a probabilistic extension of intuitionistic logic

Math. Log. Quart. 49, No. 4, 415 – 424 (2003) / DOI 10.1002/malq.200310044

A probabilistic extension of intuitionistic logic

Zoran Markovic∗1, Zoran Ognjanovic∗∗1, and Miodrag Raskovic∗∗∗2

1 Matematicki Institut, Kneza Mihaila 35, 11000 Beograd, Yugoslavia2 Prirodno-matematicki fakultet, R. Domanovica 12, 34000 Kragujevac, Yugoslavia

Received 13 May 2002, revised 31 January 2003, accepted 4 February 2003Published online 15 April 2003

Key words Probabilistic logic, intuitionistic logic, completeness, decidability.MSC (2000) 03B60, 68T27, 03B70

We introduce a probabilistic extension of propositional intuitionistic logic. The logic allows making statementssuch as P≥sα, with the intended meaning “the probability of truthfulness of α is at least s”. We describe thecorresponding class of models, which are Kripke models with a naturally arising notion of probability, and givea sound and complete infinitary axiomatic system. We prove that the logic is decidable.

1 Introduction

The aim of probabilistic logic is clearly to capture the rules of reasoning about uncertain knowledge. The classical(boolean) logic, on the other hand, may be regarded as starting from a position of omniscience, in so far as itconsiders that each proposition must be either true or false. Therefore, it is not surprising that combining thesetwo approaches, e. g. by adding probability operators to classical propositional logic as it was done in [4, 5, 9,10, 11], would be sometimes difficult.

There is a popular view of intuitionistic logic as pertaining to growth of knowledge, which is especiallyconvincing when talking about Kripke models. Namely, in addition to propositions which are proved to be trueand those which are proved to be false (contradictory), there is a third class of propositions which may turn outeither way and intuitionism allows us to reason also about them.

In this paper we combine probabilistic operators with intuitionistic logic. There are two possible approachesto do that. We may treat probabilistic operators classically or we may assume that they behave intuitionistically.The latter approach was analyzed in [2, 3], while we consider here the former one. At the syntax level we addprobabilistic operators to the propositional intuitionistic language which enables making formulas such as P≥sα.The intended meaning of the formula is “the probability of truthfulness of α is at least s”. In our logic nesting ofprobabilistic operators, i. e., higher order probabilities, will not be allowed.

As a semantics we introduce a class of models that combine properties of intuitionistic Kripke models andprobabilities. Since nesting of probabilistic operators is not allowed, it is possible to give a simple and naturalinterpretation of probabilistic formulas, quite in line with Boole’s original ideas, based on the ‘size’ of the setof possible worlds in which a proposition is true. We propose an infinitary axiomatic system which we proveis sound and complete with respect to the mentioned class of probabilistic intuitionistic models. In this paperthe terms finitary and infinitary concern meta language only. Object language is countable, formulas are finite,while only proofs are allowed to be infinite. The need for this infiniteness comes from the failure of compactnesstheorem for this type of logics, as will be explained in Section 7.

We may try to motivate this combination of intuitionistic and probabilistic logics through the following exam-ple. It is well known that (p → q) ∨ (q → p) is a classical, but not intuitionistic, tautology. Since tautologiesshould have probability one, starting with classical logic makes P≥1((p → q) ∨ (q → p)) valid in probabilisticlogic. If we take now p to be “it rains” and q to be “the sprinkler is on”, it is clear that the sprinkler should not

∗ e-mail:[email protected]∗∗ Corresponding author: e-mail: [email protected]∗∗∗ e-mail: [email protected]

c© 2003 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0942-5616/03/0407-0415 $ 17.50+.50/0

416 Z. Markovic, Z. Ognjanovic, and M. Raskovic: A probabilistic extension of intuitionistic logic

be on when it rains, i. e., p → q should have probability zero (or very low, say less than ε). But, this yields, withclassical logic, P≥1−ε(q → p) (since the measure of the union of two sets is less or equal than the sum of themeasures of those sets) although it does not seem highly probable that it should rain when the sprinkler is on.On the other hand, in our logic we may construct models in which both p → q and q → p have low probabil-ity. As for the classical treatment of probabilistic operators, we may say that once we determine (somehow) theprobability of an uncertain proposition α, it should be either greater or equal to some s ∈ [0, 1] or not, so it is notunreasonable to assume P≥sα ∨ ¬P≥sα (even if we reject α ∨ ¬α).

The rest of the paper is organized as follows. In Section 2 the syntax of the logic is given. Section 3 reviewssome basic definitions and theorems from intuitionistic logic and measure theory and contains a description ofthe corresponding Kripke-style models and satisfiability and validity notions (for more details about intuitionisticlogic the reader should consult [6, 8, 12, 13, 14], while [1] can be seen as a good source for the part of measuretheory used in the sequel). In that section, as well as in the next ones, we sketch proofs of some well knowntheorems to make the paper selfcontained. A formulation of a sound and complete axiomatic system can befound in Section 4. The completeness theorem is proved in Section 5. In Section 6 decidability of the logicis proved. We conclude in Section 7 with some remarks about similar approaches, open questions and furtherinvestigations.

2 Syntax

Let S be the set of all rational numbers from [0, 1]. The language of the logic consists of a denumerable setVar = {p, q, r, . . . } of propositional letters, connectives ¬, ∧, ∨, → and a list of unary probabilistic operators(P≥s)s∈S .

