An infinitary propositional probability logic

We introduce a logic for a class of probabilistic Kripke structures that we call type structures, as they are inspired by Harsanyi type spaces. The latter structures are used in theoretical economics and game theory. A strong completeness theorem for an associated infinitary propositional logic with probabilistic operators was proved by Meier. By simplifying Meier’s proof, we prove that our logic is strongly complete with respect to the class of type structures. In order to do that, we define a canonical model (in the sense of modal logics), which turns out to be a terminal object in a suitable category. Furthermore, we extend some standard model-theoretic constructions to type structures and we prove analogues of first-order results for those constructions.


Introduction
In the first part of this work we provide a different and, in our opinion, simpler proof of a completeness result originally proved in [13]. Such result concerns an infinitary propositional probability logic that arises from the formalization of certain structures known as Harsanyi type spaces. The latter were originally introduced and partly formalized in [7] to model beliefs arising from the interactions within a set of agents and to deal with hierarchies of beliefs. Later, the study of type spaces from a logical perspective began. In this paper we are concerned with the latter.
Among the works that investigate logical systems inspired by Harsanyi type spaces, we mention [13,21]. In this paper we build on [13]. In turn, major sources of inspiration of [13] are [8] (where a weak completeness result is established for a finitary language); B Stefano Baratella stefano.baratella@unitn.it 1 Dipartimento di Matematica, Università di Trento, via Sommarive 14, I-38123 Povo TN, Italy [9] (where a universal type space is constructed in purely measure-theoretic terms, thus removing the topological assumptions needed in a previous construction proposed in [14]) and also [3] (where the relation between the semantic and the syntactic approach in game theory and economics is discussed).
We refer the reader to the exhaustive introduction and bibliography in [13,21] for a thorough discussion on the import of Harsanyi type spaces in theoretical economics and game theory, as well as for an historical account on the development of the subject. It is also worth mentioning [15], where the authors propose a categorical approach.
Harsanyi type spaces can be formalized as probabilistic variants of Kripke structures. We deal with structures which are weaker than Harsanyi type spaces. We call them type structures.
We stress that, with the only exception of Sect. 5 (where we take a short categorical detour), we are concerned exclusively with the logical and model-theoretic aspects of type structures. In short, we have a class of structures (the type structures) and we introduce a probability logic for that class, with the aim of proving a completeness result. Then we investigate model-theoretic properties of the proposed logic. We focus almost exclusively on the algebraic/probabilistic content of type structures. As already mentioned, our structures and their associated deductive system are weaker than those in [13]. Hence, one cannot reasonably expect then to fully model key phenomena in economic or game theory like interaction of agents, agents introspection or hierarchies of beliefs (but, regarding introspection, see Remark 14). At this stage, our logic should be regarded as a probability logic. We believe that it can be extended to encompass the above mentioned phenomena. Such an extension goes beyond the scope this paper.
Going back to type structures, intuitively we may say that, in a type structure, a family of probability measures models the beliefs of a single agent at various states of the world. Differently from [13], who allows for multiple agents, we use a one-sorted language, as is done in [21].
Admittedly, the current framework may look quite restrictive to readers whose main interests are in theoretical economics or in game theory. On the other hand, the motivations behind this paper are of logic nature, as we stated above.
We point out that, in [21], Zhou refrains from using an infinitary language. Nevertheless he must allow for an infinitary deduction rule to mimic the crucial Archimedean property (see Sect. 3 below). Eventually, Zhou gets a weak completeness theorem. Namely, he shows that the set of logical theorems is exactly the set of logically valid formulas in his propositional probability logic.
In [13], Meier gets strong completeness (namely, the equivalence between provability and logical consequence) by working with an infinitary propositional probability logic. In the first part of this work, we take inspiration from [13] and we show that a suitable choice of an infinitary language allows for a different and, in our opinion, simpler proof of a completeness result.
Among the differences with [13], we mention: 1. a different formulation of the property of continuity at the empty set of a probability measure; 2. the use of very basic results in probability theory (for instance, we do not need to invoke Carathéodory's Extension Theorem at any point); 3. the straightforward proofs of completeness (Theorem 13 below) and of the existence of a terminal object within the category of type structures (Sect. 5).
The reader who is not interested in a comparison with the first part of [13] will find this paper basically self-contained.
