Completeness theorems for σ–additive probabilistic semantics

Journal Pre-proof Completeness theorems for σ–additive probabilistic semantics

Nebojša Ikodinovi´c, Zoran Ognjanovi´c, Aleksandar Perovi´c, Miodrag Raškovi´c

PII:

S0168-0072(19)30118-6

DOI:

https://doi.org/10.1016/j.apal.2019.102755

Reference:

APAL 102755

To appear in:

Annals of Pure and Applied Logic

Received date:

7 August 2018

Revised date:

17 June 2019

Accepted date:

11 October 2019

Please cite this article as: N. Ikodinovi´c et al., Completeness theorems for σ–additive probabilistic semantics, Ann. Pure Appl. Logic (2019), 102755, doi: https://doi.org/10.1016/j.apal.2019.102755.

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Completeness theorems for σ–additive probabilistic semantics Nebojˇsa Ikodinovi´c1 , Zoran Ognjanovi´c2 , Aleksandar Perovi´c3 and Miodrag Raˇskovi´c2

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1: Faculty of Mathematics, University of Belgrade, [email protected] 2. Mathematical Institute of SASA, {zorano, miodragr}@mi.sanu.ac.rs 3. Faculty of Transport and Traffic Engineering, University of Belgrade, [email protected]

Abstract

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We study propositional probabilistic logics (LP P –logics) with probability operators of the form P≥r (”the probability is at least r”) with σ–additive semantics. For regular infinite cardinals κ and λ, the probabilistic logic LP Pκ,λ has λ propositional variables, allows conjunctions of < κ formulas, and allows iterations of probability operators. LP Pκ,λ,2 denotes the fragment of LP Pκ,λ where iterations of probability operators is not allowed. Besides the well known non-compactness of LP P –logics, we show that LP Pκ,λ,2 –logics are not countably com+ pact for any λ ≥ ω1 and any κ, and that are not 2ℵ0 –compact for κ ≥ ω1 and any λ. We prove the equivalence of our adaptation of the Hoover’s continuity rule (Rule (5) in [13]) and Goldblat’s Countable Additivity Rule [9] and show their necessity for complete axiomatization with respect to the class of all σ–additive models. The main result is the strong completeness theorem for countable fragments LP PA and LP PA,2 of LP Pω1 ,ω .

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1

Introduction

Probability logic is a branch of mathematical logic that studies formal systems involving application of probability quantifiers and/or operators. A seminal work on logics with probability quantifiers (more precisely on various so called generalized quantifiers, e.g. ”for uncountably many”) was done by Jerome Keisler starting in the mid seventies of the twentieth century [19, 20, 21]. The main features of Keisler–type probability logics are extensive use of admissible set theory, infinitary model theory and Barwise completeness and compactness theorems [13, 36, 37]. 1

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Other probability logics are related to applications of modal-like probability operators in reasoning about finitely-additive probabilities. The probability operators are unary operators of the form P≥s with the intended meaning that the probability is at least s. The number of published contributions is quite extensive, primarily due to natural connections with reasoning under uncertainty, subjective and objective probabilities, and Bayesian inference. Some of the results that have influenced our work are given in [1, 4, 6, 7, 8, 11, 12, 25, 26, 42]. Syntactical foundations (i.e. formalization techniques) of the present work can be found in some of our earlier work, see for instance [5, 15, 16, 40, 31, 32, 34, 38, 39, 41]. The main technical difficulty with σ-additivity lies in its infinitary nature – we have to be able to formally express the fundamental properties of σadditive measures such as the continuity condition    P rob φn = lim P rob (φ0 ∨ · · · ∨ φn ) . n∈ω

n→∞

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In other words, to properly formalize σ–additivity one needs countable conjunctions and disjunctions in the language.

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1.1

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There are many relevant books and papers for the research presented here, some are mentioned above. Here we shall focus on the most relevant for the present paper. We start with Hoover’s axiomatization of logics with probabilistic quantifiers presented in [13]. The axioms and inference rules used here are adaptations of the corresponding Hoover’s axioms and inference rules to Kripke– style semantics of probabilistic operators acting on propositional formulas. Goldblatt’s axiomatization presented in [9] inspired us to show the equivalence between Hoover’s and Goldblat’s methods for capturing σ–additivity (Theorem 4.3 in Section 4). It has also motivated us to show the necessity of one of those rules for the strong completeness with respect to σ–additive models (Subsection 3.4). The initial motivation for the research presented here was our attempt to simplify axiomatization provided by Meier in [25]. As it was noted in [25], Carol Karp has shown that if propositional logic Lκ 1 is strongly complete, then κ must be a strongly inaccessible cardinal2 . It is well known that the existence of inaccessible cardinals is not provable in ZFC.

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Related work

1 Lκ (κ is a regular cardinal) is an extension of the classical propositional logic that additionally allows infinitary conjunctions and disjunctions of the length strictly lesser than κ. The corresponding axiom system has the additional infinitary inference rule:  from the set of premises {φ → ψn : n ∈ ω} infer φ → n∈ω ψn . 2 i.e., regular (κ = cf(κ)) and strong limit ( λ < κ =⇒ 2λ < κ).

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Moreover, in [25] it is also noted that even if Lκ logic is strongly complete, in general case it cannot be used as a base for a strongly complete probabilistic logic that allows applications of probability operators on conjunctions (disjunctions) of arbitrary length < κ. The main problem lies in the fact that a σ–complete algebra B need not  to be κ–complete (i.e., for all infinite λ < κ, if {Xi : i < λ} ⊆ B, then i<λ Xi ∈ B). Consequently, κ has to be at least a measurable cardinal. It is well known that the existence of measurable cardinals is not provable in ZFC. A complete axiomatization of a σ–additive probabilities offered in [25] was obtained in the following way: • The application of probability operators is restricted to finitary formulas; • The maximal length of formulas and inferences is equal to 2λ , where λ ≥ ℵ0 is the size of the object language, i.e. the cardinality of the disjoint union of the sets of probability operators and propositional variables (if both of these sets are finite, then by definition λ = ℵ0 ); • The standard axiomatization of Carol Karp for L(2λ )+ (see [17]) is extended with suitable probabilistic axioms and infinitary inference rules that formalize Archimedean property and σ–additivity.

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The main technical result of [25] is the proof of the strong completeness theorem for theories of cardinality ≤ 2λ .

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1.2

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Organization of the paper

In Section 2 we start with the analysis of infinitary propositional probabilistic languages and define syntax and semantics of LP P –logics. For regular infinite cardinals κ and λ, the LP Pκ,λ –logic has λ propositional variables, allows iteration of probabilistic operators and conjunctions of < κ formulas. LP Pκ,λ,2 is the fragment of LP Pκ,λ where iterations of probability operators are not allowed. Here we show that probabilistic variant of the modal K axiom P≥1 (φ → ψ) → (P≥r φ → P≥r ψ) 85 86 87 88 89 90 91 92 93

is valid for all r ∈ [0, 1]. In Section 3 we prove various non-compactness phenomena of LP P logics, thus establishing axiomatization limitations for uncountable probabilistic languages. Recall that a theory T is ν–satisfiable iff each subset Γ ⊆ T of cardinality < ν is satisfiable. The ν–compactness is the statement ”ν–satisfiability implies satisfiability”. Besides the well known noncompactness (Subsection 3.1) of all LP P –logics, we prove that LP Pκ,λ,2 is + not ω1 –compact for arbitrary κ and λ ≥ ω1 and that Lκ,λ,2 is not 2ℵ0 – compact for κ ≥ ω1 and any λ (Subsection 3.3). We conclude this section 3

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with the proof of the validity of the continuity property and construct an example of theory that violates it, but has a weak (finitely additive) model (Subsection 3.4). Aside of Subsection 3.1, all results in this section are novel contributions of the present paper. In Section 4 we provide axiomatization for countable fragments LP PA and LP PA,2 of the LP Pω1 ,ω –logic. We give an integral list of all axioms and inference rules appearing in LP P –logics. There are two parameters for the choice of the appropriate set of axioms and inference rules:

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• iteration of probability operators (present or absent);

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• length of conjunctions (finite or infinite).

