2. Zero-one laws in two-valued predicate logic

In this section, we recall some facts from classical finite model theory that we aim to generalize later to the context of many-valued logics.

Consider first a purely relational vocabulary $\tau $ for the usual finitary first-order logic that we will denote by $\mathcal {L}_{\omega \omega }$ , as it is customary in abstract model theory [Reference Barwise and Feferman6] and even in some works on many-valued logics (see, e.g., [Reference Badia and Noguera4]), where the $\omega $ subscripts represent the finitary character of, respectively, conjunctions/disjunctions and quantifier strings. Sometimes, we will also refer to the stronger logics $\mathcal {L}_{\omega _1\omega }$ and $\mathcal {L}_{\infty \omega }$ which allow for conjunctions and disjunctions of, respectively, countably many or arbitarily many formulas, while keeping the finitary character of quantifier strings.

A $\tau $ -sentence is said to be parametric in the sense of [Reference Oberschelp and Jungnickel36, p. 277] (or, alternatively, it is an Oberschelp condition) if it is equivalent to a finite conjunction of first-order formulas of the form:

$$ \begin{align*} \forall^{\not=}x_{1},\dots ,x_{k}\, \phi (x_{1},\dots ,x_{k}), \end{align*} $$

which abbreviates $\forall x_{1}\dots \forall x_{k}\,(\bigwedge \nolimits _{i<j}\lnot (x_{i}=x_{j})\rightarrow \phi (x_{1},\dots ,x_{k}))$ , where $\phi (x_{1},\dots ,x_{k})$ is a quantifier-free formula such that all of its atomic subformulas $Rx_{i_{1}}\dots x_{i_{k}} $ distinct from identities have

$$ \begin{align*} \{x_{i_{1}}\dots x_{i_{k}}\}=\{x_{1},\dots ,x_{k}\}. \end{align*} $$

In the case where $k=1$ , any formula $\forall x\,\phi (x)$ where $\phi $ is quantifier-free, is parametric. For example, $\forall x\,\lnot Rxx\wedge \forall ^{\not =}x,y\,(Rxy\rightarrow Ryx)$ is parametric, whereas $\forall ^{\not =}x,y,z\,(Rxy\wedge Ryz\rightarrow Rxz)$ is not. It is not difficult to check that any universal formula of the form $\forall x_{1}\dots \forall x_{k}\,\phi (x_{1},\dots ,x_{k})$ where each atomic subformula of $\phi $ distinct from identities contains all the variables $x_{1},\dots ,x_{k}$ is parametric. A class of structures is called an Oberschelp class if it can be axiomatized by parametric sentences.

Oberschelp’s extension [Reference Oberschelp and Jungnickel36, Theorem 3] of the classical zero-one law says: on finite models and finite purely relational vocabularies, for any class $\mathbb {K}$ definable by a finite set of parametric sentences, any first-order sentence $\phi $ is almost surely satisfied by finite members of $\mathbb {K}$ or almost surely not satisfied. More precisely, if $\mathbb {K}_{n}$ is the set of models in $\mathbb {K}$ with domain $[n]=\{1,\ldots ,n\},$ then

$$ \begin{align*} \mu _{n}(\phi )=\frac{|\{\mathfrak{M}\in \mathbb{K}_{n}: \mathfrak{M\models } \phi \}|}{|\mathbb{K}_{n}|} \ \textit{converges to}\ 0 \ \textit{or}\ 1. \end{align*} $$

Oberschelp’s result actually holds for sentences of the variable-bounded infinitary logic $\mathcal {L}_{\infty \omega }^{\omega }$ , i.e., the fragment of the infinitary logic $\mathcal {L}_{\infty \omega }$ determined by formulas containing finitely many variables only. This is a generalization of a well-known result by Kolaitis and Vardi for the class of all finite models [Reference Kolaitis and Vardi29]. An accessible presentation of these results can be found in [Reference Ebbinghaus and Flum15].

If there are no Oberschelp conditions, the counting probability measure $\mu _{n}$ in the above zero-one law corresponds to assigning probability $\frac {1}{2}$ to the probabilistically independent atomic events $R(i_{1},\ldots ,i_{k})$ , $i_{1},\ldots ,i_{k}\in [n]$ , in the random model $\mathfrak {M}\in \mathbb {K}_{n}$ . The zero-one law holds, with the same asymptotic probability for each $\phi $ , if the random model is obtained by assigning a fixed probability $p_{R}\in (0,1)$ to the atomic event $R(x_{i_1},\ldots ,x_{i_k})$ for each $R\in \tau $ .Footnote ¹ There is a wealth of positive and negative results pertaining to the case in which $p_{R}$ varies with n (cf. [Reference Spencer37]) or the class $\mathbb {K}$ is given by non-parametric conditions (cf. [Reference Grove, Halpern and Koller24]).

3. The case of finitely valued predicate logics

Let $\boldsymbol {A}={\langle {A,\wedge ^{\boldsymbol {A}},\vee ^{\boldsymbol {A}},\ldots } \rangle }$ be an algebra (which can be finite or infinite) with a lattice reduct ${\langle {A,\wedge ^{\boldsymbol {A}},\vee ^{\boldsymbol {A}}}\rangle }$ and possibly with additional operations. We will call $\boldsymbol {A}$ a lattice algebra. The underlying lattice may be bounded, in which case $0$ and $1$ will denote, respectively, the bottom and top element (which may, but need not, be denoted by constants of the signature of $\boldsymbol {A}$ ).

For the associated $\boldsymbol {A}$ -valued logic $\mathcal {L}^{\boldsymbol {A}}$ and a first-order relational vocabulary $\tau $ , the first-order logic $\mathcal {L}^{\boldsymbol {A}}(\tau )$ is built in the usual way from variables $x_{1},x_{2},\ldots $ , relation symbols in $\tau $ , connectives $\wedge , \vee , \ldots $ (corresponding to the operations of $\boldsymbol {A}$ ), and quantifiers $\forall $ and $\exists $ .

Models are $\boldsymbol {A}$ -valued $\tau $ -structures $\mathfrak {M}={\langle {M,R^{\mathfrak {M}}}\rangle }_{R\in \tau }$ with domain $M\not =\emptyset $ , where $R^{\mathfrak {M}}\colon M^{\tau (R)}\longrightarrow A$ , and $\tau (R)$ denotes the arity of $R\in \tau $ ; $\mathrm {Mod}_{\tau }^{\boldsymbol {A}}$ will denote the class of all $\boldsymbol {A}$ -valued $\tau $ -structures.

Define inductively for each formula $\varphi (x_{1},\ldots ,x_{k})$ in $\mathcal {L}^{\boldsymbol {A}}(\tau )$ with k free variables, a mapping $||\varphi ||^{\mathfrak {M}}\colon M^{k}\longrightarrow A,$ interpreting the atomic formulas according to $\mathfrak {M}$ , the connectives according to $\boldsymbol {A}$ , and the quantifiers as suprema and infima in $\boldsymbol {A}$ ( $\mathfrak {M}$ is supposed to be safe, that is, the required infima and suprema exist, which in the context of finite structures is always guaranteed).

For a finite (or complete) algebra $\boldsymbol {A}$ , we may extend the semantics to the infinitary logic $\mathcal {L}_{\omega _{1}\omega }^{\boldsymbol {A}}$ by setting $||\bigwedge \nolimits _{n}\varphi _{n}||^{\mathfrak {M}}:=\inf _{n}||\varphi _{n}||^{\mathfrak {M}}$ (all formulas $\varphi _{n}$ having free variables in a common finite set) and similarly for $\bigvee _{n}\varphi _{n}$ .

The classical logics $\mathcal {L}_{\omega \omega }$ and $\mathcal {L}_{\omega _{1}\omega }$ may be, respectively, identified with $\mathcal {L}^{\boldsymbol {B}_{2}}$ and $\mathcal {L}_{\omega _{1}\omega }^{\boldsymbol {B}_{2}}$ , where $\boldsymbol {B}_{2}$ is the two-element Boolean algebra.

