related to the classical logical analysis of sentences. Any word in a given dictionary is a unary predicate, a function from individuals to truth values. Twenty logical operators express the logical aggregations underlying the main linguistic constructions.

The functional types for the predicates, individuals, substantives, propositions, hyper-predicates, and ad-predicates are introduced. Hyper-predicates are predicates that apply to predicates and give new propositions, whereas ad-predicates apply to predicates and give new predicates.

The following sections are devoted to specific linguistic phenomena. Direct and indirect complementations are reduced to the application of a complementation operator that, in the case of direct complementation, takes a substantive, producing a new predicate, while in the case of indirect complementation, it takes an atomic proposition, again giving a new predicate. Modification is the operator transforming a predicate into a predicate modifier.

Operators of predication, complementation, modification, and specification represent the main predicative constructs. Descriptive operators apply to predicates and provide substantives, and equations introduce constants. Predicative abstraction expresses predicates over predicates, and finally, other operators deal with performatives.

Chatbots, preconceived in [15], are a frontier of artificial intelligence; their acquisition of complex and articulated competencies in dialogic activity with humans confirms the essential role of natural language in constructing the conceptual organization of cognitive systems. Namely, Greeks used the same word, “Logos“, meaning either language or reason. This consideration suggests that FLL could be a strategic tool in teaching chatbots to acquire sophisticated competencies in logical analysis [16,27].

A topic for further research is related to the relationship between FFL logical representation of meanings and the embedding vectors of LLM (Large Language Model) transformers in modern conversational systems. Namely, in LLMs [28,29,30] of the ChatGPT family of OpenAI, since version 4 (2023) [31], the invariance of many-dimensional semantic spaces emerged. If N is the dimension of word-embedding vectors, sentence and discourse meanings maintain the same dimensionality. By using “contextualization” matrices (Query and Key matrices) defined during the learning phase, the semantic influence of any words concerning others is established so that, for any word, a new vector arises (Value vector), expressing the meaning of the word in the context of the sentence. This method is called the attention mechanism, (with different variants based on the number of attention heads), which provides a $(k+1)$-gon for any sentence $S_k$ of k linguistic units (tokens) in a real hyperspace of many thousand dimensions. For the sake of simplicity, we can assume that the barycenter of the $(k+1)$-gon corresponds to the embedding vector expressing the meaning of $S_k$.

On the other hand, in FLL, meaning composition relies on the theory where operators are defined. The logical structure in which predicates are related provides the semantics of a whole sentence. In a sense, logical operators play the role of contextualization matrices. This analysis sheds new light on the role of theories as tools for adding semantic value to previously determined semantic units. A deeper comprehension of the relationship between chatbot contextualization mechanisms and logical integration could be an important achievement in understanding the processes of knowledge organization.

2. Material and Methods

2.1. λ-Abstraction and Logical Symbols

Given an expression $E\left(x\right)$ built with operations applied to constant and variables, where the variable x ranges on the class A, and taking values in the class B, we denote by $x.E\left(x\right)$ the function from A to B associating with any element $a\in A$ the value $E\left(b\right)$ assumed by $E\left(x\right)$ when x takes the value a. Alonzo Church introduced the lambda notation $\lambda x.E\left(x\right)$ to stress that the function does not depend on the chosen variable; we omit the symbol $\lambda$ for a shorter notation. If in $x.E\left(x\right)$ the variable x is replaced by y (which takes the same values as x does), for every a:

$$(x.E(x\left)\right)\left(a\right)=(y.E(y\left)\right)\left(a\right)=E\left(a\right)$$

therefore:

$$x.E\left(x\right)=y.E\left(y\right)$$

the two $\lambda$-expressions are different names for the same function.

A predicate is a symbol denoting a function from a set of individuals to a set of two truth values, which we denote by $\top ,\perp$ (True, False). In the following, we will use predicates with only one argument (monadic) or zero arguments. Monadic predicates are called properties, while predicates with no arguments are propositions, which can also be considered symbols for truth values. Letters $P,Q,R,\dots$, possibly with indices, will denote predicates. Symbols $a,b,c,\dots$, possibly with indices, are individual constants (names of particular individuals), and symbols $x,y,z,\dots$, possibly with indices, are individual variables. Letters $X,Y,Z,\dots$, possibly with indices, are predicate variables.

The symbols $\neg ,\to ,\wedge ,\vee ,\leftrightarrow$, called connectives, are operations on truth values with the following meanings (where $P,Q$ are propositions): $\neg \top =\perp ,\neg \perp =\top$; $P\to Q=\perp$ iff $P=\top ,Q=\perp$; $P\wedge Q=\top$ iff $P=\top ,Q=\top$; $P\vee Q=\perp$ iff $P=\perp ,Q=\perp$ (whence $\top =\neg P\vee P$); $P\leftrightarrow Q$ iff $P=Q$.

