Modulated logics and uncertain reasoning

Modulated logics and uncertain reasoning

Walter Carnielli

Department of Philosophy and CLE

State University of Campinas–Campinas, SP, Brazil carniell@cle.unicamp.br

Maria Cl´audia C. Gr´acio

School of Philosophy and Science State University of S˜ao Paulo–Mar´ılia, SP, Brazil

cabrini@marilia.unesp.br

Abstract

This paper studies a family of monotonic extensions of first-order logic which we call modulated logics, constructed by extending classical logic through generalized quantifiers called modulated quantifiers. We give an uniform treatment of modulated logics, obtaining some general results in model theory. Besides carefully reviewing the “Logic of Ultrafilters” and the “Logic of Most”, two new monotonic logical systems are introduced here: the “Logic of Many” and the “Logic of Ubiquity”, which formalize inductive assertions of the kind “many” and “almost everywhere” through new modulated quantifiers Zand f, respectively. Although the notion of

“most” can be captured by means of a modulated quantifier semantically interpreted by cardinal measure on sets of evidences, it is proven that this system, although sound, cannot be complete if checked against the intended model. This justifies the interest on a purely qualitative approach to this kind of quantification, what is guaranteed by interpreting the modulated quanti- fiers, respectively, as families of upper closed sets and pseudo-topologies.

Modulated logics can be used to provide alternative foundations for fuzzy concepts and fuzzy reasoning, for reasoning on social choice theory, and for gaining a new regard on certain problems in philosophy of science.

1 A softened approach to quantification

A long-standing question, which has received a modern formulation, concerns the formalization of reasoning and argumentation based upon assertions or sentences

which are not absolutely true, but are instead supported by favorable evidences.

Such assertions give support to a more wide-scoped form of reasoning and argu- mentation called uncertain reasoning or flexible reasoning, which by its turn help to support inductive reasoning (in the sense of [4] and [24]).

This form of inductive reasoning and argumentation is still waiting for a logic foundation; even if Popper (in [24]) claimed that induction is never actually used by the scientists, and deflated the role of inductive argumentation while distin- guishing between science and non-science in favor of falsifiability, it is not said that inductive reasoning and argumentation are useless. On the contrary, uncer- tain or flexible reasoning is commonly used, and in the majority of cases has to do with assertions and sentences involving vague quantification of the kind “almost all”, “most”, “many”, etc. This kind of statement occurs often, not only in ordi- nary language, but also in new attempts to formalizing abductive reasoning and heuristics.

Some approaches as in [4] and [25] concern the question of rational justifica- tion of inductive reasoning from the logical viewpoint. However, all solutions pro- posed to rational justification of inductive reasoning have shown to be insufficient (for different reasons) in the sense that they could not offer a broadly satisfactory or definitive answer to the question.

One of the most popular proposals for formalizing some aspects of this kind of reasoning is the so-called default logic (cf. [25]). In that paper, the author proposes a non-monotonic logical system which formalizes sentences of the kind

“almost always” in terms of the notion of “in the absence of any information to the contrary, assume...”. According to [25], it is clear to anyone acquainted with the tools of formal logic that what we have in these situations is a question of quantification, namely how to represent and understand generalized quantifiers¹ of the kind “most”, “almost all”, etc. Although, [25] insists upon the necessity of treating such form of reasoning through generalized quantification, he never- theless chooses to formalize sentences of the kind “almost always” or “generally”

by using extra-logical operators. Several impediments and problems on the for- malization of sentences through the operator proposed in [25] have been already advanced (e.g. [15], [23] and [8]). Other criticisms to the non-monotonic ap- proach to the formalization of reasoning under uncertainty have been presented by [29].

In the same vein of Reiter’s argument about the need of representing general- ized quantifiers in formal language, it is suggested in [2] that a semantic theory of natural language needs to incorporate generalized quantifiers expressing notions

1Mathematically interesting quantifiers which cannot be defined in terms of universal and ex- istential quantifiers were firstly investigated in [20], and have been since then extensively studied;

see [9] and [11].

such as “most”, “many”, “more than half”, etc., since quantification in natural languages is clearly not limited to universal and existential quantifiers.

In the present paper we develop some perspectives initiated in [6], and present a family of (monotonic) logical systems called modulated logics to treat some kinds of inductive assertions, namely notions of “most”, “many” and “almost everywhere” (which can also be interpreted as “almost always”, under certain assumptions concerning similarity of time and space). These systems are char- acterized by adding generalized quantifiers (here denoted by Q) called modulated quantifiers to their syntax, and are semantically interpreted by expanding classical models to include subsets of the power set of the universe called complexes (de- noted by q), defined by certain mathematical structures (as filters and ultrafilters, topological spaces and their generalizations, and son on).

Modulated structures naturally generalize the use of ultrafilters and its abstrac- tions to express notions of “largeness” or “majority”. For example, in some papers on social choice theory ultrafilters are employed to express notions of “largeness”

(see [10]). A generalization of ultrafilters called “majority spaces” are also used in [21] within the context of modal logic to express a concept of weak majority (see discussion in Section 8).

Intuitively, classical models are endowed with sets of sentences which repre- sent positive evidences with respect to a knowledge basis. Such expanded models are called modulated models or modulated structures.

Modulated structures and modulated quantifiers are the mathematical realiza- tion of the intuitive idea of understanding logical constants in a softened way, in the sense of making them more apt to express concepts and reasoning that only seem to be possible in natural language. This is the case of linguistic modifiers like “many” and “few”, “much” and “little”, “generally” and “rarely”, as well as the ability of reasoning by means of them. If the classical quantifiers are sufficient to express reasoning when the subject matter is well-behaved, as in the universe of numbers, sets, groups and so on - that is, mathematical objects and clear-cut re- lations on them - natural reasoning is of course much more wide-scoped, dealing with not so tamed objects and not so limpid relations on them. For example, “all numbers that divide 10¹⁰+1 are odd” is a clear sentence with an uncontrovertial interpretation (since odd numbers cannot have even divisors) while, all numbers greater than 10¹⁰ + 1 are big numbers” has a controversial meaning since “big number” is a vague predicate. Also, “most prime multiples of powers of 10¹⁰+1 are odd” is also controversial, but now for a different reason: although the numer- ical predicate “odd” is uncontroversial, as well as all other numerical concepts involved in this sentence, the modifier “most” complicates everything; it is true that “only” 2.(10¹⁰+1)ⁿwill be even, while p.(10¹⁰+1)ⁿwill be odd for infinitely many odd primes p. On the one hand one may feel inclined to say that there are more odd than even, but on the other hand the sets have exactly the same cardi-

nality, although this is not true in initial segments. This plastic, rather qualitative than quantitative, capacity of linguistic modifiers convey reasoning and informa- tion, and part of our intention here is to investigate how much of this reasoning and information can be expressed in expanded logical structures, while keeping as much as possible the nice and familiar features of logic. We propose to ap- proach the question by means of modulated models, which permit to talk about modulated logics.