3.1. The zero-one law for finitely valued predicate logics

Let $\boldsymbol {A}$ be a finite lattice algebra, $\tau \,$ a finite relational vocabulary, and $\mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}}$ be the class of $\boldsymbol {A}$ -valued $\tau $ -structures with domain $[n]=\{1,\ldots ,n\}$ .Footnote ² Given $\varphi (x_{1},\ldots ,x_{k})\in \mathcal {L}^{\boldsymbol {A}}(\tau )$ , $i_{1},\ldots ,i_{k}\leq n,$ and $a\in A$ , define the probability that a randomly chosen model $\mathfrak {M} \in \mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}}$ assigns the value a to $\varphi (i_{1},\ldots ,i_{k})$ as:

$$ \begin{align*} \mu _{n}(\varphi (i_{1},\ldots,i_{k})=a):=\frac{|\{\mathfrak{M}\in \mathrm{Mod}_{\tau,n}^{\boldsymbol{A}}: ||\varphi ||^{\mathfrak{M}}(i_{1},\ldots,i_{k})=a\}|}{|\mathrm{Mod}_{\tau ,n}^{\boldsymbol{A}}|}. \end{align*} $$

This definition assigns the same weight to all models in $\mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}},$ and the same probability

$$ \begin{align*} \mu _{n}(R(i_{1},\ldots,i_{\tau (R)})=a):=\frac{1}{|A|}, \end{align*} $$

to the event that an atomic statement $R(i_{1},\ldots ,i_{\tau (R)})$ takes the value $a\in A,$ for any a and numbers $i_{1},\ldots ,i_{\tau (R)}\in [n]$ .

More generally, and reciprocally, we may fix, for each $R\in \tau $ a probability distribution $p_{R} \colon A\longrightarrow [0,1]$ (that is, $\sum _{a\in A}p_{R}(a)=1$ ) such that $p_{R}(a)>0$ for any $a\in A$ , and define a probability distribution Pr $_{n}\colon \mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}}\longrightarrow [0,1]$ in the class of $\boldsymbol {A}$ -valued models with domain $[n]$ :

$$ \begin{align*} \text{Pr}_{n}(\mathfrak{M}):=\prod\{p_{R}(R^{\mathfrak{M}}(i_{1},\ldots,i_{\tau (R)})) : R\in \tau ,\ i_{1},\ldots,i_{\tau (R)}\in [n]\}, \end{align*} $$

obtained by identifying $\mathfrak {M}$ with its atomic $\boldsymbol {A}$ -valued diagram. Then, we may define for any formula $\varphi (x_{1},\ldots ,x_{k})\in \mathcal {L}^{\boldsymbol {A}}(\tau )$ , $a\in A,$ and $i_{1},\ldots ,i_{k}\in [n],$

$$ \begin{align*}\mu _{n}(\varphi (i_{1},\ldots,i_{k})=a):=\sum \{Pr_{n}(\mathfrak{M}) : \mathfrak{M} \in \mathrm{Mod}_{\tau ,n}^{\boldsymbol{A}}, ||\varphi ||^{\mathfrak{M}}(i_{1},\ldots,i_{k})=a\}.\end{align*} $$

Notice that, for the sake of clarity, we are using the simplified notation $\varphi (i_{1},\ldots ,i_{k})=a$ to denote the event $ ||\varphi ||^{\mathfrak {M}}(i_{1},\ldots ,i_{k})=a$ for an arbitrary model $\mathfrak {M}\in \mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}}$ .

It is not hard to check that, under this definition,

$$ \begin{align*} \mu _{n}(R(i_{1},\ldots,i_{\tau (R)})=a)=p_{R}(a), \end{align*} $$

and the atomic events $R(i_{1},\ldots ,i_{k})=a$ , $R^{\prime }(j_{1},\ldots ,j_{k})=b$ are (probabilistically) independent if $R \neq R'$ or ${\langle {i_{1},\ldots ,i_{k}}\rangle } \neq {\langle {j_{1},\ldots ,j_{k}}\rangle }$ .

The next theorem generalizes the classical zero-one law to the finitely-valued context.

Theorem 1. For any sentence $\varphi \in \mathcal {L}^{\boldsymbol {A}}(\tau )$ and $a\in A$ , $\lim _{n}\mu _{n}(\varphi =a)=1$ or $\lim _{n}\mu _{n}(\varphi =a)=0$ . Equivalently, for any sentence $\varphi \in \mathcal {L}^{\boldsymbol {A}}(\tau )$ , there is a (unique) $a\in A$ such that $\lim _{n}\mu _{n}(\varphi =a)=1$ .

Proof. The equivalence of both statements follows from the fact that, for each $\varphi $ , the events $\varphi =a$ , with $a\in A$ , are exhaustive and mutually incompatible. The first formulation follows from the classical zero-one laws via the following multi-translation $\theta \longmapsto {\langle {\theta ^{a}}\rangle }_{a\in A}$ , defined by simultaneous induction, from $\mathcal {L}^{\boldsymbol {A}}(\tau )$ into classical $\mathcal {L}_{\omega \omega }(\tau \times A)$ , $\tau \times A=\{R^{a}:R\in \tau ,a\in A\}$ :

$$ \begin{align*} R(x_{1},\ldots,x_{\tau (R)})^{a}:=R^{a}(x_{1},\ldots,x_{\tau (R)}), \text{ for } R\in \tau \end{align*} $$

$\circ (\varphi _{1},\ldots ,\varphi _{k})^{a}:=\bigvee _{\circ ^{\boldsymbol {A}}(b_{1},\ldots ,b_{k})=a}\varphi _{1}^{b_{1}}\wedge \ldots \wedge \varphi _{k}^{b_{k}}$ , for any connective $\circ $ of arity k in the signature of $\boldsymbol {A}$

$$ \begin{align*} &(\forall x\varphi )^{a}:=(\bigvee\nolimits_{b_{1}\wedge \ldots\wedge b_{m}=a,\text{ }m\leq |A|}\exists x\,\varphi^{b_{1}}\wedge \ldots\wedge \exists x\,\varphi^{b_{m}})\wedge \forall x\,(\bigvee\nolimits_{b\geq a}\varphi ^{b})\\ &(\exists x\,\varphi )^{a}:=(\bigvee\nolimits_{b_{1}\vee \ldots\vee b_{m}=a,\text{ }m\leq|A|}\exists x\,\varphi^{b_{1}}\wedge \ldots\wedge \exists x\, \varphi^{b_{m}})\wedge \forall x\,(\bigvee\nolimits_{b\leq a}\varphi ^{b}), \end{align*} $$

and the corresponding model transformations from $\mathrm {Mod}_{\tau }^{\boldsymbol {A}}$ into $\mathrm {Mod}_{\tau \times A}$ :

$$ \begin{align*} \mathfrak{M}\longmapsto \mathfrak{M}_{\tau \times A}:={\langle{M;(R^{a})^{\mathfrak{M}_{\tau \times A}}}\rangle}_{R^{a}\in \tau \times A}, \end{align*} $$

where $(R^{a})^{\mathfrak {M}_{\tau \times A}}:=\{\overline {i}\in M : R^{\mathfrak {M}}(\overline {i})=a\}$ , which are easily seen to satisfy:

$$ \begin{align*} ||\theta||^{\mathfrak{M}}(i_{1},\ldots,i_{k})=a \ \ \Leftrightarrow \ \mathfrak{M}_{\tau \times A}\models \theta^{a}[i_{1},\ldots,i_{k}]. \end{align*} $$

The model transformations establish a domain-preserving bijection between $\mathrm {Mod}_{\tau }^{\boldsymbol {A}}$ and the class $\mathbb {K}\subseteq \mathrm {Mod}_{\tau \times A}$ of classical models of the parametric $(\tau \times A)$ -sentences

$$ \begin{align*} &\forall x_{1}\ldots x_{\tau (R)}\,\bigvee\nolimits_{a\in A}R^{a}(x_{1},\ldots,x_{\tau (R)}),\\ &\forall x_{1}\ldots x_{\tau (R)}\,\bigwedge\nolimits_{a\not=b}\lnot (R^{a}(x_{1},\ldots,x_{\tau(R)})\wedge R^{b}(x_{1},\ldots,x_{\tau (R)})). \end{align*} $$

Being an Oberschelp class, $\mathbb {K}$ satisfies the classical zero-one law if we assign probability $\mu _{n}^{\mathbb {K}}(R^{a}(x_{1},\ldots ,x_{k})):=p_{R}(a)$ to the classical atomic events $R^{a}(x_{1},\ldots ,x_{k})$ . The latter assignment yields $\Pr _{n}(\mathfrak {M}_{\tau \times A})=\Pr _{n}(\mathfrak {M)}$ for $\mathfrak {M} \in \mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}}$ and the transformation is bijective from $\mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}}$ onto $\mathbb {K}_n$ (models in $\mathbb {K}$ with domain $[n]$ ). Then, $\mu _{n}(\varphi =a)=\\ \mu _{n}\{\mathfrak {M}\in \mathrm {Mod}_{\tau ,n}^{\boldsymbol {A}} : ||\varphi ||^{\mathfrak {M}}=a\}=\mu _{n}^{\mathbb {K}}\{\mathfrak {M}_{\tau \times A}\in \mathbb {K}_{n} : \mathfrak {M}_{\tau \times A}\models \varphi ^{a}\}=\mu _{n}^{\mathbb {K}}(\varphi ^{a})$ . Thus, $\mu _{n}(\varphi =a)$ converges to $0$ or $1$ .