Symbols $\forall ,\exists$, called quantifiers, are operations such that $\forall P\left(x\right)=\top$ iff $x.P\left(x\right)=x.\top$; $\exists P\left(x\right)=\perp$ iff $x.P\left(x\right)=x.\perp$.

Connectives and quantifiers can also be easily seen as operators over predicates, for example, if $P,Q$ are monadic predicates, $P\to Q=x.\left(P\right(x)\to Q(x\left)\right)$, $\forall P=\forall x.P\left(x\right)$.

The following examples express basic facts about arithmetic by using usual numerals, which denote numbers; the unary predicate $Even$ is satisfied by even numbers; the predicate $Prime$ is satisfied by prime numbers (which are not products of two numbers strictly smaller than them); + denotes the sum operation; and the binary predicate $Greater(x,y$) means that x follows y in the natural counting sequence. Some arithmetic facts follow, where the last two propositions say that the sum of two odd numbers is even and that twin prime numbers are infinite.

$$Even\left(0\right),Even\left(2\right),Even\left(4\right),\dots$$

$$\neg Even\left(1\right),\neg Even\left(3\right),\neg Even\left(5\right),\dots$$

$$\forall x\left(Even\right(x)\vee \neg Even(x\left)\right)$$

$$\forall x(\neg y=0\to Greater(x+y,x\left)\right)$$

$$\forall xy\left(\right(\neg Even\left(x\right)\vee \neg Even\left(y\right))\to Even(x+y\left)\right)$$

$$\forall z\exists xy\left(Greater\right(x,z)\wedge Greater(y,z)\wedge Prime(x)\wedge Prime(y)\wedge (x+2)=y).$$

2.2. Class and Predicate Abstraction Operators

The operator of class abstraction transforms a monadic predicate P in the class A of the values on which the predicate holds (gives truth value ⊤). This operator was initially defined by Georg Cantor and formalized by Bertrand Russed with the notation $\widehat{x}.P\left(x\right)$. Nowadays, the commonly used notation for classes is $A=\left\{x\right|P\left(x\right)\}$.

The notion of type is analogous to that of a class; it is used in many contexts and with many specific senses. We write $a:t$ to denote that a has a type of t. Of course, the elements of a given type provide a class, and analogously, having a type is a property. Therefore, a type can be assimilated into the concepts of class and property, even if its meaning is more related to that one of a symbolic mark to attach to objects for categorizing them.

However, it is useful to distinguish similar concepts because, in many complex analyses, these notions refer to different levels of a discourse that are useful to consider separately. For example, the names of things are different from things in themselves, and operating with names can provide useful possibilities better dealt with in specific contexts by avoiding any confusion with things and operations on them. If $x:s$ and $E\left(x\right):t$, we denote by $(s\mapsto t)$ the type of $x.E\left(x\right)$; if A is a class of elements of type t, then we denote by $\left\{t\right\}$ the type of A. Therefore, types can be arranged in expressions of increasing complexity: $t,(s\mapsto t),\left\{s\right\},\left(\right\{s\}\mapsto t),\dots$. The maximum number of nested pairs of parentheses or brackets in the expression of a type provides the logical order of that type.

In 1901, Bertrand Russel discovered a logical paradox related to the intuitive notion of class. Namely, some autoreferential conditions (the class of classes that do not belong to themselves) are contradictory. Axiomatic set theories were developed, which define sets as special classes regulated by axioms, avoiding paradoxes. Type theory, elaborated by Bertrand Russel and Alfred Whitehead, overcomes paradoxes by assigning types for dealing with high-order predicates that apply to predicates as arguments [32]. In natural language, expressions such as $Past\left(P\right)$ or $Yesterday\left(P\right)$ are typical examples of predicates taking predicates as arguments.

Two expressions denoting individuals are equal when they denote the same individual. We can express equality by using monadic properties according to the following trick. We add for every individual a a predicate $E_a$ that holds on an individual b if it denotes the individual denoted by a:

$$E_a\left(b\right)\leftrightarrow a=b.$$

Analogously, a binary operation, such as +, can be expressed by using the monadic operation $Su{m}_3=x.3+x$, namely:

$$Su{m}_3\left(5\right)=8.$$

In this way, an operation’s argument becomes a parameter embedded in a unary operation, giving the same result as a binary operation on two arguments. This phenomenon is crucial in natural language, allowing for a monadic representation of all predicates occurring in linguistic constructions.

