2 The propositional case: an overview

This section presents a brief summary of the key results and notions from [Reference Fagin, Riegel and Gray24]. Following that article, we take a (propositional) multi-dimensional sentence (in symbols, an MD-sentence) to be an expression of the form $\langle \sigma _1, \dots , \sigma _k; S\rangle $ where $S \subseteq [0,1]^k$ . For a fixed k, we may speak of k-dimensional sentences.

The semantics of MD-sentences is as follows. By a model, we mean an assignment ${\mathfrak {{M}}}$ from atomic sentences (propositional variables) of a propositional language $\mathcal {L}$ to truth values from $[0,1]$ . The usual real-valued logics (Łukasiewicz, Product, Gödel, etc.) all have inductive definitions indicating how to assign values to all formulas and hence the notion of the value of an arbitrary formula in the language $\mathcal {L}$ in a given model ${\mathfrak {{M}}}$ is well-defined. Fixing one such semantics (which means we will get different outcomes depending on the real-valued logic being considered), for an MD-sentence $\langle \sigma _1, \dots , \sigma _k; S\rangle $ , we say that ${\mathfrak {{M}}}$ satisfies this sentence (in symbols, ${\mathfrak {{M}}}\models \langle \sigma _1, \dots , \sigma _k; S\rangle $ ) if $\langle {s_1, \dots s_k}\rangle \in S$ where $s_i$ ( $1 \leq i \leq k$ ) is the value in ${\mathfrak {{M}}}$ of $\sigma _i$ according to the semantics of the real-valued logic under consideration. Finally, given a set $\Gamma \cup \{\gamma \}$ of MD-sentences, we write $\Gamma \vDash \gamma $ if every model that satisfies all the sentences in $\Gamma $ also satisfies $\gamma $ ; in this case, we call “ $\Gamma \vDash \gamma $ ” a valid inference.

Given these definitions, one can consider Boolean combinations of MD-sentences. For example, take $\gamma _1 := \langle \sigma ^1_1, \dots , \sigma ^1_n; S_1\rangle $ and $\gamma _2 := \langle \sigma ^2_1, \dots , \sigma ^2_m; S_2\rangle $ . Then, we may say that ${\mathfrak {{M}}}\models \gamma _1 \wedge \gamma _2$ iff ${\mathfrak {{M}}}\models \gamma _1$ and ${\mathfrak {{M}}}\models \gamma _2$ . An interesting result from [Reference Fagin, Riegel and Gray24] is that MD-sentences are closed under Boolean combinations, in the sense that for any Boolean combination of such sentences there is an MD-sentence equivalent to such combination. Hence, the collection of MD-sentences is expressively quite robust.

Now it is natural to try to build a calculus that will capture exactly the valid finitary inferences involving MD-sentences. This is what we do next.

3 The logic of a fixed domain

Throughout this section, let M be any fixed set, finite or infinite. Observe that for finite fixed domains, by means of eliminating quantifiers (turning a universal quantifier into a big conjunction and turning an existential quantifier into a big disjunction), we could use an approach that essentially reduces the problem to what was done in [Reference Fagin, Riegel and Gray24]. We work with a first-order relational vocabulary $\tau $ to simplify things (but everything we do can be adjusted to accommodate function and constant symbols).

3.1 First-order case (the logic of a fixed domain)

This part is devoted to provide an axiomatization of the logic of a fixed domain M (of any cardinality), in the sense of the valid inferences over all models with domain M.

Let us first give the basic notions for the semantics of real-valued first-order logics.

Definition 3. Given a vocabulary $\tau $ , a real-valued first-order model ${\mathfrak {{M}}}$ is a structure $\langle {M,\left \langle {R_{\mathfrak {{M}}}}\right \rangle _{R\in \tau }}\rangle $ , where $M \neq \emptyset $ is called the domain and for an n-ary predicate $R \in \tau $ , its interpretation in ${\mathfrak {{M}}}$ is a mapping $R_{\mathfrak {{M}}}\colon M^n \longrightarrow [0,1]$ .

Inductively, using the semantics of the real-valued logic in question, one can define the truth value of any formula for a sequence $\overline {a}$ of elements from M and write it as $\left \|{\varphi [\overline {a}]}\right \|_{\mathfrak {{M}}}$ :

• • $\|P[\overline {a}]\|_{{\mathfrak {{M}}}}=P_{{\mathfrak {{M}}}}(\overline {a})$ , for each $P\in Pred_{\tau }$ ;

• • $\|\circ ( \varphi _1, \ldots , \varphi _n)[\overline {a}]\|_{{\mathfrak {{M}}}}= \hat \circ (\|\varphi _1[\overline {a}]\|_{{\mathfrak {{M}}}}, \ldots , \|\varphi _n[\overline {a}]\|_{{\mathfrak {{M}}}})$ , for n-ary connective $\circ $ ;

• • $\left \|{(\forall x)\varphi [\overline {a}]}\right \|_{{\mathfrak {{M}}}}=\inf \{\|\varphi [\overline {a}, e]\|_{{\mathfrak {{M}}}}\mid e\in M\}$ ;

• • $\|(\exists x)\varphi [\overline {a}]\|_{{\mathfrak {{M}}}}=\sup \{\|\varphi [\overline {a}, e]\|_{{\mathfrak {{M}}}}\mid e\in M\}$ .Footnote ³

Whenever the vocabulary includes the equality symbol $\approx $ , its semantics is defined in the following way:

• • $\| (x\approx y) [d,e] \|_{{\mathfrak {{M}}}}= 1$ iff $d =e$ , for any $d,e \in M$ .

• • $\| (x\approx y) [d,e] \|_{{\mathfrak {{M}}}}= 0$ iff $d \neq e$ , for any $d,e \in M$ .

The definition of the truth value of a quantified formula as the infimum or the supremum of the truth values of its instances is customary in many-valued logics as a natural generalization of the semantics of quantifiers in classical logic.

A formula $\varphi (x_1, \ldots , x_n)$ can be said to be interpreted in the model ${\mathfrak {{M}}}$ by the mapping $f_\varphi \colon M^n \longrightarrow [0, 1]$ defined as $\langle {a_1, \ldots , a_n}\rangle \mapsto \left \|{\varphi [a_1, \ldots , a_n]}\right \|_{\mathfrak {{M}}}$ (we also say that $\varphi (x_1, \ldots , x_n)$ defines the mapping $f_\varphi $ in the model ${\mathfrak {{M}}}$ ).

Now we can define the set $\mathrm {MD}(M)$ of MD-sentences with domain M. Given a natural number n, we denote by $[0,1]^{M^{n}}$ the set of all functions from $M^{n}$ to $[0,1]$ . Let $\mathrm {MD}(M)$ contain all sentences of the form $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ where $\overline {x}_{\varphi _i}:= x_{i_{1}}, \ldots , x_{i_{n_i}}$ , and $S \subseteq [0,1]^{M^{n_1}} \times \ldots \times [0,1]^{M^{n_k}}$ . In the expression $\varphi _i(\overline {x})$ , the free variables of $\varphi _i$ (if any) will be exactly those in the list $\overline {x}_{\varphi _i}$ . When $\overline {x}_{\varphi _i}$ is empty, $\varphi _i$ is a sentence and what it gets assigned in a given S is simply a nullary function, in other words, an element of $[0,1]$ , as in the propositional case. If none of the formulas $\varphi _i$ in the MD-sentence $\langle {\varphi _1, \ldots , \varphi _k; S}\rangle $ contains free variables, then the situation is exactly as in the propositional case [Reference Fagin, Riegel and Gray24] and there is no need to mention in S the set M.

Example 4. Take a vocabulary $\tau $ with only two unary predicates P and U. Then, we can build the sentence $\langle {Px, (\forall x) Ux; S}\rangle $ where $S =\{\langle {f, r}\rangle \mid {r \in [0.5, 0.8)}, f$ is a mapping with domain M and range included in the set $[0,1]\}$ . Intuitively, we want this sentence to be satisfied in a model ${\mathfrak {{M}}}$ with domain M if the truth value of $(\forall x) Ux$ is a real number in the interval $[0.5, 0.8)$ and the interpretation of the predicate P is a mapping from M into $[0,1]$ .