We begin the second part of this work with some comments on the above mentioned category of type structures, which is the one usually studied in the literature. Such category apparently lacks some natural constructions. For this reason, we focus on algebraic rather than categorical properties of type structures. More precisely, we naturally extend products and ultraproducts of first-order structures to arbitrary families of type structures. Among others, we prove an Upward Löwenheim-Skolem-like theorem and, under an admittedly very strong set-theoretic assumption, a Łoś-like theorem. Such results hint that further model-theoretic investigations of type structures might be of interest.
By taking inspiration from the literature, we extend the inverse limit construction from systems of probability spaces to certain systems of type structures. Borrowing from the literature, we also deal the direct limit of systems of probability spaces, by pointing out the difficulties in extending such construction to systems of type structures.
We do not know whether the above mentioned constructions have economic or game-theoretic meaning.
Eventually, we point out that most logics are compositional, but probability is not. The general framework that we introduce in the first part of this work seems necessary to ensure a fruitful interplay between the logic and the probability components of the system.
First of all we introduce the formulas of our logic, to be named L ν P . We do it in a rather informal way: all the definitions below can be easily formalized.
The alphabet of L ν P contains denumerably many propositional variables q i , i ∈ N, and the symbol ⊥ for "falsum" (these also play the role of atomic formulas); the usual finitary propositional connectives; the infinitary connectives and ; the probability quantifiers P ≥r , r ∈ Q ∩ [0, 1] ; the auxiliary symbols ( and ).
We denote by V the set of propositional variables. The set P of probability formulas is the least set S with the following properties: S contains the atomic formulas; S is closed under application of the finitary propositional connectives and under conjunctions and disjunctions of countable sequences of formulas; for all ϕ ∈ S and all r ∈ Q ∩ [0, 1] , the string P ≥r ϕ is in S.
The restriction on the cardinality of conjunctions and disjunctions in P is a technical requirement which is necessary to make sure that the satisfiability relation is welldefined. See below. Each formula in P is a countable string of symbols from a countable alphabet. It follows that |P| = ‫ג‬ 1 .
Notice that, differently from the set L 0 of finitary formulas in [13], we allow for infinitary formulas in P. This will simplify axioms, deduction rules and also most of the proofs of the results in [13]. It should also be noted that a disavantage of allowing for infinitary formulas in P is that an easily computable upper bound ‫ג(‬ 2 ) for the cardinality of the canonical structure defined in Sect. 4 is higher than Meier's ‫ג(‬ 1 -when set I of agents is countable).
As is customary in set theory, throughout this paper initial letters of the greek alphabet (α, β, γ, . . . ) denote ordinals.
The set F of formulas is the least set S that contains P and is closed under application of the finitary connectives and under the following formation rule: for all α < ν and all sequences A ∈ S α the strings A 0 . . . A β . . . and A 0 . . . A β . . . , β < α, are in S. For notational simplicity, the latter will be denoted by β<α A β and β<α A β respectively. We also write for the conjunction of the formulas in set , assuming that some well-ordering on has been fixed. Similarly with . Regularity of ν and the cardinality limitation in the definition of F imply that each formula in F is a string of length < ν.
We will use the terms P-formula and F-formula, with the obvious meaning. When writing just "formula", we mean "F-formula".
In order to reduce the use of brackets, we assume that the propositional connectives (finitary or infinitary) satisfy the same binding priority as in propositional logic, with the probability quantifiers having the highest one.
From now on, letters r , s, t always denote rationals in the closed real unit interval. When we write P ≥r ϕ, it is understood that ϕ ∈ P.
We write P >r ϕ as abbreviation for ¬P ≥1−r ¬ϕ. For better readability, we will also use the expressions P ≤r ϕ and P <r ϕ, which can be easily defined in terms of the primitive probability quantifier: they stand for ¬P >r ϕ and ¬P ≥r ϕ respectively.
Next, we introduce the structures that we will deal with and their semantics. We take inspiration from the formalization of the Harsanyi type spaces first explicitly and completely given in [14].
Let ( , F) be a measurable space. On the family P( , F) of probability measures on ( , F) we consider the σ -algebra σ ,F generated by the sets of the form With reference to the previous definition, the triple ( , F, T ) is called a type space. Intuitively, for every q ∈ V ∪ {⊥}, v(q) is the set of ω ∈ at which q is true. The latter is required to be a measurable subset of .
If there is no ambiguity, we identify a type structure ( , F, T , v) with its support . Let ( , F, T , v) be a type structure. We recursively define when a formula ϕ is true at ω ∈ (notation: , ω | ϕ) as follows: (1) , ω | ⊥; (2) , ω | q if and only if ω ∈ v(q), for q ∈ V ; (3) the finitary and infinitary propositional cases relative to P-formulas are the same as in propositional logic; (4) , ω | P ≥r ψ if and only if T (ω)({ω ∈ : , ω | ψ}) ≥ r , for a P-formula ψ; (5) the cases relative to the formulas in F \ P are the same as in propositional logic.