Remark 2. Theorem 1 holds if the models are restricted to any class of $\boldsymbol {A}$ -valued structures satisfying conditions that are parametrically expressible in $\mathcal {L}_{\omega \omega }(\tau \times A)$ via the translation. Just add these conditions to the definition of the class $\mathbb {K}$ in the proof. For example, $\boldsymbol {A}$ -valued irreflexive symmetric graphs:

$$ \begin{align*} &\quad \forall x\, R^{0}xx, \forall ^{\not=}xy\,\bigwedge\nolimits_{a\in A}(R^{a}xy\leftrightarrow R^{a}yx)\ (\textit{irreflexivity is expressed by using}\\ &\quad\textit{degree 0 on the relation}), \end{align*} $$

or structures with crisp identity:

$$ \begin{align*} \forall x\,(x\approx^{1}x), \forall ^{\not=}xy\,(x\approx^{0}y). \end{align*} $$

Remark 3. Theorem 1 holds also for sentences in finite vocabularies of the variable-bounded sublogic $(\mathcal {L}^{\boldsymbol {A}})_{\omega _{1}\omega }^{\omega }$ of $\mathcal {L}_{\omega _{1}\omega }^{\boldsymbol {A}}$ , where each formula has finitely many free and bound variables, because of the obvious extension of the translation to the infinitary logic; for example,

$$ \begin{align*}(\bigwedge\nolimits_{n}\varphi _{n})^{a}:=\big[\bigvee_{b_{1}\wedge \ldots\wedge b_{k}=a,k\leq |A|,n_{1},\ldots,n_{k}}(\varphi_{n_{1}}^{b_{1}}\wedge \ldots\wedge \varphi_{n_{k}}^{b_{m}})\big]\wedge \bigwedge\nolimits_{n}\bigvee _{b\geq a}\varphi _{n}^{b}, \end{align*} $$

sends it to classical $\mathcal {L}_{\omega _{1}\omega }^{\omega }$ , known to satisfy the zero-one law under Oberschelp’s conditions [Reference Ebbinghaus and Flum15, Reference Kolaitis and Vardi29]).

Remark 4. Zero-one laws have been studied for classical modal logics (e.g., [Reference Goranko and Kapron21, Reference Halpern and Kapron27, Reference Le Bars31, Reference Verbrugge39]). In [Reference Halpern and Kapron27], the authors prove zero-one laws for model validity for various systems of modal logic and state a similar result for frame validity. Unfortunately, the latter is refuted in [Reference Le Bars31] using ideas from [Reference Goranko and Kapron21]. In the finitely valued case, any fully modal sentence of the $\boldsymbol {A}$ -valued version S $5_{c}^{\boldsymbol {A}}$ of propositional bimodal logic S $5$ over finite universal crisp frames (cf. [Reference Caicedo, Metcalfe, Rodríguez and Rogger9, Reference Caicedo and Rodríguez10]) satisfies a zero-one law for model validity, because this logic may be interpreted as the one-variable fragment of $\mathcal {L}^{\boldsymbol {A}}$ on unary predicates via the translation $\varphi \mapsto \widehat {\varphi }$ :

$$ \begin{align*} &\widehat{p_{i}}:=P_{i}(w)\\ &\widehat{\square \varphi}:=\forall w\widehat{\varphi }(w)\\ &\widehat{\vphantom{A^,}\Diamond \varphi}:=\exists w\widehat{\varphi }(w)\\ &\widehat{t(\varphi _{1},\ldots,\varphi _{k})}:=t(\widehat{\varphi }_{1}(w),\ldots,\widehat{\varphi }_{k}(w)), \end{align*} $$

which sends any fully modal formula to a sentence.

3.2. Finitely valued random models and extension axioms

In the above proof, the finitely valued zero-one law lurks a countable $\boldsymbol {A}$ -valued random model, corresponding to the classical random model that one obtains for the language of the translation.

Theorem 5. There is a countable $\boldsymbol {A}$ -valued $\tau $ -model $\mathfrak {R}_{\tau }^{\boldsymbol {A}}$ such that, for any sentence $\varphi \in \mathcal {L}^{\boldsymbol {A}}(\tau )$ , we have $\lim _n \mu _{n}(\varphi =a)= 1$ iff $||\varphi ||^{\mathfrak {R}_{\tau }^{\boldsymbol {A}}}=a $ .

Proof. Take the $\boldsymbol {A}$ -model $\mathfrak {R}_{\tau }^{\boldsymbol {A}}$ corresponding to the countable $\omega $ -categorical random model $\mathfrak {R}_{\tau \times A}$ of the class $\mathbb {K}$ introduced in the proof of Theorem 1. Then, $\lim _n\mu _{n}(\varphi =a)= 1$ iff $\lim _n \mu _{n}^{\mathbb {K}}(\varphi ^{a})= 1$ iff $\mathfrak {R}_{\tau \times A}\models \varphi ^{a}$ iff $||\varphi ||^{\mathfrak {R}_{\tau }^{\boldsymbol {A}}}=a$ .

The classical extension axioms characterizing $\mathfrak {R}_{\tau \times A}$ , and indirectly $\mathfrak {R}^{\boldsymbol {A}}_{\tau }$ , are rarely expressible in $\mathcal {L}^{\boldsymbol {A}}$ , but they are expressible in ( $\mathcal {L}_{\omega \omega })_{\tau \times A}$ as:

$$ \begin{align*} Ext_{A}(k,f):\forall^{\not=}x_{1},\ldots,x_{k}\,\exists x_{k+1}\not=x_{1},\ldots,x_{k}\,\bigwedge\nolimits_{\varphi \in F_{k+1}}\varphi^{f(\varphi )}(x_{1},\ldots,x_{k+1}), \end{align*} $$

where $F_{k+1}$ is the set of atomic $\tau $ -formulas with variables in $x_{1},\ldots ,x_{k},x_{k+1}$ , truly containing the variable $x_{k+1}$ , and $f\colon F_{k+1}\longrightarrow A$ . These axioms are automatically consistent with the set of parametric sentences defining the class $\mathbb {K}$ in the proof of Theorem 1. If there are additional parametric conditions, only the extension axioms consistent with them are considered.

Example 6. According to Remark 2, the random $\boldsymbol {A}$ -valued irreflexive symmetric graph may be described as the unique countable $\boldsymbol {A}$ -weighted irreflexive symmetric graph $\mathfrak {R}={\langle {G,R}\rangle }$ such that, for any distinct $x_{1},\ldots ,x_{k}\in G$ and $a_1,\ldots , a_k\in A$ , there is an $x_{k+1}\not =x_{1},\ldots ,x_{k}$ in G such that $R(x_{i},x_{k+1})=a_i$ . This structure was studied as a Fraïssé limit in [Reference Badia and Noguera3].

Since the extension axioms and thus the random model are independent of the probabilities $\{p_{R}\}_{R\in \tau }$ , we can use Theorem 5 to obtain the following corollary.

Corollary 7. For any sentence $\varphi \in \mathcal {L}^{\boldsymbol {A}}$ , the value $a\in A$ such that $\lim _n \mu _{n}(\varphi =a)= 1$ depends only on $\varphi $ and not on the probability distributions $p_{R}$ (provided that $p_{R}(a)>0$ for all $a\in A)$ .

We call the value a in Corollary 7 the almost sure or asymptotic truth-value of $\varphi $ . This corollary entails that assigning the same probability $\frac {1}{|A|}$ to all truth-values has the same effect in the limit as assigning a positive probability to each one.

Remark 8. If $p_{R}$ is allowed to take value 0 at some $a\in A$ , this corresponds to asking the Oberschelp conditions $\forall x_{1}\ldots \forall x_{n}\,\lnot R^{a}(x_{1},\ldots ,x_{n})$ . Thus, the asympotic value of $\varphi $ still exists and depends on the support of $p_{R}$ only.