The set ForI of intuitionistic propositional formulas is the smallest set X containing Var and closed under theformation rules: if α and β belong to X , then ¬α, α ∧ β, α ∨ β, and α→ β are in X . Elements of ForI will bedenoted by α, β, . . . .

The set ForP of probabilistic propositional formulas is the smallest set Y containing all formulas of the formP≥sα for α ∈ ForI, s ∈ S, and closed under the formation rules: if A and B belong to Y , then ¬A and A ∧ Bare in Y . Probabilistic literals are formulas of the form P≥sα or ¬P≥sα. Formulas from ForP will be denotedby A, B, . . . . We use A ∨ B, A → B and P<sα to denote the formulas ¬(¬A ∧ ¬B), ¬A ∨ B and ¬P≥sα,respectively.

Let ForI ∪ ForP be denoted by For. We use ϕ, ψ, . . . to denote formulas from For. For α ∈ ForI andA ∈ ForP, we abbreviate both ¬(α→ α) and ¬(A→ A) by ⊥ letting the context determine the meaning.

Note a difference between the sets ForI and ForP. Namely, according to the previous definitions, propositionalconnectives are independently introduced in the set ForI, while ∨ and → are defined from¬ and ∧ in the set ForP.

3 Semantics

We propose a possible-world approach to give semantics to formulas from the set For. According to the structureof For, there are two levels in the definition of models. At the first level there is the notion of intuitionistic Kripkemodels [8], while probability comes in the picture at the second level.

3.1 Intuitionistic Kripke models

Definition 1 An intuitionistic (propositional) Kripke model for the language ForI is a structure 〈W,≤, v〉,where

· 〈W,≤〉 is a partially ordered set of possible worlds which is a tree, and

· v is a valuation function, i. e., v maps the set W into the powerset P(Var), which satisfies the followingcondition: for all w, w′ ∈W , if w ≤ w′, then v(w) ⊆ v(w′).

Note: We find it convenient to work with Kripke models in which the ordering of worlds is a tree. It is wellknown that intuitionistic propositional logic is sound and complete also with respect to this restricted class ofKripke models.

Math. Log. Quart. 49, No. 4 (2003) 417

In each Kripke model we define the forcing relation �⊂W × ForI by the following definition:

Definition 2 Let 〈W,≤, v〉 be an intuitionistic Kripke model. The forcing relation � is defined by the follow-ing conditions for every w ∈ W and α, β ∈ ForI:

· if α ∈ Var, w � α iff α ∈ v(w),· w � α ∧ β iff w � α and w � β,

· w � α ∨ β iff w � α or w � β,

· w � α→ β iff for every w′ ∈ W , if w ≤ w′, then w′� α or w′ � β,

· w � ¬α iff for every w′ ∈ W , if w ≤ w′, then w′� α.

We read w � α as “w forces α” or “α is true in the world w”. Validity in the intuitionistic Kripke model〈W,≤, v〉 is defined by 〈W,≤, v〉 � α iff w � α for all w ∈ W . A formula α is valid (� α) if it is valid in everyintuitionistic Kripke model. For example:

Theorem 3 Formula ¬α↔ (α→ ⊥) is intuitionistically valid.

The second condition from Definition 1 is generalized in the next theorem.

Theorem 4 For every intuitionistic Kripke model 〈W,≤, v〉, every w ∈ W , and every α ∈ ForI the followingholds: w � α iff for every w′ ∈ W , if w ≤ w′, then w′ � α.

P r o o f. See [6, 12, 14].

Corollary 5 If 0 is the least element in 〈W,≤〉, then 〈W,≤, v〉 � α iff 0 � α.

3.2 Finitely additive measures

For easier readability we provide here a few definitions and results from [1].

Definition 6 Let W be a set. A family H of subsets of W is a lattice on W if the following conditions aresatisfied: (a) if G1, G2 ∈ H , then G1 ∪ G2 ∈ H , and (b) if G1, G2 ∈ H , then G1 ∩ G2 ∈ H . A family H ofsubsets of W is an algebra on W if the following conditions are satisfied: (a) W ∈ H , (b) if G1, G2 ∈ H , thenG1 ∪ G2 ∈ H , and (c) if G ∈ H , then W \G ∈ H .

Definition 7 Let W be a set and H an algebra of subsets of W . A map µ : H −→ [−∞,∞] is a finitelyadditive measure if the following conditions are satisfied: (a) µ(∅) = 0, and (b) for every G1, G2 ∈ H such thatG1 ∩ G2 = ∅, µ(G1 ∪ G2) = µ(G1) + µ(G2). A finitely additive measure µ is a finitely additive probability if(c) µ(W ) = 1.

Note that in [1] the words ‘charge’ and ‘field of subsets’ are used instead of ‘finitely additive measure’ and‘algebra of subsets’, respectively. Also, the notion of a finitely additive measure (and probability) can be definedas above on lattices or some other families of subsets of a set. Under some natural conditions such a measure canbe extended to any algebra containing the corresponding family.

Theorem 8 Let H be any lattice of subsets of a set W with ∅, W ∈ H . Let µ be a finitely additive probabilityon H . Let H ′ be any algebra of subsets of W containing H . Then there is a finitely additive probability µ′ onH ′ which is an extension of µ.

P r o o f. See [1, Theorem 3.2.10].

Theorem 9 Let H be an algebra of subsets of a set W and µ a finitely additive probability on H . Let H ′

be an algebra of subsets of W containing H . Then there is a finitely additive probability µ′ on H ′ which is anextension of µ.

P r o o f. See [1, Corollary 3.3.4].