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We prove the soundness, the equivalence between the axiom A8 and the Cont rule (Theorem 4.2), and the equivalence between the Hoover’s and Goldblatt’s approaches to formal expression of σ–addivity, i.e. the equivalence of rules Cont and CPr (Theorem 4.3). Both theorems are original contributions of this paper. Section 5 is dedicated to the proof of the strong completeness theorems for LP PA and LP PA,2 logics with respect to the class of σ–additive models. The simplest case of finitary formulas without iterations of probability operators (logics LP Pω,ω,2 and LP PA,2 , ω ∈ / A) was obtained by direct application of Caratheodory’s extension theorem. The most difficult cases LP PA with infinitary formulas (ω ∈ A) and finitary formulas (ω ∈ / A) are handled with great detail. The argument used here is a modification of the techniques presented in [13] and some of our earlier work. Concluding remarks are in the final section.

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Propositional probabilistic languages

In practice, it often becomes necessary to consider structures satisfying certain collections of formulas, rather than just single formulas. This consideration leads to the familiar notion of a theory in a logic. While theories are suitable to simulate infinite conjunctions, there is no apparent way to simulate infinite disjunctions. For a fixed cardinal λ, we consider probabilistic propositional languages (in short, LP P -languages) that contain a set of λ propositional letters =  V ar  {pξ : ξ < λ}, and the following logical connectives: ¬ ’not’, ’and’, ’or’, and modal-like probabilistic operators of the form P≥r , for r ∈ [0, 1], ’the probability is at least r’. The set F or of all formulas of LP P∞,λ is the smallest set X such that:

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• all propositional letters are in X (V ar ⊆ X);

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• if φ ∈ X then ¬φ and P≥r φ ∈ X, for all r ∈ [0, 1]; 4

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• if Φ ⊆ X then



Φ ∈ X.

The first subscript   ∞ means that we can join together arbitrary many formulas by or . It is important to emphasize that each LP P∞,λ -language is formally specified in a set-theoretic sense: each formula of LP P∞,λ is a set. For definiteness, we agree that each letter pξ is the ordered pair (0,ξ); formulas are  finite sequences of symbols: ¬φ is the ordered pair (1, φ), Φ is (2, Φ), Φ is (3, Φ), P≥r φ is (4, r, φ). Variables for LP P -formulas are α, β, γ, φ, ψ, indexed if necessary.   The set LP Pω,λ consists of those formulas in which and are only used to join together finitely many formulas. More generally, if κ is any regular cardinal (such as the first uncountable cardinal ω1 ), then LP Pκ,λ   is the same as LP P∞,λ except that or are only used to join together fewer than κ formulas at time. Thus we can study many smaller subclasses inside LP P∞,λ . The study of LP P∞,λ involves additional set theoretic assumptions, usually certain large cardinal axioms, making it more related to the set theory than to the probability logic. The main focus of this paper is on LP Pω1 ,ω and particularly its countable fragments since the extended (i.e. strong) completeness theorem for LP Pω1 ,ω does not hold. This is a consequence of the fact that there are unsatisfiable LP Pω1 ,ω –theories whose all proper subsets are satisfiable, see subsection 3.3. One of difficulties related to LP Pω1 ,ω is that it has uncountably many formulas. Due to various non-compactness phenomena that will be thoroughly discussed later on, it is more convenient to consider suitable countable subsets (fragments) of LP Pω1 ,ω . A particular fragment is usually constructed with respect to some natural closure properties that allow sufficiently expressive syntax. For any set A, let LP PA = LP Pω1 ,ω ∩ A. We shall call LP PA a fragment of LP Pω1 ,ω if A is a non-empty transitive set which is closed under pairs and union, i.e., the following are true:

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• ∃x(x ∈ A),

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• if y ∈ x, x ∈ A then y ∈ A,

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• if x, y ∈ A then {x, y} ∈ A and x ∪ y ∈ A.

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The following are immediate consequences:

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1. ω ⊂ A;

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2. if x, y ∈ A then (x, y) ∈ A;

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3. LP PA is closed under ¬, and under finite

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and



;

4. if φ ∈ LP PA and r ∈ A ∩ [0, 1], then P≥r φ ∈ LP PA (if ω ∈ A, we consider the reals of A to be Dedekind cuts of Q in A; if ω ∈ A, 5

we consider the reals of A to be just the rationals, coded as pairs of integers.)

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Of particular interest are fragments LP PA , where A is a countable transitive set. The most important types of such sets are admissible sets [3]. If ω∈ / A, then A = HF , i.e. the set of all hereditary finite sets. Recall that a set X is hereditary finite iff it is finite and all of its elements are hereditary finite. The corresponding set of probabilistic formulas is the one that is the most frequently studied in the relevant literature [1, 6, 7, 8, 27, 28, 29, 34]. These papers consider slightly different languages, but their formulas are finite. On the other hand, [5, 13, 18, 19, 20, 21, 37] study the case ω ∈ A, where certain infinitary conjunctions are allowed. Here, the most important examples of admissible sets A are the set HC of all hereditary countable sets and its admissible fragments. Admissible fragments introduced by Jon Barwise3 as generalization of finiteness, allow extensive applications of Barwise compactness theorem. If nesting and iterations of probability operators are not allowed, such logic will be designated by addition of 2 to the subscript of its generic label. For example, LP Pω,ω,2 contains either classical propositional formulas, or finite Boolean combinations of atomic probabilistic formulas of the form P≥r α, where α is a classical propositional formula. Through the rest of the paper, by F orC will be denoted the set of all classical propositional formulas over the given set of propositional letters V ar. Since the context is clear, ”V ar” is not used in the label.

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2.1

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Here we study probabilistic operators related to the modal operators of necessity  and possibility ♦, see [14]. Consequently, semantics for probabilistic logics is obtained by suitable adaptations of Kripke-structures. Instead of modal accessibility relations between worlds, a quantitative characterization of accessibility is given in terms of probabilities, i.e., possible worlds are connected with some probabilities. In particular, each world is equipped with a probability space, see [2, 24].

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Semantics

A probabilistic model (shortly: model) is a triple of the form (W, P r, v), where • W is a nonempty set of worlds,

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• P r is a function that associates to each world w the corresponding probability structure P r(w) = (W (w), H(w), μ(w)) with the standard Kolmogorov properties:

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Barwise often quoted Kreisel’s opinion that cardinality consideration alone is too crude for a rational choice of infinitary formulas.

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– W (w) ⊆ W and W (w) = ∅;

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– H(w) is a subalgebra of the power set algebra P(W (w));

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– μ(w) : H(w) −→ [0, 1] is a probability measure (finitely or σ– additive), and • v is an evaluation such that for each world w, v(w) : V ar −→ {0, 1}. A model will be called strong or σ-additive, if for each world w, P r(w) is a probability space, i.e. H(w) is a σ–algebra and μ(w) is a σ–additive probability measure. Otherwise, a model will be called a weak model.

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A precise description of intrinsic properties of probabilistic models by LP P -formulas is achieved by the corresponding satisfiability relation |= between worlds and formulas defined inductively on complexity of formulas as follows:

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• M, w |= p iff v(w)(p) = 1, where p is an arbitrary propositional letter;

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• M, w |= P≥r α iff [α]w = {u ∈ W (w) : μ(w)([α]w ) ≥ r;  • M, w |= i∈I αi iff M, w |= αi for all i ∈ I;

u |= α} ∈ H(w) and

• M, w |= ¬α iff M, w |= α. If the context is clear, M will be omitted. As usual, if Γ is any theory (set of formulas), then w |= Γ means that w |= γ for all γ ∈ Γ. A model (W, P r, v) is measurable if for every formula α and every w ∈ W , the set [α]w belongs to H(w). We will omit the subscript w from [α]w if it is clear from the context. Let Mσ be the class of all measurable σ-additive models. A semantic consequence relation is then defined by putting Γ |= φ iff w |= Γ implies w |= φ, for all worlds of all models M ∈ Mσ . In the case that Γ = ∅, the relation ∅ |= φ, in short |= φ, means that φ is valid, i.e., satisfied in every world of every model in Mσ . 2.1 Example. Let us show that the following probabilistic version of the modal axiom K (see [7, 8]), P≥1 (φ → ψ) → (P≥r φ → P≥r ψ) is valid in probabilistic models for all r ∈ [0, 1]Q . Suppose that w |= P≥1 (φ → ψ) and w |= P≥r φ. By definition of |= we have that, for all u ∈ W (w), u |= φ → ψ ⇔ u |= ¬φ ∧ u |= ψ ⇔ u |= φ ∧ u |= ψ,

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hence [φ → ψ] = [φ]c ∪ [ψ], so: μ(w)([φ → ψ]) = μ(w)([φ]c ∪ [ψ]) = μ(w)([φ]c ) + μ(w)([ψ]) − μ(w)([φ]c ∩ [ψ])  = 1 − μ(w)([φ]) + μ(w)([ψ]) − μ(w)([ψ]) − μ(w)([φ] ∩ [ψ]) = 1 − μ(w)([φ]) + μ(w)([φ] ∩ [ψ]). Since μ(w)([φ → ψ]) = 1, we have that 1 = 1 − μ(w)([φ]) + μ(w)([φ] ∩ [ψ]), i.e. μ(w)([φ]) = μ(w)([φ] ∩ [ψ]). Moreover, monotonicity of a measure μ(w) implies that μ(w)([φ] ∩ [ψ]) ≤ μ(w)([ψ]), thus we obtain the inequality μ(w)([φ]) ≤ μ(w)([ψ]). In particular, from μ(w)([φ]) ≥ r and the above we conclude that μ(w)([ψ]) ≥ r,

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i.e. that w |= P≥r ψ.