We show how particular modulated models defined through specific mathe- matical structures are able to capture specific kinds of uncertain or flexible state- ments. For example, considering the complex q as the class of subsets of the universe whose cardinal number is greater than their complements, one can for- malize assertions of the form “most...”. We call this system the Logic of Most.

Another example, which we call the Logic of Many, is obtained by identifying q with upper closed sets, intended to formalize assertions of the form “many...”. A third example, taking q as a weak version of a topological space, can be used to formalize assertions of the form “almost everywhere”. This system is called the Logic of Ubiquity. In particular, we will see that the Ultrafilter Logic (extensively studied in [6], [7], [29], [33], [34] and [35]) is a special case of modulated logic which formalizes assertions of the form “almost all...”.

It is proven that the Logic of Many and the Logic of Ubiquity are conservative extensions of first-order classical logic and that they preserve important properties of classical logic as soundness and completeness with respect to the defined mod- els, among others. The Logic of Most, however, even if sound, is not complete with respect to the intended models.

We also discuss some fundamental issues about relations among modulated logics, presenting examples of situations where each one is better applied than the competitors. Concluding, we comment on the particularities of modulated logics, pointing out for some possibilities of further work and applications. This paper is a companion of [36], in the sense that the essentials of modulated logics are treated here, while a deeper investigation on particular logics for the foundations of qualitative reasoning is done there².

2 Modulated logics

In this section we introduce a family of monotonic extensions of first-order logic denominated modulated logics, which provide a general representation for several kinds of inductive assertions. This family is defined by extending classical logic

2This paper is based on the Ph.D. thesis by the second author ([13]) written at the State Uni- versity of Campinas and supervised by the first author. A condensed version of this paper was presented at the 6^thKurt G¨oodel Colloquium held in Barcelona, June 1999.

through the modulated quantifiers. Such quantifiers are semantically interpreted by convenient subsets of the power set of universe; such subsets mean to represent, in intuitive terms, arbitrary sets of assertions supported by evidences related to knowledge basis.

Next section introduces the syntax, semantics and axiomatics for general mod- ulated logics L^τ_ωω(Q), which constitutes thus a general formal approach for induc- tive or uncertain reasoning.

2.1 Syntax, semantics and axiomatics for general modulated logics L

(Q)

Let L^τ_ωωbe the usual first-order language of similarity typeτcontaining symbols for predicates, functions and constants, and closed under the connectives∧,∨,→ and¬, and under the quantifiers∃and∀.

By L^τ_ωω(Q) we denote the extension of L^τ_ωωobtained by including in the formal language generalized quantifiers Q, called modulated quantifiers. The formulas (and sentences) of L^τ_ωω(Q) are the ones of L^τ_ωω plus those generated by the fol- lowing clause: if ϕis a formula in L^τ_ωω(Q) then Qxϕis a formula in L^τ_ωω(Q) too.

The notion of free and bound variables in a formula, as well as other syntactical notions, is extended in the usual way for the quantifiers Q.

The result of substituting all free occurrences of a variable x inϕby a term t is denoted byϕ(t/x). For simplicity, when there is no danger of confusion, we write simplyϕ(t) instead ofϕ(t/x).

The semantical interpretation for the formulas in the modulated logics L^τ_ωω(Q) is defined as follows, for a typeτ=< I,J,K,T₀,T₁i(where I,J and K are subsets of the set of natural numbers N, T₀is a function from I into N and T₁is a function from J into N):

A=hA,{R^A_i }_i∈I,{f_j^A}_j∈J,{c^A_k }_k∈Ki

a classical first-order structure of similarity type τ, let q be a set of subsets of the power set of the universe A such that empty set ∅does not belong to q, i.e., q ⊆ ℘(A)ⁿ − {∅}. The structure A^q formed by the pair hA,qi composed of a usual structure plus a complex is called a modulated structure for L^τ_ωω(Q). It is to be noted that [16] calls this type of structure weak structure. In formal terms a modulated structure is :

A^q= hA,qi=hA,{R^A_i }_i∈I,{f_j^A}_j∈J,{c^A_k }_k∈K,qi

whereAis a familiar structure such that A is its universe, R^A_i is a T₀-ary(i) relation defined in A (for i ∈ I), f_j^A is a function from Aⁿ in A (supposing T₁( j) = n, for j ∈ J),c^A_k is a constant of A (for k ∈ K), and q is a complex. Usually, n = 1 and

the complex is just q⊆ ℘(A); this will the case in all of our basic examples, except in the suggested applications for formalizing fuzziness in Section 1.

The interpretation of relations, functions and constant symbols is the same as in L^τ_ωω with respect to A. The notion of satisfaction of a formula of L^τ_ωω(Q) in a structure A^q is inductively defined in the usual way, by adding the following clause: letϕbe a formula which free variables set is contained in{x} ∪ {y₁, . . . ,y_n} and consider a sequence→−

a = (a₁, . . . ,a_n) in A. We define A^q |= Qxϕ[→−

a ] iff{b∈A :A^q|=ϕ[b;→− a ]} ∈q.

Usual semantical notions, such as model, validity, logical consequence, etc.

are appropriately adapted from classical logic.

The axioms of L^τ_ωω(Q) are those of L^τ_ωω, including the identity axioms (see, for example, [19]), plus the following specific axioms for the quantifier Q:

(Ax1)∀xϕ(x)→ Qxϕ(x);

(Ax2) Qx(ϕ(x))→ ∃x(ϕ(x));

(Ax3)∀x(ϕ(x)↔ ψ(x))→(Qx(ϕ(x))↔ Qx(ψ(x)));

(Ax4) Qx(ϕ(x))↔ Qy(ϕ(y)).