The operator of predicate abstraction allows us to raise the logical order of a predicate. If $Pred$ is a predicate, we denote by ^Pred its predicative abstraction, for which:

$$^Pred\left(P\right)\leftrightarrow (P\to Pred)$$

that is, ^Pred is a hyper-predicate (with respect to $Pred$), which holds over all predicates that imply $Pred$. This means that the implication $P\to Pred$ becomes the application of a hyper-predicate ^Pred(P) indicating that P is an implicant of $Love$, which is different from $Love\left(a\right)$, where a is a loving individual. Namely, if P is a predicate, then it does not love, as $Love$ is a property of individuals. Figure 1 visualizes the difference between a direct predication and a predication through a predicate constant and predicate abstraction.

2.3. Comparison with Related Logical Formalisms

Many formalisms aim at representing sentences logically. After the mentioned approach inaugurated by Richard Montague, in many fields, such as logic, linguistics, philosophy, computer science, knowledge representation, and other related fields, the search for logical representations, coupling rigor with simplicity and adequacy, was always very active and oriented to many specific requirements of some applicative contexts. In the setting of logical approaches, let us mention the works developed in the CSLI (Center for the Study of Language and Information), especially Situation Logic, the Natural Language Semantics, and Intensional Logic [33,34,35,36]. For many aspects, FLL has common features with these approaches, but two important characteristics distinguish it properly. It is directly related to the traditional logical analysis of the discourse, which is very popular in the educational curricula, especially in the context of classical dead languages (Greek, Latin); moreover, it is based on a limited number of logical symbols applied to the words of fixed dictionaries, which makes it very simple to learn, even without entering in its complex logical basis.

The formalism of FLL is concerned with the logical representations of sentences. However, for completeness, we want to mention some topics that are important in logical formalisms but outside the scope of the paper. One of these topics is concerned with formal deductions (Proof Theory), realized by suitable deductive algorithms; the other refers to the interpretations of formulas within mathematical structures (Model Theory) [5,32,37,38].

The first logical calculus was elaborated for predicate logic by Gottlob Frege [3]. This logic has constant and individual variables, with predicative constants denoting relations of any number of arguments on individuals, together with connectives and quantifiers. Frege’s predicative calculus, and many other equivalents to it, resulted to be complete, that is, it can deduce all the logical consequences deriving from a list of axioms. A logical consequence of a set of propositions $\mathbb{T}$ is a proposition true in all the models where the propositions of $\mathbb{T}$ are true.

In Model Theory, a model M is associated with a class $\mathbb{T}$ of formulae (a theory) when all the formulae of $\mathbb{T}$, according to suitable interpretation rules, are true in M. For predicate logic, a fundamental result, known as the Löwenheim–Skolem Theorem, holds according to which any coherent theory (where ⊥ cannot be deduced) can be interpreted in the domain of natural numbers [5,32].

3. Results

Let us assume that all words of a given language, in our case English (written with capital initial letters), are monadic predicates of some logical order. Sometimes, for a better reading of complex formulas, we will use the inverse parentheses notation by writing $\left)a\right(P$ instead of $P\left(a\right)$.

3.1. Categories and Functional Types

The formalism FLL has the following categories of expressions, and some of them will receive the following types:

(1)

Arguments, of type $arg$, is the category of any expression that occurs as an argument of a function;

(2)

Individuals, of type $ind$, is the category of denotations of the constants $a,b,c$, …, which can be considered as indexes or indications of objects assumed in a discourse;

(3)

Propositions, of type $bool$, is the category of denotations of truth values, also seen as predicates of zero arguments. An expression $P\left(a\right)$ is an atomic proposition, or a simple predication while $(P\wedge Q)\left(a\right)$, for example (eat and drink)(a), where the predicate is the conjunction of two predicates, or $\left(P\right(a)\wedge Q(a)$, which is the conjunction of two propositions, which are not atomic predications. We will indicate, using $atom$, the type of atomic propositions;

(4)

Predicates, of type $pred$, is the category of monadic predicates, that is, functions from arguments to truth values:

$$pred=(ind\mapsto bool)$$

in this category, predicative constants $P,Q,R$, …are included;

(5)

Hyper-predicates, of type $hyperpred$, is the category of predicates that apply to predicates and provide propositions:

$$hyperpred=(pred\mapsto bool)$$

hyper-predicates can be considered second-order predicates; analogously, third-order predicates can be considered and, in general, higher-order predicates for further levels;

(6)

Ad-predicates, of type $adpred$, is the category of functions that apply to predicates and provide predicates:

$$adpred=(pred\mapsto pred);$$

(7)

Substantives, of type $subst$, is the category of individuals, predicative constants, and any expression that can be equated to them. Also, capital letters $A,B,C$ … denoting classes (possibly with indexes) are substantives. Equating an expression to a constant provides a substantivation;

(8)

Logical operators are the symbols expressing operations on predicates;

(9)

Performatives are the symbols expressing discourse functionalities.