3.3 Probabilistic models

Let M = 〈W,≤, v〉 be an intuitionistic Kripke model. We use [α]M to denote {w ∈ W : w � α} for α ∈ ForI.Clearly, the family HI = {[α]M : α ∈ ForI} is a Heyting algebra with operations [α]M ∪ [β]M = [α ∨ β]M ,[α]M ∩ [β]M = [α ∧ β]M , [α]M ⇒ [β]M = [α→ β]M , and ∼ [α]M = [¬α]M . Thus, HI is a lattice on W , butit may not be closed under complementation.


Definition 10 A probabilistic model is a structure 〈W,≤, v,H, µ〉, where

· 〈W,≤, v〉 is an intuitionistic Kripke model,

· H is the smallest algebra on W containingHI = {[α]M : α ∈ ForI},

· µ : H −→ [0, 1] is a finitely additive probability.

Note that H contains all sets of the formW \ [α]M , even if for some α ∈ ForI it may be W \ [α]M �= [¬α]M .

Definition 11 The satisfiability relation � is defined by the following conditions for every probabilistic modelM = 〈W,≤, v,H, µ〉:

· if α ∈ ForI, then M � α iff for all w ∈ W , w � α,

· M � P≥sα iff µ([α]M ) ≥ s,

· if A ∈ ForP, then M � ¬A iff M � A does not hold,

· if A, B ∈ ForP, then M � A ∧ B iff M � A and M � B.

A formula ϕ ∈ For is satisfiable if there is a probabilistic model M such that M � ϕ; ϕ is valid if for everyprobabilistic model M , M � ϕ; a set T of formulas is satisfiable if there is a probabilistic model M such thatM � ϕ for every formula ϕ ∈ T .

4 A sound and complete axiomatization

The set of all valid formulas can be characterized by the following sound and complete set of axiom schemata

1. all ForI-instances of intuitionistic propositional tautologies;

2. all ForP-instances of classical propositional tautologies;

3. P≥0α;

4. P≥1−r¬α→ ¬P≥sα, for s > r;

5. P≥rα→ P≥sα, for r ≥ s;

6. P≥1(α→ β) → (P≥sα→ P≥sβ);7. P≥sα ∧ P≥rβ ∧ P≥1¬(α ∧ β) → P≥min(1,s+r)(α ∨ β);8. P<sα ∧ P<rβ → P<s+r(α ∨ β), for s+ r ≤ 1;

and inference rules

1. from ϕ and ϕ→ ψ infer ψ;

2. if α ∈ ForI, from α infer P≥1α;

3. from B → P≥s−1/kα, for every k ≥ 1/s, infer B → P≥sα.

Note that formulas obtained by applications of the inference rules must obey the formation rules, i. e., in theinference rules 2 and 3, α must be from ForI, s ∈ S and k ∈ N .

Axiom 1 plus Rule 1, and Axiom 2 plus Rule 2 represent a sound and complete axiomatic system for thepropositional intuitionistic logic and for the propositional classical logic, respectively. Any other equivalentsystems can be used [7, 13, 14]. Axioms 3 – 8 consider the probabilistic part of the logic. For example, Axiom3 says that every formula is satisfied by a set of worlds of the measure at least 0, i. e., that the measure is non-negative. Axiom 5 corresponds to monotonicity of probabilistic measures, while Axiom 7 says that if the sets ofworlds that satisfy α and β are disjoint, then the measure of the set of worlds that satisfy α ∨ β is the sum of themeasures of the former two sets. Rule 3 corresponds to the Archimedean axiom for real numbers. It is the onlyinfinitary inference rule in the system.

A formula ϕ ∈ For is deducible from a set T of formulas (T � ϕ) if there is an at most countable sequence offormulas ϕ0, ϕ1, . . . , ϕ, such that every formula in the sequence is an axiom or a formula from the set T , or it isderived from the preceding formulas by an application of an inference rule. If ∅ � ϕ, we say that ϕ is a theoremof the deductive system, also denoted by � ϕ.

A set T of formulas is consistent if neither T � ¬(α → α) nor T � ¬(A → A) for arbitrary α ∈ ForI,A ∈ ForP. Otherwise, T is inconsistent.

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A set T of formulas is deductively closed if for every ϕ ∈ For, if T � ϕ, then ϕ ∈ T . A set T of formulas hasthe disjunction property if for every α, β ∈ ForI, T � α ∨ β implies T � α or T � β. A disjunctive closure of aset T of formulas is a set T ′ which contains T and for every α, β ∈ ForI, if α ∨ β ∈ T ′, then α ∈ T ′ or β ∈ T ′.

Theorem 12 Let T be a consistent set of ForI-formulas, α ∈ ForI and T � α. Then, there is a consistent,deductively closed set T ′ which has the disjunctive property such that T ′

� α and T ⊆ T ′. Every consistentsubset of For has a consistent disjunctive closure.

P r o o f. See [6, 12, 14].

5 Soundness and completeness

Theorem 13 (Soundness theorem) The above axiomatic system is sound with respect to the class of proba-bilistic models.

P r o o f. Soundness of our system follows from the soundness of propositional intuitionistic and classicallogics, as well as from the properties of probabilistic measures, so we give only a sketch of a straightforward buttedious proof.