4. The set of almost sure values in finitely valued predicate logics

Call $a\in A$ an almost sure truth-value of $\mathcal {L}^{\boldsymbol {A}}$ (of $\boldsymbol {A}$ ) if there is a sentence $\varphi \in \mathcal {L}^{\boldsymbol {A}}$ such that $\lim _n\mu _{n}(\varphi =a)=1$ . Do all the elements of A arise as almost sure values? If not, which are those values? We will see that for many familiar classes of finite algebras the answer to the first question is positive (for example $\mathrm {MV}$ -chains for Łukasiewicz logic, or $\mathrm {G}$ -chains for Gödel logic), for others have a small subset of almost sure values ( $0$ or $1$ for Boolean algebras, at most four values for De Morgan algebras, at most three for the lattice-negation reducts of G-chains, etc.).

The first assertion follows from the following general lemma.

Proof. Let $R(x)$ be an atomic formula with a single variable x. We show that

$$ \begin{align*} \mu _{n}(\forall x_{1}\ldots\forall x_{k}\,t(R(x_{1}),\ldots,R(x_{k}))=m)\ \text{converges to}\ 1. \end{align*} $$

Fix tuples ${\langle {a_{j}^{1}}\rangle }_{j},\ldots ,{\langle {a_{j}^{\ell }}\rangle }_{j}\in |A|^{k}$ , such that $t^{\boldsymbol {A}}(a_{1}^{1},\ldots ,a_{k}^{1})\wedge ^{\boldsymbol {A}}\ldots \wedge ^{\boldsymbol {A}}t^{\boldsymbol {A}}(a_{1}^{\ell },\ldots ,a_{k}^{\ell })=m,$ and for each $n\geq k\ell $ find $K\subseteq [ n]^{k}$ of cardinal $|K|=\lfloor n/k\ell \rfloor $ such that all tuples in K have all their components distinct, and two different tuples in K are distinct at each coordinate. Then,

$$ \begin{align*} &\quad \mu _{n}(\forall x_{1}\ldots\forall x_{k}\,t(R(x_{1}),\ldots,R(x_{k}))=m)\\ &\quad =\Pr_{n}(\inf_{i_{1},\ldots,i_{k}\in [ n]}t^{\boldsymbol{A}}(R(i_{1}),\ldots,R(i_{k}))=m)\\ &\quad \geq \Pr_{n}\ (\text{for some } {\langle {i_{j}^{1}}\rangle}_{j},\ldots,{\langle {i_{j}^{\ell}}\rangle}_{j} \text{ in } [n]^{k}:R(i_{j}^{i})=a_{j}^{i}, \text{ for } i=1,\ldots,\ell , j=1,\ldots,k)\\ &\quad \geq \Pr_{n}\ (\text{for some }{\langle{i_{j}^{1}}\rangle}_{j},\ldots,{\langle{i_{j}^{\ell}}\rangle}_{j} \text{ in } K:R(i_{j}^{i})=a_{j}^{i}, \text{ for } i=1,\ldots,\ell , j= 1,\ldots,k)\\ &\quad =1-\Pr_{n}\ (\text{for all } {\langle {i_{j}^{1}}\rangle}_{j},\ldots,{\langle {i_{j}^{\ell}}\rangle}_{j} \text{ in } K \text{ is false: } R(i_{j}^{i})=a_{j}^{i},\\ &\qquad\qquad\text{ for } i=1,\ldots,\ell, j=1,\ldots,k)\\ &\quad =1-\prod\nolimits_{{\langle {i_{1}^{1},\ldots, i_{k}^{1}}\rangle},\ldots,{\langle {i_{1}^{\ell},\ldots, i_{k}^{\ell}}\rangle} \in K}[1-\Pr_{n}(R(i_{j}^{i})=a_{j}^{i} \text{ for } i=1,\ldots,\ell, j=1,\ldots,k))]\\ &\quad =1-(1-p_{R}(a_{1}^{1})\ldots p_{R}(a_{k}^{\ell }))^{\lfloor n/k\ell \rfloor}. \end{align*} $$

For the last two identities, we use that the $k\ell $ events $R(i_{j}^{i})=a_{j}^{i}$ form a mutually independent set for distinct $i_{1}^{1},\ldots , i_{k}^{1},\ldots ,i_{1}^{\ell },\ldots , i_{k}^{\ell }\in K$ . Since $p_{R}(a_{i}^{j})>0$ , the last quantity converges to $1$ . A similar proof shows that

$$ \begin{align*} \mu _{n}(\exists x_{1}\ldots\exists x_{k}\,(t(R(x_{1}),\ldots,R(x_{k}))=s) \,\, \text{converges to }1. \end{align*} $$

Example 16. Let $\boldsymbol {G}_{N+1}={\langle {\{0=g_{0}<\ldots <g_{N}=1\},\wedge ,\vee ,\lnot ,\rightarrow }\rangle }$ be the G-chain of $N+1$ elements, where $\lnot ,\rightarrow $ are Gödel implication and negation, then all values of $\boldsymbol {G}_{N+1}$ are almost sure values of the corresponding predicate logic. Define:

$$ \begin{align*} &t_{1}(v_{1}):=v_{1}\vee \lnot v_{1}, t_{k+1}(v_{1},\ldots,v_{k+1}):=v_{k+1}\vee (v_{k+1}\rightarrow t_{k}(v_{1},\ldots,v_{k}))\\ &\qquad \qquad =v_{k+1}\vee (v_{k+1}\rightarrow (v_{k}\vee (v_{k}\rightarrow \ldots\rightarrow (v_{2}\vee (v_{2}\rightarrow (v_{1}\vee \lnot v_{1}))\ldots))) \end{align*} $$

then the minimum possible value of $t_{1}(v_{1})$ in $\boldsymbol {G}_{N+1}$ is $g_{1}\vee \lnot g_{1}=g_{1},$ since $g_{0}\vee \lnot g_{0} =1$ and $g_{j}\vee \lnot g_{j}=g_{j}$ for $g_{j}>g_{1}$ . For $2\leq k\leq N$ :

$$ \begin{align*} \,\min_{v_{1},\ldots,v_{k}\in G_{N+1}}t_{k}(v_{1},\ldots,v_{k})=g_{k,} \end{align*} $$

obtained at $g_{1},\ldots ,g_{k},$ because

$$ \begin{align*} t_{k}(x_{1},\ldots,x_{k})=\left\{ \begin{array}{cc} 1 & \text{ if }x_{i+1}\leq x_{i}\text{ for some }i \\ x_{k} & \text{ if }x_{i}<x_{i+1}\text{ for all }i. \end{array} \right. \end{align*} $$

Remark 17. Notice that in the proof of Lemma 13, we have used a single predicate symbol and k variables to write the sentences

$$ \begin{align*} &\forall x\, t(P_{1}(x),..,P_{k}(x))\\ &\exists x\, t(P_{1}(x),..,P_{k}(x)) \end{align*} $$

having the desired almost sure values m and s. We may use instead k (unary) predicates and a single variable with a similar proof. Therefore, via the translation explained in Remark 4, the formulas $\square t(p_{1},\ldots ,p_{k})$ and $\Diamond t(p_{1},\ldots ,p_{k})$ of the S $5_{c}^{A}$ have almost sure values m and $s,$ respectively $.$ It follows then from Example 15 that all values of ${\mathrm {L}}_{N+1}$ are almost sure values of S $5_{c}^{{\mathrm {L}}_{N+1}}$ , since the formula $\square (p^{k}\rightarrow kp^{N})$ has almost sure value $\frac {k}{N},$ and it follows, from Example 16, that all values of G $_{N+1}$ are almost sure values of S $5_{c}^{G_{N+1}},$ since the modal formula $\square (p_{k}\vee (p_{k}\rightarrow (p_{k-1}\vee (p_{k-1}\rightarrow (\ldots (p_{1}\vee \lnot p_{1})\ldots ))$ takes the almost sure value $g_{k}$ .

We turn now to algebras with few almost sure values and obtain the following result inspired by the techniques in [Reference Grädel, Helal, Naaf, Wilke, Baier and Fisman22]. Notice that the proof of this theorem actually gives a new proof of the zero-one law in the cases considered.

Theorem 18. Let $\boldsymbol {A}={\langle {A,\wedge ,\vee ,\lnot }\rangle }$ be a finite distributive lattice with a negation satisfying:

1. (1) $\lnot (v\wedge w)=\lnot v\vee \lnot w$ , $\lnot (v\vee w)=\lnot v\wedge \lnot w$

2. (2) $v\leq \lnot \lnot v$

3. (3) $\lnot 1=0$ .