Remark 2
One easily shows by induction on ϕ ∈ P that ϕ ∈ F. Here it is crucial that P is closed under conjunctions and disjunctions of countable cardinality only.
Furthermore, when ϕ of the form P ≥r ψ, notice that Then measurability of T and the inductive assumption ψ ∈ F yield ϕ ∈ F. Therefore the satisfiability relation is well-defined.
We adopt a "local" notion of logical/semantic consequence. We say that ϕ ∈ F is a logical consequence of (notation: | ϕ) if ⊆ ϕ , for all type structures .

Axioms, rules and soundness
As in [11, Chapter V], we denote by P(L ν ) the infinitary propositional logic (with denumerably many propositional variables) in which conjunctions and disjunctions over sequences of formulas of length less than ν are allowed. In the following we will use, often without explicit mention, [11,Theorem 5.5.4] on the strong completeness of P(L ν ).
The following set of axioms schemas for L ν P is by no means intended to be minimal (this is certainly not the case with A0 below). Rather, we aim at an intuitive set of axioms. The axiomatization below is inspired by those in [12] and [13].

Remark 3
Axiom P8 formalizes the property of continuity (from above) at the empty set (the latter is often referred to as continuity at zero). Namely, P8 is intended to syntactically capture the property that, for a finite measure μ, whenever {A k } k∈N is a decreasing sequence of measurable sets such that k∈N A k = ∅, then lim k→∞ μ(A k ) = 0.
In order to clarify the content of P8, we argue semantically. We fix a type structure and {ϕ m } m∈N ⊆ P. By Remark 2, each ϕ m is a measurable set. Hence so is Let ω ∈ be arbitrarily chosen. By definition of truth of a formula, we have that formula P ≤1/n ( m≤k ϕ m ∧¬ m∈N ϕ m ) is true at ω if and only if T (ω)(A k ) ≤ 1/n. Therefore truth of P8 at ω amounts to requiring that the following holds: Finally, we note that the A k 's defined above form a decreasing sequence and that k∈N A k = ∅. Since, for a finite measure, property of σ -additivity is equivalent to finite additivity plus continuity at the empty set (see, for instance, [2, §1.2]) and since T (ω) is a probability measure, statement (*) is necessarily true at ω.
We stress that the previous argument shows that P8 is a valid axiom.
Notice that, differently from [13], we formulate continuity at the empty set by means of an axiom, rather than with an inference rule.
The inference rules of L ν P are the following. Albeit our formal proofs will be Hilbert style, we formulate the rules in natural deduction style for sake of better understanding.

MP (Modus Ponens
Rules MP and C are exactly those in [11, §5.1]. Rule N is reminiscent of the necessitation rule in modal logic (hence the name).
Notice that, same as in modal logic and differently from MP and C-rule, N-rule can be applied only when ϕ is provable from logical axioms and deduction rules only (i.e. when ϕ is a logical theorem, according to the terminology introduced below).
As we remarked above, differently from [13], we do not need an infinitary rule expressing the property of continuity at the empty set.
The notion of (Hilbert style) proof from a given set of assumptions is formulated as usual (see the beginning of [11,Chapter 5]). We stress that a proof is a sequence of fewer than ν formulas. We use the standard notation ϕ to say that there exists a proof of ϕ from the set of assumptions. As is usual, ϕ stands for ∅ ϕ. Formulas ϕ satisfying the latter are called logical theorems.
In order to become familiar with the deduction system, we invite the reader to verify that P ≤1 ϕ, for every P-formula ϕ. Notice also that P >1 (ϕ) ↔ ⊥.
Furthermore, if ϕ, ψ are P-formulas such that ϕ ↔ ψ then, for all r , P ≥r ϕ ↔ P ≥r ψ follows from N rule and axiom P6. In the following we will often use the latter fact without explicit mention.
A set ⊆ F is consistent if ⊥. A maximal consistent set is a set which is maximal with respect to inclusion among consistent sets.
Let be maximal consistent. It is straightforward to verify that the following hold: is closed under provability; for every formula ϕ exactly one between ϕ and ¬ϕ is in The existence of (maximal) consistent sets will be a consequence of the Soundness Theorem below.
Proof By induction on a proof of ϕ from , after verifying that axioms are valid and rules preserve validity. As for validity of P8, see Remark 3 above.
As a corollary of the Soundness Theorem, we get that every satisfiable set of sentences is consistent. Furthermore, for all type structures and all ω ∈ , the set {ϕ ∈ F : , ω | ϕ} is maximal consistent.
Same as in infinitary propositional logic, it is immediate to exhibit an inconsistent set which can be obtained as union of a chain of consistent sets. Hence the usual application of Zorn's Lemma for proving that every consistent set extends to some maximal consistent one cannot be performed in the current framework.

Completeness
We provide a construction which vaguely resembles the canonical model construction in various modal logics.
We denote by the family of sets ω of P-formulas with the following properties: (1) ω is consistent; (2) every formula in ω is an atomic or a negated atomic formula or is of the form P ≥r ϕ or ¬P ≥r ϕ for some r and some ϕ ∈ P; (3) for each propositional variable q, one between q and ¬q is in ω; (4) for every formula ϕ of the form P ≥r ψ, one between ϕ and ¬ϕ is in ω; It follows from Theorem 4 that = ∅.
We denote by ω the closure under provability of ω ∈ .
Remark 6 A straightforward proof by induction on formulas shows that, for every ω ∈ and every F-formula ψ, the set ω contains exactly one between ψ and ¬ψ. It follows that ω is maximal consistent. Furthermore, there is a one-to-one correspondence between and the family of maximal consistent sets of F-formulas.
For every F-formula η, we let where ω is the deductive closure of ω (see Remark 6).
Same as in [13,Proposition 2], but with different and possibly simpler proofs, we get the following: ω .

Proof
(1) Let R be an injective map which, restricted to V , is the identity and maps every formula of the form P ≥r ϕ into a new propositional variable q ϕ,r . Application of R transforms every ω ∈ into a set R(ω) of propositional variables or negated propositional variables of the infinitary propositional logic P(L ν ) whose set, let us call it V 1 , of propositional variables has cardinality ‫ג‬ 1 and contains V . Let T be the set of truth valuations of P(L ν ). Notice that |T | = ‫ג‬ 2 , for every v ∈ T is uniquely determined by its restriction to the set V 1 . For every v ∈ T and every propositional variable q we let (with a slight notational abuse). After recalling that ψ ∨ ⊥ ↔ ψ, we remove from the formula above all the disjuncts which are inconsistent formulas of L ν P and we conclude that ω∈ ω .
Recalling Remark 6 and applying same argument used at the end of (1), we get the required result. Let It is straightforward to verify that F is a σ -algebra. In particular, Clearly, v(⊥) = ∅.