We can show that every instance of an axiom schemata holds in every model, while the inference rules preservethe validity. For example, let α ∈ ForI be an instance of an intuitionistic propositional tautology. Obviously,for every world w from a probabilistic model M it must be w � α. Thus, for every model M , M � α. Next,let us consider Axiom 6. Suppose that P≥1(α → β) holds in a model M = 〈W,≤, v,H, µ〉. It means thatµ([α]M ∩ (W \ [α → β]M )) = 0, i. e., that the set of worlds in which α holds, but β does not hold has themeasure 0. From [α]M = ([α]M ∩ (W \ [α → β]M )) ∪ ([α]M ∩ [α → β]M ), we have that µ([α]M ) =µ([α]M ∩ [α → β]M ), and since [α]M ∩ [α → β]M ⊂ [β]M , it follows that µ([α]M ) ≤ µ([β]M ). Hence,M � P≥sα → P≥sβ, and Axiom 6 holds in M . Similarly, since for an arbitrary model M = 〈W,≤, v,H, µ〉,[α]M = ([α]M ∩ (W \ [β]M )) ∪ [α ∧ β]M , we have that µ([α]M ) ≥ µ([α]M ∩ (W \ [β]M )) for arbitraryα, β ∈ ForI. From [α ∨ β]M = ([α]M ∩ (W \[β]M )) ∪ [β]M it follows that µ([α ∨ β]M ) ≤ µ([α]M )+µ([β]M ),i. e., that Axiom 8 is valid. The other axioms can be proved to be valid in a similar way.

Rule 1 is validity-preserving for the same reason as in classical and intuitionistic logics. Consider Rule 2and suppose that a formula α ∈ ForI is valid. Then, for every model M = 〈W,≤, v,H, µ〉, [α]M = W andµ([α]M ) = 1. Hence, P≥1α is valid too. Rule 3 preserves validity because of the properties of the set of rationalnumbers.

In the proof of the completeness theorem the following strategy is applied. We start with a form of Deductiontheorem. Then, we show how to extend a consistent set T of formulas to a consistent set T ∗ which is in somesense maximal. Then, a canonical model M is constructed out of the formulas from the set T ∗ such that M � ϕiff ϕ ∈ T ∗.

Theorem 14 (Deduction theorem) If T is a set of formulas and T ∪ {ϕ} � ψ, then T � ϕ → ψ, whereeither ϕ, ψ ∈ ForI or ϕ, ψ ∈ ForP.

P r o o f. We use transfinite induction on the length of the proof of ψ from T ∪ {ϕ}. For example, we considerthe case whereψ = C → P≥sδ is obtained from T ∪ {ϕ} by an application of the inference rule 3 and ϕ ∈ ForP.Then

(1) T, ϕ � C → P≥s−1/kδ, for every integer k ≥ 1/s,

(2) T � ϕ→ (C → P≥s−1/kδ), for k ≥ 1/s, by the inductive hypothesis,

(3) T � (ϕ ∧ C) → P≥s−1/kδ, for k ≥ 1/s,

(4) T � (ϕ ∧ C) → P≥sδ, from (3) by Rule 3,

(5) T � ϕ→ ψ.

The other cases follow similarly.


We now describe a proper extension of a consistent set T of formulas. Let CnI(T ) = {α ∈ ForI : T � α}be the set of all intuitionistic propositional consequences of T , and let CnI(T ) be a consistent disjunctive closureof CnI(T ). Let A0, A1, . . . be an enumeration of all formulas from ForP. We define a sequence of sets Ti,i = 0, 1, 2, . . . , and a set T ∗ such that

1. T0 = T ∪ CnI(T ) ∪ {P≥1α : α ∈ CnI(T )};

2. for every i ≥ 0, if Ti ∪ {Ai} is consistent, then Ti+1 = Ti ∪ {Ai}, otherwise, Ti+1 = Ti ∪ {¬Ai};

3. if Ti+1 is obtained by adding a formula of the form ¬(B → P≥sγ), then for some positive integer n,B → ¬P≥s−1/nγ, is also added to Ti+1, so that Ti+1 is consistent,

4. T ∗ =⋃

i Ti.

The next theorem summarizes properties of the above sets.

Theorem 15 Let T1, T2, . . . , T∗ be as above. Then the following holds for every ϕ, ψ ∈ For and α ∈ ForI:

1. Every Ti is consistent;

2. T ∗ is consistent;

3. if ϕ ∈ T ∗, then ¬ϕ /∈ T ∗;

4. ϕ ∧ ψ ∈ T ∗ iff ϕ ∈ T ∗ and ψ ∈ T ∗;

5. if � ϕ, then ϕ ∈ T ∗;

6. if ϕ ∈ T ∗ and ϕ→ ψ ∈ T ∗, then ψ ∈ T ∗;

7. if P≥sα ∈ T ∗ and s ≥ r, then P≥rα ∈ T ∗;

8. if r is a rational number and r = sup{s : P≥sα ∈ T ∗}, then P≥rα ∈ T ∗.