Then, the almost sure values of $\mathcal {L}^{\boldsymbol {A}}$ are $0$ , $\varepsilon ,\varepsilon ^{\prime },\delta ,\delta ^{\prime }\!$ , and $1$ , where

$$ \begin{align*} &\varepsilon =\sup\nolimits_{x\in A}(x\wedge \lnot x), \varepsilon ^{\prime}=\sup\nolimits_{x\in A}(\lnot x\wedge \lnot \lnot x),\,\\ &\delta =\inf\nolimits_{x\in A}(x\vee \lnot x), \delta ^{\prime}=\inf\nolimits_{x\in A}(\lnot x\vee \lnot \lnot x). \end{align*} $$

Proof. We already know these values are almost sure by Lemma 13. In order to see that they are the only ones, notice first the following derived laws of $\boldsymbol {A}$ :

1. (4) $v\leq w$ implies $\lnot w\leq \lnot v$

2. (5) $\lnot \lnot \lnot v=\lnot v$

3. (6) $\lnot 0=\lnot \lnot 1=1$ , $\lnot \lnot 0=0$ .

4. (7) $0\leq \varepsilon \leq \varepsilon ^{\prime }\leq \delta \leq \delta ^{\prime }\leq 1$ , $\lnot \varepsilon =\lnot \varepsilon ^{\prime }=\delta ^{\prime }$ , $\lnot \delta =\lnot \delta ^{\prime }=\varepsilon ^{\prime },$ thus $E=\{0,\varepsilon ,\varepsilon ^{\prime }, \delta ,\delta ^{\prime },1\} $ is a subalgebra of $\boldsymbol {A}$ .

Expand $\boldsymbol {A}$ to the algebra $\boldsymbol {A}^{+}={\langle {A,\wedge ,\vee ,\lnot ,0,\varepsilon ,\varepsilon ^{\prime },\delta ,\delta ^{\prime },1}\rangle }$ and use the same symbols in the signature to denote the constants in E. By distributivity, (1), (5), (6), and (7), any term $t(v_{1},\ldots ,v_{k})$ in the signature of $\boldsymbol {A}^{+}$ may be put in disjunctive normal form: $\bigvee \nolimits _{s}C_{s}$ with $C_{s}=e\wedge \bigwedge \nolimits _{r}\lnot ^{n_{r}}v_{i_{r}},$ where $e\in E$ , $v_{i_{r}}\in \{v_{1},\ldots ,v_{k}\}$ and $n_{r}\leq 2$ . Moreover, by (2), we may assume no pair $\{v_{i},\lnot \lnot v_{i}\}$ appears in $C_{s}$ , since $v_{i}\wedge \lnot \lnot v_{i}=v_{i}$ . By idempotency and commutativity of $\wedge $ , we may assume that each variable appears at most once in $C_{s}$ as: $v_{i}$ , $\lnot v_{i},$ or $\lnot \lnot v_{i},$ or at most twice as: $v_{i}\wedge \lnot v_{i},$ or $\lnot v_{i}\wedge \lnot \lnot v_{i}$ . Similarly, $t(v_{1},\ldots ,v_{k})$ may be put in conjunctive normal form $\bigwedge \nolimits _{s}D_{s}$ with $D_{s}=e\vee \bigvee \nolimits _{r}\lnot ^{n_{r}}v_{i_{r}}$ and no occurrence of both $v_{i},\lnot \lnot v_{i}$ . We prove now a result on elimination of quantifiers. Let $Ext(\ell )$ denote the conjunction of the extension axioms $Ext_{\boldsymbol {A}}(k,f)$ for $k\leq \ell ,$ and define for formulas $\varphi ,\psi \in \mathcal {L}_{\tau }^{\boldsymbol {A}^{+}}$ : $\varphi \equiv _{Ext(\ell )}\psi $ iff $||\varphi (m_{1},\ldots ,m_{k})||^{\mathfrak {M}}=||\psi (m_{1},\ldots ,m_{k})||^{\mathfrak {M}}$ for any model $\mathfrak {M}$ of $\mathcal {L}_{\tau }^{\boldsymbol {A}^{+}}$ satisfying $Ext(\ell )$ , and any $m_{1},\ldots ,m_{k}\in M$ .

Claim. Any formula $\varphi \in (\mathcal {L}_{\tau }^{\boldsymbol {A}^{+}})^{(\ell )}$ is $Ext(\ell )$ -equivalent to a quantifier-free formula in the same free variables as $\varphi $ .

We proceed by induction on the complexity of $\varphi $ . Assume by induction hypothesis that $\varphi (x_{1},\ldots ,x_{k},x_{k+1})\equiv _{Ext(\ell )}\psi (x_{1},\ldots ,x_{k},x_{k+1})$ quantifier-free, and $k+1\leq \ell ,$ we must show the same for $\exists x_{k+1}\,\varphi (x_{1},\ldots ,x_{k},x_{k+1})$ and $\forall x_{k+1}\,\varphi (x_{1},\ldots ,x_{k},x_{k+1})$ . By the previous algebraic remarks, we may assume $\psi $ is in disjunctive normal form, i.e., $\bigvee \nolimits _{s}C_{s}(x_{1},\ldots ,x_{k},x_{k+1})$ where each $C_{s}$ is a conjunction:

$$ \begin{align*} e\wedge C_{s}^{\prime }(x_{1},\ldots,x_{k})\wedge \bigwedge\nolimits_{r}\lnot^{n_{r}}R_{r}(x_{1},\ldots,x_{k},x_{k+1}),\ n_{r}\in \{0,1,2\}, \end{align*} $$

$C_{s}^{\prime }(x_{1},\ldots ,x_{k})$ being the part not containing the variable $x_{k+1}$ , such that each atomic $R_{r}(x_{1},\ldots ,x_{k},x_{k+1})$ appearing in $C_{s}$ does it in one and only one of the following forms:

$$ \begin{align*} R_{r} \text{ alone}, \lnot R_{r} \text{ alone}, \lnot \lnot R_{r} \text{ alone}, R_{r}\wedge \lnot R_{r}, \lnot R_{r}\wedge \lnot \lnot R_{r}. \end{align*} $$

Substitute the first three types of formulas with (the symbol) $1$ , the fourth with $\varepsilon ,$ and the fifth with $\varepsilon ^{\prime }; $ and let c be the minimum of e and the substituted constants to obtain:

$$ \begin{align*} C_{s}^{+}(x_{1},\ldots,x_{k}):=c\wedge C_{s}^{\prime }(x_{1},\ldots,x_{k}). \end{align*} $$

Since we have substituted the highest possible values in each case, we have for any model $\mathfrak {M}$ and $x_{1},\ldots ,x_{k},i\in M$ :

$$ \begin{align*} ||C_{s}(x_{1},\ldots,x_{k},i)||^{\mathfrak{M}}\leq ||C_{s}^{+}(x_{1},\ldots,x_{k})||^{\mathfrak{M}}. \end{align*} $$

On the other hand, if $\mathfrak {M}$ satisfies $Ext(\ell )$ and $k+1\leq \ell ,$ there is, for any $x_{1},\ldots ,x_{k}$ in M, an $i_{0}\in M$ such that for each $R_{r}$ appearing in $C_{s}(x_{1},\ldots ,x_{k},x_{k+1}),$ the atomic formula $R_{r}(x_{1},\ldots ,x_{k},i_{0})$ takes a value making the configuration in which it appears to take the highest possible value; that is,

$$ \begin{align*} ||C_{s}(x_{1},\ldots,x_{k},i_{0})||^{\mathfrak{M}}=||C_{s}^{+}(x_{1},\ldots,x_{k})||^{\mathfrak{M}}. \end{align*} $$

Therefore, for any $x_{1},\ldots ,x_{k}\in M$

$$ \begin{align*} ||\exists x_{k+1}\,C_{s}(x_{1},\ldots,x_{k},x_{k+1})||^{\mathfrak{M}}&=\sup_{i\in M}||C_{s}(x_{1},\ldots,x_{k},i)||^{\mathfrak{M}}\\ &=||C_{s}^{+}(x_{1},\ldots,x_{k})||^{\mathfrak{M}} \end{align*} $$

and, finally,

$$ \begin{align*} ||\exists x_{k+1}\,\varphi (x_{1},\ldots,x_{k},x_{k+1})||^{\mathfrak{M}}&=||\bigvee\nolimits_{s}\exists x_{k+1}\,C_{s}(x_{1},\ldots,x_{k},i)||^{\mathfrak{M}}\\ &=||\bigvee\nolimits_{s}C_{s}^{+}(x_{1},\ldots,x_{k})||^{\mathfrak{M}}, \end{align*} $$

showing that $\exists x_{k+1}\,\varphi (x_{1},\ldots ,x_{k},x_{k+1})\equiv _{Ext(\ell )}\bigvee \nolimits _{s}C_{s}^{+}(x_{1},\ldots ,x_{k})$ .