Next, take a sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ . Then, we may write

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$

if $\langle {f_{\varphi _1}, \ldots , f_{\varphi _k}}\rangle \in S$ . Notice that, if any of the $\varphi _i$ s is a sentence, then the corresponding $f_{\varphi _i}$ is a constant function. If all the $\varphi $ s are sentences, this definition basically boils down to what appears in [Reference Fagin, Riegel and Gray24].

We introduce now a proof system associated to the domain M, called the MD-system of M, by considering the axioms and inference rules given in [Reference Fagin, Riegel and Gray24] for the propositional case and modifying only what is needed:

Axioms. We have only one axiom schema:

1. (1) $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), [0,1]^{M^{n_1}} \times \ldots \times [0,1]^{M^{n_k}}}\rangle $ for all formulas

   $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})$ .

Inference rules.

1. (2) From

   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle{\varphi_{\pi (1)}(\overline{x}_{\varphi_{\pi (1)}}), \ldots, \varphi_{\pi (k)}(\overline{x}_{\varphi_{\pi (k)}}); S'}\rangle,\end{align*} $$
   where $S' = \{\langle {f_{\pi (1)}, \ldots , f_{\pi (k)}}\rangle \mid \langle {f_{1}, \ldots , f_{k}}\rangle \in S\}$ and $\pi $ is a permutation of $1, \ldots , k$ .

1. (3) From

   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}), \varphi_{k+1}(\overline{x}_{\varphi_{k+1}}), \ldots, \varphi_m(\overline{x}_{\varphi_m}); S \times [0,1]^{M^{n_{k+1}}} \times \ldots \times [0,1]^{M^{n_{m}}}\rangle.\end{align*} $$

1. (4) From

   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S_1}\rangle,\end{align*} $$
   and
   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S_2}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S_1 \cap S_2}\rangle.\end{align*} $$

1. (5) For $0< r < k$ , from

   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_{k-r}(\overline{x}_{\varphi_{k-r}}); S'}\rangle,\end{align*} $$
   where $S'= \{\langle {f_1, \ldots , f_{k-r}}\rangle \mid \langle {f_1, \ldots , f_{k}}\rangle \in S\}$ .

1. (6) From

   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S'}\rangle,\end{align*} $$
   where $S \subseteq S'$ .

Finally, before we introduce the last rule, let us define a piece of notation. Consider an arbitrary domain M and functions $f_1, \ldots , f_j$ from some Cartesian products of M into $[0,1]$ . Then, for any j-ary connective $\circ $ from a real-valued logic, we can define the function $\circ (f_1, \ldots , f_j)$ as taking arguments componentwise as indicated by the output of the $f_i$ s ( $i \in \{1, \ldots , j\}$ ) and giving as output what $\circ $ indicates. Also, we need to generalize also the notion of good tuple. Indeed, given an MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ , we say that a tuple $\langle {f_1, \ldots , f_{k}}\rangle \in S$ is good if

1. (a) $f_m = \circ (f_{m_1}, \ldots , f_{m_j})$ whenever $\varphi _m(\overline {x}_{\varphi _m}) = \circ (\varphi _{m_1}(\overline {x}_{\varphi _{m_1}}), \ldots , \varphi _{m_j} (\overline {x}_{\varphi _{m_j}})),$

2. (b) $f_i(e_{1}, \ldots , e_{n_j})= \inf \{f_j(e_{1}, \ldots , e_{n_j}, e) \mid e \in M\}$ whenever

   $\varphi _i(\overline {x}_{\varphi _i}) = \forall y\, \varphi _j(\overline {x}_{\varphi _j}),$ for all $e_{1}, \ldots , e_{n_j} \in M^{n_j}$ ,

3. (c) $f_i(e_{1}, \ldots , e_{n_j})= \sup \{f_j(e_{1}, \ldots , e_{n_j}, e) \mid e \in M\}$ whenever

   $\varphi _i(\overline {x}_{\varphi _i}) = \exists y\, \varphi _j(\overline {x}_{\varphi _j})$ , for all $e_{1}, \ldots , e_{n_j} \in M^{n_j}$ .

1. (7) From

   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S'}\rangle,\end{align*} $$
   where $S'$ is the set of good tuples in S.

The following result establishing the soundness of the formal system is a simple exercise but it helps in building intuition on how the formalism works.

Lemma 5. The axioms and rules of the system are sound with respect to the semantics.

Proof For axiom schema (1), given a model ${\mathfrak {{M}}}$ with domain M, and formulas $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})$ , evidently, $f_{\varphi _k} \in [0, 1]^{M^{n_k}} $ by definition. Thus, the first-order MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), [0,1]^{M^{n_1}} \times \ldots \times [0,1]^{M^{n_k}}}\rangle $ holds in ${\mathfrak {{M}}}$ .

For rule (2), if ${\mathfrak {{M}}}$ is a model and $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ holds in ${\mathfrak {{M}}}$ , clearly for any permutation $\pi $ of $1, \ldots , k$ , if $S' = \{\langle {f_{\pi (1)}, \ldots , f_{\pi (k)}}\rangle \mid \langle {f_{1}, \ldots , f_{k}}\rangle \in S\}$ , we also have that $\langle {\varphi _{\pi (1)}(\overline {x}_{\varphi _{\pi (1)}}), \ldots , \varphi _{\pi (k)}(\overline {x}_{\varphi _{\pi (k)}}); S'}\rangle $ holds in ${\mathfrak {{M}}}$ .

For rule (3), if ${\mathfrak {{M}}}$ is a model of $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ , it immediately follows that $\langle {f_{\varphi _1}, \ldots , f_{\varphi _k}}\rangle \in S$ , and taking formulas $\varphi _{k+1}(\overline {x}_{\varphi _{k+1}}), \ldots , \varphi _m(\overline {x}_{\varphi _m})$ , it is also obvious that $\langle {f_{\varphi _{k+1}}, \ldots , f_{\varphi _m}}\rangle \in [0,1]^{M^{n_{k+1}}} \times \ldots \times [0,1]^{M^{n_{m}}}$ . Thus, the first-order MD-sentence

$$ \begin{align*}\langle\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}), \varphi_{k+1}(\overline{x}_{\varphi_{k+1}}), \ldots, \varphi_m(\overline{x}_{\varphi_m}); S \times [0,1]^{M^{n_{k+1}}} \times \ldots \times [0,1]^{M^{n_{m}}}\rangle\end{align*} $$

holds in ${\mathfrak {{M}}}$ .

For rule (4), if we have both

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S_1}\rangle,\end{align*} $$

and

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S_2}\rangle,\end{align*} $$

then $\langle {f_{\varphi _1}, \ldots , f_{\varphi _k}}\rangle \in S_1$ and $\langle {f_{\varphi _1}, \ldots , f_{\varphi _k}}\rangle \in S_2$ . Thus,

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S_1 \cap S_2}\rangle,\end{align*} $$

as desired.

We leave the proofs of the soundness of rules (5)–(7) to the reader. The key observation for rule (7) is that $S'$ retains only the elements of S corresponding to formulas that respect the semantics of the real-valued logic in question.

A proof of an MD-sentence $\gamma $ from a set $\Gamma $ of MD-sentences in this system consists, as usual, of a finite sequence of MD-sentences such that the last member is $\gamma $ and every element of the sequence is either an axiom, one of the member of $\Gamma $ , or it follows from previous elements by one of the inference rules. We write $\Gamma \vdash _M \gamma $ to indicate that there exists a proof of $\gamma $ from $\Gamma $ .

Before stating Lemma 8, let us introduce some terminology.

Definition 6. Given a set A of first-order formulas, we will say that A is closed under subformulas if for any formula $\varphi \in A$ , every subformula of $\varphi $ is also in A.

Definition 7. We say that an MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ is minimized, i.e., whenever $\langle {f_1, \ldots , f_k}\rangle \in S$ , there is a model of $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ such that for $1\leq i \leq k$ the interpretation of $\varphi _i(\overline {x}_{\varphi _i})$ is $f_i$ .