P r o o f. 1. According to the above construction and Theorem 12, T0 is a consistent disjunctive closure of aset of consequences of a consistent set. Suppose that Ti is obtained by the Step 2 of the above construction. Itfollows from Deduction theorem that it is not possible that both Ti ∪ {Ai} and Ti ∪ {¬Ai} are inconsistent.Consider the Step 3 of the construction. Suppose that ¬(B → P≥sγ) is added to Ti and that Ti+1 cannot beconsistently extended as above. Then

(1) Ti, ¬(B → P≥sγ), B → ¬P≥s−1/kγ � ⊥, for every k ≥ 1/s, by the hypothesis,

(2) Ti, ¬(B → P≥sγ) � ¬(B → ¬P≥s−1/kγ), for k ≥ 1/s, by Deduction theorem,

(3) Ti, ¬(B → P≥sγ) � B → P≥s−1/kγ, for k ≥ 1/s, from (2), by the classicaltautology ¬(α→ γ) → (α→ ¬γ),

(4) Ti, ¬(B → P≥sγ) � B → P≥sγ, from (3), by Rule 3,

which contradicts consistency of Ti ∪ {¬(B → P≥sγ)}.2. We can show that T ∗ is a deductively closed set that neither contains all formulas from ForI nor all formulas

from ForP, and as a consequence that T ∗ is consistent.Since T is a consistent set, there is an α ∈ ForI such that T � α, α /∈ T0, and α /∈ T ∗. Also, if A ∈ ForP, it

cannot be A = Ak ∈ T ∗ and ¬A = Am ∈ T ∗, because Tmax{k,m}+1 is consistent.Next, note that for every ϕ ∈ For, if Ti � ϕ, then it must be ϕ ∈ T ∗. If ϕ ∈ ForI, then from Ti � ϕ it follows

T � ϕ and ϕ ∈ T0. If ϕ = Ak ∈ ForP, Ti � ϕ and ϕ /∈ T ∗, then Tmax{i,k}+1 � ϕ and Tmax{i,k}+1 � ¬ϕ, acontradiction.

Suppose that T ∗ � ϕ. If ϕ = α ∈ ForI, it means that T0 � α, and reasoning as above, we conclude thatα ∈ T ∗. On the other hand, let ϕ = A ∈ ForP. Suppose that the sequence ϕ1, ϕ2, . . . , A of formulas whichforms the proof of A from T ∗ is countably infinite (otherwise there must be some k such that Tk � A, and itfollows that A ∈ T ∗). We can show that for every i, if ϕi is obtained by an application of an inference rule andall the premises of ϕi belong to T ∗, then ϕi ∈ T ∗. Suppose that ϕi is obtained by Rule 1 (modus ponens) andits premises ϕ1

i and ϕ2i belong to T ∗. There must be some k such that ϕ1

i , ϕ2i ∈ Tk. From Tk � ϕi, it follows

ϕi ∈ T ∗. If ϕi is obtained by Rule 2, its premise belongs to T0 and the same holds for ϕi. Finally, suppose thatϕi = B → P≥sγ is obtained by the infinitary inference rule 3 and that the premises ϕ1

i = B → P≥s−1/kγ,ϕ2

i = B → P≥s−1/(k+1)γ, . . . belong to T ∗. If ϕi /∈ T ∗, by the Step 3 of the construction of T ∗, there is ann > 1/s, such that B → ¬P≥s−1/nγ ∈ T ∗. There must be some m such that B → P≥s−1/nγ ∈ Tm and

Math. Log. Quart. 49, No. 4 (2003) 421

B → ¬P≥s−1/nγ ∈ Tm. It follows that Tm ∪ {B} is not consistent. Thus, Tm � B → C, for every C ∈ ForP.It follows that Tm � B → P≥sγ andB → P≥sγ ∈ T ∗, a contradiction. Hence, from T ∗ � A, it followsA ∈ T ∗.

3. – 7. These cases can be proved as usual.8. If P≥rα /∈ T ∗, by the Step 3 of the construction of the set T ∗, there is a positive integer n such that

¬P≥r−1/nα ∈ T ∗. But, then r cannot be the supremum of the set {s : P≥sα ∈ T ∗}.

The set T ∗ is used to construct a canonical probabilistic model M . First, we construct an intuitionistic Kripkemodel. Let w0 denote the set CnI(T ).

Theorem 16 Let W be the set of all consistent, deductively closed extensions of w0 having the disjunctionproperty. For every w ∈ W and every α, β ∈ ForI the following holds:

1. α→ β /∈ w iff there is some w′ ∈W such that w ⊆ w′, α ∈ w′ and β /∈ w′;2. ¬α /∈ w iff there is some w′ ∈W such that w ⊆ w′ and α ∈ w′.

P r o o f. 1. Suppose that there is some w′ ∈ W such that w ⊆ w′, α ∈ w′ and β /∈ w′. If α → β ∈ w, thenit must be α → β ∈ w′ and, since w′ is deductively closed, β ∈ w′ which contradicts consistency of w′. Forthe other direction, suppose that α → β /∈ w. Then, w � α → β and, by Deduction theorem of intuitionisticpropositional logic, w ∪ {α} � β. Thus, w ∪ {α} is consistent and, by Theorem 12, there is a w′ ∈ W suchthat w ∪ {α} ⊆ w′ and w′

� β. Since α ∈ w′, it must be β /∈ w′.2. This follows from 1., since intuitionistically ¬α↔ (α→ ⊥) and ⊥ /∈ w′.

Theorem 17 Let W be as above and for every w ∈W , v(w) = {α ∈ Var : α ∈ w}. Then

1. 〈W,⊆, v〉 is an intuitionistic Kripke model;

2. for every w ∈W and every α ∈ ForI, w � α iff α ∈ w.

P r o o f. 1. Since ⊆ is a partial ordering, and v(w) ⊆ v(w′) follows from w ⊆ w′, it is easy to see that〈W,⊆, v〉 is an intuitionistic Kripke model.