Symmetrically, one eliminates the quantifier in $\forall x_{k+1}\,\varphi (x_{1},\ldots ,x_{k},x_{k+1})$ using a conjunctive normal form of $\psi $ and substituting in the elementary disjunction each configuration by its lowest possible value:

$$ \begin{align*} &R_{r} \text{ alone}, \lnot R_{r} \text{ alone}, \lnot \lnot R_{r} \text{ alone go to } 0,\\ &R_{r}\vee \lnot R_{r} \text{ goes to } \delta,\\ &\lnot R_{r}\vee \lnot \lnot R_{r} \text{ goes to } \delta ^{\prime }. \end{align*} $$

Finally, any sentence $\varphi \in (\mathcal {L}_{\tau }^{\boldsymbol {A}^{+}})^{(\ell )}$ is $Ext(\ell )$ -equivalent to a quantifier-free sentence $\psi $ in $\mathcal {L}_{\tau }^{\boldsymbol {A}^{+}},$ necessarily a $\wedge ,\vee ,\lnot $ combination of elements of E and thus an element e of E. Hence, $\lim _n \mu _{n}(\varphi =e)\geq \lim _n \mu _{n}(Ext(\ell ))= 1$ .

Clearly, the conditions of the theorem are satisfied by De Morgan algebras, which include Boolean algebras and the $\{\wedge ,\vee ,\lnot \}$ -reducts of $\mathrm {MV}$ -algebras. They are satisfied also by the $\{\wedge ,\vee ,\lnot \}$ -reduct of any G-algebra since the law $\lnot (v\wedge w)=\lnot v\vee \lnot w$ fails in arbitrary Heyting algebras but holds in prelinear ones.

Notice that no new almost sure values are added by $(\mathcal {L}^{\boldsymbol {A}})_{\omega _{1}\omega }^{(\omega )}$ because any sentence of this logic is asymptotically equivalent to a finitary one, as we prove next using our translation. Let $(\mathcal {L}^{\boldsymbol {A}})_{\omega _{1}\omega }^{(k)}$ denote the formulas in $(\mathcal {L}^{\boldsymbol {A}})_{\omega _{1}\omega }^{(\omega )}$ utilizing at most the variables $\mathbf {x}=x_{1},\ldots ,x_{k}$ , free or bound. Define $(\mathcal {L}^{\boldsymbol {A}})^{(k)}$ similarly, and denote by $\theta _{k}$ the conjunction of the classical extension axioms $Ext_A(i,f)$ for $i\leq k$ .

Proof. We do it by induction on the complexity of $\varphi $ . We prove the inductive step for infinite conjunctions; the other inductive steps are similar or trivial. Assume $\varphi _{n}\in (\mathcal {L}_{\tau }^{\boldsymbol {A}})_{\omega _{1}\omega }^{(k)}$ , $\tau $ finite, and $\theta _{k}\models \forall \mathbf {x}[\varphi _{n}^{a}\leftrightarrow \varphi _{n}^{\prime }{}^{a}]$ with $\varphi _{n}^{\prime }\in (\mathcal {L}_{\tau }^{\boldsymbol {A}})^{(k)}$ for each $a\in A$ and $n\in \omega $ . Then the classical random $\tau $ -model $\mathfrak {R}_{\tau \times A}$ (cf. Theorem 5) satisfies for each $a\in A$ :

$$ \begin{align*} (\bigwedge_{n\in \omega}\varphi_{n})^{a}\leftrightarrow \lbrack \underset{b1\wedge \ldots\wedge b_{k}=a,k\leq |A|,n_{1},\ldots, n_{k}\in \omega }{\bigvee}\varphi_{n_{1}}^{\prime}{}^{b_{1}}\wedge \ldots\wedge \varphi_{n_{k}}^{\prime}{}^{b_{k}}]\wedge \bigwedge_{b\geq a}\bigvee\nolimits_{n\in \omega }\varphi_{n}^{\prime}{}^{b}. \end{align*} $$

As $Th(R)$ is $\omega $ -categorical, by the Ryll–Nardzewski theorem, there is a finite subset $S\subseteq \omega $ such that any formula $\varphi _{n}^{\prime }(\mathbf {x})^{b}$ (with $n\in \omega $ , $b\in A$ ) is equivalent to one in the set $\{\varphi _{n}^{\prime }(\mathbf {x})^{b}:n\in S, b\in A\}$ . Thus, $(\bigwedge _{n\in \omega }\varphi _{n})^{a}$ is equivalent in $\mathfrak {R}_{\tau \times A}$ to

$$ \begin{align*} \lbrack \underset{b1\wedge \ldots\wedge b_{k}, k\leq |A|,n_{1},\ldots, n_{k}\in S}{\bigvee}\varphi _{n_{1}}^{\prime}{}^{b_{1}}\wedge \ldots\wedge \varphi_{n_{k}}^{\prime}{}^{b_{k}}]\wedge \bigwedge_{b\geq a}\bigvee_{n\in S}\varphi _{n}^{\prime}{}^{b}, \end{align*} $$

clearly equivalent to $(\bigwedge _{n\in S}\varphi _{n}^{\prime })^{a}$ . Then $\theta _{k}\models \forall \mathbf {x}[(\bigwedge _{n\in \omega }\varphi _{n})^{a}\leftrightarrow (\bigwedge _{n\in S}\varphi _{n})^{a}]$ ; otherwise, by [Reference Ebbinghaus and Flum15, Claim (4), in Section 4.2], we obtain $\theta _{k}\models \lnot \forall \mathbf {x}[(\bigwedge _{n\in \omega }\varphi _{n})^{a}\leftrightarrow (\bigwedge _{n\in S}\varphi _{n})^{a}]$ , contradicting that the equivalence holds in $\mathfrak {R}_{\tau \times A}$ .

5. A zero-one law for infinitely valued Łukasiewicz predicate logic

Proving zero-one laws for logics given by infinite lattice algebras seems challenging. The translation yields in this case infinitary formulas with essentially infinite vocabularies, and it is known that in the case of vocabularies with an infinite supply of unary predicate symbols, the zero-one law for classical $\mathcal {L}_{\omega _{1}\omega }^{\omega }$ fails [Reference Ebbinghaus and Flum15, Exercise 4.1.9.]. Moreover, we cannot expect positive probabilities $p_{R}(a)$ for all the truth-values (except for countable A).

We will deal mainly with infinitely valued Łukasiewicz logic, determined by the standard $\mathrm {MV}$ -algebra $[0,1]_{\mathrm {L}}={\langle {[0,1],\wedge ,\vee ,\rightarrow ,\lnot }\rangle }$ with crip identity (cf. [Reference Hájek26]).

Consider $\sigma $ -additive Borel probability measures $\{p_{R}\}_{R\in \tau }$ over $[0,1]$ which assign positive measure to all non-trivial intervals of $[0,1]$ , say, $p_{R}=$ Lebesgue measure. These induce a product measure $\rho _{n}$ in the infinite space of models with domain $[n]$ : $\mathrm {Mod}_{\tau ,n}^{[0,1]}=\prod _{R\in \tau }[0,1]^{[n]^{\tau (R)}}\!$ , which permits to define for any measurable set $S \subseteq [0,1]$ the probability that a sentence $\varphi $ takes a value in $S:$

$$ \begin{align*} \mu_{n}(\varphi \in S)=\rho _{n}\{\mathfrak{M}\in \mathrm{Mod}_{\tau,n}^{[0,1]}:\varphi ^{\mathfrak{M}}\in S\}, \end{align*} $$

(the map $\mathfrak {M}\longmapsto \varphi ^{\mathfrak {M}}$ is measurable because it may be shown to be continuous).

It is important to stress that for each pair of predicates R and $R^{\prime }$ , and $i_{1},\ldots ,i_{k},i_{1}^{\prime },\ldots ,i_{k}^{\prime }\in [ n]$ , we have that $R(i_{1},\ldots ,i_{k}) \in S$ and $R^{\prime }(i_{1}^{\prime },\ldots ,i_{k}^{\prime }) \in S'$ become independent events provided that the tuples ${\langle {R,i_{1},\ldots ,i_{k}}\rangle }$ and ${\langle {R^{\prime },i_{1}^{\prime },\ldots ,i_{k}^{\prime }}\rangle }$ are distinct.

For simplicity, we will use the notation $\mathfrak {M}\models \varphi $ as an equivalent of $||\varphi ||^{\mathfrak {M}} = 1$ .