Axiom (Ax2) asserts that an inductive statement, formalized in L_ωω^τ (Q), cannot be supported by an empty evidence set. Axiom (Ax1) means that if all individuals of the universe support a statement, then it is an inductive statement in L^τ_ωω(Q).

Axiom (Ax3) states that, if two statements are universally equivalent, then they are equivalent inductive statements.

The basic logic rules of the system L^τ_ωω(Q) are the usual rules of classical logic: Modus Ponens (MP) and Generalization (Gen). Usual syntactical concepts such as proof, theorem, non-contradictoriness (or consistency), etc., for L^τ_ωω(Q) are also appropriately adapted from classical first-order logic.

As it can be seen bellow, abstract modulated logics preserve several syntactical properties of classical first-order logic.

Theorem 2.1. The system L^τ_ωω(Q) is consistent.

Proof. Using the same steps of the proof for classical logic L_ωω^τ (see [19]), chang- ing the definition of the “forgetting function” by means of the inclusion of the

condition h(Qxϕ(x))=h(ϕ(x)).

ForΣ∪ {ϕ, φ}a sentences set andψa formula in L^τ_ωω(Q), the following prop- erties of theories in L^τ_ωω(Q) can be easily proved by means of simple adaptation of the classical proofs:

Theorem 2.2. ForΣ∪ {ϕ}sentences, ifΣ∪ {ϕ} `ψthenΣ `ϕ→ψ.

Theorem 2.3. (a)Σis consistent iffevery finite subsetΣ0of Σis consistent.

(b)Σ∪ {ϕ}is inconsistent iffΣ` ¬ϕ.

(c) IfΣis maximal consistent, then:

(i)Σ` ϕiffϕ∈Σ; (ii)ϕ<Σiff¬ϕ∈Σ;

(iii)ϕ∧φ∈Σiffϕ∈Σandφ∈Σ.

Theorem 2.4. Any consistent theory in L^τ_ωω(Q) can be extended to a maximal

consistent theory.

2.2 Rudimentary model theory for L

(Q)

Considering that [16] proves soundness and completeness for an axiomatic system analogous to L^τ_ωω(Q) with respect to weak models, we are interested in establish- ing in this section some other prime results of model theory for modulated logics.

We thus discuss here an analogous result of Ło´s ultraproduct theorem for mod- ulated models, and a counterexample to the problem of interpolation in L^τ_ωω(Q).

A similar result for topological models was given in [30].

A modulated ultraproduct for L^τ_ωω(Q) is defined by extending the definition of ultraproduct in classic first-order logic (see [19]), by means of the procedure below.

Let J be a nonempty set, and for each j in J, let M_j =hA_j,q_jibe a modulated model for L^τ_ωω(Q). Let F an ultrafilter on J. For each j in J, let Aj denote the universe of the model M_j. The cartesian productΠjA_j is defined as the set of all functions f with domain J such that f ( j) ∈ A_j, for all j ∈ J. In the cartesian product ΠjAj, the following equivalence relation ³ is defined: f =F g iff {j :

f ( j)= g( j)} ∈ F.

On the basis of this equivalence relation,ΠjA_jcan be splitten into equivalence classes: for any f in ΠjAj, its equivalence class fF is defined as {g : f =F g}. Denote the set of all equivalence classes f_F by ΠFA_j, and define a model M = hA,qifor L^τ_ωω(Q), with universeΠFA_j, in the usual way, including the clause:

• Let ε_j be any element of q_j, for all j in J. Then, ε_j ⊆ A_j, from where we have that if a ∈εj, then a= f ( j) for some f ∈ΠjAj. The subset q⊆ ΠFAj

is generated by:

{[ΠFε_j]_F :{j :ε_j ∈q_j} ∈ F}

where [ΠFε_j]_F ={f_F ∈ΠFA_j :{β: f (β)∈ε_β} ∈ F}

We call the model M just defined a modulated ultraproduct for L^τ_ωω(Q) and denote it byΠFM_j. The definition allows us to obtain the following theorem.

3It is easy to see that f =F g is indeed an equivalence relation ([19]).

Theorem 2.5. Let F be an ultrafilter on a set J and{hA_j,q_ji, j ∈ J}be a family of modulated models for L^τ_ωω(Q). Let M = ΠFMj be a modulated ultraprod- uct. Then, for any formulaϕ, which free variables among v₁, . . . ,v_n, and any se- quence (g₁)_F, . . . ,(g_n)_F of elements ofΠFA_j,M |= ϕ[(g₁)_F, . . . ,(g_n)_F] if and only if{j∈ J : Mj |= ϕ[g1( j), . . . ,gn( j)]} ∈ F.

Proof. Using induction on the number of connectives and quantifiers in ϕ, it is simple to carry the proof through the same steps of the proof for classical logic L^τ_ωω. The only case that remains to prove is for formulasϕof the form Qyψ. The other cases are analogous to that of classical first-order logic (see [19], p.105).

Let ϕ(→−x ) be a formula of the form Qyψ(→−

x ) whose free variables belong to

−

→x = {x₁, . . . ,x_n}. Suppose, firstly, that{j∈ J : M_j |= Qyψ[g₁( j), . . . ,g_n( j)]} ∈ F.

Then, by definition,{j∈ J :{a∈ Aj : Mj |= ψ[a; g1( j), . . . ,gn( j)]} ∈ F. Denoting {a ∈ A_j : M_j |= ψ[a; g₁( j), . . . ,g_n( j)]}by ε^ψ_j,{j ∈ J : ε^ψ_j} ∈ q_j} ∈ F from where [ΠFε^ψ_j]_F = {f_F ∈ ΠFA_j : {β : f (β) ∈ ε^ψ_j} ∈ F} and [ΠFε^ψ_j]_F ∈ q. Therefore, M |= Qyψ[(g1)F, . . . ,(gn)F].