Lemma 8. Let $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ be the premise of Rule (7) and assume that $G=\{\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})\}$ is closed under subformulas. Then, the conclusion $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S'}\rangle $ is minimized and this is witnessed by models with domain M.

Proof Assume that $\langle {f_1, \ldots , f_k}\rangle \in S'$ . Since G is closed under subformulas, there is a subsequence of $\langle {f_1, \ldots , f_k}\rangle $ that determines interpretations on the domain M for the atomic formulas appearing in G, i.e., interpretations for the predicates of $\tau $ . But this subsequence then defines a model ${\mathfrak {{M}}}$ based on the domain M where the interpretations of $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})$ are as indicated by $\langle {f_1, \ldots , f_k}\rangle $ . This is because Rule (7) is designed to select only those sequences $\langle {f_1, \ldots , f_k}\rangle $ that respect the semantics of the underlying real-valued logic.

Remark 9. Observe that Lemma 8 does not claim that any MD-sentence has a model. It is rather telling us that if the set $\{\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})\}$ of traditional formulas in the MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ used as a premise in an application of Rule (7) is closed under subformulas, then if $S'\neq \emptyset $ , the MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S'}\rangle $ coming from Rule (7) has a model.

Remark 10. Lemma 8 plays an important role in the completeness argument in this general framework. Roughly speaking, it relies on the fact that the set $S'$ can encode a model for a series of formulas $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})$ with domain M by a sequence of interpretations to the finite list of predicates appearing in such formulas in a way that is consistent with the semantics of the underlying real-valued logic. It is not difficult to see that, for a finite vocabulary $\tau $ , we can find a set S encoding all possible models with domain M. For example, if $\tau $ is the set $\{P_1, \dots , P_k\}$ of predicates, then we can take S to be the set of all sequences $\langle {f_1, \ldots , f_k}\rangle $ of possible interpretations of the predicates from our list on the domain M.

Similarly to [Reference Fagin, Riegel and Gray24, Lemma 5.3], we obtain:

Lemma 11. The conclusion and premises of rules (2), (3), (4), and (7) are logically equivalent.

Proof The equivalence of the premise and conclusion of Rule (2) is clear. For Rules (3) and (7), the fact that the premise logically implies the conclusion follows from soundness of the rules, as does the fact that the conjunction of the premises of Rule (4) logically implies the conclusion. We now show that for Rules (3) and (7), the conclusion logically implies the premise. For Rule (3), the equivalence follows from the soundness of Rule (5). For Rule (4), the conclusion logically implies the each of the premises, and hence the conjunction of the premises, because of the soundness of Rule (4).

We will sketch the argument for Rule (7). Let ${\mathfrak {{M}}}$ be a model such that

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle.\end{align*} $$

But then the interpretations $f_1, \ldots , f_{k}$ of the formulas $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})$ in the model ${\mathfrak {{M}}}$ respect the semantics of the connectives and quantifiers according to the real-valued logic in question. Since $\langle {f_1, \ldots , f_{k}}\rangle \in S$ by hypothesis, we must have that $\langle {f_1, \ldots , f_{k}}\rangle \in S'$ where $S'$ is as in Rule (7). Hence,

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S'}\rangle,\end{align*} $$

as desired. On the other hand, if

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S'}\rangle,\end{align*} $$

given the soundness of Rule (6), it follows that

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle.\end{align*} $$

The following lemma is straightforward to show.

Lemma 12. Minimization is preserved by the rules (2) and (4), i.e., if the premises of the rules are minimized, then their conclusions are too.

Let $\Gamma \vDash _{M} \gamma $ mean that for each model ${\mathfrak {{M}}}$ with domain M, if ${\mathfrak {{M}}} \models \Gamma $ then ${\mathfrak {{M}}} \models \gamma $ . We call the relation $\vDash _{M}$ the MD-logic of M. We can now reconstruct the soundness and completeness argument from [Reference Fagin, Riegel and Gray24] and obtain the following theorem that the MD-system of M is actually an axiomatization of the MD-logic of M.

Theorem 13 (Completeness of the logic of a fixed domain).

Let $\Gamma $ be a finite set of MD-sentences and $\gamma $ an MD-sentence. Then, $\Gamma \vdash _M \gamma $ iff $\Gamma \vDash _M \gamma $ .

Proof To see that $\Gamma \vdash _M \gamma $ only if $\Gamma \vDash _{M} \gamma $ , one proceeds, as usual, by induction on the length of the proof, i.e., we start by showing that the axiom schema is sound and that the rules preserve the truth of the MD-sentences. For example, every instance of the axiom schema is sound since every formula in the usual first-order sense is interpreted by some mapping on a given model based on the domain M.

To show completeness, we follow the argument on [Reference Fagin, Riegel and Gray24, p. 12] and thus only provide a sketch. The strategy is to transform $\Gamma $ into an equivalent MD-sentence from which $\gamma $ can be deduced. We may assume without loss of generality that $\Gamma $ is non-empty, for otherwise we could replace it by an instance of Axiom (1).

Indeed, assume that we have a finite set $\Gamma = \{\gamma _1, \ldots , \gamma _n \}$ of MD-sentences in which, for each $i \in \{1, \ldots , n\}$ , $\gamma _i$ is the MD-sentence $\langle {\varphi ^i_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^i_k(\overline {x}_{\varphi _{k_i}}); S_i}\rangle $ . Suppose further that $\gamma $ is $\langle {\varphi ^0_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^0_k(\overline {x}_{\varphi _{k_0}});} $ $ {S_0}\rangle $ . Then, take the sets $\Gamma _i=\{\varphi ^i_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^i_k(\overline {x}_{\varphi _{k_i}})\}$ and $\Gamma _0=\{\varphi ^0_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^0_k(\overline {x}_{\varphi _{k_0}})\}$ . We take G to be the usual closure under subformulas of the set $\bigcup _{j\geq 0}\Gamma _j$ .

G is a finite set and then we can follow step by step the argument in [Reference Fagin, Riegel and Gray24], applying our slightly modified Rules (3) and (7). In particular, we make use of Lemma 8 instead of [Reference Fagin, Riegel and Gray24, Lemma 5.2].

For each i such that $1\leq i \leq n$ , we set $H_i= G \setminus \Gamma _i$ . Let $r_i$ be the cardinality of $H_i$ and suppose that $H_i =\{\theta _1(\overline {x}_{\theta _1}), \ldots , \theta _{r_i}(\overline {x}_{\theta _{r_i}})\}$ . Then, by applying Rule (3), we can deduce the MD-sentence

$$ \begin{align*}\langle\varphi^i_1(\overline{x}_{\varphi_1}), \ldots, \varphi^i_k(\overline{x}_{\varphi_{k_i}}), \varphi_{k+1}(\overline{x}_{\varphi_{k+1}}), \ldots, \varphi_m(\overline{x}_{\varphi_m}); S \times [0,1]^{M^{n_{k+1}}} \times \ldots \times [0,1]^{M^{n_{m}}}\rangle,\end{align*} $$

from $\langle {\varphi ^i_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^i_k(\overline {x}_{\varphi _{k_i}}); S}\rangle $ , i.e., $\gamma _i$ , where the sequence $ \varphi _{k+1}(\overline {x}_{\varphi _{k+1}}), \ldots , \varphi _m(\overline {x}_{\varphi _m})$ is $\theta _1(\overline {x}_{\theta _1}), \ldots , \theta _{r_i}(\overline {x}_{\theta _{r_i}})$ . Now let $\psi _i$ be the MD-sentence that results from applying Rule (7) to the conclusion of Rule (3) displayed above.

Let $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _p(\overline {x}_{\varphi _{p}})$ be some ordering of the formulas in G; then, since the set of first-order formulas that appear in $\psi _i$ is exactly G, we may use Rule (2) to turn $\psi _i$ into an equivalent MD-sentence of the form $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _p(\overline {x}_{\varphi _{p}}); T_i}\rangle $ , which we may denote by $\chi _i$ . Furthermore, since in deriving $\chi _i$ , we only appealed to rules (2), (3), and (7), by Lemma 11, this MD-sentence is logically equivalent to $\gamma _i$ .