In the following, we provide examples of FLL representations for small texts in the natural language (English). Twenty logical operators emerge from these representations that can describe the meaning of these texts by composing the meanings of the single words. In this reduction, we understand texts from the predicates associated with the lemmas of a dictionary.

We want to recall that the usual grammatical categories of verbs, nouns, and adjectives are based on spatiotemporal features. Adverbs realize hyper-predicates on predicates (negation, intensification, modality, …). At the same time, the other linguistic units are “empty words”, with their meanings driven by the contexts in which they play roles that correspond to logical operators of FLL.

3.2. Predication and Complementation

Let us start with a simple sentence: “John is good.” Its FLL representation is given in

Table 1. “John is good.”

John(a)

Good(a)

. John is a person’s name, and John(a) means that there is an individual a who satisfies the property of having the name “John”, and a satisfies the property “Good”.

The sentence “John loves Mary” has the FLL representation seen in

Table 2. “John loves Mary.”

John(a)

Mary(b)

Love_(b)(a)

.

Here a new operator appears, with postfix notation, indicated by _ and called complementaton, which transforms a predicate, such as love, into the predicate Love_(b), completing the meaning of love with the object b. When Love_(b) applies to the individual a, we obtain (Love_(b))(a), or simply Love_(b)(a), expressing “a love b”. Therefore, complementing a monadic predicate, we express a binary predicate. In general, given a predicate $Pred$, the expression $Pred\_$ is a function of two possible types:

$$Pred\_:(subst\mapsto pred)$$

$$Pred\_:(atom\mapsto pred)$$

the example in Table 1 falls in the first case and is called direct complementation, while that in Table 2 corresponds to the second case, and represents indirect complementation.

We avoid parentheses after _ for better reading and assume that the complementation operator applies with left priority. Firstly, the leftmost operator applies, then the operator _ following it on the right, and so on, up to the rightmost complementation operator. In this way, in the usual notation $P\left(a\right)$, the subject of the predicate is at the end, on the right, while in the inverse parentheses notation $\left)a\right(P$, the subject is at the beginning on the left.

The sentence “John goes home with a bike” has the representation given in

Table 3. “John goes home with a bike.”

John(a)

Home(b)

Bike(c)

)a( Go_Place(b)_Instrument(c)

.

In this representation, the operator _ transforms Go into Go_, a function taking an atomic proposition and producing a predicate. Analogously, Go_Place(b) is a predicate to which the operator _ applies, and Go_Place(b)_ takes as an argument Instrument(c) and becomes Go_Place(b)_Instrument(c). In conclusion, the resulting predicate applies to the individual a providing a proposition. Atomic propositions Place(b) and Instrument(c) define the roles of complements $b,c$ which complete the basic predicate Go.

Figure 2 visualizes the representation of Table 3 using a labeled graph, in which constants or words are labeled as nodes. Simple arrows denote predication, and bigger circles enclose the atomic propositions of indirect complementations. We remark that the graph is a second-order graph (in more complex cases, third or fourth orders are necessary) because there are nodes that include subgraphs (internal to bigger closed curves).

A different way of expressing complementation is through arguments that are sequences, as in

Table 4. “John goes home with the bike.”

John(a)

Home(b)

Bike(c)

u = (a,b,c)

Go(u) ∧ Subject(a) ∧ Place(b) ∧ Instrument(c)

. However, the method based on the complementation operator is more adherent to the linguistic mechanism of complementation. Therefore, in the sequel, we follow it.

Traditional linguistic analysis is focused on a long list of possible complements: object, specification, place, time, instrument, …. In a list used in schools, it is possible to find fifty different types of complements. However, such lists are, to a large extent, arbitrary and incomplete. The linguistic form of complementation depends on specific syntactic features. In a logical representation, it is important only to identify the elements completing a predicate by distinguishing each one from the others. Let us consider the sentence “John gives a pen to Mary.” The following FLL representation of this sentence is given in

Table 5. “John gives a pen to Mary.”

John(a)

Pen(b)

Mary(c)

)a( Give_(b)_Receive(c)

.

However, different predicates (take, accept, destination, target) could be used instead of ”Receive” to adequately express the role of constant c, apart from specific syntactical realizations of the sentence.

3.3. Modification

The FLL representation of the sentence “People elected John as major” is given in

Table 6. “People elected John as major.”

A = People John(b)

^Elect(P)

Past(P)

Major[P]_(b)(A)

.

In Table 6, the modification operator [ ] transforms a predicate into an ad-predicate. The notation Pred[Pred’] represents the predicate Pred’ modified by the predicate Pred.

In typical cases, ad-predicates are the adverbs modifying verbs, or special verbs, such as begin, finish, interrupt, can, will, must, appear, seem, …, are modifiers of other verbs, as can be observed in the representation of

Table 7. “John wanted to speak.”