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Axiomatization issues

Our main objective is to axiomatize the relation |= by constructing a deducibility relation  and showing that Γ  φ iff Γ |= φ. The choice of axioms and inference rules is significantly influenced by the absence of compactness of semantical consequence.

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Recall that a theory Γ is κ–satisfiable iff each subtheory T ⊆ Γ of cardinality < κ is satisfiable (κ is a regular infinite cardinal). In particular, ω–compactness is called compactness, while ω1 –compactness is called countable compactness. The κ–compactness theorem (or just κ–compactness) is the following statement:

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κ–satisfiability implies satisfiability.

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We say that particular LP P –logic is κ–compact if it satisfies the κ– compactness theorem. The general status of the compactness for LP P – logics is given below: 8

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• All LP P –logics are not compact;

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• LP Pκ,λ and LP Pκ,λ,2 for λ ≥ ω1 are not countably compact;

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+

• LP Pω1 ,ω is not 2ℵ0 –compact. Recall that κ+ is the smallest cardinal + + strictly greater than κ. For example, ℵ+ 0 = ℵ1 , ℵ1 = ℵ2 , ℵω = ℵω+1 etc. In the subsection 3.1 we provide a generic example that not only shows the non-compactness of LP P –logics, but also isolates the main reason for this phenomena - so called non-Archimedean order types, i.e. ability to construct finitely satisfiable LP P –theories that force non-Archimedean probabilities. In the subsection 3.2 we show that LP Pω,ω1 ,2 is not countably compact, hence the same is true for any LP Pκ,λ such that λ ≥ ω1 . In the + subsection 3.3 we show that Lω1 ,ω is not 2ℵ0 –compact. Consequently, all LP Pκ,ω logics have the same status for all κ ≥ ω1 . There are two ways to handle these issues. The one presented in [25] is to apply set theoretical techniques for infinitary propositional logics developed by Carol Karp in [17]. The price is the fact that probability operators can be applied only to finitary formulas. The main approach was initiated by Keisler and Hoover [13, 18, 19, 20, 21] for infinitary model theory, logics with generalized quantifiers and logics with probabilistic quantifiers, which was later adapted for logics with probabilistic operators [1, 5, 7, 6, 8, 11, 12, 34, 42]. The key feature of this line of research is the focus on countable fragments of LP Pω1 ,ω and application of the Barwise compactness. Our aim is to provide an integral approach to axiomatization of the mention fragments that is strongly complete with respect to the class of the σ–additive models. The following well known example from the undergraduate measure theory (binary measure on the finite–cofinite Frechet algebra F(N) on N) shows existence of the finitely additive probability space that cannot be extended to a σ–additive probability space. 3.1 Example. The Frechet algebra F(N) is the subalgebra of the powerset algebra P(N) defined by X ∈ F(N) ⇐⇒ |X| < ω ∨ |N \ X| < ω, i.e. F(N) contains nothing else but all finite and cofinite subsets of N. Define binary probability measure μ : F(N) −→ {0, 1} by

1 , |N \ X| < ω μ(X) = . 0 , |X| < ω

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Note  that μ is not extendable to a σ–additive measure since μ(N) = 1, N= {n}, μ({n}) = 0 for all n ∈ ω and {n} ∩ {m} = ∅ for all m = n.  n∈ω

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The subsection 3.4 starts with the lemma 3.3, where we show that each σ–additive probability measure satisfies continuity property CP. Then we show that the example 3.1 can be coded in LP Pω,ω , hence in all LP P –logics that allow iterations of probability operators.

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3.1

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Non-compactness

As said above, here we shall show that all LP P –logics are not compact. Note that it is enough to prove this in the case for the LP Pω,ω,2 –logic. Let T = {P>0 p} ∪ {P< 291 292 293 294

1 n+1

p : n ∈ ω},

where p is an arbitrary propositional letter. Let T0 be an arbitrary finite subset of T and let k be the maximal integer such that P< 1 p ∈ T0 . We k+1

define the model M = (W, P r, v) as follows: • W = {w1 , w2 };

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• Fix v(w1 ), v(w2 ) : V ar −→ {0, 1} so that v(w1 )(p) = 1 and v(w2 )(p) = 0;

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• P r(w1 ) = P r(w2 ) = (W, H, v, μ), where:

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– H = P(W ); – μ(w1 )({w1 }) = μ(w1 )({w1 }) = k k+1 .

1 k+1 ,

μ(w1 )({w2 }) = μ(w1 )({w1 }) =

It is easy to see that w1 |= T0 . On the other hand, let M = (W, P r, v) be any model, w ∈ W an arbitrary world and let μ(w)([p]) = a. If a = 0, then w |= P>0 p. If a > 0, then w |= P< 1 p for any n > a1 . Thus, T is not satisfiable and the compactness n theorem fails. Note that the standard ordering properties of [0, 1] imply the following semantical property: • If w |= P≥r α for all r < s, then w |= P≥s α. This is inspiration for the Archimedean rule that shall be used to formally make inconsistent theories that force nonstandard (non-Archiedean) probabilities.

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3.2

Countable non-compactness

Here we shall show that LP Pκ,λ –logics are not countably compact for all λ ≥ ω1 . Note that it is sufficient to prove this in the case of the LP Pω,ω1 ,2 – logic. Let {pi : i < ω1 } ⊆ V ar. For 0 ≤ i < j < ω1 define the formula φi,j as follows: φi,j =def P=0 (pi ∧ ¬pj ) ∧ P>0 (¬pi ∧ pj ). It is easy to that the following holds: if a world w |= φi,j , then μ(w)([pi ]) < μ(w)([pj ]). Next, we define the theory T = {φi,j : i < j < ω1 }. Note that T is not satisfiable due to separability4 of [0, 1]. Namely, if ai is the measure of [pi ] then the sequence ai : i ∈ ω1  is isomorphic (as ordering) to the ordinal ω1 . In particular, {(ai , ai+1 ) : i ∈ ω1 }

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is an uncountable family of pairwise disjoint nonempty open intervals in [0, 1] (ai < ai+1 for all i ∈ ω1 ), which contradicts the separability of [0, 1]. In order to prove the countable satisfiability of T it is sufficient to show that for any λ ∈ ω1 the theory Tλ = {φi,j : i < j < λ}

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is satisfiable. Since λ is a countable ordinal, it can be embedded into (0, 1)Q . Let ai : i < λ be any such embedding where i → ai . Define the model M = (W, P r, v) where P r(w) = W, H, μ for each w ∈ W , and W , H, v and μ are defined as follows:

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1. W = {0, 1}V ar ;

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2. v(w)(p) = 1 iff w |= p;

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3. [α] = {w ∈ W : w |= α};

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4. H = {[α] : α ∈ F or};

ai , i<λ 5. μ([pi ]) = ; 0 , λ ≤ i < ω1 4

each open interval in [0, 1] contains a rational number.

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 6. μ([pi1 ∧ · · · ∧ pik ]) = min μ([pi1 ]), . . . , μ([pik ]) , i1 < · · · < ik . In [35] we have shown that conditions (5) and (6) determine a unique probability measure on H. It remains to prove that w |= φi,j for all i < j < λ, where s ∈ W is arbitrary. It is sufficient to show that μ([pi ∧ ¬pj ]) = 0 and μ(¬pi ∧ pj ) > 0 for all i < j < λ. μ([pi ∧ ¬pj ]) = μ([pi ]) − μ([pi ∧ pj ])  = ai − min μ([pi ]), μ([pj ] = ai − min(ai , aj ) = a i − ai = 0; μ([¬pi ∧ pj ]) = μ([pj ]) − μ([pi ∧ pj ])  = aj − min μ([pi ]), μ([pj ] = aj − min(ai , aj ) = aj − a i > 0.