Assume that $T= T_1 \cap \ldots \cap T_n$ and define $\chi :=\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _p(\overline {x}_{\varphi _{p}}); T}\rangle $ . From Lemma 8, each $\psi _i$ is minimized since it comes from Rule (7) and

$$ \begin{align*}\{\varphi^i_1(\overline{x}_{\varphi_1}), \ldots, \varphi^i_k(\overline{x}_{\varphi_{k_i}}), \varphi_{k+1}(\overline{x}_{\varphi_{k+1}}), \ldots, \varphi_m(\overline{x}_{\varphi_m})\}\end{align*} $$

is closed under subformulas. Moreover, by Lemma 12, each $\chi _i$ is minimized and, hence, $\chi $ is minimized.

The MD-sentence $\chi $ can be derived from the MD-sentences $\chi _i$ by repeated applications of Rule (4). In fact, by Lemma 11, $\chi $ and $\{\chi _1, \ldots , \chi _n\}$ have the same logical consequences, and since $\chi _i$ is equivalent to $\gamma _i$ , we have that $\{\chi _1, \ldots , \chi _n\}$ and $\{\gamma _1, \ldots , \gamma _n\} = \Gamma $ have the same logical consequences. Hence, $\chi \vDash \gamma $ given that $\Gamma \vDash \gamma $ by hypothesis. Furthermore, in order to show that $\Gamma \vdash \gamma $ we simply need to show that $\chi \vdash \gamma $ since $\Gamma \vdash \chi $ by the above reasoning.

Recall that $\gamma $ is $\langle {\varphi ^0_1(\overline {x}_{\varphi _1} ), \ldots , \varphi ^0_k(\overline {x}_{\varphi _{k_0}} ); S_0}\rangle $ and $\chi $ is $\langle \varphi _1(\overline {x}_{\varphi _1} ), \ldots , \varphi _p(\overline {x}_{\varphi _{p}} ); T\rangle $ , so by applying Rule (2) we can rearrange the order of the formulas $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _p(\overline {x}_{\varphi _{p}})$ so they start with $\varphi ^0_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^0_k(\overline {x}_{\varphi _{k_0}})$ and infer from $\chi $ the MD-sentence $\chi ':=\langle {\varphi ^0_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^0_k(\overline {x}_{\varphi _{k_0}})\ldots; T'}\rangle $ . Using Lemma 11, we may see that $\chi $ and $\chi '$ are logically equivalent. Hence, $\chi ' \vDash \gamma $ since $\chi \vDash \gamma $ . Given that $\chi $ is minimized, it follows that $\chi '$ is too by Lemma 12. Using Rule (5), from $\chi '$ we may infer an MD-sentence $\chi "$ of the form $\langle {\varphi ^0_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^0_k(\overline {x}_{\varphi _{k_0}}); T"}\rangle $ .

The final step in the proof is to show that $T" \subseteq S_0$ (which uses minimization in a fundamental manner) for then we can use Rule (6) to infer $\gamma $ from $\chi "$ , and hence we would have $\chi \vdash \chi '\vdash \chi " \vdash \gamma $ , which means that $\chi \vdash \gamma $ as desired.

Assume now that $\langle {f_1, \ldots , f_k}\rangle \in T"$ to show that $\langle {f_1, \ldots , f_{k_0}}\rangle \in S_0$ . By definition of $T"$ , there is a $\langle {f_1, \ldots , f_{k_0}, \ldots , f_p}\rangle \in T'$ . Given that $\chi '$ is minimized, there is a model ${\mathfrak {{M}}}$ of $\chi '$ such that the interpretations of the formulas $\varphi ^0_1(\overline {x}_{\varphi _1}), \ldots , \varphi ^0_k(\overline {x}_{\varphi _{k_0}})$ are $f_1, \ldots , f_{k_0}$ , respectively. Since $\chi ' \vDash \gamma $ , then ${\mathfrak {{M}}}\models \gamma $ , and so $\langle {f_1, \ldots , f_{k_0}}\rangle \in S_0$ .

There are some subtle points to consider around what we have done, which we will discuss in the next remarks. It is important to stress that we have axiomatized the logic of all models based on the set M, not the logic of one particular model ${\mathfrak {{M}}}$ based on M.

Remark 14. Let us look at the case of two-valued logic with equality (i.e., the classical first-order logic which, of course, is covered by our approach). Let M be a finite set (say of size n). Now, enumerate all the first-order validities of the form $(|M|=n) \rightarrow \varphi $ where $\varphi $ is any first-order formula and $|M|=n$ is the first-order formula saying that the size of the domain M is exactly n. In the case of finite domains, one might modify the approach here by allowing only MD-sentences that are interval-based (in the sense of [Reference Fagin, Riegel and Gray24], that is, where the sets of truth values involved in S are unions of finitely-many rational intervals) or that come from such sentences by an application of Rule (7), making the set $\mathrm {MD}(M)$ countable, and then it is possible to show by essentially the argument in [Reference Fagin, Riegel and Gray24, Theorem 6.1] that validity is not only recursively enumerable but decidable on such domains for Łukasiewicz and Gödel logic.

Remark 15. Recall that satisfiability on countably infinite models is not recursively enumerable in two-valued first-order logic. Now take a first-order sentence $\varphi $ and let $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), \varphi $ be the list of all its subformulas. Fixing a countably infinite domain M, we may consider now the MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), \varphi; S}\rangle $ (call it $\psi $ ) where $S:= \{0,1\}^{M^{n_1}}\times \ldots \times \{0,1\}^{M^{n_k}} \times \{1\}$ . Take now the MD-sentence obtained by applying our Rule (7) to this sentence, $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), \varphi; S'}\rangle $ (call it $\psi '$ ). Observe that $\psi $ and $\psi '$ are equivalent. Furthermore, $\varphi $ has a countably infinite model iff $\psi $ is satisfiable iff $\psi '$ is satisfiable. Finally, by minimization and the semantics of MD-sentences, $\psi '$ is satisfiable iff $S'$ is non-empty. Hence, the problem of whether an arbitrary $S'$ is non-empty is not recursively enumerable.

Rule (7) implies that our formal system is not finitistic in the sense of metamathematics [Reference Kleene30] since when infinite domains are involved it cannot all be formalizable in arithmetic, it goes into the realm of infinitary mathematics. In this sense, it is akin to an infinitary proof system (although it does not involve infinitary formulas in the usual sense). Thus, the system we have presented here is by necessity less “usable” in practice than a finitary one but not than an infinitary one.

3.2 Propositional modal logic (of a fixed frame)

Expansions of propositional many-valued logics with modalities are a topic of lively research (see, e.g., [Reference Badia, Caicedo and Noguera3, Reference Bou, Esteva, Godo and Rodríguez10, Reference Caicedo, Metcalfe, Rodríguez and Rogger11, Reference Cintula, Menchón and Noguera14, Reference Fitting25, Reference Fitting26, Reference Vidal46] due to their richer expressive power that makes them more amenable for a variety of applications, as compared to purely propositional logics. Thus, it is natural to extend them to the setting of multi-dimensional sentences too.

For this subsection, fix a frame $\mathfrak {F}:= \langle {M, R}\rangle $ where $R \subseteq M^2$ is a binary relation on a non-empty set M (finite or infinite, where we may call the elements M worlds).Footnote ⁴ Consider now a vocabulary $\tau $ consisting only of propositional variables as in modal logic and a base modal language with $\Box $ and $\Diamond $ (unlike classical logic, many-valued logics do not allow in general to define these two operators from one another). Now the set $\mathrm {MD}(M)$ of MD-sentences contains all the expressions of the form $\langle {\varphi _1, \ldots , \varphi _k; S}\rangle $ where each $\varphi _i$ is a modal formula and $S \subseteq ([0,1]^{M})^k$ .