John(a)

^Want[Speak](P)

Past(P)

Progressive(P)

P(a)

.

An analogous modification occurs when a noun or adjective modifies an adjective, as shown in

Table 8. “John is a good policeman.”

John(a)

Good[Policeman](a)

, and Figure 3.

Of course, “John is good” and “John is a good policeman” use “good” in two completely different ways, making it evident that the same word, in other contexts, can exhibit different logical types. Namely, in the first case, good is a predicate, while in the second one, it is an ad-predicate. Figure 3 introduces the graphical notation of two arrows for modification.

3.4. Specification

The specification is a special type of indirect complementation, denoting a relationship (of membership, inclusion, pertinence, possess, …) with a substantive. We introduce the operator < >; <a> gives a predicate “of a”, that is, the property of being relative to the substantive a.

The sentence “John was going home with his bike” is given in

Table 9. “John was going home with his bike.”

John(a)

Home(b)

<a>(b)

Bike(c)

<a>(c)

^Go(P)

Past(P)

Progressive(P)

)a( P_Place(b)_Instrument(c)

, where a predicate constant P and predicate abstraction are used. Figure 4 visualizes this FLL representation, where predicate abstraction is expressed by a double arrow and specification by a circle intersecting the circle of the specified substantive.

The sentence “John asked Mary for information on the train timetable” is represented in

Table 10. “John asked Mary for information on the train timetable.”

John(a)

Train[Timetable](b)

<b>[Information](c)

Mary(d)

^Ask(P)

Past(P)

)a( P_d_c

.

The specification allows us to represent all cases of indirect complementation (in some languages, such as Arabic, there are only the object complement and the specification). For example, “John goes home with his bike” is represented in

Table 11. “John goes home with his bike.”

John(a)

Home(b)

<a>(b)

Bike(c)

<a>(c)

^Go(P)

<P>[Direction](b)

<P>[Instrument](c)

P(a)

using the specification operator for expressing complementation.

The sentence “Yesterday I was walking without shoes”, in

Table 12. “Yesterday I was walking without shoes.”

Me(a)

^Walk(P)

^Without[2[Shoe]](P)

Past(P)

Progressive(P)

Yesterday(P)

P(a)

, has a complex ad-predicate realized by modifications. Equivalent representations are given in

Table 13. “Yesterday I was walking without shoes.”

Me(a)

^Walk(P)

Past(P)

Progressive(P)

Yesterday(P)

c = $(a_1,a_2)$

<a>[Shoe]($a_1$)

<a>[Shoe]($a_2$)

Shoe[Pair](c)

)a( P_Without(c)

and

Table 14. “Yesterday I was walking without shoes.”

Me(a)

^Walk(P)

Past(P)

Progressive(P)

Yesterday(P)

)a( P$\_\neg$Wear$\_\epsilon$(2[Shoe])

.

The expression $\epsilon$(2[Shoe]) of Table 14 means “some pair of shoes”, as will be clarified in the next section. These examples clearly show that the same sentence can be represented in many ways. Each representation has advantages or inadequacies concerning the others. The right choice depends on the kind of the intended application of the representation.

The FLL representations given in this and in the previous sections underscore the capability of FLL in dealing with all the phenomena of ambiguity and vagueness in natural language. In FLL, ambiguity can be avoided, and when many possible meanings can be given to a sentence, the formal representation can select the most appropriate in the given context of use.

Material errors (misspelling, grammatical mistakes) are outside the scope of the FLL formalism. Therefore, sentences to which FLL applies are intended to be previously processed to check their morphological reliability according to the language considered in a given application.

3.5. Descriptive Operators

The $\iota$ operator of determination was introduced by Giuseppe Peano [39]. Let P be a predicate that is satisfied only by one individual; then, this individual is denoted by $\iota P$. Hence:

$$\iota P=a\leftrightarrow P\left(a\right)\wedge (\neg (a=b)\to \neg P(b\left)\right)$$

The expression $\iota P$ corresponds to the definite article of natural languages. If we write $a=\iota Boy$, we mean that in the given context, a boy is univocally determined and is identified by the individual constant a. If more than one value satisfies P, all the propositions where $\iota P$ occurs are false.

The $\epsilon$ operator choice has been introduced by David Hilbert [32]; it provides a chosen indefinite value that satisfies P. If no argument satisfies P, all the propositions where $\epsilon P$ occurs are false. Hence:

$$a=\epsilon P\leftrightarrow P\left(a\right)$$

and:

$$P\left(\epsilon P\right)\leftrightarrow \forall xP\left(x\right).$$

Using $\epsilon P$ we can put:

$$\left\{\epsilon P\right\}=\left\{x\right|P\left(x\right)\}.$$

Operators $\iota$ and $\epsilon$ have both type $(pred\mapsto subst)$.