In the following we often omit brackets and we write T (ω) [ϕ] for T (ω)([ϕ]).
Next, we verify that T is a measurable function. Let ϕ ∈ P. As is well known, it suffices to prove that, for all r ∈ Q∩[0, 1] , the set is measurable. We have: and s < r and the latter intersection is in F. Actually, by axiom P7 (Archimedean property), the intersection above is equal to P ≥r ϕ , but we point out that P7 is not really necessary here.
Next, we prove that, for all ω ∈ , T (ω) is continuous at the empty set. This is the point where we use axiom P8. Let {ϕ n } n∈N ⊆ P. We show that Being an instance of axiom P8, formula 0<n∈N k∈N is in the deductive closure ω of ω. It follows that, for all 0 < n ∈ N, the P-formula Since ψ n is provably equivalent to the negation of η n , the consistency of ω would be contradicted.
Therefore, for every 0 < n there exists k n such that formula By N rule and by P6, we immediately get that, for all 0 < n and k n < k, We summarise the results above in the following:
Proof For every ω ∈ the measure T (ω) is finite, finitely additive and continuous at the empty set. As we already remarked just after introducing the axioms, it follows that T (ω) is σ -additive. As for the second part, we have shown above that map T is measurable and map v satisfies the required properties.
We call canonical structure the type structure of Proposition 9.

Proposition 10
With reference to the canonical structure, it holds that, for every Fformula ψ, Proof By induction on formulas. The cases relative to P-formulas are straightforward, possibly with the exception when ψ of the form P ≥r ϕ. We deal with the latter case in some detail. Let us assume that the statement of the theorem holds for ϕ ∈ P. Then, for every ω ∈ , where the left-to-right implication in the bottom right equivalence holds by axiom (P7). Then the statement holds for ψ.
Cases relative to formulas in F \ P are trivial.
Notice that, throughout the paper, we use axiom P7 in the previous proof only.

Corollary 11
Let be a consistent set of F-formulas. Then is satisfiable in the canonical structure.

Corollary 12
Every consistent set of F-formulas extends to some maximal consistent set.
Notice that in Corollaries 11 and 12 there is no limitation on the cardinality of the set of F-formulas. The same holds for the strong completeness theorem below. Differently from [13, Theorem 1], the proof of strong completeness is straightforward.
Eventually, we summarize the technical reasons for choosing ν = ‫ג(‬ 2 ) + . As remarked in Sect. 2, so doing we limit the cardinality of P-formulas to ‫ג‬ 1 . It follows that the type structure defined at the very beginning of this section has cardinality ≤ ‫ג‬ 2 . See Remark 5. In the latter remark, we also explain why we need to go up to ‫ג(‬ 2 ) + . Furthermore, in the background, there is another reason for dealing a regular cardinal (as ν is): regularity of ν is a necessary assumption in the proof of strong completeness of logic P(L ν ) which is mentioned at the beginning of Sect. 2). Such completeness result is needed in the proof of key Proposition 8.

Remark 14
The following axiom schemas axiomatize the phenomenon of agent introspection that Harsanyi type spaces want capture (in this regard, see [13,Definition 8]): for ϕ ∈ P I2 ¬P ≥r ϕ → P ≥1 (¬P ≥r )ϕ, for ϕ ∈ P Clearly, expanding our set of axioms amounts to restricting the class of type structures to those satisfying the additional axioms.
Quite interestingly, if include I1 and I2 on our list of axioms, the above defined canonical structure validates those two schemas. We provide the details for I1, the same argument applies to I2. To begin with, we notice that, by Proposition 10 and by definition of [ψ], for each formulas ψ, it holds that Let ω ∈ and let us assume that , ω | P ≥r ϕ. It follows from ( ) that P ≥r ϕ ∈ ω . Being I1 an axiom, it belongs to ω as well. Hence ω P ≥1 P ≥r ϕ and, by closure under provability of ω , (P ≥1 P ≥r ϕ) ∈ ω . From ( ), we finally get , ω | P ≥1 P ≥r ϕ.
Being ω ∈ arbitrarily chosen, we conclude that the canonical structure validates schema I1.
Hence, even if it may not satisfy the characteristic property of a Harsanyi type space (see [13,Definition 8]), the above defined canonical structure validates the instances of introspection that are formalized by schemas I1 and I2.