Conversely, suppose now that M |= Qyψ[(g1)_F, . . . ,(g_n)_F]. Then, by the defi- nition of satisfaction, {f_F ∈ ΠFA_j : M |= ψ[ f_F; (g₁)_F, . . . ,(g_n)_F]} ∈ q and hence, by inductive hypothesis,ε^ψ = {fF ∈ ΠFAj :{j : Mj |= ψ[ f ( j); g1( j), . . . ,gn( j)]} ∈ F} ∈q. But, by definition of q,

∀j∈ {j : M_j |= ψ[ f ( j),g₁( j), . . . ,g_n( j)]}

and

{f ∈ε^ψ : M_j |=ψ[ f ( j); g₁( j), . . . ,g_n( j)]}

are elements of qj. So,{j ∈ J : {f ∈ ε^ψ : Mj |= ψ[ f ( j); g1( j), . . . ,gn( j)]} ∈ qj} ∈ F. Therefore, by definition,{j∈ J M_j |= Qyψ[g₁( j), . . . ,g_n( j)]} ∈ F.

We conclude this section by presenting a counterexample to the validity of the interpolation theorem in L^τ_ωω(Q). The problem of interpolation in L^τ_ωω(Q) can be formulated in the following way: Let ϕ andψ be sentences of L^τ_ωω(Q) such that ϕ |=ψ: Is there a sentenceθof L^τ_ωω(Q) such thatϕ |=θ, θ |=ψand, with exception of equality, all extra-logical symbols (relations, functions and constants) occurring inθoccur both inϕandψ? The sentenceθis called an interpolate ofϕandψ.

Let ϕ and ψ be the sentences of L^τ_ωω(Q). Suppose L^τ_ωω(Q) is the language containing just symbols occurring inϕorψ. Let:

L₁=sublanguage of L^τ_ωω(Q) containing just extralogical symbols inϕ;

L₂=sublanguage of L^τ_ωω(Q) containing just extralogical symbols inψ;

L₀ = L₁∩L₂.

We will construct sentencesϕandψand modulated modelshA,q₁iandhA,q₂i in such a way thatϕ|= ψ,hA,q1i |=ϕ,hA,q2i |=¬ψandh<,q1i ≡ h<,q2i, where

< is the reduct ofAto L₀. This construction violates the interpolation property, since any interpolate is in L₁ ∩L₂ and the models above are elementarily equiv- alents in L1 ∩ L2. So, if hA,q2i |= ¬ψ, then it is true that hA,q2i |= ¬θ; but hA,q₁i |= θtoo, what generates a contradiction (sinceh<,q₁i ≡ h<,q₂i) ([30]).

Let L₁ = {B(x),C(x),P(x)}and L₂ = {B(x),C(x),R(x)}. Letϕbe the sentence QxP(x)∧ ∀x(P(x)↔ ((B(x)∨C(x)))∧ ¬QxC(x) andψbe the sentence Qx(B(x)∨ C(x))∨ ¬∀x(C(x)→ R(x)).

It can be verified, on the basis of Axiom 1, that|= ϕ→ ψ(or, equivalently, by the deduction theorem,ϕ|=ψ).

Let N be the set of natural numbers, P^A = {2n : n ∈ N},C^A = {4n : n ∈ N},B^A = {n : n∈ P^A−C^A or n= 8k for some k ∈ N}and R^A = {n : n∈C^A or n= 2k+1 for some k ∈N}.

Define

hA,q₁i=hN,B^A,C^A,P^A,R^A,{P^A}i and

hA,q₂i= hN,B^A,C^A,P^A,R^A,{P^A}i;

thenh<,q₁i= hN,B^A,C^A,{P^A}i andh<,q₂i = hN,B^A,C^A,{R^A}i. It can be easily checked thathA,q₁i |=ϕandhA,q₂i |= ¬ψ. The above construction represents a counterexample to the interpolation property in L^τ_ωω(Q), sinceh<,q1i ≡ h<,q2i⁴. In the following section we present some specific forms of uncertain or flexible reasoning formalized by means of the notion of modulated logics.

3 The Logic of Most

This section is devoted to particularizing L^τ_ωω(Q) for a system which formalizes inductive statements of the type “most...”. Under the approach to be formalized in this section, we say that most individuals satisfy a sentence ϕ when the set composed by the individuals satisfying it is greater than the set composed by the individuals not satisfying it. This non-qualitative approach intends to explain the notion of “most” in terms of frequency (majority) or set cardinality.

Under this account for “most”, when an assertion represents most individuals in a universe (a large set of individuals), the assertion expressing its complement represents the minority of individuals (a small set of individuals). Also, when a sentence ϕ represents most individuals and it forms a subset of an assertion expressed byψ, then, of course most individuals satisfyψtoo. Moreover, ifϕand ψrepresent most individuals of a universe, then there exists at least one individual

4Result due to [30], p.188.

of this universe which satisfies both assertions. So, this notion take into account the size of the set.

Considering the exposed above, we present some criteria for subsets of a set U to be considered large (in the sense of majority). Let X,Y be arbitrary subsets of U:

(1) if X is large then its complement X^cwith respect to U is not large;

(2) if X is large and X⊆ Y then Y is also large;

(3) if X and Y are large, then X∩Y ,∅;

(4) U is large.

These criteria are naturally based in the concept of the cardinal number |X|

associated with a set X. In this way, we say that most individuals satisfy X if and only if|X|< |X^c|.

Other approaches to this notion in the literature are [22] (which analyzes the relationship between “few”, “many”, “most”, “all” and “some” by means of Aristotelian squares of opposition), and [28] and [31], which treat the notion of

“most”, in the sense of majority, through a new quantifier.

As an illustration on the application of this account of “most” for uncertain reasoning, we recall the famous example of birds given in [25]. Let B be the universe of birds and F(x) be any property applicable to birds, as “flying” for example. We say that “the property F applies to most birds” if for the set F = {x∈B : F(x)}we have|F|< |F^c|.

Syntactically, we will include a new quantifier]in the language, and denote the fact that the property R applies to most individuals by means of the sentence ]xR(x).

The semantics and syntax of this modulated quantifier ], intended to capture the notion of “most”, were firstly introduced in [28]. Although some papers (as those mentioned above) have discussed this notion, its formal properties have never been investigated.

In next section we present the formal system L^τ_ωω(]) to represent this notion and investigate some of its basic syntactical and semantical properties.

3.1 Incompleteness of the Logic of Most L

(])

The system L^τ_ωω(]) constitutes, thus, a particular modulated logic intended to cap- ture the notion of “most”. In formal terms, as we have seen, the language of L^τ_ωω(]) is obtained by particularizing the quantifier Q in L^τ_ωω(Q) to the quantifier] of most.