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Hence, w |= Tλ .

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3.3

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Non-completeness of LP Pω1 ,ω

Here we shall construct an example of an unsatisfiable LP Pω1 ,ω -theory T + whose each proper subset is satisfiable. Consequently, LP Pκ,ω is not 2ℵ0 – compact for any κ ≥ ω1 . For any a ∈ [0, 1] the operator P=a is definable by P≥an α ∧ P≤bn α, P=a α ⇐⇒def n∈ω

n∈ω

where an : n ∈ ω and bn : n ∈ ω are monotone sequences of rational numbers from [0, 1]Q so that an ↑ a and bn ↓ a. It is easy to see that the theory T = {P=a p : a ∈ [0, 1]} 337

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is not satisfiable. Let us show that each proper subset of T is satisfiable. It is sufficient to prove that the theory Ta = T \ {P=a p} is satisfiable. We define the model (W, P r, v) as follows: • W = {0, 1}V ar ; 12

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• W (w) = W ;

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• v(w)(p) = 1 iff w |= p;

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• H(w) = {[α] : α ∈ F or};

0 , q = p • μ(w)([q]) = ; a , q=p • μ(w)([q1 ∧ · · · ∧ qn ]) = min(μ[q1 ], . . . , μ[qn ]), where q1 , . . . , qn are pairwise distinct propositional letters.

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By a straightforward application of the definition of the satisfiability relation we obtain that w |= Ta for any w ∈ W such that w |= p.

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3.4

346

349 350 351

352 353 354

Continuity property CP

Here we shall show that the example 3.3 can be represented in all LP P – logics that allow iterations of probability operators. Note that it is sufficient to show this in the case of the LP Pω,ω –logic. 3.2 Definition. Let M = (W, P r, v) be any LP P –model. We say that the satisfiability relation |= has the continuity property (abbreviated as CP) at the world w iff w |= P≥r P≥s φ iff (∀t < s)w |= P≥r P≥t φ

355 356

357

(CP)

for all φ. We say that the model M has the CP iff the above is true for all w ∈ W.  Let su show that the CP is valid in the class of σ–additive models. 3.3 Lemma. Let (W, P r, v) be a σ–additive model. Then (W, P r, v) satisfies the CP, i.e., for all w ∈ W , w |= P≥r P≥s φ iff (∀t < s)w |= P≥r P≥t φ. Proof of the lemma. Suppose that w |= P≥r P≥s φ. Since P≥s φ → P≥t φ is a valid formula for all t < s, we have that [P≥s φ → P≥t φ] = W (w), so μ(w)[P≥s φ → P≥t φ] = 1 for all t < s. Hence, w |= P≥1 (P≥s φ → P≥t φ) for all t < s. Moreover, by the example 2.1, w |= P≥1 (P≥s φ → P≥t φ) → (P≥r P≥s φ → P≥r P≥t φ) 13

for all t < s, so 358 359 360 361

w |= P≥r P≥s φ → P≥r P≥t φ

for all t < s. By the initial assumption, w |= P≥r P≥s φ, so w |= P≥r P≥t φ for all t < s, which concludes the proof of the ”if” part. Note that for the proof of the converse implication it is sufficient to prove the following:

362

1. {P≥t φ : t < s} |= P≥s φ;

363

2. {P≥r P≥t φ : t < s} |= P≥r P≥s φ.

364 365 366 367 368 369

370

The first item is a consequence of the Archimedean property of the reals, which is independent of the properties of a measure, so it remains to prove the second item. The nontrivial case is when s > 0, because the formula P≥0 φ is valid, hence so is P≥r P≥0 φ. The proof of this part substantially depends on the continuity of a σ–additive measure, see [22]. Assume that w |= P≥r P≥t φ, for all t < s, i.e., μ(w)({u ∈ W (w) : u |= P≥t φ}) ≥ r, for all t < s.  Since {u ∈ W (w) : u |= P≥s φ} = n≥ 1 {u ∈ W (w) : u |= P≥s− 1 φ}, and s

μ(w)({u ∈ W (w) : u |= P≥s φ}) =

n

lim μ(w)({u ∈ W (w) : u |= P≥s− 1 φ})

n→∞

n

≥ r, 371 372 373

we have w |= P≥r P≥s φ, which concludes the proof of the lemma.

Now we shall show how to formally interpret the example 3.1. More precisely, for the theory Γ = {P≥1 P>0 p} ∪ {P≥1 P≤

374 375



1 2n

p : n ∈ N}

we shall show that it is not satisfiable in any world of any strong model, and construct a weak model and show that it is satisfied in one of its worlds. Suppose that Γ is satisfiable in a σ-additive model. According to the CP property {P≥1 P≤ 1n p : n ∈ N} |= P≥1 P=0 p 2

376 377 378 379 380 381

However, P≥1 P=0 p and P≥1 P>0 p are inconsistent, a contradiction. Hence, Γ has no σ-additive model. It remains to show that Γ has a finitely–additive model. Let W = {wn : n ∈ ω}, H(wi ) = P(W ), v(w2i )([p]) = 1 and v(w2i+1 )([p]) = 0 for all i ∈ N. The sequence of measures μ(wi ) is defined as follows: μ(w0 )(X) = lim

n→∞

|X ∩ {wi : i ≤ n}| , X ⊆ W. n 14

382 383

It is shown in [4] that μ(w0 ) is a finitely-additive probability on H(w0 ) which is not σ-additive. Note that for all i ∈ N we have that |{i, i + 1, . . .} ∩ {0, 1, . . . , n}| n→∞ n n−i+1 = lim n→∞ n = 1,

μ(w0 )({wk : k ≥ i}) =

384 385 386

lim

a property that will be referred by (∗). Note also that μ(w0 ) can be any measure determined by an arbitrary non–principal ultrafilter on ω. For i ≥ 1 the probability μ(wi ) is determined by the distribution

 w0 w1 w2 w3 w4 w5 w6 w7 · · · 1 1 1 0 2i+2 0 2i+3 0 0 1 − 21i 2i+1 By the definition of the satisfiability relation we have that [p] = {w2k : k ∈ N}.

387

For i ≥ 1 we have that μ(wi )([p]) =

∞  1 2k

k=i+1

=

1 , 2i

388

Moreover, wi |= P≤ 1 p, for all j ≤ i. Thus, {wj , wj+1 , wj+2 , . . .} ⊆ [P≤ 1 p].

389

Clearly [P>0 p] = W . By (∗) we have that

2j

2j

μ(w0 )([P>0 p]) = μ(w0 )(P≤ 1 p) = 1. 2j

390

Hence, w0 |= Γ.

391

4

392 393 394 395 396 397 398 399 400



Axiomatization

In order to obtain complete axiomatization, it is necessary to address the compactness issues stated in the section 3. This will be achieved by introduction of the adequate infinitary inference rules. The presented axioms and inference rules are adaptations of the ones presented in [13, 22, 20, 33, 34]. The plan is to provide an integral list of axioms and inference rules. The particular LP P –logics will be axiomatized by the specific choice of the listed items (not all of them). All of logics studied here are countable fragments of the logic LP Pω1 ,ω . Through the rest of the paper, indices in probability operators P≥r are restricted to rational numbers from the real unit interval.

401 402

In infinitary logics, it is convenient to define φ¬: 15

403

• p¬ = ¬p for all p ∈ V ar;

404

• φ¬¬ = φ;

405 406

• (P≥r φ)¬ = ¬P≥r φ = P
411

Now we shall state all axioms and inference rules. Keep in mind that this is not a description of a single formal system, but a list of available axioms and inference rules that will be used to achieve strong completeness in different settings: LP Pω,ω , LP Pω,ω,2 , LP PA and LP PA,2 . Here LP PA is a countable transitive fragment of LP Pω1 ,ω .