A short categorial detour
In the following we recall the notion of morphism in the category of type structures given in [13]. Such definition is a natural extension of the notion of morphism in the category, to be denoted by P, of probability spaces and measure preserving maps. More precisely, a morphism in P between the probability spaces ( 1 , F 1 , P 1 ) and ( 2 , F 2 , P 2 ) is a measurable map f : 1 → 2 with the property that The following is taken from [13].
(1) f is a measurable function; (2) for all ω 1 ∈ 1 and all q ∈ V , (3) for all ω 1 ∈ 1 and all E ∈ F 2 , Notice that all conditions imposed on f in Definition 15, with the exception of (2), refer to the inverse image map f −1 : P( 2 ) → P( 1 ).
Let us denote by id 1 the identity morphism on a type structure ( 1 , and f be as in Definition 15. By a straightforward induction on formulas one shows that for all ϕ ∈ F and all ω 1 ∈ 1 , Set-theoretic composition of morphisms yields a morphism and the identity map is a morphism. Therefore type structures and morphisms form a category that we denote by T S. A morphism f from the type structure ( 1 , F 1 , T 1 , v 1 ) to the type structure ( 2 , F 2 , T 2 , v 2 ) is an isomorphism if there exists a morphism g : 2 → 1 such that f • g = id 2 and g • f = id 1 . In particular, an isomorphism of type structures is a bijection between the underlying spaces of events, but this condition alone does not suffice in general.
Next we show that the canonical structure is a terminal object in T S. In our opinion, our proof is simpler than Meier's in [13].

Theorem 16 The canonical structure ( , F, T , v) defined before Proposition 9 is a terminal object in the category T S.
Proof Let ( 1 , F 1 , T 1 , v 1 ) be an arbitrary type structure. We claim that the map is a morphism. Let ϕ ∈ P. From Proposition 10 we get In particular, for all ω 1 ∈ 1 and all q ∈ V , Moreover, f −1 ([ϕ]) = f −1 (ϕ ) = ϕ 1 and the latter is in F 1 (see Remark 2). Hence f is measurable.
It remains to verify that (3) of Definition 15 is satisfied by f . Let ϕ ∈ P and ω 1 ∈ 1 . From (•) above we get, for all r ∈ Q ∩ [0, 1], Hence F, T , v) be a morphism and let ω 1 ∈ 1 . From ( * * ) above we get From Proposition 10 it follows that and so f (ω 1 ) = g(ω 1 ), by Remark 6. Therefore f is the only morphism from ( 1 ,

We conclude that ( , F, T , v) is a terminal object in the category T S.
Notice that morphism f defined in the proof of Theorem 16 identifies any two points in 1 that satisfy the same formulas. This admittedly strong property reflects the economic and game-theoretic intuition behind type indistinguishability.

A model-theoretic perspective
We begin with some considerations which may lead to an investigation of type structures from an algebraic perspective.
We notice that (3) of Definition 15 is the counterpart of ( * ) in the category P of probability spaces (for the latter, see Sect. 5). It is well-known that P does not have products in categorical sense, simply because, in general, projections do not preserve probability. Also, category P has a very restrictive notion of subobject: a natural candidate for a subobject of a probability space turns out to be isomorphic mod 0 to the original space (see, for instance, [5] for the latter notion). We explain what we mean: let ( , F, P) be a probability space and let B ∈ F. We let Assuming P(B) = 0, we let where P(A/B) is the conditional probability of A given B. Probability space (B, F B , P B ) is a natural candidate for being a subobject of (X , F, P). This requires the inclusion map ι : B → X being a (mono)morphism. Measurability of ι is straightforward. Condition ( * ) above becomes In particular, we get P(B) = 1 and so the spaces (B, F B , P B ) and ( , F, P) are isomorphic mod 0. As pointed out by [5], in ergodic theory the latter is the standard notion of isomorphism between probability spaces. Thus, in P, the notion of subobject is extremely restrictive.
For this reason, one may want to drop condition ( * ) in the definition of morphism, but then two probability spaces ( , F, P 1 ) and ( , F, P 2 ), with P 1 = P 2 , would be indistinguishable.
Similar considerations apply to the category T S. From property ( * * ) above we see that a morphism of type structures behaves "locally" like an isomorphism, even if "globally" it is not injective, in general. A natural question is whether there is a weaker notion of morphism with related notions of embedding, type substructure, etc…Even if existence of a universal type structure possibly fails, the resulting framework might be of interest. In this regard, we make a parallel with the category of sets where each singleton is a terminal object, but such property is not as relevant as other set-theoretic properties.
In light of the previous considerations, we suggest shifting the focus from categorical to algebraic properties of type structures. We first introduce a natural notion of type substructure.

Products
Despite category T S does not have products in categorical sense, we can also form the product of an arbitrary family of type structures. For simplicity, we deal with the product of two structures ( 1 , F 1 , T 1 , v 1 ) and ( 2 , F 2 , T 2 , v 2 ). The product of finitely many type structures can be defined by induction on natural numbers. Later we will deal with infinite products. We denote the product as follows: (1) 1 × 2 is the cartesian product of the two sets; (2) F 1 ×F 2 is the product σ -algebra, namely the σ -algebra generated by the rectangles B × C, with B ∈ F 1 and C ∈ F 2 ; (3) for all ω 1 ∈ 1 and all ω 2 ∈ 2 , (T 1 × T 2 )(ω 1 × ω 2 ) is the product probability of T 1 (ω 1 ) and T 2 (ω 2 ); (4) for all q ∈ V , (v 1 × v 2 )(q) is the cartesian product of v 1 (q) and v 2 (q) and ( We only need to prove that T 1 × T 2 is a measurable map. It suffices to verify that, for all B ∈ F 1 , C ∈ F 2 and r ∈ Q∩ ]0, 1], Case r = 0 follows by taking a countable union: r > 0 ⇔ (r > 1/n, for some n > 0).
For B, C, r as above, we have: and the latter countable union is in of type structures, we proceed as in the construction of a countably infinite product of probability spaces. See, for instance, [6, §38]. In particular, for all (ω i ) ∈ we define T ((ω i )) as the unique probability measure S such that, for each generator i∈N A i of F, Measurability of T follows as in the finite product case after recalling that, for all generators i∈N A i of F, A i = i for finitely many i's only.
Regarding the generalization to uncountable products, see Exercise (2) at the end of [6, §38].