The semantics of the formulas in L^τ_ωω(]) is defined by means of modulated structures, identifying, in this case, the complex q with the following subsets:

q={B⊆ A :|B|< |B^c|}.

For a sequence→−

a =(a₁, . . . ,a_n) in A, the satisfaction of a formula of the form ]xϕ, whose free variables are contained in{y1, . . . ,yn}, is defined by

A |= ]xϕ[→−

a ] iff{b∈A :A |=ϕ[b;→−

a ]} ∈ q, i.e., A |= ]xϕ[→−

a ] iff|{b∈A :A |=ϕ[b;→−

a ]}|< |{b∈A :A |=¬ϕ[b;→− a ]}|. (I) Intuitively,]xϕ(x) is true inAiffmost individuals in A satisfyϕ(x).

The following examples illustrate assertions expressed in L^τ_ωω(]).

Example 3.1. Let I(x),R(x),E(x) and G(x,y) be predicates in L^τ_ωω(]) representing the properties of being an irrational number, a rational number, an even number and the relation “x is greater than y”, respectively.

Considering the universe of real numbers, one can, naturally, formalize:

(a) “most numbers are irrational” by: ]xI(x);

Considering the universe of natural numbers, one can formalize:

(b) “most numbers are not even” by:]x¬E(x);

(c) “it is not the case that most numbers are even” by:¬]xE(x);

(d) “for any natural number y, most natural numbers are greater than y” by:

∀y]xG(x,y).

The axioms of L^τ_ωω(]) are the ones of L^τ_ωω(Q) augmented by the following specific axioms for the quantifier]:

(Ax5)∀x(ϕ(x)→ ψ(x))→(]x(ϕ(x))→]x(ψ(x));

(Ax6)]x(ϕ(x))→ ¬]x(¬ϕ(x));

(Ax7) (]x(ϕ(x))∧]x(ψ(x))→ ∃x(ϕ(x)∧ψ(x)).

In intuitive terms, given an interpretation with universe A and formulasϕand ψwith exactly one free variable, then for sets [ϕ]= {a∈A :ϕ[a]}and [ψ]={a ∈ A :ψ[a]}, axioms (Ax5) to (Ax7) assert that:

(Ax5) if [ϕ] ⊂ [ψ] and most individuals belong to [ϕ], then most individuals belong to [ψ];

(Ax6) if [ϕ] is composed by most individuals, then [¬ϕ] is not composed by most individuals;

(Ax7) if [ϕ] and [ψ] are composed by most individuals, then their intersection is not empty.

The basic logic rules and usual syntactical concepts such as proof, theorem, logical consequence, consistency, etc., for L^τ_ωω(]) are the same of L^τ_ωω(Q).

Returning to Example 3.1 one can deduce, for instance, by means of axioms (Ax3) and (Ax6) and the sentences (a) and∀x(¬I(x)↔R(x)) that “rational num- bers do not constitute most of the numbers”, that is,: ¬]xR(x). By means of the established axioms for L^τ_ωω(]), it is easy to prove the following general theorems.

Theorem 3.2. The following formulas are theorems of L^τ_ωω(]) : (a) ]x(ϕ(x)∧ ψ(x)) → (]x(ϕ(x))∧]x(ψ(x))), (b) ]x(ϕ(x)) → ]x(ϕ(x)∨ψ(x)), (c)]x(¬ϕ(x)) →

¬]x(ϕ(x)), (d) (∀x(ϕ(x))∧]x(ϕ(x)→ψ(x)))→]x(ψ(x)).

It can be also easily verified that Theorems 2.1, 2.2 and 2.4 and Theorem 2.3 for general modulated logics remain valid in the Logic of Most.

It can be proven, however, that this logic, although preserving soundness, is not complete with respect to the defined models, that is, not every valid for- mula can be derived in L^τ_ωω(]). This justifies the interest on purely qualitative approaches to this type of quantifiers, as opposed to such cardinality-based ap- proach.

Theorem 3.3 (Soundness). Ifϕis a theorem in L^τ_ωω(]), thenϕis valid.

Proof. Since the inference rules MP and Gen in L^τ_ωω(]) preserve validity of for- mulas and the axioms are valid in such structures, soundness of L^τ_ωω(]) is imme-

diate.

However, since the completeness problem for a language containing quanti- fiers Q¹, . . . ,Q^srelies on whether the set of true formulas is recursively denumer- able ([20]), the converse of this theorem (i.e, completeness) cannot hold, as it can be seen by the following argument.

Let S = {(κ, θ) : exist R ⊆ A such that κ = |R| and θ = |R^c|}. Given A (a classical first-order structure), for each function T : S → {0,1},R ⊆ Aⁿ⁺¹ and

−

→b =(b₁, . . . ,b_n)∈Aⁿ, we define, following [20] : Q_T(R)=T (|{a∈A : (a,→−

b )∈R}|,|{a∈A : (a,→−

b )∈R}^c|).

Therefore, for a formulaϕin L^τ_ωω(]), with exactly one free variable, the defin- ition in (I) is equivalent to:

A |=]xϕ(x) iffT^M(κ, θ)= 1, in which T^M(κ, θ)= 1 iffκ < θ.

The following definition, also from [20], provides the basis for our proof.

Definition 3.4. Let m,n be non-negative integers and T⁰,T⁰⁰ functions such that {T⁰(κ, θ) = 1} ≡ (κ = m) and {T⁰⁰(κ, θ) = 1} ≡ (θ = n). Quantifiers Q_T⁰ and QT ” will be denoted by Σ^(m) andΠ⁽ⁿ⁾, respectively. A numerical quantifier is any Boolean polynomials of quantifiersΣ^(m),Π⁽ⁿ⁾(m,n= 0,1,2,· · ·).

The quantifier]is clearly not a numeric quantifier, since it cannot be expressed by a Boolean polynomial of quantifiersΣ^(m)andΠ⁽ⁿ⁾; the following theorem proves that the completeness problem in L^τ_ωω(]) has a negative solution as shown in [20]:

Theorem 3.5. Let the quantifiers∃and∀occur among Q¹, . . . ,Q^sand let A be a denumerable universe. A necessary and sufficient condition for the completeness problem for quantifiers Q¹, . . . ,Q^s to have a positive solution is that the quanti-

fiers Q¹, . . . ,Q^sbe numerical.