Theorem 25. For any sentence $\varphi \in \mathcal {L }^{[0,1]_{\mathrm {L}}}$ , there is a unique $a\in [ 0,1]$ such that

$$ \begin{align*} \lim_{n}\mu _{n}(\varphi \in V)=1 \ \ \text{for any open interval} \ V\text{ containing }a. \end{align*} $$

Equivalently,

$$ \begin{align*} \lim_{n}\mu _{n}(|\varphi -a|<\varepsilon )=1{ \ \ } \textit{for any}\text{ }\varepsilon>0. \end{align*} $$

The two formulations are clearly equivalent. They say that $\varphi $ takes approximately value a with probability $1$ , for any degree of approximation. To prove the theorem, we will need the following observations on the expressivity of the logic (see [Reference Belluce7, Reference Chang13, Reference Mundici34]). For each fraction $r=\frac {n}{m} \in [0,1]$ , there are connectives of Łukasiewicz logic $( )_{\geq r}$ , $( )_{\leq r}$ such that

$$ \begin{align*} \mathfrak{M} &\models \varphi _{\geq r} \text{ iff } ||\varphi||^{\mathfrak{M}}\geq r,\\ \mathfrak{M} &\models \varphi _{\geq r} \text{ iff } ||\varphi||^{\mathfrak{M}}\leq r. \end{align*} $$

More generally and precisely (see [Reference Caicedo and Iovino8]),

$$ \begin{align*} &||\varphi_{\geq r}||^{\mathfrak{M}}\geq 1-\varepsilon \text{ iff } ||\varphi||^{\mathfrak{M}}\geq r-\frac{\varepsilon }{m}\\ &||\varphi_{\leq r}||^{\mathfrak{M}}\geq 1-\varepsilon \text{ iff } ||\varphi||^{\mathfrak{M}}\leq r+\frac{\varepsilon }{m} \end{align*} $$

We will assume also that the logic $\mathcal {L}^{[0,1]_{\mathrm {L}}}$ has a ‘crisp identity’ $\approx $ taking only the values $0$ or $1$ .Footnote ³

Given $k,N\in \omega ,$ let $F_{k+1}$ consist of all the atomic formulas distinct from identities truly containing the variable $x_{k+1},$ and $A_{N}=\{[\frac {j}{N},\frac {j+1}{N}]$ , $j=0,\ldots , N-1\}$ . For each choice of intervals $g \colon F_{k+1}\longrightarrow A_{N}$ , define the extension axiom $Ext_{A}(k,N,g)$ :

$$ \begin{align*} \forall x_{1}\ldots x_{k}\,(\bigvee\nolimits_{i\not=j}x_{i}&\approx x_{j}\vee \exists x_{k+1}\,\bigwedge\nolimits_{i\leq k}\lnot x_{k+1}\\&\approx x_{j}\wedge \bigwedge\nolimits_{\varphi \in F_{k+1}}\varphi _{g(\varphi)}(x_{1},\ldots, x_{k},x_{k+1})) \end{align*} $$

Then define the sentence $Ext_{A}(k,N)$ :

$$ \begin{align*} \bigwedge\nolimits_{g \colon F_{k+1}\longrightarrow A_{N}}Ext_{A}(k,N,g) \end{align*} $$

and the theory

$$ \begin{align*} Ext:=\{Ext_{A}(k,N)\}_{k,N}. \end{align*} $$

Due to the semantical interpretation of quantifiers as infima and suprema, we have that $Ext_{A}(k,N,g)^{M}\geq 1-\varepsilon $ if and only if for any distinct $i_{1},\ldots ,i_{k}$ in M there is an i in $M\setminus \{i_{1},\ldots ,i_{k}\}$ such that $||\varphi _{g(\varphi )}||^{\mathfrak {M}}(i_{1},\ldots ,i_{k},i)\geq 1-\varepsilon $ for each $\varphi \in F_{k+1}$ . More precisely, if $g(\varphi )=[\frac {j}{N},\frac {j+1}{N}],$

$$ \begin{align*} ||\varphi ||^{\mathfrak{M}}(i_{1},\ldots ,i_{k},i)\in \left[ \frac{j}{N}-\frac{\varepsilon }{m},\frac{j+1}{N}+\frac{\varepsilon }{m}\right], \end{align*} $$

for each $\varphi \in F_{k+1}$ , which is a relaxation of the intended condition

$$ \begin{align*}||\varphi ||^{\mathfrak{M}}(i_{1},\ldots ,i_{k},i_{k+1})\in \left[ \frac{j}{N},\frac{j+1}{N}\right].\end{align*} $$

In the following, we will denote by $g(\varphi )_{\pm \varepsilon }$ the displayed relaxed interval.

Proof. Fix to natural numbers k and N, a mapping $g \colon F_{k+1}\longrightarrow A_{N}$ and recall that according to the previous observation $Ext_{A}(k,g,N)^{M}<1-\varepsilon $ means that there are distinct $i_{1},\ldots ,i_{k}$ in $[n]$ such that:

for all $i\in [ n]\setminus \{i_{1},\ldots ,i_{k}\}$ , it is false that $\bigwedge \nolimits _{\varphi \in F_{k+1}}||\varphi ||^{\mathfrak {M}}(i_{1},\ldots ,i_{k},i)\in g(\varphi )_{\pm \varepsilon }$ .

Let $S(M,\varepsilon ,i_{1},\ldots ,i_{k})$ be the above classical statement for fixed $\varepsilon $ and $i_{1},\ldots ,i_{k}$ , and $\mathfrak {M}$ chosen randomly in $\mathrm {Mod}_{\tau ,n}^{[0,1]}$ . Then, since the set of events

$$ \begin{align*} \{||\varphi ||^{\mathfrak{M}}(i_{1},\ldots ,i_{k},i)\in g(\varphi )_{\pm \varepsilon }:\varphi \in F_{k+1},\text{ }i\in [ n]\setminus \{i_{1},\ldots ,i_{k}\}\}, \end{align*} $$

is independent,

$$ \begin{align*} \mu_n(S(M,\varepsilon ,i_{1},\ldots ,i_{k}))=\prod\nolimits_{i\in [ n]\setminus \{i_{1},\ldots ,i_{k}\}}(1-\prod\nolimits_{\varphi \in F_{k+1}}\mu_n (\varphi(i_{1},\ldots ,i_{k},i)\in g(\varphi )_{\pm \varepsilon })) \end{align*} $$

But $\mu _n (\varphi (i_{1},\ldots ,i_{k},i)\in g(\varphi )_{\pm \varepsilon })=p_{R}(g(\varphi )_{\pm \varepsilon })\geq p_{R}([\frac {j}{N},\frac {j+1}{N}])>0,$ for some relation symbol R and some j. Then, taking $\delta _{N}=\min \{p_{R}([\frac {j}{N},\frac {j+1}{N}]):R\in \tau $ , $0\leq j\leq N-1\},$ we have

$$ \begin{align*} \mu _{n}(S(M,\varepsilon ,i_{1},\ldots ,i_{k}))\leq (1-\delta _{N})^{n-k}, \end{align*} $$

Therefore,

$$ \begin{align*} \mu _{n}(Ext_{A}(k,g,N)<1-\varepsilon )\leq \sum\nolimits_{i_{1},\ldots ,i_{k,}}\mu _{n}(S(M,\varepsilon ,i_{1},\ldots ,i_{k}))\leq n^{k}(1-\delta _{N})^{n-k} \end{align*} $$

which clearly converges to $0,$ since $1-\delta _{N}<1$ . Therefore, $\mu _{n}(Ext_{A}(k,g,N)\geq 1-\varepsilon )$ converges to $1$ .

Recall that $Ext_{A}(k,N,g)^{\mathfrak {M}}\geq 1-\varepsilon $ iff $M\models Ext_{A}(k,N,g)_{\geq 1-\varepsilon }$ . Therefore, the previous lemma shows that any finite subset of the theory $Ext^{\ast }=\{\psi _{\geq 1-\varepsilon }:\psi \in Ext$ , $\varepsilon>0)$ is satisfiable. Hence, by satisfiability compactness of Łukasiewicz logic it has a model, which will be necessarily a model of $Ext$ . As $Ext$ is countable, this model may be chosen to be countable by the downward Löwenheim–Skolem theorem.

Considering tuples enumerating $\mathfrak {A}$ and $\mathfrak {B}$ respectively, we may define, similarly, the notion of $\varepsilon $ -isomorphism. Partial $\varepsilon $ -isomorphisms between models of $Ext$ have the extension property, more precisely.

We will utilize the following observation: $|(r\oplus s)-(r^{\prime }\oplus s^{\prime })|\leq 2\max \{|r-r^{\prime }|,|s-s^{\prime }|\}$ . For any formula # $\varphi$ , let $(\varphi )$ be the number of occurrences of $\oplus $ in $\varphi $ .