Different occurrences of $\epsilon P$ may denote different individuals. If we say Any man who loves a woman is happy, we refer to an indefinite man. If we say that Any man who loves a woman is happy, but any man who does not love any woman is searching for a woman whom he can love, clearly, the two occurrences of “any man” have to denote different people. Otherwise, the sentence is meaningless.

Proposition $Q\left(\epsilon P\right)$ implies the following propositions, where constants cover all the values satisfied by P:

$a_1=\epsilon P\to Q\left(a_1\right)$

$a_2=\epsilon P\to Q\left(a_2\right)$

…

Therefore, the choice operator $\epsilon$ provides universal quantification and the constructions distributing the values of a predicate over other predicates (every man is mortal).

We can extend $\epsilon$ notation with numeric indexes so that $\epsilon$ expressions with the same index denote the same individual. Therefore, expressions such as ${\epsilon }_{i}P$ can be used as usual variables. For example, lambda expressions can be expressed by:

$$x.E\left(x\right)={\epsilon }_1.E\left({\epsilon }_1\right).$$

Indefinite values expressed by $\epsilon$ expressions are different from generic indeterminate values that, in many languages, correspond to indefinite articles. In FLL, particular values are denoted by individual constants. Namely, when we write $P\left(a\right)$, we mean that there exists a value that satisfies P, and we call it a.

Relative clauses are of two kinds: descriptive e restrictive. If we say “John, who lives in Rome, will not come to the meeting”, the relative clause (introduced by who) adds information, can be equivalently given by saying: “John will not come to the meeting, he lives in Rome”.

Conversely, “John is searching for a pen that writes green” is a restrictive relative clause because it characterizes what John is searching for. The FLL representation of

Table 15. “John is searching for a pen that writes green.”

John(a)

^Green[Write][Pen](P)

^Search-for(Q)

Present(Q)

Progressive(Q)

b = $\epsilon$P

⊧ )a(Q_(b)

is obtained using the choice Hilbert operator.

Figure 5 is the visualization of the representation of Table 15. The choice operator is expressed by a line exiting from the predicate and ending with a small circle. A double line connects circles denoting the same substantive. A double line attached to an orthogonal bar expresses the asserted predicate, that is, the verb of the main clause (when it is useful to stress it).

Now, we give an example using the $\epsilon$ operator to express a consecutive construction.

Table 16. “The bag is so heavy that I cannot bring it.”

Me(a)

b = $\iota$Bag

<b>[Weight[Quantity]](c)

c’ = $\epsilon$(x.Can[Bring]_Quantity[Weight](x)(a))

⊧ Greater_c’(c)

provides the FLL representation of “The bag is so heavy that I cannot bring it”.

In the last equation, $\epsilon$ applies to the predicate within parentheses, and Greater_c’(c) means that: “The weight c (of the bag) overcomes $c^{\prime }$, which is any weight that a can bring”. Figure 6 visualizes the representation of Table 16.

In FLL numerals: 0, 1, 2, …(in decimal notation) and ordinals: $1^o,2^o,3^o,\dots$, with the usual symbols of arithmetic operations and relations, are available.

Modification with numerals (0, 1, 2, …) allows for a simple representation of plurals. Given a predicate $Pred$, the expression $2\left[Pred\right]$ means a couple of individuals that satisfy $Pred$. Analogously, $(>1)\left[Pred\right]$ realizes a descriptive operator denoting a plurality of individuals satisfying $Pred$.

Modifications such as $2^o\left[Pred\right]$ denote ordinals (“the second which satisfies Pred”), assuming an order specified by the context or previously given. For example, the following is a representation that refers to two boys; the first speaks, and the second listens:

2[Boy](a)

1o[<a>](b)

2o[<a>](b)

Speak(b)

Listen(c)

3.6. Contexts, Distributions, and Performatives

Deixis (Greek etymology) refers to all the aspects of a sentence’s spatiotemporal context. Words such as this, that, now, I, and you are deictic words assuming meanings that refer to their specific context. A situation consists of all elements necessary to establish the correct meaning of a sentence, including deixis and other aspects, such as presuppositions that a speaker assumes about the persons, things, facts, and habits on which a specific communication is based. Moreover, other aspects regarding the persons involved in communication can be relevant, and in many languages, these aspects can remarkably influence the expressions used. The register (familiar, formal, institutional, …), especially in some languages, can determine even the choice of the words of sentences.

Anaphora (Greek etymology) refers to the linguistic elements pointing to words and expressions already occurring in sentences in the linear order of their generation. Pronouns are the typical elements playing this role. The concordance is the mechanism on which anaphora is based. Moreover, the same mechanism is also responsible for the aggregation of linguistic expressions in bigger units, including them as components.