412

Propositional axioms:

413 414

(A0) Tautology instances;  (A1) Φ → φ, where φ ∈ Φ;

415

(A2) ¬φ ↔ φ¬;

416

Bookkeeping axioms:

417

(A3) P≥0 φ;

418

(A4) P≥r φ → P>s φ, where r > s and r, s ∈ [0, 1]Q ;

419

Additivity axioms:

420

(A5) P≥r φ ∧ P≥s ψ ∧ P=0 (φ ∧ ψ) → P≥r+s (φ ∨ ψ), r + s ≤ 1;

421

(A6) P≤r φ ∧ P≤s ψ → P≤r+s (φ ∨ ψ), r + s ≤ 1;

422

σ–additivity axioms:

423

(A7)

407 408 409 410

424 425 426 427

428

429 430 431





n Ψ⊂fin Φ

P< 1 ( n



Ψ∧¬



Φ),

where Ψ ⊂fin Φ means that Ψ is a finite subset of Φ. The purpose of this axiom is to ensure that the probability of an infinite conjunction can be estimated by its finite sub-conjunctions, which is crucial for the Loeb’s construction;    (A8) P< 1 P≥r− 1 φ ∧ P
n

m

The purpose of this axiom is to formally express the CP. It will be relevant for the logics with infinitary formulas, i.e. for logics LP PA where ω ∈ A. 16

432

Inference rules:

433

(Nec) From a theorem φ infer P≥1 φ (necessitation);

434

(MP) From φ and φ → ψ infer ψ (modus ponens);

435 436 437 438

439 440 441 442 443 444

445 446 447 448 449 450 451 452 453 454 455 456

457 458 459 460 461 462 463 464 465 466 467 468 469

(Inf) From the set of premises {θ → φ : φ ∈ Φ} infer θ → (conjunction rule);



Φ

(Arch) From the set of premises {θ → P≥t φ : t < s} infer θ → P≥s φ (Archimedean rule);   (Cont) From the set of premises θ → P≥ 1 (P≥t φ ∧ P
470 471 472 473 474 475 476 477 478 479 480 481 482

1. LP P2 = LP Pω,ω,2 and LP PA,2 logics where ω ∈ / A. This is the simplest case (no additional axioms and inference rules). Formulas are finitary and neither iterations, nor nesting of probability operators is allowed; 2. LP Pω,ω and logics LP PA where ω ∈ / A: we add Cont. Here the formulas are finitary and probability operators can be applied without any restrictions; 3. LP PA,2 , where ω ∈ A: we add A7 and Inf. Here the formulas are infinitary, but neither nesting, nor iterations of probability operators is allowed; 4. LP PA , where ω ∈ A: we add A7, A8 and Inf. Here the formulas are infinitary and probability operators can be applied without any restrictions.

486

For each of the above logics we shall prove the strong completeness theorem with respect to the class of the σ–models (i.e. σ–additive or strong models). The most challenging cases are LP Pω,ω and LP PA where ω ∈ A (items 2 and 4).

487

4.1

483 484 485

Soundness

491

By soundness we assume the property T  φ ⇒ T |= φ. The standard proof uses induction on the length of the inference. So, it remains to show validity of all axioms and that inference rules preserve truth. Recall that, for any model M = (W, P r, v) and any w ∈ W :

492

1. w |= p iff v(w)(p) = 1, where p is an arbitrary propositional letter;

488 489 490

493 494 495

496 497 498 499 500 501

2. w |= P≥r α iff [α] = {u ∈ W (w) : u |= α} ∈ H(w) and μ(w)([α]) ≥ r;  3. w |= i∈I αi iff M, w |= αi for all i ∈ I; 4. w |= ¬α iff M, w |= α. Validity of propositional axioms A0–A2 is an immediate consequence of the items 1, 3 and 4 of the above definition. Validity of A3 is a consequence of the fact that the range of any μ(w) is the real unit interval [0, 1], while validity of A4 is a consequence of the linearity of the standard ordering of reals. Validity of A5 and A6 is a consequence of the fact that μ(w) is additive: μ(w)([φ ∨ ψ]) = μ(w)([φ] ∪ [ψ]) = μ(w)([φ]) + μ(w)([ψ]) − μ(w)([φ] ∩ [ψ]) = μ(w)([φ]) + μ(w)([ψ]) − μ(w)([φ ∧ ψ]), 18

502 503 504 505 506

so derived inequality μ(w)([φ])+μ(w)([ψ]) ≥ μ(w)([φ∨ψ]) implies validity of A6, while μ(w)([φ])+μ(w)([ψ]) = μ(w)([φ∨ψ]) in the case when μ(w)([φ∧ψ]) = 0 implies validity of A5. Validity of A7 is a direct consequence of the continuity of σ–additive measure μ(w):      μ(w) φn [φn ] = μ(w) n∈ω

n∈ω



=

=

lim μ(w)

n→∞

lim μ(w)

n 

 [φk ]

k=0  n

n→∞

 φk

.

k=0

To see that A8 is valid, first note that (P
m≥ r1

⎛⎡

so

⎤⎞

⎜⎢ ⎥⎟ μ(w) ⎝⎣ (P
m≥ r1

On the other hand, ⎛⎡ ⎤⎞ ⎛⎡ ⎤⎞ m ⎜⎢ ⎥⎟ μ(w) ⎝⎣ (P
507 508

m

m→∞

k=m0

m

where m0 is the smallest integer greater than 1r . In particular, by definition of convergence in reals we get the validity of A8.

514

Rules Inf and MP clearly preserve validity. If φ is a valid formula, then [φ] = W (w) for all w ∈ W , hence μ(w)([φ]) = μ(w)(W (w)) = 1. In particular, Nec preserves validity. Note that Cont is a reformulation of A8 in terms of inference rules, so Cont also preserves validity. Finally, soundness of Arch is a direct consequence of the Archimedean property of the reals.

515

4.2

509 510 511 512 513

516 517

518 519

Equivalence of cont and CP

Here we shall prove the equivalence of CP and Cont. We start with the basic proof theoretical properties of . 4.2 Theorem. Let  be the inference relation of an LP P –logic. Then the following holds: 19

520 521 522 523 524 525 526 527 528 529

1. Deduction theorem: T  φ → ψ iff T, φ  ψ; 2. Structural properties and introduction and elimination of connectives:  has the same structural and introduction/elimination properties as the classical inference relation. For instance: (a) T  φ implies T ∪ Γ  φ (weakening); (b) T, φ, ¬φ  θ;  (c) T, Φ  θ iff T ∪ Φ  θ (left introduction/elimination of conjunction);  (d) T  Φ iff T  φ for all φ ∈ Φ (right introduction/elimination of conjunction);

530

3. Probabilistic K:  P≥1 (φ → ψ) → (P≥r φ → P≥r ψ);

531

4. Monotonicity:  P≥r φ → P≥s φ, for r ≥ s;

532 533

534 535 536 537 538

5. In presence of the axioms A0–A6, and the rules Nec, MP, Inf, Arch, Axiom A8 and the continuity rule Cont are equivalent. Proof. The first item can be proved by the standard induction on the length of inference. The second item is a straightforward consequence of the definition of the inference relation, while the third and the fourth item can be proved in the same way as it was presented in [34] (Lemma 3.1). Let us prove the final item. Firstly we shall derive the Cont. Fix a positive integer n. By A1, A8 and MP,    P< 1 P≥s− 1 φ ∧ P
m

539 540

m

Now, applying the standard introduction/elimination and structural properties of  we obtain the following chain of equivalences:     P< 1 P≥s− 1 φ ∧ P
m

⇐⇒

m

⇐⇒

 P≥ 1 P≥s− 1 φ ∧ P
 m

m



m



θ → P≥ 1 P≥s− 1 φ ∧ P
m



θ→⊥

#   1 ⇐⇒ θ → P≥ 1 P≥s− 1 φ ∧ P
541

which is Cont.

20

For the converse implication, using the above chain of equivalences we get 

 m

  P< 1 P≥s− 1 φ ∧ P
m

for all positive n, so by Rule Inf, we obtain     P< 1 P≥s− 1 φ ∧ P
n m

542 543 544 545

m

i.e., A8 is a consequence of Cont.



In the following theorem we prove the equivalence between the continuity rule and the CP. Similar rules, closely related to Rule CPr, are considered in [9, 10, 22]. 4.3 Theorem. The rule (Cont) is equivalent to the rule (CPr): from the set of premises {θ → P≥r P≥t φ : t < s} infer θ → P≥r P≥s φ.

546 547 548 549 550

Proof. (Cont) ⇒ (CPr). The case r = 0 is trivial, so let us assume that r > 0. Let S = {θ → P≥r P≥t φ : t < s}. Let n and t be such that n > 1r and t < s. By transitivity of implication and |= (α → β) → [(α ∧ γ) → (β ∧ γ)], we obtain: S  (θ ∧ P≤r− 1 P≥s φ) → (P≥r P≥t φ ∧ P≤r− 1 P≥s φ). n

551

n

By  P≥l α ↔ P≤1−l ¬α, it follows that S  (θ ∧ P≤r− 1 P≥s φ) → (P≤1−r P
552

n

By Axiom A6 we have S  (θ ∧ P≤r− 1 P≥s φ) → P≤1− 1 (P
553

n

which is equivalent to S  (θ ∧ P≤r− 1 P≥s φ) → P≥ 1 (P≥t φ ∧ P
554

n

This is true for all t < s, so by (Cont) we deduce S  ¬(θ ∧ P≤r− 1 P≥s φ) n

which holds for every n ≥

1 r.