Ultraproducts
We develop a notion of ultraproduct of type structures. For the pertinent set-theoretic and topological notions we refer the reader to [1].
The notation and conventions previously introduced are in force.
Let U be an ultrafilter on a set I .
be a family of type structures and let be the set-theoretic ultraproduct of the i 's with respect to U . We write ω U for the equivalence class with respect to U of the sequence ω = (ω i ) i∈I ∈ i∈N i . Let (E i ) i∈I be a sequence such that, for all i ∈ I , E i ⊆ i . We let Notice that (E i ) U is just a convenient notation for the set-theoretic ultraproduct It is easy to verify that the family is a boolean algebra on . We denote by F be the σ -algebra generated by F • and by ( , F • ) the family of pre-measures on ( , F • ).
We define a map T : where lim U stands for the limit with respect to the ultrafilter U of the bounded sequence (T i (ω i )(E i )) i∈I (see [1]). By [1, Proposition 1.3, Ch.2], for all ω ∈ i , map T (ω) is well-defined and σ -additive, hence it can be uniquely extended to a probability measure T (ω) on F.
Furthermore, if ω, ψ ∈ i∈N i are such that ω U = ψ U , by the filter properties we get T (ω) = T (ψ). Therefore T induces a well-defined map We turn ( , F) into a measurable space as described before Definition 1 and we prove the following:

Proposition 17 The map T defined above is measurable.
Proof It suffices to show that, for all E = (E i ) U ∈ F • and all r ∈ [0, 1] ∩ Q, the set First of all we notice that the following are equivalent for ω U = (ω i ) U : For all 0 < k ∈ N and all i ∈ I , by measurability of and the latter intersection is in F.
Summing up: the 4-tuple ( , F, T , v) defined above is a type structure. We call it the ultraproduct of the family Next we investigate whether a Łoś-like theorem holds for the ultraproduct of type structures. We recall that an ultrafiter U on a set I is ω 1 -complete if any countable intersection of elements from U belongs to U . Principal ultrafilters are ω 1 -complete, but the existence of a nonprincipal ω 1 -complete ultrafilter is equivalent to the existence of a measurable cardinal (see [4,Theorem 1.11,Ch.6]). The latter is unprovable in ZFC. Actually, it cannot even be proved that the existence of a measurable cardinal is consistent with ZFC. See [10].
In classical first-order logic, ultraproducts with respect to principal ultrafilters are trivial, in a sense made precise by [4,Corollary 2.3,Ch.6]. An analogous result holds in the framework of type structures, as we show next.
Let be the set theoretic ultraproduct of the family of sets { i } i∈I with respect to the principal ultrafilter U generated by {k}, for some k ∈ I . An easy verification shows that the map where ω k = η and ω i ∈ i , for all i = k, is a well-defined bijection. In particular, h(η) does not depend on the choice of the coordinates ω i ∈ i , for i = k. Let ( , F, T , v) be the ultraproduct of the family {( i , F i , T i , v i )} i∈I of type structures with respect to the principal ultrafilter U on I generated by {k}, for some k ∈ I . Let h : k → be the bijection defined above. The following hold:
and, for i = k, X i = i (equivalently, by (1), any X i ∈ F i would do).
Notice that h × (E) = {h(η) : η ∈ E}. It follows from the proof of (2) that h × respects countable intersections. The rest of the proof that h × is a morphism of σ -algebras is straightforward.  ( , F, T , v) is an isomorphic copy of the fiber Finally, we prove the following Łoś-like theorem: Then, for all F-formulas ϕ and all ω ∈ i∈I i , Proof By induction. We begin with P-formulas. If ϕ is ⊥, the equivalence is trivial. If ϕ is the variable q, we have Cases relative to negation and to countable conjunctions are straightforward (countably infinite conjunctions are taken care by the assumption of ω 1 -completeness). If ϕ is of the form P ≥r η, for some η ∈ P, the inductive assumption on η is equivalent to the following: The following are equivalent: where the equivalence of (4) and (5) holds by ω 1 -completeness of U .
We are left with the formulas in F. The only nontrivial case is that of an infinite conjunction of formulas (recall that each conjunction involves less than ν-many formulas). We just notice that, if U is principal, then it is κ-complete for any cardinal κ, namely an arbitrary intersection of elements from U lives in U . If U is nonprincipal, then |I | is a measurable cardinal. Clearly, |I | > ν and so the intersection over an arbitrary family of cardinality < ν of elements from U lives in U .
Admittedly, the assumption of existence of a nonprincipal ω 1 -complete ultrafilter in Theorem 19 is very strong. It might be interesting to investigate whether weaker set-theoretic assumptions do ensure validity of that theorem.