Concordance is realized using grammatical marks expressing features (gender, number, person, time, …). The system of grammatical features can change in different languages (form, color, localization, distribution, consistency, …). A pronoun can be seen as an aggregation of marks. In this way, it refers to the closest linguistic expression preceding it and having the same marks. Grammatical features alter linguistic forms using inflection and conjugation so that elements with the same marks are aggregated in bigger units.

In the FLL representations, the individual constants realize pronouns, while parentheses realize aggregation. Considering the complexity of phenomena realizing anaphora and concordance, we can appreciate FLL’s great advantage over natural languages.

A class of sentences widely analyzed by logicians since the Middle Ages are donkey sentences, titled as such given an example reported in an ancient treatise of logical analysis of language (“Every man who owns a donkey sees it", Walter Burley (1328), De puritate artis logicae tractatus longior). The problem with these sentences is the pronoun reference in the context of a universal quantification.

The sentence “Every man loves the woman who loves him”. In predicative logic becomes:

$$\forall x,y\left(\right(Man\left(x\right)\wedge Woman\left(y\right)\wedge Love\left(y\right)\_\left(x\right))\to Love(x)\_(y\left)\right)$$

where a reference dictates the distributive nature of the referred term ($Every\_man/who$).

If we express universal quantification with the $\epsilon$ operator, we obtain:

$$Love\_\epsilon Man\left(\epsilon Woman\right)\to Love\_\iota Woman\left(\iota Man\right)$$

where iota operator refers to the individuals chosen on the left of implication. Using $\epsilon$ with indexes:

$$Love\_{\epsilon }_{1}Man\left({\epsilon }_{1}Woman\right)\to Love\_{\epsilon }_{1}Woman\left({\epsilon }_{1}Man\right)$$

however, a form more adherent to the linguistic form and using $\epsilon$ once is the following:

$$\left(\right(a=\epsilon Man)\wedge Woman(b)\wedge Love\_a(b)\to Love\_b(a).$$

“Any man loved by a woman loves her”, which we can also represent by

Table 17. “Any man loved by a woman loves her.”

a = $\mathit{\epsilon }$Man

(Woman(b) ∧ Love_a(b)) → Love_b(a)

and

Table 18. “Any man loved by a woman loves her.”

a = $\mathit{\epsilon }$Man

^Love(P)

^Love(Q)

(Woman(b) ∧ P_(b) ) → Q_b(a)

(the choice is intended in the class of substantives).

Let us consider the sentence “The boys were entering two at a time.” Traditional logical analysis tells us that “two at a time” is a complement of ”distribution.” However, this does not completely clarify its underlying logical mechanism, which is completely represented in

Table 19. “The boys were entering two at a time.”

A = $\mathit{\iota }$(Class[2[Boy]]

a $=\epsilon$A

<a>[Time](b)

$\vDash \mathrm{Enter}\_\mathrm{Time}$(b)(a)

.

Class[2[Boy]] is the property of the classes of pairs of boys. The operator $\iota$ provides a determinate class (in the discourse context), and the equation a $=\epsilon$A introduces the substantive a for denoting a choice in this class, that is, a pair of boys. The predication <a>[Time](b) tells us that b is a time associated with the pair a, and the last predication tells us that a enters at the time b.

The values of $\epsilon A$ change with the choices within the class A, and for each pair, there is an entrance time. The assertion symbol ⊧ expresses the principal proposition and the fact that it holds for a generic chosen pair of boys implies its universal validity.

We can further elucidate the distribution process. Let $a_1,a_2,\cdots$ be the choices $\epsilon P$ and $b_1,b_2,\cdots$ the choices $\epsilon Q$ (covering the boys and the times). Then, the FLL representation is equivalent to the sequence of propositions:

$Enter\_Time\left(b_1\right)\left(a_1\right)$

$Enter\_Time\left(b_2\right)\left(a_2\right)$

…

It is important to remark on the continuative characteristics of the verb form “were entering”, because it tells us that the process is developed in a time interval along a sequence of steps. Therefore, the distribution expresses a modality of the realization of the process associated with the verb enter.

Table 20. “The boys were entering two at a time.”

A = $\mathit{\iota }$(>1)[Boy]

^Enter(P)

Continuative(P)

P_Distribution(2)(A)

shows the associated FLL representation.

The representation of Table 20 associates A with a determinate class of boys (in the context of the discourse), which is a plurality (>1). The constant A is the argument of P complemented by the atomic proposition ”Distribution(2)”. We can read: “The boys were entering distributing in two”. In this way, we are very close to the linguistic form of the sentence through an analysis of the deep structure of the sentence.

Coordination and subordination between propositions consist of predications over propositions. In the sentence: “While the boys were entering the classroom, the teacher was writing on the blackboard”, a relationship expressed by while occurs between propositions $P_1,P_2$, representable by the predication:

$$While\_P_1\left(P_2\right)$$

where:

$P_1=$ The boys were entering the classroom;

$P_2=$ The teacher was writing on the blackboard.