, i.e.,

S  θ → P>r− 1 P≥s φ n

Finally, by Arch it follows that

S  θ → P≥r P≥s φ.

21

555 556

(CPr) ⇒ (Cont) Let n be a positive integer, and T = {θ → P≥ 1 (P≥t φ ∧ n P
557

by Nec,MP and the probabilistic K, we have  P≥ 1 (P≥t φ ∧ P
558

n

and using transitivity of implication we obtain T  θ → P≥ 1 P
559

By  P≥l α ↔ P≤1−l ¬α, we derive T  θ → P≤1− 1 P≥s φ. n

560

Since  P≤1− 1 α → P<1 α, by transitivity of implication, it follows that n

T  θ → P<1 P≥s φ. 561 562

(1)

On the other hand, from  P≥s φ → P≥t φ, t < s, using |= (α → β) → [α → (α ∧ β)], Nec,MP and the probabilistic K, we have  P≥v P≥s φ → P≥v (P≥t φ ∧ P≥s φ),

563 564

for every t < s, and every v ∈ [0, 1]Q . Since T  θ → P≥ 1 (P≥t φ ∧ P
565

for every t < s, and every v ∈ [0, 1]Q . It follows that T  (θ ∧ P≥v P≥s φ) → (P≥ 1 (P≥t φ ∧ P
566

By additivity, T  (θ ∧ P≥v P≥s φ) → P≥v+ 1 P≥t φ n

567

for every t < s, and every v ∈ [0, 1]Q . By R we obtain T  (θ ∧ P≥v P≥s φ) → P≥v+ 1 P≥s φ n

568

for every v ∈ [0, 1]Q . Let v = 0. Then we have T  (θ ∧ P≥0 P≥s φ) → P≥ 1 P≥s φ n

22

569

and since  P≥0 P≥s φ, it follows that T  θ → P≥ 1 P≥s φ. n

Moreover, for v =

1 n

we have T  (θ ∧ P≥ 1 P≥s φ) → P≥ 2 P≥s φ n

570

n

and since T  θ → P≥ 1 P≥s φ, we deduce n

T  θ → P≥ 2 P≥s φ. n

571

Repeating the previous inference v =

2 n,

...,

n−1 n

we get:

T  θ → P≥1 P≥s φ and together with (1) we finally deduce T  ¬θ. 

572

573

5

574

5.1

575

We start with the simplest case where the direct application of the Caratheodory’s theorem is possible.

576

577 578

579 580 581 582

Strong canonical models Logics without iterations

5.1 Theorem. Logic LP P2 is strongly complete with respect to the class of all σ–additive models. Proof. In [34] it is shown that LP P2 is strongly complete with respect to the class of weak models (μ(w) needs not to be a σ–additive). Let T be any consistent LP P2 theory, T ∗ its maximal consistent extension, and let N = (W, P r, v) be the canonical model of T ∗ . More precisely:

584

• W is the set of all evaluations that satisfy all classical propositional formulas from T ∗ ;

585

• v(w, p) = 1 iff p ∈ T ∗ ;

586

• P r(w) = (W, H, μ) for all w ∈ W , where:

583

588

– H = {[α] : α ∈ F orC }, where [α] is the set of all evaluations from W that satisfy α;

589

– μ([α]) = sup{r ∈ [0, 1]Q : T ∗  P≥r α};

587

23

590

591 592 593 594

• w |= T ∗ for all w ∈ W . Note that H is the propositional Lindenbaum’s algebra of T ∗ . Let us showthat μ satisfies the conditions of Caratheodory’stheorem. Assume that n∈ω [βn ] ∈ H. Then, there is α∈ F orC such that n∈ω [βn ] = [α]. We claim that there is m ∈ ω such that n∈ω [βn ] = [β0 ] ∪ · · · ∪ [βm ]. Indeed, if [β0 ] ∪ · · · ∪ [βm ] is a proper subset of [α] for all m ∈ ω (note  that the immediate consequence of n∈ω [βn ] = [α] is the fact that [β0 ] ∪ · · · ∪ [βm ] ⊆ [α] for all m ∈ ω), then there is a world w ∈ WN such that w |= ¬β0 ∧ · · · ∧ ¬βm ∧ α, and the following propositional theory {¬β0 ∧ · · · ∧ ¬βm ∧ α : m ∈ ω}

595 596 597 598 599 600 601

is finitely satisfiable, so by compactness theorem for classical propositional logic, it is satisfiable. It means that there is a world w ∈ W such that: • w ∈ [α],  • w ∈ n∈ω [βn ].

 As a consequence we get a contradiction: n∈ω [βn ] is a proper subset of [α]. Thus, H does not contain nontrivial countable unions and the continuity condition of Caratheodory’s theorem   $ μ [βn ] = lim μ([β0 ] ∪ · · · ∪ [βm ]) whenever

m→∞

n∈ω



n∈ω [βn ]

∈ H is satisfied. Indeed, if $



n∈ω [βn ]

∈ H, then

[βn ] = [β0 ] ∪ · · · ∪ [βk ]

n∈ω 602

for some k ∈ ω, so  μ

$ n∈ω

 [βn ]

=

lim μ([β0 ] ∪ · · · ∪ [βm ])

m→∞

= μ([β0 ] ∪ · · · ∪ [βk ]). 603 604 605 606

By Caratheodory’s theorem, there exist a σ–algebra H ∗ that extends H and a σ–additive measure μ∗ on H that extends μ∗ . In particular, for the strong model N = (W, P r∗ , v) where P r∗ (w) = (W, H ∗ , μ∗ ) we have that N, W |= T ∗ for all w ∈ W . Hence the completeness of LP P2 . 

24

607

5.2

608

In this subsection we will prove strong completeness for our most general infinitary logic LP PA with iterations (we will consider both cases: ω ∈ A, and ω ∈ A). The corresponding axiom systems are given in Section 4. There is a well known correspondence between models and maximal consistent theories: each model (W, P r, v) generates the maximal consistent theories of the form

609 610 611 612 613

Logics with iterations

Γw = {γ : w |= γ}, w ∈ W.

620

The main method in obtaining the strong completeness theorem is to prove the satisfiability of maximal consistent theories and to show the corresponding Lindenbaum’s theorem, i.e. to show that each consistent theory can be extended to a maximal consistent theory. This is achieved via so called canonical models, i.e. models whose worlds are maximal consistent theories, and the corresponding semantical notions are defined by the inference relation.

621

5.2.1

614 615 616 617 618 619

622 623 624 625

Weak and strong canonical models

5.2 Definition. A weak canonical model is a structure of the form M = (W, P r, v), where W is the set of all maximal consistent theories, and for each w ∈ W the corresponding W (w), H(w), v(w) and μ(w) are defined as follows:

626

• W (w) = W ;

627

• H(w) = {[φ] : φ ∈ LP PA }, where [φ] = {u ∈ W : φ ∈ u};

628

• v(w)(p) = 1 iff p ∈ w for any propositional letter p;

629

• μ(w)([φ]) = sup{t ∈ [0, 1]A : P≥t φ ∈ w}.

630 631 632 633 634 635

Note that H is a Boolean algebra with respect to the usual set operations. Indeed, [φ]∩[ψ] = [φ∧ψ], [φ]∪[ψ] = [φ∨ψ], [φ]\[ψ] = [φ∧¬ψ], so H is closed for the intersection, union and set difference. Moreover, ∅ = [p ∧ ¬p] ∈ H and W = [p ∨ ¬p] ∈ H, so H is a field of sets (Boolean algebra). Let us show that μ(w) is a finitely additive probability. The first step is to prove that (r is a rational number from the unit interval): μ(w)([φ]) ≥ r iff P≥r φ ∈ w.