2. We use the induction on complexity of formulas. Let α ∈ Var. Then, w � α iff α ∈ v(w) iff α ∈ w. Letα = β ∨ γ. Then, α ∈ w iff β ∈ w or γ ∈ w (since w is closed for disjunction) iff w � β or w � γ iff w � α.Let α = ¬β. If w � α, then for every w′ ∈ W such that w ⊆ w′ it must be w′

� β and β /∈ w′. It followsfrom Theorem 16.2 that α = ¬β ∈ w. On the other hand, let α = ¬β ∈ w. Then, for every w′ ∈ W such thatw ⊆ w′ it must be α ∈ w′ and β /∈ w′. It follows that for every w′ ∈ W such that w ⊆ w′, w′

� β and w � ¬β.Finally, let α = β → γ. If w � β → γ, then, by the definition of �, for every w′ ∈ W such that w ⊆ w′ ifw′ � β, then w′ � γ, i. e., by the inductive assumption, if β ∈ w′, then γ ∈ w′. It follows, by Theorem 16.1, thatβ → γ ∈ w. If β → γ ∈ w, then the same holds for every w′ ∈ W such that w ⊆ w′. If for such an w′, w′ � β,then by the inductive assumption β ∈ w′ and, since w′ is deductively closed, γ ∈ w′ and w′ � γ. It follows thatw � β → γ.

In the next step we define a canonical probabilistic model M .

Theorem 18 Let W , ⊆, and v be as above, let HI = {[α]M}α∈ForI and µI([α]M ) = sup{s : P≥sα ∈ T ∗}for every α ∈ ForI. Then, the following holds:

1. If [α]M = [β]M , then µI([α]M ) = µI([β]M );2. µI([α]M ) ≥ 0;

3. µI(W ) = 1;

4. µI([α]M ) ≤ 1 − µI([¬α]M );5. µI([α]M ∪ [β]M ) = µI([α]M ) + µI([β]M ), for all disjoint sets [α]M , [β]M ∈ HI.

P r o o f. 1. It is enough to prove that [α]M ⊂ [β]M implies µI([α]M ) ≤ µI([β]M ). But, [α]M ⊂ [β]M impliesthat for every w ∈W from w � α it follows that w � β. Since 〈W, v〉 is an intuitionistic Kripke model, it meansthat w0 � α → β. By the construction of T ∗ we have that P≥1(α → β) ∈ T ∗. It follows from Axiom 6 that forevery s ∈ S, P≥sα→ P≥sβ ∈ T ∗, and µI([α]M ) ≤ µI([β]M ).

2. It follows from Axiom 3 that for every α ∈ ForI, P≥0α ∈ T ∗ and µI([α]M ) ≥ 0.3. Since α→ α, P≥1(α→ α) ∈ T ∗ for every α ∈ ForI, we have that W = [α→ α]M and µI(W ) = 1.


4. Let µI([¬α]M ) = 1 − r0 = sup{1 − r : P≥1−r¬α ∈ T ∗}. If r0 = 1, the statement trivially holds sinceµI([α]) ≤ 1. Next, suppose that r0 < 1. It means that for every s ∈ S ∩ (r0, 1], P≥1−s¬α ∈ T ∗, and byAxiom 4, P≥sα /∈ T ∗. From the assumption that there is some r′0 = sup{s : P≥sα ∈ T ∗} > r0 it follows thatfor every s ∈ S ∩ (r0, r′0], P≥sα ∈ T ∗, a contradiction.

5. Let [α]M ∩ [β]M = ∅. It means that for every w ∈ W , w � α or w � β does not hold, i. e. thatit is not w � α ∧ β. Since 〈W, v〉 is an intuitionistic Kripke model we have w0 � ¬(α ∧ β), and from� ¬(α ∧ β) → (β → ¬α) it follows w0 � β → ¬α. From Theorem 17.2 and the construction of T ∗,P≥1(β → ¬α) ∈ T ∗. Using Axiom 6, we obtain µI([β]M ) ≤ µI([¬α]M ). Let µI([α]M ) = r and µI([β]M ) = s.Then, r + s = µI([α]M ) + µI([β]M ) ≤ µI([α]M ) + µI([¬α]M ), and according to Theorem 18.4, r + s ≤1 − µI([¬α]M ) + µI([¬α]M ) = 1. By the well known properties of the supremum for every r′ ∈ [0, r] ∩ S andevery s′ ∈ [0, s] ∩ S we have P≥r′α, P≥s′β ∈ T ∗. Since ¬(α ∧ β), P≥1¬(α ∧ β) ∈ T ∗, it follows by Axiom 7that P≥r′+s′(α ∨ β) ∈ T ∗. Thus, r + s ≤ sup{t : P≥t(α ∨ β) ∈ T ∗}. If r + s = 1, then the assertion triviallyholds. Suppose r + s < 1. If r + s < t0 = sup{t : P≥t(α ∨ β) ∈ T ∗}, then for every t′ ∈ (r + s, t0] ∩ S wehave P≥t′(α ∨ β) ∈ T ∗. We can choose rational numbers r′′ > r and s′′ > s such that ¬P≥r′′α, ¬P≥s′′β ∈ T ∗

and r′′ + s′′ = t′ ≤ 1. By Axiom 8, we have P<r′′+s′′(α ∨ β) ∈ T ∗, i. e. ¬P≥t′ (α ∨ β), a contradiction withconsistency of T ∗. Hence, µI(w)([α]M ∪ [β]M ) = µI(w)([α]M ) + µI(w)([β]M ).

Theorem 19 LetW , ⊆, v, HI, and µI be as above. Let H be the smallest algebra onW containingHI. Then,there is a finitely additive probability µ on H which is an extension of µI such that M = 〈W,⊆, v,H, µ〉 is aprobabilistic model.