For each real-valued model ${\mathfrak {{M}}}$ -based on $\mathfrak {F}= \langle {M, R}\rangle $ , i.e., a structure where each propositional variable $p \in \tau $ is interpreted as a mapping $p_{{\mathfrak {{M}}}}\colon M\longrightarrow [0,1]$ , we can define a notion of truth value at a world $w \in M$ :

• • $\|p[w]\|_{{\mathfrak {{M}}}}=p_{{\mathfrak {{M}}}}(w)$ , for each $p\in \tau $ ;

• • $\|\circ ( \varphi _0, \ldots , \varphi _n)[w]\|_{{\mathfrak {{M}}}}=$

• $\circ (\|\varphi _0[w]\|_{{\mathfrak {{M}}}}, \ldots , \|\varphi _n[w]\|_{{\mathfrak {{M}}}})$ , for n-ary connective $\circ $ ;

• • $\left \|{ \Box \varphi [w]}\right \|_{{\mathfrak {{M}}}}=\inf \{\|\varphi [v]\|_{{\mathfrak {{M}}}}\mid v\in M, \langle {w, v}\rangle \in R\}$ ;

• • $\left \|{ \Diamond \varphi [w]}\right \|_{{\mathfrak {{M}}}}=\sup \{\|\varphi [v]\|_{{\mathfrak {{M}}}}\mid v\in M, \langle {w, v}\rangle \in R\}$ .

Every formula $\varphi $ can be said to be interpreted in the model ${\mathfrak {{M}}}$ by the mapping $f_\varphi \colon M \longrightarrow [0, 1]$ defined as $w \mapsto \left \|{\varphi [w]}\right \|_{\mathfrak {{M}}}$ (we also say that $\varphi $ defines the mapping $f_\varphi $ in the model ${\mathfrak {{M}}}$ ). Given an MD-sentence $\langle {\varphi _1, \ldots , \varphi _k; S}\rangle $ , we write

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1, \ldots, \varphi_k; S}\rangle,\end{align*} $$

if the formulas $\varphi _1, \ldots , \varphi _k$ , respectively, define mappings $f_1, \ldots , f_k$ in the model ${\mathfrak {{M}}}$ and $\langle {f_1, \ldots , f_k}\rangle \in S$ .

As with the first-order case, from the axioms and inference rules from [Reference Fagin, Riegel and Gray24] we need to modify only the following:

Axioms.

1. (1) $\langle {\varphi _1, \ldots , \varphi _k; [0,1]^{M} \times \ldots \times [0,1]^{M}}\rangle $ for any formulas $\varphi _1, \ldots , \varphi _k$ .

Inference rules.

1. (3) From

   $$ \begin{align*}\langle{\varphi_1, \ldots, \varphi_k; S}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle \varphi_1, \ldots, \varphi_k, \varphi_{k+1}, \ldots, \varphi_m; S \times [0,1]^{M}\times \ldots \times [0,1]^{M}\rangle,\end{align*} $$

and we also need to modify the notion of good tuple for Rule (7). Indeed, given an MD-sentence $\langle {\varphi _1, \ldots , \varphi _k; S}\rangle $ , now we say that a tuple $\langle {f_1, \ldots , f_{k}}\rangle \in S$ is good if

1. (a) $f_m = \circ (f_{m_1}, \ldots , f_{m_j})$ whenever $\varphi _m = \circ (\varphi _{m_1}, \ldots , \varphi _{m_j})$ ,

2. (b) $f_i(w)= \inf \{f_j( e) \mid e \in M, \langle {w,e}\rangle \in R\}$ whenever $\varphi _i = \Box \varphi _j,$ for all $w \in M$ ,

3. (c) $f_i(w)= \sup \{f_j( e) \mid e \in M, \langle {w,e}\rangle \in R\}$ whenever $\varphi _i = \Diamond \varphi _j,$ for all $w \in M$ .

As before, we get the following (since the interpretations of the propositional variables in $\tau $ is what determines a model over $\mathfrak {F}$ ):

Lemma 16. Let $\langle {\varphi _1, \ldots , \varphi _k; S}\rangle $ be the premise of Rule (7) and assume that $G=\{\varphi _1, \ldots , \varphi _k\}$ is closed under subformulas in the usual sense. Then, the conclusion $\langle {\varphi _1, \ldots , \varphi _k; S'}\rangle $ is minimized.

Once more, closely following the argument from [Reference Fagin, Riegel and Gray24], we may show that:

Theorem 17 Completeness of the logic of a fixed frame

For $\Gamma $ a finite set of MD-sentences and $\gamma $ an MD-sentence, $\Gamma \vdash _{\mathfrak {F}} \gamma $ iff $\Gamma \vDash _{\mathfrak {F}} \gamma $ .

The proofs of Lemma 16 and Theorem 17 are very similar (modulo some trivial modifications) to those of Lemma 8 and Theorem 13, respectively, and thus we omit them. One might think of modal formulas as first-order formulas in one variable, and then it is easy to see how the same arguments work.

Remark 18. An interesting topic of research would be to extend this multidimensional approach to many-valued first-order modal logics. This can be done for a fixed frame and a fixed domain.

4 Axiomatizations of prominent first-order (and propositional modal) real-valued logics

Recall that, in the context of classical first-order logic, by the Löwenheim–Skolem theorem, the first-order sentences which are true in all countably infinite models coincide with the sentences that are true in all infinite models. For if $\varphi $ is true in all countably infinite models, then $\neg \varphi $ cannot have any infinite model since otherwise $\neg \varphi $ would have a countably infinite model by the Löwenheim–Skolem theorem. Moreover, the class of infinite models is axiomatizable in first-order logic: consider the theory formed by the sentences “there are at least n elements” for all natural numbers $n>0$ . Hence, the first-order sentences which are true in all infinite models are recursively enumerable.

Let us analyze now what happens in the real-valued case. In this section, we will consider only the case of languages without equality. This is a very standard practice in mathematical fuzzy logic (e.g. [Reference Baaz, Hájek, Krajíček and Švejda1, Reference Belluce and Chang7, Reference Hay29, Reference Montagna, Aguzzoli, Ciabattoni, Gerla, Manara and Marra34, Reference Scarpellini43, Reference Takeuti and Titani44]). It is well-known that neither Łukasiewicz nor Product first-order logic have a recursively enumerable set of validities with the semantics given on $[0,1]$ (see [Reference Scarpellini43] and [Reference Baaz, Hájek, Krajíček and Švejda1], respectively). In contrast, Gödel first-order logic is recursively axiomatizable [Reference Takeuti and Titani44], and both Łukasiewicz and Product logics can be axiomatized by the addition of an infinitary rule (see [Reference Belluce and Chang7, Reference Hay29] and [Reference Montagna, Aguzzoli, Ciabattoni, Gerla, Manara and Marra34], respectively).

From this proposition and Theorem 13, we immediately obtain that consequence from finite sets of premises in Łukasiewicz, Product, and Gödel first-order real-valued logic (without equality) is complete with respect to the MD-system of a countable domain:

Observe that Corollary 20 would fail in the presence of equality in the vocabulary. This is because general validity cannot be reduced to truth in any particular infinite (even if only countable) model. The reason is that, if $\psi $ is the first-order sentence expressing that the size of the domain is 3 then $\neg \psi $ would hold in every infinite domain M, whereas this cannot be a valid sentence in any of the logics we are considering here since $\psi $ holds in models with universes of size 3. Thus, we would have that $\not \vDash _{\mathcal {L}} \neg \psi $ but $\vdash _M \neg \psi $ .

The purpose of any completeness theorem is to obtain the equivalence between a universal statement (about validity) and an existential statement (about the existence of a proof). The claim of existence of a proof is a $\Sigma _1$ claim on the natural numbers when the proof system is arithmetizable. By Corollary 20 and since neither Łukasiewicz nor Product first-order logic has a recursively enumerable set of validities, our proof systems are not arithmetizable when the domain is infinite.

Part of the interest of the present approach is the uniformity it provides in axiomatizing the previously mentioned logics (which were known to be axiomatizable by other infinitary methods). We are essentially giving one recipe to deal with all cases. Moreover, none of our rules are explicitly infinitary and the infinitary component of our formulas is hidden in the sets S.