636 637 638 639 640



(2)

Since the converse implication (⇐) is true by definition of μ(w), it remains to prove the ”if” part. Suppose that μ(w)([φ]) ≥ r. Then for all s < r, by definition of μ(w) there is t > s such that P≥t φ ∈ w, which implies that P≥s φ ∈ w, since  P≥t φ → P≥s φ for all t > s. Thus, {P≥s φ : s < r} ⊆ w. Moreover, w is closed for the Arch rule, hence P≥r φ ∈ w. 25

Similarly we can prove that

641

μ(w)([φ]) ≤ r iff P≤r φ ∈ w 642

(3)

where r is a rational number from the unit interval. Now we are ready to prove the additivity of μ. Let μ(w)([φ]) = a, μ(w)([ψ]) = b, μ(w)([φ] ∩ [ψ]) = 0 and let ai,j : j ∈ N and bi,j : j ∈ N (i ∈ {0, 1}) be sequences of rational numbers from the real unit interval so that a0,n ≤ a ≤ a1,n and b0,n ≤ b ≤ b1,n (n ∈ N) and lim ai,n = a and lim bi,n = b.

n→∞ 643

n→∞

By (2) and (3) we have that P≥a0,n φ, P≤a1,n φ, P≥b0,n ψ, P≤b1,n ψ ∈ w

(4)

for all n ∈ N. Furthermore, (P≥r φ ∧ P≥s ψ ∧ P=0 (φ ∧ ψ)) → P≥r+s (φ ∨ ψ) and (P≤r φ ∧ P≤s ψ) → P≤r+s (φ ∨ ψ) (r + s ≤ 1) are valid formulas, so by (4) we obtain P≥a0,n +b0,n (φ ∨ ψ), P≤a1,n +b1,n (φ ∨ ψ) ∈ w 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662

for all n ∈ N. By definition of μ(w) we finally obtained μ(w)([φ]∪[ψ]) = a+b, which concludes the proof of additivity. The proofs of remaining properties μ(w)(∅) = 0 and μ(w)(W ) = 1 are straightforward, so they are omitted. Now we will apply Loeb’s construction on the above weak (finitely additive) model in order to obtain a σ–additive model. Roughly speaking, the Loeb technique usually consists in picking up a finitely additive measure and converting the corresponding nonstandard ∗-finite measure to the standard measure via the standard part mapping and the Caratheodory Extension theorem [23]. As it is usual in nonstandard analysis, by ∗ we denote the elementary embedding of the superstructure V (W ) into the ω1 –saturated superstructure ∗ V (W ). Recall that V (W ) = W , V 0 n+1 (W ) = Vn (W ) ∪ P(W ) and V (W ) =  Vn (W ). Note that ∗ V (W ) is a proper subset of V (∗ W ). The standard n∈ω

part function that maps finite hyperreal numbers into real numbers will be denoted by ◦ . The application of an elementary embedding ∗ on the weak canonical model (W, P r, v) gives the model (∗ W , ∗ P r, ∗ v), where for each w ∈ ∗ W the corresponding probability space (∗ W , ∗ H, ∗ μ(w)) has the following additional properties: 26

663 664 665

∗ • Boolean algebra ∗ H is closed for hyperfinite unions: n if n ∈ ∗ N is any ∗ hyperfinite number and X0 , . . . , Xn ∈ H, then k=0 Xk ∈ H. Note that for any n ∈ ∗ N \ N this union has continuum terms;

• ∗ μ(w) is hyperfinitely additive, i.e. for any n ∈ ∗ N and any pairwise disjoint X0 , . . . , Xn ∈ ∗ H, ∗

μ(w)(X0 ∪ · · · ∪ Xn ) =

n 

∗

μ(w)(Xk ).

k=0 666 667 668 669 670

By Loeb’s process for each w ∈ ∗ W we obtain the corresponding Loeb’s probability space (∗ W , Lo(∗ H), Lo(◦∗ μ(w))). Here ”Lo” designates the Loeb’s transformation of the underlying object. Instead of Lo(◦∗ μ(w)) we shall write λ(w). The algebra Lo(∗ H) is the smallest σ–algebra extending ∗ H and containing the sets of the form [φ]∗ , where:

671

• [p]∗ = ∗ [p] for any propositional letter p;

672

• [P≥s θ]∗ = {v ∈ ∗ W : λ(v)([θ]∗ ) ≥ s};

673 674 675 676 677

• [¬φ]∗ = ∗ W \ [φ]∗ ;   • [ Φ]∗ = φ∈Φ [φ]∗ for any at most countable set of formulas Φ. Clearly, for each finite classical formula φ we have that ∗ [φ] = [φ]∗ . However, this is generally not true for classical infinitary formulas and for probabilistic formulas. The strong canonical model is the structure (∗ W , Lo(P r), ∗ v), where Lo(P r(w)) = (∗ W , Lo(∗ H), λ(w)).

678 679 680 681 682 683 684 685 686 687

5.2.2

Strong Completeness

The first step towards the strong completeness for the logic LP PA is to prove the adequate variant of the Lindenbaum’s theorem for LP PA . 5.3 Lindenbaum’s theorem for LP PA . Suppose that T is a consistent LP PA –theory. Then it can be extended to a maximal consistent LP PA – theory wT . Proof. Assume that ω ∈ / A. Let {φk : k ∈ N} be an arbitrary enumeration of LP PA –formulas and let (χk , sk ) : k ∈ ω be an enumeration of the pairs of formulas and real numbers in [0, 1]A listed so that each pair occurs infinitely often. We define wT by $ wn , wT = n∈N

688

where the sequence wn is inductively defined as follows: 27

689

• w0 = T ;

690

• n = 2k + 1:

691

– If wn , φk  ⊥, then wn+1 = wn ∪ {φk };

692

– Let wn , φk  ⊥. Then we have the following cases: ∗ φk = θ → P≥s ψ. Then, there is r < s such that wn , θ → P
693 694

• n = 2k: We chose a positive integer m such that   wn+1 = wn ∪ P< 1 (P≥sk − 1 χk ∧ P
m

is consistent. Let us prove the existence of m. Suppose that, for all m, wn  ¬P< 1 (P≥sk − 1 χk ∧ P
m

i.e. wn   → P≥ 1 (P≥sk − 1 χk ∧ P
695

696 697 698 699 700 701 702 703

704 705 706

m

By Cont, wn  ⊥, which contradicts the consistency of wn . By a straightforward, albeit lengthy induction on the length of inference it can be shown that wT is deductively closed, i.e. that wT  φ iff φ ∈ wT . Now consistency of wT is a consequence of the consistency of all wn . Indeed, if ⊥ ∈ wT , then ⊥ ∈ wn fore some n, which contradicts the consistency of wn . Next, assume that ω ∈ A. In this case the proof of the theorem is simpler than above. We only need an enumeration of all LP PA –formulas and odd steps of the above construction with the following addition:  • If φk = Ψ and wn , φk  ⊥, then, there is ψ ∈ Ψ such that wn ∪ {¬ψ, ¬φk } is consistent. In this case, wn+1 = wn ∪ {¬ψ, ¬φk }. The existence of ψ is provided by Inf.  Note that in the weak canonical model M = (W, P r, v), for each w ∈ W and for each formula φ: w |= φ iff φ ∈ w

707 708

(the proof can be found, for example, in [34]). Let the structure Ms = (∗ W , Lo(P r), ∗ v) be the corresponding strong canonical model. 28

709 710 711 712 713 714

The following lemma is closely related to Main lemma in [13]. It provides a key property for the completeness proof. Although, [φ]∗ and ∗ [φ] could be very different, they have the same probability, λ(w)([φ]∗ = ∗ [φ]), i.e. that their symmetric difference has Loeb’s measure zero: λ(w)([φ]∗ ∗ [φ]) = 0. As a consequence of this fact, we draw the conclusion that any formula θ being true at a world w of the weak model will be true at w in Ms . 5.4 Lemma. For any formula φ and all w ∈ W the following is true: λ(w)([φ]∗

715 716 717 718

∗

[φ]) = 0

Proof of Lemma. By induction on complexity of φ. If φ is a propositional letter, then by definition [φ]∗ = ∗ [φ], so λ(w)([φ]∗ ∗ [φ]) = λ(w)(∅) = 0. Let φ = ¬θ. Then, [¬θ]∗ = ∗ W \ [θ]∗ and ∗ [¬θ] = ∗ (W \ [θ]) = ∗ W \ ∗ [θ]. By induction hypothesis, λ(w)([θ]∗ ∗ [θ])=0, so   λ(w) [¬θ]∗ ∗ [¬θ] = λ(w) (∗ W \ [θ]∗ ) (∗ W \ ∗ [θ])  = λ(w) [θ]∗ ∗ [θ] = 0;

719 720

∗ ∗ ∗ ∗ If φ = θ∧ψ,  then [θ∧ψ]∗ = [θ]∗ ∩[ψ]∗ and [θ∧ψ] = ([θ]∩[ψ]) = [θ]∩ [ψ]. Let φ = Φ. Then   λ(w)(∗ [φ] [φ]∗ ) φ∈Φ

≤ λ(w)(

∗



φ∈Φ

[φ]

φ∈Φ 721 722 723



[φ]

φ∈Φ 724

[φ]) + λ(w)(

φ∈Φ



∗



[φ]

φ∈Φ

∗



[φ]) = μw (∗

φ∈Φ0

[φ]∗ ).