P r o o f. First, note that HI is a lattice on W . It follows from Theorem 18 that µI is a finitely additive proba-bility on HI. According to Theorem 8, there is a finitely additive probability µ on H which is an extension of µI.Then, the statement trivially holds.

Theorem 20 (Completeness theorem) Every consistent set of formulas is satisfiable.

P r o o f. Let T be a consistent set of formulas. Reasoning as above we construct the set T ∗ and the canonicalprobabilistic model M = 〈W,⊆, v,H, µ〉. By the induction on the complexity of formulas we can prove that forevery formula ϕ, M � ϕ iff ϕ ∈ T ∗, which implies that T is satisfiable in M . Let ϕ = α ∈ ForI. Then, α ∈ T ∗

iff α ∈ w0 iff for every w ∈ W , α ∈ w iff for every w ∈ W , w � α iff M � α. Let ϕ = P≥sα. If P≥sα ∈ T ∗,then µ([α]M ) = sup{r : P≥rα ∈ T ∗} ≥ s and M � P≥sα. For the other direction, suppose that M � P≥sα.By Theorem 15.8, P≥sα ∈ T ∗. The other cases follows similarly.

Note that, thanks to Theorem 9 we can extend the probability µ of the canonical model M to be defined onH = 2W . In that way it is possible to prove completeness of our axiomatic system with respect to the class of allprobabilistic models in which every subset of possible worlds is measurable.

6 Decidability

It is well known that a formula α ∈ ForI is intuitionistically satisfiable iff it is forced in the root of a tree-likemodel which is decidable [6, 8, 12]. It follows from Corollary 5 that satisfiability problem of ForI-formulas inour probabilistic logic is decidable. Thus, to prove decidability of our logic it is enough to show that satisfiabilityproblem for probabilistic formulas is decidable. In the rest of this section we will use a kind of filtration, as wellas linear programming theory to show that.

Let A ∈ ForP and SubI(A) = {α ∈ ForI : α is a subformula of A}. Let |A| and |SubI(A)| denote the lengthof A and the number of formulas in |SubI(A)|, respectively. Obviously, |SubI(A)| ≤ |A|.

Theorem 21 A probabilistic formula A ∈ ForP is satisfiable iff it is satisfiable in a finite probabilistic modelcontaining at most 2|A|2 worlds.

P r o o f. Let M = 〈W,≤, v,H, µ〉 � A. For every w ∈ W , we use SubI(w) to denote the set of all formulasfrom SubI(A) forced in w, i. e., SubI(w) = {α ∈ SubI(A) : w � α}. In the sequel we will follow the idea from[12, Theorem 5.3.4], and select some of the worlds from W to construct a finite model M∗ satisfying A. Let w0

be the least element from W . We define the worlds of M∗ (indexed by finite sequence) in the following way:

Math. Log. Quart. 49, No. 4 (2003) 423

· u〈〉 = w0, where 〈〉 denotes the empty sequence;

· given uσ let uσ∗〈1〉, . . . , uσ∗〈k〉 be the maximal set of worlds w(1), . . . , w(k) from W such that for everyi, j ∈ {1, . . . , k}: uσ ≤ w(i), SubI(uσ) �= SubI(w(i)), if uσ ≤ w ≤ w(i), then either SubI(uσ) = SubI(w) orSubI(w) = SubI(w(i)), and if i �= j, then SubI(w(i)) �= SubI(w(j)).

Let W ∗ = {σ : uσ is defined}, ≤∗ be the usual ordering of finite sequences and α ∈ v∗(σ) iff α ∈ v(uσ)for every σ ∈ W ∗ and α ∈ Var. Using induction on complexity of formulas we can prove that for everyα ∈ SubI(A) and σ ∈ W ∗, σ � α in 〈W ∗,≤∗, v∗〉 iff uσ � α in 〈W,≤, v〉. If α ∈ Var, the statement holdsby the definition of v∗. Let α = β → γ. Suppose that σ � β → γ. Then there exists some ∈ W ∗ such thatσ ≤∗ , � β, and � γ. By the induction hypothesis, u� � β and u� � γ, uσ ≤ u�, and uσ � β → γ. On theother hand, suppose that uσ � β → γ. Then, there are two possibilities. First, if uσ � β, it must be uσ � γ, andby the induction hypothesis σ � β and σ � γ, which means that σ � β → γ. In the second case there is somew ∈ W such that uσ ≤ w, w � β, and w � γ. Since uσ � β, obviously SubI(uσ) �= SubI(w). According to theabove construction, there must be some uθ ∈ W such that uσ ≤ uθ ≤ w, σ ≤∗ θ, uθ � β, and uθ � γ. By theinduction hypothesis, θ � β, θ � γ, and σ � β → γ. The other cases follow similarly.

Let µ′ be the finitely additive probability defined on {{w ∈ W ∗ : w � α} : α ∈ SubI(A)} such thatµ′({w ∈ W ∗ : w � α}) = µ([α]M ). Since for every α ∈ SubI(A), [α]M �= ∅ iff {w ∈ W ∗ : w � α} �= ∅,it is easy to see that µ′ is correctly defined. Let M∗ = 〈W ∗,≤∗, v∗, H∗, µ∗〉 be the probabilistic model suchthat H∗ is the smallest algebra on W ∗ containing the family {[α]M∗ : α ∈ ForI}, while µ∗ is a finitely additiveprobability on H∗ which is an extension of µ′. Note that it follows from Theorem 8 that such an extensionalways exists. Since probabilities of ForI-subformulas of A remain the same, M∗ � A. Finally, note that the setW ∗ is finite because every world has at most 2|SubI(A)| immediate successors and every branch contains at most|SubI(A)| worlds. Thus |W ∗| ≤ (2|SubI(A)|)|SubI(A)| ≤ 2|A|2 .