Finally, in general, we are clearly axiomatizing more levels of formal reasoning than it could be done before, for preservation of value $1$ is a mere fraction of the possibilities that the present system actually handles. The system axiomatizes genuine real-valued reasoning in all of Gödel, Łukasiewicz, and Product first-order (and modal) logics.

5 A zero-one law for MD-logics

Beginning with [Reference Erdős20] in the context of graph theory, a natural question that one can consider in general is: what is the probability that a structure satisfies P when randomly selected among finite structures with the same domain for a suitable probability measure? Or, more interestingly, what do these probabilities converge to (if anything) as the size of the domain of the structures grows to infinite? Well-known and highly celebrated results show that when the properties under consideration are expressible by formulas of a certain logic the probabilities converge to either $0$ or $1$ (and so we say that the formula is either almost surely false or almost surely true, respectively). After an early result for monadic predicate logic [Reference Carnap12], the topic of logical zero-one laws was properly started independently in the papers by Glebskiĭ et al [Reference Glebskii, Kogan, Liogon’kii and Talanov27] and Fagin [Reference Fagin23] for first-order classical logic on finite purely relational vocabularies.

In this section, we want to establish a zero-one law for certain MD-logics, namely those based on suitable finite subalgebras of $[0,1]$ (of the form $\langle {A,\wedge ^{\boldsymbol A},\vee ^{\boldsymbol A},\mathbin {\&}^{\boldsymbol A},\to ^{\boldsymbol A},\overline {0}^{\boldsymbol A},\overline {1}^{\boldsymbol A}}\rangle $ ). For example, both Gödel and Łukasiewicz logic have multiple finitely-valued versions (though Product logic does not), and we will list some examples below. This restriction to the finite setting is because we wish to have, when our vocabularies are relational and finite, only a finite number of possible models on a given finite domain, in analogy to what happens in classical logic in [Reference Fagin23] (or in the finitely-valued case already considered in [Reference Badia and Noguera5]). Regarding infinitely-valued logics, the recent paper [Reference Badia, Caicedo and Noguera4] contains a zero-one law for infinitely-valued Łukasiewicz logic and related systems.

Example 24 The algebra of Łukasiewicz 3-valued logic

The algebra

such that

• •

• •

• •

• •

More generally, we may consider any Łukasiewicz n-valued logic by using the algebra on the carrier set $\{0,{\frac {1}{n-1}}, {\frac {2}{n-1}}, \dots , {\frac {n-2}{n-1}}, 1\}$ and with the same definitions of operations.

Example 25 The algebra of Gödel 4-valued logic

The algebra

$$ \begin{align*} \text{G}_4=\langle{\{0, {\frac{1}{3}}, {\frac{2}{3}}, 1\},\wedge^{\text{G}_4},\vee^{\text{G}_4},\mathbin{\&}^{\text{G}_4},\to^{ \text{G}_4}, 0,1}\rangle, \end{align*} $$

such that

• • ${\wedge ^{\text {G}_4}}(x, y) = \mathbin {\&}^{ \text {G}_4}(x, y) = \min \{x,y\}$

• • ${\vee ^{\text {G}_4}}(x,y) = \max \{x,y\}$

• • and for $ \to ^{\text {G}_4}$ :

  $$ \begin{align*} \to^{\text{G}_4}(x, y) = \begin{cases} 1 &\text{if} \ x \leq \ y\\ y & \text{otherwise}. \end{cases} \end{align*} $$

As in the previous example, we may also consider any Gödel n-valued logic by using the algebra $\text {G}_n$ on the carrier set $\{0,\frac {1}{n-1}, \frac {2}{n-1}, \dots , \frac {n-2}{n-1}, 1\}$ and with the same definitions of operations.

Let us now recall some facts from classical finite model theory. Consider a purely relational vocabulary. A sentence is said to be parametric in the sense of Oberschelp in [Reference Oberschelp and Jungnickel40, p. 277] if it is a conjunction of sentences of the form

$$ \begin{align*}\forall x_1, \ldots, x_k (\neq(x_1, \ldots, x_k) \rightarrow \phi(x_1, \ldots, x_k )),\end{align*} $$

where $\neq (x_1, \ldots , x_k)$ is the conjunction of negated equalities expressing that $x_1, \ldots , x_k$ are pairwise distinct, and $\phi (x_1, \ldots , x_k )$ is a quantifier-free formula where in all of its atomic subformulas $Rx_{i_1} \ldots x_{i_k}$ we have that

$$ \begin{align*}\{x_{i_1}, \ldots, x_{i_k}\} = \{x_1, \ldots, x_k \}.\end{align*} $$

Moreover, for $k=1$ , any formula $\forall x_1 \phi (x_1)$ , where $\phi $ is a quantifier-free formula, is parametric. For example,

$$ \begin{align*}\forall x \neg Rxx \land \forall x \forall y (x \neq y \to (Rxy \to Ryx)),\end{align*} $$

is a parametric sentence, whereas

$$ \begin{align*}\forall x\forall y\forall z (\neq(x,y,z) \to (Rxy \land Ryz \to Rxz)),\end{align*} $$

is not.

Oberschelp’s extension [Reference Oberschelp and Jungnickel40, Theorem 3] of Fagin’s zero-one law [Reference Fagin23] says: on finite models and finite purely relational vocabularies, for any class K definable by a parametric sentence, any first-order sentence $\varphi $ will be almost surely true in members of K or almost surely false. By “almost surely true” here we mean that the limit as n goes to $\infty $ of the fraction of structures in K with domain $\{1, \dots , n\}$ that satisfy a given sentence $\varphi $ is $1$ (and “almost surely false” is defined analogously). Naturally, these fractions are well defined because there is only a finite number of possible structures on finite vocabulary on the domain $\{1, \dots , n\}$ . As we mentioned earlier, this fact is what motivates our restriction to finitely valued logics in this section. A very accessible presentation of Oberschelp’s result is [Reference Ebbinghaus and Flum19, Theorem 4.2.3].

An appropriate translation for our purposes from finitely-valued first-order logics into classical first-order logic is introduced in [Reference Badia, Caicedo and Noguera3]. Namely, for any sentence $\phi $ of a first-order logic based on a finite set $A \subseteq [0,1]$ of truth values and element $a \in A$ , we have a first-order sentence $T^a(\phi )$ such that, for a certain theory $\Sigma $ (which can be written as a parametric sentence in the sense of Oberschelp [Reference Oberschelp and Jungnickel40]), $T^a(\phi )$ is satisfied by a classical first-order model ${\mathfrak {{M}}}$ model of $\Sigma $ iff there is a corresponding first-order real-valued model ${\mathfrak {{M}}}^*$ where $\phi $ takes value exactly a.

The idea is that, starting with a relational vocabulary $\tau $ containing countably many predicate symbols $P^n_1, P^n_2, P^n_3, \ldots $ for each arity n, we can introduce a vocabulary $\tau ^*$ containing predicate symbols $P_i^{na}$ for each $a \in A$ and each n (the intuition here is that $P_i^{na}$ will hold of those objects for which $P_i^{n}$ takes truth value a in a given model), and the following translation from [Reference Badia, Caicedo and Noguera3] (where $\circ \in \{\vee , \wedge , \&, \rightarrow \}$ ):

Observe how the translations of quantified formulas exactly describe the semantics of quantifiers in these finitely-valued logics (i.e., existential as maximum of the truth values of instances of the formula and, dually, universal as minimum). We use classical disjunctions to run over all the possible choices of values $b_1, \ldots , b_k \in A$ that would give value a as their maximum (resp. minimum) and then write the conjunction of the necessary conditions that make sure that these $b_i$ ’s are indeed values of instances of $\psi $ and any other instance would give a value smaller (resp. bigger) than a.