From the above theorem, thus, we conclude that there are valid formulas which cannot be proved in L^τ_ωω(]). This is an interesting case of elementary incomplete- ness having on its genetic constitution a numerical approach, which is, on the other hand the essence of the celebrated incompleteness arguments of [12]. It is well known that his first theorem is general, as much as it can be applied to any axiomatic theory which is ω consistent, whose proof procedure is effective, and is strong enough to represent basic arithmetic. However, pace G¨odel, the theorem loses force as much as we should be able to avoid reasoning in numerical terms.

This can of course be regarded as a motivation to think in other than numerical terms. Of course, big questions remain, as for example how effective or interest- ing this deviation could be. We do not try to answer to this, but at least sketch on the essentials of what can be done from the qualitative side.

4 The Logic of Many

This section now considers a qualitative form of modulated logic intended to for- malize inductive assertions of the kind “many...”. The vague notion of “many”

here approached is associated to the concept of relevant set of evidences, but not necessarily linked to the notion of majority (in terms of cardinality) nor to the invariant thresholds (in the sense of something not changing from a model to an- other).

So, for instance, considering the universe of Brazilians, when we assert that

“many Brazilians wear skirts”, we have associated to the assertion a set of evi- dences (people) considered to be large. On the other hand, when we assert “many Brazilians love soccer”, we have associated to this statement another set of peo- ple, also supposed to be large. However, not necessarily the two sets of people are of comparable size. Besides, the evidences observed in favor of these asser- tions do not necessarily represent the majority of cases in the universe, nor their intersection is necessarily large.

In this way the notion of “many” here treated and expressing “a great amount of evidence” is more abstract than the one treated in the Logic of Most.

The notion of “many” can be considered, as well as the notions of “most”

and “almost all”, as an intermediary notion between the maniqueist vision of the universe conveyed by mere “there exists” and “for all”. However, it distinguishes itself of those (“most” and “almost all”) by the fact that the set of instances that

do not satisfy the assertions within the scope of “many” is not necessarily small.

In the notions of “most” and “almost all”, the set of individuals that not satisfying the assertions is necessarily small (in the particular case of “almost all”, they are termed exceptions). There are situations, however, in which the set of evidences in favor of the assertion as well as of its complement (in relation to a particular universe) can be considered large. For example, “many Brazilians wear skirts” is large, but it seems intuitive that “many Brazilians don’t wear skirts”, is (hopefully) a large set too.

As another example of statements formalized under the notion of “many”, consider the universe of natural numbers. It seems intuitive that one can assert that “many natural numbers are odd” and “many natural numbers are even”. In the Logic of Most, however, we cannot assert either of them, and in the Ultrafilter Logic (that treats the notion of “almost all”), stating one of them impeaches stating the other.

Besides the properties presented for general modulated logics, the following property is clearly identified to the notion of “many”: if many individuals satisfy a sentenceϕ, andϕimpliesψ, then ψis also satisfied by many individuals of the universe.

The notion of “many”, exposed above, can be captured by the mathematical concept of families of upper closed sets over an universe A, defined as collections of subsets of A such that:

(i) if B∈F and B⊆C, then C ∈F;

(ii) A∈F;

(iii)∅<F.

The notion of large sets in a universe is thus identified with the concept of upper closed sets over this universe. In this way, a property is true for many individuals in a universe (i.e., the set of evidence is large) if it belongs to the family of upper closed sets associated to the model.

By an argument analogous to the well-known result that shows that families enjoying the finite intersection property can be extended to principal filters (see [3]), it is easy to show that any family of subsets of A not containing the empty set can be extended to a family of upper closed sets:

Theorem 4.1. Let B be a family of subsets of A such that∅ < B; then B can be extended to a family of upper closed sets.

Proof. Take F = {C : X j C j A : X ∈ B}; F is obviously a family of upper closed sets, since:

(i) Suppose that Y ∈ F and Y ⊆ Z ⊆ A; then by definitions of F there exists some X ∈B such that X j Y j A. Since Y ⊆ Z ⊆ A, then Z ∈F;

(ii) A∈F, by definition of F;

(iii)∅<F, by definition of F.

Syntactically, we define a new modulated quantifier, Z, called quantifier of many, in modulated logics language given by

Zxϕ(x) meaning “for many x,ϕ(x) holds”.

In the following section we present the formal system L^τ_ωω(Z) and its seman- tics, which formalize the notion of “many” interpreted by the concept of families of upper closed sets.

4.1 Syntax, semantics and axiomatics for the Logic of Many L

( Z )

The language of L_ωω^τ (Z) is obtained by particularizing in L^τ_ωω(Q) the quantifier Q to the quantifierZfor many. The semantical interpretation of formulas in L^τ_ωω(Z) is carried out in modulated structures where the complexes q are identified with families of upper closed sets. Formally, a modulated structure for L^τ_ωω(Z), called a structure of upper closed sets, is constructed by endowing Awith a family of upper closed sets F^A over A given by:

A^F =hA,F^Ai=hA,{R^A_i }_i∈I,{f_j^A}_j∈J,{c^A_k }_k∈K,F^Ai.

The notion of satisfaction of a formula of the formZxϕ, whose free variables belong to{y₁, . . . ,y_n}, by a sequence→−

a = (a₁, . . . ,a_n) in A, is defined by A^F |=Zxϕ[→−

a ] iff {b∈A :A^F |=ϕ[b;→−

a ]} ∈ F^A for F^A a family of upper closed sets in A.

In intuitive terms,Zxϕ(x) is true inA^F, i.e., the set of individuals in A satisfy- ingϕ(x) belongs to F^A, if and only if many individuals of A satisfyϕ(x).

Again, the usual semantical notions like model, validity and semantical con- sequence for this system are the general ones in modulated logics.

The following examples illustrate assertions that can be naturally expressed in L^τ_ωω(Z).

Example 4.2. Let E(x),O(x) and G(x,y) be predicates in L^τ_ωω(Z), standing for “x is even”, “x is odd” and “x is greater than y”, respectively.

Considering the universe of natural numbers, we represent the assertions:

(a) “many natural numbers are even” by: ZxE(x);

(b) “many natural numbers are odd” by:ZxO(x);

(c) “for any natural number, many natural numbers are greater than it” by:

∀yZxG(x,y).