Connections of a temporal and situational nature (concessive, adversative, consecutive, final, causal, …) express relationships in typical subordinative clauses.

An FLL representation is centered around a principal atomic proposition such as $P\left(a\right)$, where all the specific information about P and a is given in the remaining part of the representation. In a sense, all the components of the sentence representation converge into $P\left(a\right)$. We use the assertion symbol ⊧ to stress this special role. In the linguistic terminology, $\vDash P\left(a\right)$ means that $P\left(a\right)$ is the principal proposition of the sentence, to which the other propositions refer in determining their subordinative relationships.

Languages allow us to describe facts but also give commands, as well as ask questions. Performatives are linguistic elements determining the specific functionality of statements. We give only two examples in FLL here: “Go home!” and “Where do you go?”, in

Table 21. “Go home.”

You(a)

b = $\iota$Home

^Go(P)

$!a\vDash P\_b\left(a\right)$

and

Table 22. “Where do you go?”

You(a)

Place(b)

^Go(P)

?b ⊧ P_Place(b)(a)

, respectively.

The interrogative symbol before the assertion symbol tells us that the following expression is a question and the constant corresponds to an interrogative pronoun. Analogously, the exclamation mark expresses orders and, before the constant, it indicates the individual at which the order is directed.

In conclusion, the FLL logical representation of language is based on predication. The presence of different logical orders provides the main complexity of linguistic expressions. When people learn to speak, they implicitly acquire the capability of analysis and synthesis that allows for correct and efficient use of the integrated system of predications underlying FLL representations. Three of four logical orders are often present in the ordinary discourse (“Your beauty fascinates me”).

Logical symbols of FLL can be reduced to 20. No variable symbols are present, but individual constants $a,b,c$ …possibly indexed, and predicative constants $P,Q,R,\dots$ or class constants $A,B,C,\dots$ (possibly indexed).

Table 23. FLL operators.

$\to \neg \wedge \vee$	Implication, Negation, Conjunction, Disjunction
$\left(\right)^\iota \epsilon$	Predication, Pred. Abstraction, Determination, Choice
$\_\_\left(\right)\left[\right]<>$	Complementation, Indir. Compl., Modif., Specification
$\vDash =!?$	Assertion, Equation, Exhortation, Interrogation
$\mapsto \left\{\right\}indbool$	Typing

summarizes the FLL operators. Figure 7 will show the interlingual character of FLL representations, and Figure 8 provides a visual representation of FLL operators.

4. Conclusions

The adequacy of FLL in representing the logic of natural languages highlights, in terms of mathematical logic, the role of traditional logical analysis developed within the classical linguistic tradition, linked to the study of ancient languages and based on Aristotile’s schema of predication. Namely, FLL gives a rigorous logical basis for the main concepts of logical linguistic analysis by using twenty logical operators (Table 23). In this sense, the logic of the natural language results in a link between mathematical logic and linguistics and a natural way to approach the first using the latter. The monadic nature of FLL is a crucial aspect concerning variable elimination, coupled with high-order predicates.

In previous work, conversations with ChatGPT were reported, which show the ability of these systems to learn a logical formalism similar to FLL and acquire capability in terms of providing correct logical representations of given texts. However, we know these chatbots are based on transformers, and linguistic meanings are reduced to embedding vectors, as numerical vectors of many thousands of components.

The idea of an embedding vector is rooted in a long linguistic tradition [40], which emerged in the 20th century, with its origin in structural linguistics, concerning phonology, where a phoneme is a set of pertinent features that exclusively identify it in opposition to all the other phonemes of a language. The same intuition can be exported to word semantics because, given a document corpus, a word can be identified by all the documents in which it occurs and by all the positions in which it can be found.

A further investigation could be focused on the relationship between embedding vectors for sentences and discourses and their corresponding FLL representations. The two methods correspond to (geometric) synthetic versus (logical) analytic comprehension. Specific aspects of a detailed analysis of their comparison could provide crucial elements for a deep understanding of the related cognitive process on which knowledge is based [16].

Mathematical logic, with the notions of class, symbol, number, variable, operation, equation, relation, function, predicate, set, type, proposition, truth value, connective, variable abstraction, and predicate abstraction, provides a powerful and universal system of conceptualization, which surely is one of the most relevant successes of mathematics. Teaching chatbots mathematical logic could improve their semantic mechanisms by acquiring theoretical competencies to assist in the organization of their internal knowledge.

A future direction for the development of this paper could be an analysis of chatbot interactions in learning and exhibiting FLL representations, along with the experience presented in [27]. The levels, times, and strategies of FLL training could provide tools for evaluating the logical competencies of future conversational systems.