Next, we define the theory $\Sigma $ given by:

$$\begin{align*}&\forall x_1, \ldots, x_n (\bigvee_{\substack{a \in A }}P_i^{na}x_1 \ldots x_n),\\&\forall x_1, \ldots, x_n (\neg (P_i^{na}x_1 \ldots x_n\land P_i^{nb}x_1 \ldots x_n)), \\&\text{for } a, b \in A, a \neq b, P_i^{n} \in \tau. \end{align*}$$

For any A-valued model ${\mathfrak {{M}}}$ for the vocabulary $\tau $ , we can introduce a classical model ${\mathfrak {{M}}}^*$ for the vocabulary $\tau ^*$ such that for any $a \in A$ , the value of $\phi $ in ${\mathfrak {{M}}}$ is a iff ${\mathfrak {{M}}}^*\models T^a(\phi )$ . ${\mathfrak {{M}}}^*$ is built by taking the same domain, M, as in ${\mathfrak {{M}}}$ and letting the interpretation of $P_i^{na}$ be the set of all elements from $M^n$ such that the interpretation of $P_i^{n}$ in ${\mathfrak {{M}}}$ assigns them value a. Observe that ${\mathfrak {{M}}}^*$ is a model of the theory $\Sigma $ . By a similar process, from any model ${\mathfrak {{N}}}$ of $\Sigma $ , we can extract an A-valued model ${\mathfrak {{M}}}$ such that ${\mathfrak {{N}}} = {\mathfrak {{M}}}^*$ .

Rewriting the theory $\Sigma $ with some care, one can turn it into a parametric sentence when $\tau $ is finite. For example, suppose that $\tau $ contains only a binary predicate R. Then, $\Sigma $ would have the form (for $a, b \in A, a \neq b$ ):

$$\begin{align*}\forall x_1 \forall x_2 (\bigvee_{\substack{a \in A }}R^{a}x_1 x_2), \end{align*}$$

$$\begin{align*}\forall x_1 \forall x_2 (\neg (R^{a}x_1 x_2 \land R^{b}x_1 x_2)). \,\,\,\,\,\, \,\,\,\,\,\, \end{align*}$$

This can be put into parametric form by considering instead (for $a, b \in A, a \neq b$ ):

$$\begin{align*}\forall x_1 (\bigvee_{\substack{a \in A }}R^{a}x_1 x_1), \end{align*}$$

$$\begin{align*}\forall x_1 \forall x_2 (x_1\neq x_2 \rightarrow \bigvee_{\substack{a \in A }}R^{a}x_1 x_2), \end{align*}$$

$$\begin{align*}\forall x_1 ( \neg (R^{a}x_1 x_1 \land R^{b}x_1 x_1)), \,\,\,\,\,\, \,\,\,\,\,\, \end{align*}$$

$$\begin{align*}\forall x_1 \forall x_2 (x_1\neq x_2 \rightarrow \neg (R^{a}x_1 x_2 \land R^{b}x_1 x_2)). \,\,\,\,\,\, \,\,\,\,\,\, \end{align*}$$

6 The logic of all domains

In this section, we will be using the same notion of model as in Definition 3 and we will allow the presence of equality in the vocabulary. Now, for any given domain M, let us denote by $\mathcal {L}_{\mathrm {MD}}(M)$ the finitary part of $\vDash _{M}$ , that is, the set of all pairs $\langle {\Gamma , \theta }\rangle $ where $\Gamma $ is a finite set of MD-sentences, $\theta $ is an MD-sentence, and every model over M of $\Gamma $ is a model of $\theta $ . In this section, we intend to take the next natural step and axiomatize the finitary part of the MD-logic of all domains, i.e., the logic $\bigcap _{M \, \text {a domain}} \mathcal {L}_{\mathrm {MD}}(M)$ . Let us denote this consequence relation simply as $\vDash $ .

What kinds of inferences can appear in $\bigcap _{M \, \text {a domain}} \mathcal {L}_{\mathrm {MD}}(M)$ ? Clearly, only those not mentioning any of the domains M, since otherwise the inference could be rather specific to a particular M. For example, an MD sentence where a domain $M'\neq M$ is mentioned in the set S does not make sense in models based on the domain M, or rather it is always false. Thus, we set the goal of axiomatizing all the valid inferences $\Gamma \vDash \theta $ where $\Gamma \cup \{\theta \}$ is a finite set of MD-sentences of the form $\langle {\varphi _1, \ldots , \varphi _k; S}\rangle $ with each $\varphi _i$ being sentences in the usual sense of a first-order predicate language and, hence, S is simply a set of suitable tuples of truth values (thus without a mention of any domain).

Focusing on logical entailments between this kind of MD-sentences, we can restrict attention (without loss of generality) to the models based in the following countable list of domains (let us call these the legal domains):

1. (i) the infinite domain of natural numbers $\{1, 2, \ldots \}$ ,

2. (ii) for each natural number n, a domain $D_n$ of size n (making sure that they are pairwise disjoint and also disjoint from $\{1, 2, \ldots \}$ ).

This is because we have the following:

Consequently, if we denote the finitary part of the consequence relation over legal domains by $\vDash ^{\text {legal}}$ , using Proposition 30, we can see that $\Gamma \vDash ^{\text {legal}} \theta $ iff $\Gamma \vDash \theta $ (where $\Gamma \cup \{\theta \}$ is a finite set of MD-sentences of the form $\langle {\varphi _1, \ldots , \varphi _k; S}\rangle $ with each $\varphi _i$ being sentences in the usual sense of a first-order predicate language). This means that we can focus on axiomatizing $\vDash ^{\text {legal}}$ for the class of MD-sentences that we have described in Proposition 30 (even though proofs may involve manipulating all kinds of MD-sentences, like those we will introduce in the next paragraph). Therefore, in what follows, we will restrict ourselves to consider legal models, i.e., those based on a legal domain.

The idea is to assume MD-sentences to have the form $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , }$ ${\varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ where each $\varphi _i$ is a first-order formula whose free variables are $\overline {x}_{\varphi _i}= x_{i_{1}}, \ldots , x_{i_{n_i}}$ (for some $n_i \geq 0$ ), and $S \subseteq [0,1]^{\bigcup _{M \ \text {is legal}} M^{n_1}} \times \ldots \times [0,1]^{\bigcup _{M \ \text {is legal}} M^{n_k}}\kern-1.2pt$ .

Example 31. Take a vocabulary $\tau $ with one binary predicate R. Then, we can build the MD-sentence $\langle {Rxy, \forall x \forall y (Rxy \rightarrow Ryx); S}\rangle $ where

$$ \begin{align*}S =\{\langle{f, 0.5}\rangle \mid f \colon \bigcup_{M \ \text{is legal}} M^2 \longrightarrow [0, 1]\}.\end{align*} $$

We want this sentence to be satisfied in a legal model ${\mathfrak {{M}}}$ with domain M if the truth value of $\forall x \forall y (Rxy \rightarrow Ryx)$ is $0.5$ and, furthermore, the interpretation of R in the model ${\mathfrak {{M}}}$ is the restriction to M of one of the functions f described in the definition of S (which in this case, happens trivially).

As expected, we may then write

$$ \begin{align*}{\mathfrak{{M}}} \models \langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$

if the formulas $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k})$ , respectively, define functions $f_1, \ldots , f_k$ on the domain M such that there are $\langle {f_1', \ldots , f_k'}\rangle \in S$ for which $f_1, \ldots , f_k$ are the respective restrictions to the domain M.

We transform Axiom (1) into (1) $^*$ :

$$ \begin{align*}\langle\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}), [0,1]^{\bigcup_{M \ \text{is legal}} M^{n_1}} \times \ldots \times [0,1]^{\bigcup_{M \ \text{is legal}} M^{n_k}}\rangle,\end{align*} $$

for all formulas $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}\kern-1pt)$ .