Example 4.3. Let B(x,y) be a predicate in L^τ_ωω(Z), standing for “x likes drinks of type y”. Examples of formalized assertions in L^τ_ωω(Z) are:

(a) “many people like some type of drink” by:Zx∃yB(x,y);

(b) “many people like many species of drink” by: ZxZyB(x,y).

The axioms for L^τ_ωω(Z) are the ones of L^τ_ωω(Q), augmented by the following specific axiom for the quantifierZ:

(Ax5)∀x(ϕ(x)→ ψ(x))→(Zx(ϕ(x))→ Zx(ψ(x)).

Given an interpretation with a universe A andϕ, ψformulas, with exactly one free variable, axiom (Ax5) intuitively asserts, for sets [ϕ] = {a ∈ A : ϕ[a]}and [ψ] = {a ∈ A : ψ[a]}, that if [ϕ] contains many individuals and [ϕ] is a subset of [ψ], then [ψ] also contains many individuals.

The inference rules, usual syntactical notions like sentence, proof, theorem, logic consequence, consistence, etc., for L^τ_ωω(Z) are the same as defined for gen- eral modulated logics.

Looking to the definition of satisfaction and the axiomatic system in L^τ_ωω(Z), we see that, syntactically, the quantifiers Z and ∃ have similar logical conse- quences. However, the modulated quantifier Z differs qualitatively from ∃ by offering a free choice about the sets representing large sets (i.e., sets that represent many individuals). In this way, under the notion of “many”, we establish for each situation (or model) a different measure of largeness. Under the classical notion of “there exists” such measure is invariant for all structures (and models): the ex- istential quantifier is but a limit case, interpreted in all structures by the complex q= ℘(A)− {∅}, for each universe A .

The following logical consequence of L^τ_ωω(Z), for example, can be easily proven:

Theorem 4.4. The following formulas are theorems in L^τ_ωω(Z) : (a)Z xϕ(x)∧

Zxψ(x)→Zx(ϕ(x)∨ψ(x)).

So, on the basis of the axiomatic system of the Logic of Many, we can deduce for example, from (a): “many people wear skirts” and (b): “many people wear trousers”, that “many people wear skirts or wear trousers”. But the “many people wear skirts and wear trousers” is not deducible. This fact is in agreement with our intuition that, although assertions (a) and (b) may represent large sets, the set interpreting their intersection is not necessarily non-empty. Besides, as already cited in previous examples, we may not deduce in this case that the negation of assertions (a) and (b) necessarily constitute small sets.

We can easily verify that Theorems 2.1, 2.2 and 2.4 and Theorem 2.3 for modulated logics remain also valid in the Logic of Many.

4.2 Soundness and Completeness for L

( Z )

This section shows that the Logic of Many is sound and complete with respect to structures of upper closed sets.

Theorem 4.5 (Soundness). Ifϕis a theorem of L^τ_ωω(Z), thenϕis valid.

Proof. This is immediate, since rules MP and Gen preserve the validity of for- mulas, and the axioms of L^τ_ωω(Z) are true in every structure of upper closed sets.

The proof of the Completeness Theorem for L^τ_ωω(Z) uses an analogue of the well-known method of constructing models by adding witnesses (known as Henkin’s method) to modulated models.

Theorem 4.6 (Extended Completeness). Let T a set of sentences of L^τ_ωω(Z). Then, T is consistent if and only if T has a model.

Proof. We sketch here only the crucial steps of the proof. Given a set T of sen- tences in L^τ_ωω(Z) and C a set of new constants, we extend T to a consistent set T^∗ such that T^∗has C as the witness set. We define a canonical model of a family of upper closed sets A^F = hA,{R^A_i }_i∈I,{f_j^A}_j∈J,{c^A_k }_k∈K,F^Aiin a similar way to the completeness proof for classical logic, including the definition of an appropriate family of upper closed sets F^Aover A as follows:

(1) Define for each formulaϕ(→−

v ), with free variables→−

v = {v₁, . . . ,v_n}, ϕ(v)^T ={(c^∼₁, . . . ,c^∼_n)∈Aⁿ : T `ϕ(c₁, . . . ,c_n)}

and considering the formulasθ(x), with only one free variable, B^T ={θ(x)^T ⊆ A : T `Zxθ(x)}.

(2) in view of axiom (Ax5) in L^τ_ωω(Z), and using Theorem 4.1,B^T ⊆ ℘(A) can be extended to a family of upper closed sets F^A.

It can be shown by induction on the length ofϕ that for every sentenceϕ of T,A^F |=ϕiffT ` ϕ. The only interesting step is whereϕis a sentence of the form Zxϕ(x).

Let θ(x) ≡ Zxϕ(→−

x ). Then A^F |= Zxϕ(x) iff {c^∼ ∈ A : A^F |= ϕ(c^∼)} ∈ F^A iff, by inductive hypothesis, {c^∼ ∈ A : T ` ϕ(c^∼)} ∈ F^A iff (ϕ(x))^T ∈ B^T iff

T ` Zxϕ(x).

Since L^τ_ωω(Z) preserves the Completeness Theorem, then Compactness and L¨owenheim-Skolem theorems can be adapted to hold for L^τ_ωω(Z). It is worth not- ing that though there may be an apparent conflict involving such results and the well-known Lindstr¨om’s theorems which characterize classical logics (see, e.g., [11]), this can be easily explained taking into consideration that ours is a non- standard notion of models, due to the presence of complexes in the models.

5 The Logic of Ubiquity

This section is interested in formalizing inductive statements of the kind “almost everywhere” or “almost allways”. This is used here to mean a significantly wide- spread set of positive evidences, but not necessarily large with respect to a uni- verse. The sense of significant set employed here is that, although it can be small, it represents a characteristic that it is ubiquitously present in the universe.

For example, consider again, the universe of Brazilians and the assertion “there exists unemployed people almost everywhere”. What we mean in this situation is that, even considering that the set formed by unemployed Brazilians may be not large (with respect to the universe size), it represents a property present almost everywhere among the country. In other words, even if we do not know a single person who is not unemployed, we can find someone in a closer neighborhood who is unemployed; so this logic formalizes a notion of “ϕis ubiquitous” or “ϕis valid almost everywhere”, for a sentenceϕ.

The concept we want to formalize is, in this way, independent of the notion of large sets of evidences, but akin to the notion of “significant set of evidences”

ascribed to an assertion. A smaller set may be more significant than a larger set, or just as significant.