Upward Löwenheim-Skolem theorem
We prove the following: Proposition 20 Let ( , F, T , v) be a type structure and let κ > | | be a cardinal. Then there exists a probability structure ( 1 , F 1 , T 1 , v 1 ) such that: (3) for all ω ∈ and all ϕ ∈ F, Proof We pick a set X of cardinality κ disjoint from and we let 1 = ∪ X ; v 1 = v. Let F 1 be the σ -algebra generated by F ∪ {X }. We fixω ∈ and we define T 1 : 1 → ( 1 , F 1 ) as follows: -for all ω ∈ and all E ∈ F 1 we let It is straightforward to verify that each T 1 (ω) is a probability measure on ( 1 , F 1 ).
To complete the proof that ( 1 , F 1 , T 1 , v 1 ) is a type structure it remains to prove that T 1 is a measurable map. It suffices to show that, for all E ∈ F ∪ {X } and all r ∈ [0, 1] ∩ Q, the set In any case A ∈ F 1 . If E = X then A = 1 for r = 0 and A = ∅ for r > 0. It remains to prove that (3) holds. We proceed by induction on ϕ. Since the other cases are trivial, we assume that ϕ is of the form P ≥r ψ and we work under the inductive assumption ψ 1 ∩ = ψ , for all ω ∈ .
Let ω ∈ . Then We leave as an open problem whether a Downward Löwenheim-Skolem theorem hold for type structures.
In the next two sections we provide partial results relative to the existence of inverse and direct limits of systems of type structures. We begin each section by dealing with limits of systems of probability spaces. Surprisingly for this author, their construction seems to require very specific assumptions.

Inverse limits
Inverse limits of systems of probability spaces are known to exist only under suitable topological assumptions. See, for instance, the Introduction in [17]. In the latter work, the author introduces a purely measure-theoretic condition, called -completeness, that suffices for existence and uniqueness of inverse limits when the index set is the set of natural numbers. We exploit -completeness to extend the inverse limit construction to inverse families of type structures.
We suitably extend the framework in [17]. For coherence, we stick to the notation used so far. Let be an inverse system of sets and maps. Namely, for all i ≤ j ≤ k in N, the following hold: The inverse limit ( , ( p i : → i ) i∈N ) of the system above is defined as follows: = ω ∈ i∈N i : ω i = f i j (ω j ), for all i ≤ j ∈ N and the p i 's are the projection maps.
Next, we consider an inverse system (( i , F i , T i , v i ), ( f i j )) i≤ j,i, j∈N of type structures. By this we mean that, for all i ≤ j ≤ k, (2) f i j is a measurable function; i j denotes the restriction to F i of the pre-image function associated to f i j .
(Compare with [17,Definition 2.2].) Since we are mostly concerned with the measure-theoretic features of the inverse limit construction, we forget about the assignments of values to the propositional Thus we deal with the inverse system We will see that the v i 's play no role in the construction below.
It can be shown that We let F be the σ -algebra generated by F • . Clearly, F is the coarsest σ -algebra on for which all p i 's are measurable.
For all i ∈ N and all ω ∈ i , let T i (ω) * : P( i ) → [0, 1] be the outer measure defined by We give the following, inspired by [17,Definition 3.1]: Hence, -completeness of (( i , F i , T i ), ( f i j )) i≤ j,i, j∈N is equivalent to the condition that, for all ω ∈ , the inverse system of probability spaces is -complete according to [17,Definition 3.1].
From now on, we assume that (( i , F i , T i ), ( f i j )) i≤ j,i, j∈N is an -complete inverse system of type pre-structures.
From [17,Theorem 3.2] we get that, for all ω ∈ , there exists a unique probability measure T (ω) on ( , F) such that, for all i ∈ N, We recall the definition of T (ω) from [17]: for some n ∈ N and someÂ ∈ F n such that p −1 n (Â) = A. (It follows from [17, Theorem 3.2] that T (ω) is well-defined.) We fix n andÂ as above. By measurability of T n , we have: Hence We call ( , F, T ) the inverse limit of the -complete inverse system of type pre-structures. Eventually, we may also want to define an assignment of values v to the propositional variables so to get a type structure. A natural choice is to let, for q ∈ V , Notice that v(q) ∈ F, as required by the definition of type structure.