Theorem 22 The satisfiability problem for probabilistic formulas is decidable.

P r o o f. For A ∈ ForP let us denote by DNF(A) the formula∨

i

∧j ±P≥si,jαi,j , where ±P≥si,jαi,j’s are

probabilistic literals, which is equivalent to A. For every A ∈ ForP there is at least one DNF(A), becausepropositional connectives behave classically at the probabilistic level. A is satisfiable iff at least one disjunctD from DNF(A) is satisfiable. Since D is a conjunction of probabilistic literals, without loss of generality wecan assume that A is of the same form. It follows from Theorem 21 that A is satisfiable iff it is satisfiable in aprobabilistic model with at most kA = 2|A|2 worlds. Thus, we can check satisfiability of A in the following way.For every l, 1 ≤ l ≤ kA, there is only finitely many intuitionistic models with different valuations with respect tothe set of propositional letters that occur in A. For every such intuitionistic model MI = 〈W,≤, v〉 we can findthe algebra H generated by the set {[α]MI : α ∈ SubI(A)}. Thanks to Theorem 9, we can suppose that every{w} (w ∈ W ) belongs to H as well, and consider the following linear system:

∑w∈W µ(w) = 1,

µ(w) ≥ 0, for w ∈ W,∑

w∈[α]MIµ(w) ≥ r, for every P≥rα which appears in A,

∑w∈[α]MI

µ(w) < r, for every ¬P≥rα which appears in A.

Obviously, if the above system is solvable, M = 〈W,≤, v,H, µ〉 � A. There is a finite number of modelsand linear systems we have to check. Since linear programming problem is decidable, the same holds for theconsidered satisfiability problem.

Finally, since A is valid iff ¬A is not satisfiable, the validity problem is also decidable.

7 Conclusion

In this paper we have investigated a logic which combines probabilistic and intuitionistic reasoning. The com-pactness theorem does not hold for our logic. To see that, consider the set

T = {¬P≥1α} ∪ {P≥1−1/nα : n a positive integer}.


Although every finite subset of T is satisfiable, the set T itself is not. Since the compactness theorem followseasily from the extended completeness theorem (‘every consistent set of formulas is satisfiable’), we cannot hopefor the extended completeness when we have a finitary axiomatic system. However, including an infinitary rulein our axiomatization (Rule 3) we obtain the extended completeness theorem (Theorem 20).

As we have already noted this is not the first paper which considers combinations of intuitionistic and proba-bilistic logics. In [2, 3] two kinds of probabilistic operators for upper and lower probabilities were added to Heyt-ing propositional logic. The corresponding models were Kripke models with two families of measures that weresubadditive and superadditive, respectively, and monotone with respect to the order of the worlds. Our approachis more similar to [4, 5, 9, 10, 11]. However, as the basic logic in those papers is classical propositional logic,there are formulas that are valid in the mentioned logics, but not in ours. For example, as discussed in Section 1,P≥1((p → q) ∨ (q → p)) is valid in logics from [4, 5, 9, 10, 11], but does not hold in the probabilistic modelM = 〈W,≤, v,H, µ〉, where W = {w0, w1, w2}, ≤ = {(w0, w0), (w0, w1), (w0, w2), (w1, w1), (w2, w2)},v(w0) = ∅, v(w1) = {p}, v(w2) = {q}, H = 2{w0,w1,w2}, and µ({w0}) = µ({w1}) = µ({w2}) = 1/3. Obvi-ously, w1 � q → p, w2 � p → q and w0 � (p → q) ∨ (q → p). Then, µ([(p → q) ∨ (q → p)]M ) = 2/3and M � P≥1((p → q) ∨ (q → p)). Moreover, we can construct a model in which both p → q and q → p willhave probability 1/n, by simply adding n− 3 linearly ordered new worlds below w0 in M .

In [9, 10, 11] probabilistic operators of the forms P≤s and P>s were defined as P≤sα = P≥1−s¬α andP>sα = ¬P≥1−s¬α, respectively. It can not be done in our logic. Namely, P≤s¬α follows from P≥1−sα, butit is easy to see that the other direction does not hold. It is enough to consider the model M = 〈W,≤, v,H, µ〉,where W = {w0, w1, w2}, ≤ = {(w0, w0), (w0, w1), (w0, w2), (w1, w1), (w2, w2)}, v(w0) = v(w1) = ∅,v(w2) = {p}, H = 2{w0,w1,w2}, µ({w0}) = µ({w1}) = µ({w2}) = 1/3. Then, µ({w ∈ W : w � ¬p}) =µ({w1}) = 1/3 and µ({w ∈W : w � p}) = µ({w2}) = 1/3. It follows that M � P≤1/2¬p, M � P≥1/2pand M � P≤1/2¬p → P≥1/2p. Instead, we may extend our logic, in the spirit of intuitionistic systems, byhaving two kinds of probabilistic operators of the forms P≥s and P≤s, that are not inter-definable, and by addingcorresponding duals of the axioms and inference rules from Section 4.

Acknowledgements This work was supported by the Ministarstvo za nauku, tehnologije i razvoj Republike Srbije, throughMatematicki Institut Beograd, grant number 1379.

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a probabilistic extension of intuitionistic logic

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