On the one hand, those statements express a more “vague” form of inductive reasoning. On the other, sentences of this kind (“ significant positive evidences”) represent assertions close to that used in statistical inference, in which the set of evidences (sample) considered sufficient to the establishment of inferences, al- though significant, is generally small in relation to the universe size. In this the- ory, in general, a small non-biased (random) set is more significant than another, perhaps larger but biased sample.

This suggests a connection between modulated logics and Bayesian inference, which permits interpretations of probabilities as degrees of belief, contrary to strict frequentism, and also with Bayesian epistemology, a contemporary theory inter- ested in the proposal of a formal apparatus for inductive logic⁵.

Thinking about the features of the notion of “almost everywhere”, it seems naturally intuitive that if almost everywhere there exist individuals for which ϕ is true and for which ψ is true, then ϕ or ψ is also almost everywhere satisfied.

For instance, back to our example, if almost everywhere you find people that love soccer and people that love samba, then it is natural to assert that almost everywhere you find people that love soccer or love samba.

It also seems completely intuitive that, if we know that the whole universe satisfies an assertion ϕ, thenϕ is also “almost everywhere” satisfied. Inversely,

5Both Bayesian inference and Bayesian epistemology are based upon certain presuppositions inherited from Thomas Bayes in the 18th Century

if no individual in the universe satisfies an assertion ϕ, then it is certainly not satisfied “almost everywhere” .

Another apparently intuitive property of this notion is that, if two sets are not significant in the sense of not representing assertions of the type “almost every- where” , then their union is also not significant. For example, if neither the the set of Brazilians that play baseball, nor the set of Brazilians that play golf, are significant, then it seems intuitive that the set of Brazilians that play baseball or play golf is not significant.

By duality, we can assert that if two sets are significant in the sense of repre- senting assertions of the type “almost everywhere”, then their intersection is sig- nificant too. In [29] similar considerations are used to justify that the intersection of (qualitatively) large sets is large.

The notion of ubiquity sketched above has the following structural properties:

(1) if two assertions are ubiquitous, then so is their conjunction and disjunction;

(2) if every individual of the universe satisfies the assertion, it is ubiquitous; (3) if no individual of the universe satisfies the assertion, it is not ubiquitous. Such properties lead us to the idea of topology.

However, the usual clause in the definition of topology which asserts that “ar- bitrary unions of families of open sets are open sets” has no counterpart in uncer- tain reasoning, since reasoning and argument (as deductions) are assumed to be of finite character. Moreover, although the empty set belongs to every topologi- cal space, no form of inductive reasoning can infer assertions without supporting evidences (cf. axiom (Ax2) of general modulated logics).

For such reasons, in the formalization of inductive assertions of type “al- most everywhere” we employ a more abstract notion of topology called pseudo- topology.

A pseudo-topology is a family = of subsets of a set X, called pseudo-open subsets, which satisfies the following conditions:

(a) The intersection of two pseudo-open subsets is a pseudo-open subset;

(b) The union of two pseudo-open subsets is a pseudo-open subset;

(c) X is a pseudo-open subset;

(d) The empty set∅is not a pseudo-open subset.

The notions of pseudo-topological space and pseudo-closed subset are defined by analogy with those for usual topology. Also, by analogy with corresponding notions in topology, we can introduce a pseudo-topology in a set by describing just the ‘basic’ pseudo-open sets, as follow.

A pseudo-open basis, or, simply, a basis in a pseudo-topological space (X,=), is a collection B of pseudo-open subsets of X, called basic pseudo-opens, with the following property: every pseudo-open subset A ⊆ X is expressed as a non-void finite union A= ∪β_λof pseudo-open subsets which belongs to B.

The following theorem can be proven in an analogous way as the correspond- ing theorem for topological spaces (see [17]):

Theorem 5.1. A family B of sets is a basis for some pseudo-topology over X =

∪{β:β∈B}if and only if all members of B are pairwise non-disjoint and, for any two members U,V of B and each point x in U∩V, there is W in B such that x∈W and W ⊆U∩V.

This last condition is fulfilled, in particular, when U∩V ∈B.

It can be easily noted that the notion of pseudo-topology does not refer to a new concept of neighborhood, but rather to the concept of dense neighborhood, since, as it will be shown, every pseudo-topology defines a basis of dense opens of a topology (in another space). This fact is a consequence of the following result:

Theorem 5.2. Every pseudo-topology defines a basis of dense opens of a topo- logical space.

Proof. Let =a pseudo-topology. Given U ∈ =, we define U^→ = {V ∈ = : U ⊆ V},U^→ ⊆ =. The set B = {U^→ : U ∈ =}is a basis of a topology in =. In fact, if U^→₁ , . . . ,U^→_n ∈B, then U₁^→∩ · · · ∩U_n^→=(∪ⁿ_i=1U)^→∈B. So,=^∗= {∪_i∈IU_i^→: U^→_i ∈ B}is a topology over =. Moreover, U^→ is dense. In fact, let V = ∪_i∈IU_i^→ , ∅ be a open subset in =^∗,U^→∩ ∪_i∈IU_i^→ = ∪_i∈IU^→∩U_i^→ = ∪_i∈I(U ∪U_i)^→. But, U ∪Ui ∈ (U∪Ui)^→, then U ∪Ui ∈ ∪_i∈I(U∪Ui)^→, whence U^→∩ ∪_i∈IU_i^→ , ∅.

Then, U^→is dense.

Therefore, every pseudo-topology defines a basis of dense opens of a topolog-

ical space.

In this way, every pseudo-topology=defines a topology=^∗(in another space) such that the pseudo-open sets of=are identified with elements of a basis of dense subsets of=^∗.

The previous result justifies the interpretation of assertions of type “almost everywhere” by means of th notion of pseudo-topology: although the evidence set may not be large (in relation to the universe), individuals satisfying that property are densely widespread in the universe.

As an example of this assertion, consider the universe of real numbers and let R(x) be the unary predicate standing for “x is rational”. We can assert “the rational numbers are ubiquitous”, since in any open neighborhood of a real number we find a rational. We remind, however, that the set of rational numbers is not large (in relation of size of real numbers).

We can still easily verify the following property about pseudo-topologies.

Proposition 5.3. Every family of dense opens in a topological space is a pseudo-

topology.