Direct limits
As inverse limits, direct limits of systems probability spaces are known to exist under suitable (mostly topological) assumptions. See, for instance, [18,Chapter III]. As for their existence under purely measure-theoretic assumptions, we only know the partial results obtained in [20]. Actually, [20] contains significant additional assumptions, not just "remarks and alterations", of results previously claimed by the same author in [19]. A major problem already lies with the definition of the measurable space underlying the direct limit. For such reason, in [20], the author is forced to introduce the notions of pseudo σ -algebra and pseudo probability space.
In this section, we first define the direct limit of a system of probability spaces. We do it under stronger assumptions than those in [20]. The additional assumptions are quite natural and allow to get rid of the above mentioned pseudo-notions. Eventually, we discuss the extension of the direct limit to systems of type structures.
It seems quite awkward to refer the reader directly to [20], which, in turns, makes systematic reference to [19]. For this reason, we set up the framework quite in detail.
Let (I , ≤ ) be a directed set and let (( i , ( f i j )) i≤ j,i, j∈I be an injective direct system of sets, namely, for all i ≤ j ≤ k, When writing ω i , we implicitly assume that ω i ∈ i . The direct limit ( , ( f i ) i∈I ) of the above system is defined as follows: is the quotient set of i∈I ( i × {i}) with respect to the equivalence relation ∼ defined by We denote by ω ∼ the equivalence class of ω ∈ i∈I i .
It can be easily verified that the f i 's are injective. Furthermore, Next, we give the following: Definition 22 An injective direct system of probability spaces is a structure such that (( i , ( f i j )) i≤ j,i, j∈I is an injective direct system of sets and, for all i ≤ j ≤ k and all A ∈ F i , Definition 22 captures the setting in [20]. We point out the strength of condition (3). For instance, it implies that, whenever i ≤ j, Albeit rather strong, the previous assumptions do not suffice to define a σ -algebra on (as wrongly claimed in [19]). Hence the notion of pseudo probability space and the statement of [20,Theorem 3].
From now on, we further assume that -I is countable; -for all i ≤ j, f i j is a measurable function.
The latter is a quite natural assumption on maps between probability spaces. Then we endow with a structure of probability space.
We claim that R is a ring of subsets of . Clearly, ∅ ∈ R. Let X = f i (A), Y = f j (B), for some A ∈ F i and B ∈ F j and let i, j ≤ k. By ( ) above, we have: Similarly, where the rightmost equality holds by injectivity of f k .
We need a preliminary result before proving that μ is actually a pre-measure on R.
Lemma 23 Under the assumptions above, let i, j ∈ I and let C ∈ F i , B ∈ F j be such that f i (C) ⊆ f j (B). Then there exists D ∈ F j such that f i (C) = f j (D).
Proof Let D = {ω j ∈ B : there exists ω i ∈ C with (ω i , i) ∼ = (ω j , j) ∼ }. Clearly, f i (C) = f j (D). It remains to prove that D ∈ F j . Let i, j ≤ k ∈ I . From f i (C) = f j (D), we get f k ( f ik (C)) = f k ( f jk (D)). Hence f ik (C) = f jk (D), by injectivity of f k . Therefore f −1 jk ( f ik (C)) = D. Moreover, f ik (C) ∈ F k , by Definition 22 (2). Hence D = f −1 jk ( f ik (C)) ∈ F j , by measurability of f jk . Let { f i n (A n ) : n ∈ N} ⊆ R be a family of pairwise disjoint sets such that n∈N f i n (A n ) = f k (A), for some A ∈ F k . By Lemma 23, for each n ∈ N there exists B n ∈ F k such that f i n (A n ) = f k (B n ). Notice that the B n 's are pairwise disjoint. We have: where the second-last equality holds by ( ).
Hence μ is a pre-measure on R. By the Carathéodory's extension theorem, μ extends to a measure, that we denote by the same name, on the σ -algebra F generated by R. We claim that μ is a probability measure.
First of all, since = i∈I f i ( i ) and I is countable, belongs to F. Hence F is actually a σ -algebra on . Therefore, in order to prove that μ is a probability measure on F, it suffices to verify that the outer measure μ * associated to μ satisfies the condition μ * ( ) ≤ 1.
The latter follows by fixing an enumeration {i n } n∈N of I and by noticing that is the disjoint union of the family Moreover, μ( f i 0 ( i 0 )) = 1 and, for all 0 < n, μ( f i n ( i n ) \ 0≤ j<n f i j ( i j )) = 0.
We call (( , F, μ), ( f i ) i∈I ) the direct limit of (( i , F i , μ i ), ( f i j )) i≤ j,i, j∈I . Eventually, we briefly comment on the extension of the direct limit construction to type pre-structures. Hence, as in Sect. 6.5, we forget about the assignments of values to propositional variables. The notation introduced in the first part of this section is in force.
We fix a direct system (( i , F i , T i ), ( f i j )) i≤ j,i, j∈I of type pre-structures, where ( i , ( f i j )) i≤ j,i, j∈I is a countable injective direct system. We form the set-theoretic direct limit ( , ( f i ) i∈I ) and, as above, we denote by F the σ -algebra generated by i∈I f i (F i ). For all ω ∈ , let I ω = {i ∈ I : there exists ω i ∈ i such that (ω i , i) ∼ = ω}.
The set I ω is a directed subset of I .
We may consider the direct subsystem (( i , F i , T i (ω i ), ( f i j )) i≤ j,i, j∈I ω of probability spaces. Assuming that the latter satisfies conditions (2) and (3) in Definition 22 and that the f i j 's are measurable, we denote its direct limit by (( ω , F ω , μ ω ), ( f ω i ) i∈I ω )). Therefore ω is the quotient set of i∈I ω ( i × {i}) with respect to the restriction ∼ ω of ∼ to i∈I ω ( i × {i}). We define the map T : → ω∈ ( ω , F ω ) by letting T (ω) = μ ω for all ω ∈ .
We notice that the structure ( , F, T ) is not a type pre-structure, simply because each T (ω) is a probability measure on F ω , not on F. Apart from the very special case when, for all ω ∈ , I ω = I (the latter implies that ( , F) = ( ω , F ω )), there seems to be no intuitive way of obtaining a direct limit pre-structure.
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