φ∈Φ

 φ∈Φ

[φ]



∗

[φ]) <

φ∈Φ0

The second term is equal to 0 by induction hypothesis. Let φ = P≥s θ, where s ∈ [0, 1]Q . Note that ∗

726

∗

The first term is equal to 0 by Axiom A7, or more precisely, by the fact that all instances of A7 belong to w. This means that for each n there exists a finite Φ0 ⊆ Φ such that μw ( ∗

725



as well as

[P≥s θ] = ∗ {v ∈ W : μ(v)[θ] ≥ s},

[P≥s θ]∗ = {v ∈ ∗ W : λ(v)([θ]∗ ) ≥ s}.

29

1 . n

727

Then: λ(w)([P≥s θ]∗

∗

[P≥s θ])

= λ(w) (∗ {v ∈ W : μ(v)([θ]) ≥ s} {v ∈ ∗ W : λ(v)([θ]∗ ) ≥ s}) ≤ λ(w) (∗ {v ∈ W : μ(v)([θ]) ≥ s} {v ∈ ∗ W : λ(v)(∗ [θ]) ≥ s}) + λ(w) ({v ∈ ∗ W : λ(v)(∗ [θ]) ≥ s} {v ∈ ∗ W : λ(v)([θ]∗ ) ≥ s}) . 728 729 730

The second term is equal to 0 by induction hypothesis. Indeed, for all v ∈ ∗ W we have that λ(v) (∗ [θ] [θ]∗ ) = 0, so λ(v)(∗ [θ]) = λ(v)([θ]∗ ), hence for all s ∈ [0, 1]Q : λ(v)(∗ [θ]) ≥ s iff λ(v)([θ]∗ ) ≥ s.

731

Let us focus on the first term λ(w) (∗ {v ∈ W : μ(v)([θ]) ≥ s} {v ∈ ∗ W : λ(v)(∗ [θ]) ≥ s}) .

732

Note that for all v ∈ ∗ W we have that λ(v)(∗ [θ])

= ⇔

◦∗ ∗

μ(v)(∗ [θ]) ≥ s ∗

∗

∗

μ(v)( [θ]) ≥ s or μ(v)( [θ]) ∈

 m∈N

1 s − ,s m



1 ⇔ μ(v)( [θ]) ≥ s − , for all m ∈ N m 1 ⇔ v ∈ ∗ {v ∈ W : μ(v)([θ]) ≥ s − }, for all m ∈ N m  1 ∗ ⇔ v∈ {v ∈ W : μ(v)([θ]) ≥ s − }. m ∗

∗

m∈N

733

Hence, ∗

{v ∈ W : μ(v)([θ]) ≥ s} {v ∈ ∗ W : λ(v)(∗ [θ]) ≥ s}    1 ∗ = {v ∈ W : μ(v)([θ]) ≥ s − } \ ∗ {v ∈ W : μ(v)([θ]) ≥ s} m m∈N

  1 ∗ = {v ∈ W : μ(v)([θ]) ≥ s − } \ ∗ {v ∈ W : μ(v)([θ]) ≥ s} m m∈N #

 1 ∗ = ∧ ¬μ(v)([θ]) ≥ s . v ∈ W : μ(v)([θ]) ≥ s − m m∈N

  For all n there exists m such that P< 1 P≥s− 1 θ ∧ P
m

# 1 1 μ(w) v ∈ W : μ(v)([θ]) ≥ s − ∧ ¬μ(v)([θ]) ≥ s < . m n 30

734 735 736 737 738

If ω ∈ A, the last inequality holds because all instances of (A8) belongs to w (remember, w is a maximal consistent set of formulas). If ω ∈ A, the inequality is true by the construction, i.e. its step related to Rule Cont, given in the proof of Lemma 5.3. Using some quite common nonstandard and measure-theoretic arguments, we obtain that  # 

1 ∗ ◦∗ ∧ ¬μ(v)([θ]) ≥ s = 0, v ∈ W : μ(v)([θ]) ≥ s − μ(w) m m∈N

739

that is λ(w) (∗ {v ∈ W : μ(v)([θ]) ≥ s} {v ∈ ∗ W : λ(v)(∗ [θ]) ≥ s}) = 0,

740 741 742



which concludes the proof of the lemma.

The last step in the proof of the strong completeness for the logic LP PA is to show satisfiability of maximal consistent LP PA –theories. 5.5 Theorem. Let Ms = (∗ W , Lo(P r), ∗ v) be the strong canonical model for LP PA . For any formula φ and any maximal consistent theory w ∈ W the following is true in Ms : Ms , w |= φ iff φ ∈ w.

743 744

Proof. The proof of the theorem is obtained by induction on the complexity of φ. Here we shall focus only on the nontrivial case when φ = P≥s θ: Ms , w |= P≥s θ ⇔ λ(w)([θ]∗ ) ≥ s ⇔ λ(w)(∗ [θ]) ≥ s  1 ∗ ⇔ w∈ {v ∈ W : μ(v)([θ]) ≥ s − } m m∈N

⇔ μ(w)([θ]) ≥ s − ⇔ μ(w)([θ]) ≥ s

1 , for all m ∈ N m

⇔ w |= P≥s θ ⇔ P≥s θ ∈ w, 745

which concludes the proof of the theorem.

746

5.3

747 748

749 750 751



Completeness of LP PA,2

The proof of completeness of LP PA,2 is obtained by a simplification of the corresponding proof for the LP PA –logic presented above. More precisely: • Firstly, in adaptation of the proof of Theorem 5.3, since neither A8 nor Cont are part of the axiom system of LP PA,2 , we do not need the even steps, so they are omitted; 31

754

• Secondly, in the proof of Lemma 5.4 we omit the case φ = P≥s θ, since it is redundant in LP PA,2 (iteration of probability operators is not allowed);

755

• The rest of the proof is identical as in the case of the LP PA –logic.

752 753

756

757 758 759 760 761 762 763 764 765 766 767 768 769 770

6

Conclusion

In this paper we give an integral approach to axiomatization of propositional probabilistic logics with the aim to obtain a strong completeness theorem with respect to the class of σ–additive models. Axioms and inference rules are adaptations of Hoover’s rules for logics with probabilistic quantifiers [13], Goldblat’s rule for σ–additivity [9] and some of our earlier work [34]. We prove the equivalence of Hoover’s and Goldblat’s approaches to capturing σ–additivity (Theorem 4.3), various non-compactness phenomena (Section 3) that has significant impact on completeness of LP Pκ,λ and LP Pκ,λ,2 logics and strong completeness theorem for LP PA –logic for the countable admissible A and its various fragments. In particular, in the subsection 3.4 we show that either A8 or Cont are essential for the axiomatization of the class of σ–additive models. Table 1 summarizes the completeness status. We use the following abbreviations in the table:

771

• λ is the cardinality of the set of propositional letters, while

772

• lengths of conjunctions of formulas are less than κ.

784

In the subsection 3.2 we show that even in the simplest case LP Pω,ω1 and LP Pω,ω1 ,2 , when the formulas are finite, but with an uncountable set of propositional letters, strong completeness cannot be obtained. In the subsection 3.3 shows the same for LP Pω1 ,ω and LP Pω1 ,ω,2 . It means that there is no difference between these two kinds of logics (with and without iterations of probabilistic operators), at least when completeness is considered. The cases when we study admissible fragments of LP Pω1 ,ω are denoted by LP PA . Theorem 5.5 provides strong completeness for these admissible fragments. Note that we do not consider admissible fragments of finitary logics (i.e., κ = ω), and the corresponding cells in the table are empty. The signs + and − in the cells of the table denote the status of strong completeness for the corresponding logics.

785

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773 774 775 776 777 778 779 780 781 782 783

786 787

[1] N. Alechina. Logic with probabilistic operators. In Proc. of the ACCOLADE ’94, 121 – 138. 1995.

32

κ, λ = ω κ=ω λ>ω κ = ω1 λ=ω

LP Pκ,λ +

LP Pκ,λ,2 +

−

−

−

−

LP PA

LP PA,2

+

+

Table 1: Overview of completeness results

789

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790

[3] J. Barwise. Admissible sets and structures. Springer, 1975.

791

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35