Rules ( $2$ ), ( $4$ ), ( $5$ ), and ( $6$ ) from the original system are modified analogously into ( $2$ ) $^*$ , ( $4$ ) $^*$ , ( $5$ ) $^*$ , and ( $6$ ) $^*$ . Rule (3) needs to be modified as:

1. (3)^* From

   $$ \begin{align*}\langle{\varphi_1(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}); S}\rangle,\end{align*} $$
   infer
   $$ \begin{align*}\langle\varphi_1&(\overline{x}_{\varphi_1}), \ldots, \varphi_k(\overline{x}_{\varphi_k}), \varphi_{k+1}(\overline{x}_{\varphi_{k+1}}), \ldots, \varphi_m(\overline{x}_{\varphi_m}); S \times\\&\quad[0,1]^{\bigcup_{M \, \text{is legal}} M^{n_{k+1}}} \times \ldots \times [0,1]^{\bigcup_{M \, \text{is legal}} M^{n_{m}}}\rangle. \end{align*} $$

Finally, Rule (7) is modified into Rule (7) $^*$ by changing the notion of good tuple. Indeed, given an MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ , we will say that a tuple $\langle {f_1, \ldots , f_{k}}\rangle \in S$ is good if for some legal domain M

1. (a) $f_m\restriction M = \circ ((f_{m_1}\restriction M), \ldots , (f_{m_j}\restriction M))$ whenever

   $\varphi _m(\overline {x}_{\varphi _m}) = \circ (\varphi _{m_1}(\overline {x}_{\varphi _{m_1}}), \ldots , \varphi _{m_j}(\overline {x}_{\varphi _{m_j}})),$

2. (b) $(f_i\restriction M)(e_{1}, \ldots , e_{n_j})= \inf \{(f_j\restriction M)(e_{1}, \ldots , e_{n_j}, e) \mid e \in M\}$ whenever $\varphi _i(\overline {x}_{\varphi _i}) = \forall y\, \varphi _j(\overline {x}_{\varphi _j}),$ for all $e_{1}, \ldots , e_{n_j} \in M^{n_j}$ ,

3. (c) $(f_i\restriction M)(e_{1}, \ldots , e_{n_j})= \sup \{(f_j\restriction M)(e_{1}, \ldots , e_{n_j}, e) \mid e \in M\}$ whenever $\varphi _i(\overline {x}_{\varphi _i}) = \exists y\, \varphi _j(\overline {x}_{\varphi _j})$ , for all $e_{1}, \ldots , e_{n_j} \in M^{n_j}$ .

Rule (7) $^*$ is clearly sound with respect to the relation $\vDash ^{\text {legal}}$ since we are only considering models based on legal domains.Footnote ⁷ Given this system, we denote the corresponding provability relation simply as $\vdash $ .

We will say that $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S' }\rangle $ is minimized if when $\langle {f_1, \ldots , f_k}\rangle \in S' $ , then there is a legal model of $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S' }\rangle $ , ${\mathfrak {{M}}}$ , such that for $1\leq i \leq k$ the interpretation of $\varphi _i(\overline {x}_{\varphi _i})$ is $f_i\restriction M$ .

With these key facts at hand, the soundness and completeness proof goes through basically as before:

Another interesting consequence of our approach is that we can provide a finitary axiomatization of the valid inferences on finite models for any real-valued logic. Let the class of legal $^*$ domains be that of the legal domains minus the one countably infinite domain (so we are keeping only the finite domains). One can then modify the axiomatization given above by replacing the legal domains by the legal $^*$ ones. Clearly, $\Gamma \vDash ^{\text {legal}^*} \theta $ iff $\Gamma \vDash ^{\text {finite}} \theta $ , where $\vDash ^{\text {finite}}$ is the obvious logical consequence over all finite domains (notice that the legal domains are just a specific subset of all finite domains). Exactly as we did previously, we can obtain:

By a well-known theorem of Trakhtenbrot [Reference Trakhtenbrot45], the validities of classical first-order logic on finite models are not recursively enumerable. In the real-valued setting, the result was generalized in [Reference Bianchi and Montagna8] to a large class of logics. This entails that, once more, our axiomatization cannot possibly be recursive. In fact, we can observe that the problem of determining whether $S' = \emptyset $ in Rule (7) $^*$ of our axiomatization is not recursively enumerable, which explains why our system is not recursive. This is because we can reduce the problem of whether a sentence of classical first-order logic is valid in the finite to whether $S' = \emptyset $ . Take a first-order sentence $\varphi $ and let $\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), \varphi $ be the list of all its subformulas. Consider now the MD-sentence $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), \varphi; S}\rangle $ (call it $\psi $ ) where $S:= \{0,1\}^{\bigcup _{M\, \text{is legal}} M^{n_1}}\times \ldots \times \{0,1\}^{\bigcup _{M\, \text{is legal}} M^{n_k}}\times \{0\}$ . Take now the MD-sentence obtained by applying our Rule (7) to this sentence, $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}), \varphi; S'}\rangle $ (call it $\psi '$ ). Observe that $\psi $ and $\psi '$ are equivalent. Furthermore, $ \varphi $ is valid on all finite models iff $\neg \varphi $ has no finite model iff $\psi $ is not satisfiable in a finite domain iff $\psi '$ is not satisfiable in a finite domain. Finally, by minimization and the semantics of MD-sentences, $\psi '$ is not satisfiable in a finite domain iff $S'= \emptyset $ .

Remark 40. An alternative approach to the one followed in this section would have been to take instead of MD-sentences, “MD-formulas” to be objects of the form $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ where S is a set of tuples of truth values. Then, given a first-order model ${\mathfrak {{M}}}$ and assignment variable v to the free individual variables in $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ , we say that ${\mathfrak {{M}}}$ satisfies $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $  under the assignment v if

$$ \begin{align*}\langle{\|\varphi_1[v(\overline{x}_{\varphi_1})]\|_{{\mathfrak{{M}}}}, \dots, \|\varphi_k[v(\overline{x}_{\varphi_k})]\|_{{\mathfrak{{M}}}} }\rangle \in S.\end{align*} $$

With this modification, everything we have done in this section would work in a very similar manner manner as long as we modify Rule (7) $^*$  appropriately: given an MD-formula $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ , we will say that a tuple $\langle {s_1, \ldots , s_{k}}\rangle \in S$ of truth values is good if for some model ${\mathfrak {{M}}}$ and variable assignment v for the signature of $\langle {\varphi _1(\overline {x}_{\varphi _1}), \ldots , \varphi _k(\overline {x}_{\varphi _k}); S}\rangle $ based on a legal domain M,

1. (a) $s_m = \circ (s_{m_1}, \ldots , s_{m_j})$ whenever $\|\varphi _m[v(\overline {x}_{\varphi _m})]\|_{{\mathfrak {{M}}}} = s_m, \|\varphi _{m_1}[v(\overline {x}_{\varphi _{m_1}})] \|_{{\mathfrak {{M}}}} = s_{m_1}$ , etc., and $\varphi _m(\overline {x}_{\varphi _m}) = \circ (\varphi _{m_1}(\overline {x}_{\varphi _{m_1}}), \ldots , \varphi _{m_j}(\overline {x}_{\varphi _{m_j}})),$

2. (b) $s_i = \inf \{\|\varphi _{j}[v_{y \mapsto e}(\overline {x}_{\varphi _{j}})]\|_{{\mathfrak {{M}}}} \mid v_{y \mapsto e}, e \in M\}$ whenever $\varphi _i(\overline {x}_{\varphi _i}) = \forall y\, \varphi _j(\overline {x}_{\varphi _j}),$ and $v_{y \mapsto e}$ is an assignment just like v except that the value of variable y is made e, and if $\varphi _j(\overline {x}_{\varphi _j})$ appears on the left-hand-side of our MD-formula, $\|\varphi _{j}[v(\overline {x}_{\varphi _{j}})]\|_{{\mathfrak {{M}}}} = s_j$ ,

3. (c) $s_i = \sup \{\|\varphi _{j}[v_{y \mapsto e}(\overline {x}_{\varphi _{j}})]\|_{{\mathfrak {{M}}}} \mid v_{y \mapsto e}, e \in M\}$ whenever $\varphi _i(\overline {x}_{\varphi _i}) = \exists y\, \varphi _j (\overline {x}_{\varphi _j}),$ and $v_{y \mapsto e}$ is an assignment just like v except that the value of variable y is made e, and if $\varphi _j(\overline {x}_{\varphi _j})$ appears on the left-hand-side of our MD-formula, $\|\varphi _{j}[v(\overline {x}_{\varphi _{j}})]\|_{{\mathfrak {{M}}}} = s_j$ .

With this new rule, once can reproduce the proof of the Minimization Lemma and the rest works in an analogous way.