We will write $\mathfrak {A}\equiv _{\mathcal {L }^{[0,1]_{\mathrm {L}}}}\mathfrak {B}$ whenever $\varphi ^{\mathfrak {A}}=\varphi ^{\mathfrak {B}}$ for each sentence $\varphi $ .

It is easy to see that a theory T is complete iff for any sentence $\varphi $ and rational r, we have $T\models \varphi _{\geq r}$ or $T\models \varphi _{\leq r}.$

6. Further results for infinitely valued predicate logics

Although we cannot expect a complete description of the infinitely valued case, in this section we still obtain some results generalizing those in Section 4. In particular, we obtain that all rational numbers in $[0,1]$ are almost sure values of infinitely valued Łukasiewicz predicate logic, and we prove a zero-one law for a large family of logics satisfying De Morgan laws with values in closed sublattices of $[0,1]$ .

Observe that the analogue of Lemma 13 holds for infinitely valued Łukasiewicz logic with a similar proof, due to the continuity of McNaughton functions.

Proof. Notice first that m and s exist by compactness of $[0,1]$ and the continuity of $t^{\boldsymbol {A}}$ . Let $R(x)$ be an atomic formula in one variable. We show that for any open interval V of $[0,1]$ containing m:

$$ \begin{align*} \mu _{n}(\forall x_{1}\ldots \forall x_{k}\,(t(R(x_{1}),\ldots,R(x_{k})))\in V)\ \text{converges to}\ 1. \end{align*} $$

By definition of m, there is a tuple ${\langle {a_{j}}\rangle }_{j}\in |A|^{k}$ such that $t^{\boldsymbol {A}}(a_{1},\ldots ,a_{k})\in V$ and by continuity of $t^{\boldsymbol {A}}$ , there are open intervals $V_{j}$ such that $a_{j}\in V_{j}$ and $v_{j}\in V_{j}$ implies $t^{\boldsymbol {A}}(v_{1},\ldots ,v_{k})\in V.$

For each $n\geq k$ , find $K\subseteq [ n]^{k}$ of cardinal $|K|=\lfloor n/k\rfloor $ such that all tuples in K have all their components distinct, and two different tuples in K are distinct at each coordinate. Then,

$$ \begin{align*} &\quad \mu _{n}(\forall x_{1}\ldots\forall x_{k}\,(t(R(x_{1}),\ldots,R(x_{k}))\in V)\\ &\quad =\Pr (\inf\limits_{i_{1},\ldots,i_{k}\in [n]}t^{\boldsymbol{A}}(R(i_{1}),\ldots,R(i_{k}))\in V)\\ &\quad =\Pr (\text{for some } (i_{j})_{j} \text{ in } [n]^{k}:t^{\boldsymbol{A}}(R(i_{1}),\ldots,R(i_{k}))\in V) (\text{since } V \text{ is an open}\\ &\qquad\quad\text{interval containing } m)\\ &\quad \geq \Pr (\text{for some } (i_{j})_{j} \text{ in } [n]^{k}:R(i_{j})\in V_{j}, \text{ for } j=1,\ldots,k).\\ &\quad \geq \Pr (\text{for some } (i_{j})_{j} \text{ in } K:R(i_{j})\in V_{j}, \text{ for } j=1,\ldots,k)\\ &\quad =1-\Pr (\text{for all } (i_{j})_{j} \text{ in } K \text{ is false: } R(i_{j})\in V_{j}, \text{ for } j=1,\ldots,k)\\ &\quad =1-\prod\nolimits_{(i_{1}\ldots i_{k})\in K}[1-\Pr (R(i_{j})\in V_{j} \text{ for } j=1,\ldots,k)]\\ &\quad =1-(1-p_{R}(V_{1})\ldots p_{R}(V_{k}))^{\lfloor n/k\rfloor}. \end{align*} $$

For the last two identities, we use that the events $R(i_{j})\in V_{j}$ form a mutually independent set for the coordinates of the k-tupes in K. Since $p_{R}(V_{i})>0$ , the last quantity converges to $1$ when n goes to $\infty $ .

A similar argument shows that, for any open set V containing s,

$$ \begin{align*}\mu _{n}(\exists x_{1}\ldots \exists x_{k}\,(t(R(x_{1}),\ldots,R(x_{k}))\in V)\end{align*} $$

converges to $1$ .

It is easy to see that Theorem 32 generalizes to any bounded linearly ordered lattice algebra for which the additional operations are continuous with respect to the order topology because its proof only uses the continuity of the functions interpreting the terms.

Finally, notice that the analogue of Lemma 18 holds for infinite closed sublattices of $[0,1]$ with a continuous negation and actually gives a proof of the zero-one law in this case.

Theorem 37. Let $\boldsymbol {A}={\langle {A,\wedge ,\vee ,\lnot }\rangle }$ be an infinite closed sublattice of $[0,1]$ containing $0$ and $1$ with a continuous negation satisfying:

1. (1) $\lnot (v\wedge w)=\lnot v\vee \lnot w$ , $\lnot (v\vee w)=\lnot v\wedge \lnot w$

2. (2) $v\leq \lnot \lnot v$

3. (3) $\lnot 1=0$

Then, $\mathcal {L}^{\boldsymbol {A}}$ satisfies the zero-one law and its only almost sure values are $0$ , $1$ , and

$$ \begin{align*} &\varepsilon = \sup\limits_{x\in A}(x\wedge \lnot x), \varepsilon ^{\prime}=\sup\limits_{x\in A}(\lnot x\wedge \lnot \lnot x),\\ &\delta =\inf\limits_{x\in A}(x\vee \lnot x), \delta ^{\prime}=\inf\limits_{x\in A}(\lnot x\vee \lnot \lnot x). \end{align*} $$

7. Final remarks

The translation we have used in the case of finite algebras has appeared in [Reference Badia, Caicedo and Noguera1, Reference Badia, Fagin and Noguera2] and permits to transfer straightforwardly from classical to finitely valued logics most relevant model-theoretic facts and concepts: compactness, ultraproducts, elementary embeddings, type omission, etc.

Grädel et al. [Reference Grädel, Helal, Naaf, Wilke, Baier and Fisman22] consider certain semirings $\boldsymbol {A}={\langle {A,+,\cdot ,0,1}\rangle }$ as algebras of truth values in which $\vee $ and $\wedge $ are interpreted as $+$ and $\cdot $ , and the quantifiers $\exists $ and $\forall $ are interpreted as iterated $+$ and $\cdot $ , respectively. Negation is allowed at the atomic level only, and no further operations (connectives) are allowed. In this setting, they obtain zero-one laws for finite distributive lattices and “absorptive” semirings such as ${\langle {{\mathrm {L}}_{N+1},\vee ,\odot ,0,1}\rangle }$ . Our results in the present paper generalize those on lattice semirings because we do not require distributivity, and we allow arbitrary additional operations. Moreover, the treatment of negation in [Reference Grädel, Helal, Naaf, Wilke, Baier and Fisman22] can be expressed in terms of Oberschelp conditions.

Notice that the zero-one law fails for the simplest non-lattice semiring, namely the two-element field $\mathbb {F}_{2}={\langle {\{0,1\},+,\cdot ,0,1}\rangle }$ , as witnessed by the sentence $\exists x(Px\vee \lnot Px)$ whose value in universes of cardinal n is $\Sigma _{x\in [n]}1$ , which oscillates between $0$ and $1$ with the parity of n (negation is interpreted by the function $v+1$ so that $Pi\vee \lnot Pi$ takes the value $v+v+1 = 1$ for any $i \in [n]$ , and $\exists $ is interpreted by $\Sigma $ ).

It is worth noting that, in the context of continuous model theory, a zero-one law is proved in [Reference Goldbring, Hart and Kruckman20] for the theory of metric spaces, which yields a zero-one law for Łukasiewicz of pure identity under certain restrictions.

We conclude with some questions that merit further investigation:

1. (1) Does $\mathcal {L}^{[0,1]_{\mathrm {L}}}$ have almost sure irrational values?

2. (2) Does full Łukasiewicz logic over $[0,1]$ with product and its (discontinuous) residuated implication have a zero-one law?

3. (3) Does full Gödel logic over $[0,1]$ have a zero-one law?

4. (4) The set $A_{as}$ of almost sure values of a logic $\mathcal {L}^{\boldsymbol {A}}$ is an invariant of $\boldsymbol {A}$ , in fact, a subalgebra. Is there a characterization of $A_{as}$ in purely algebraic terms?