Algebras
As usual in logic, if \(\Lambda \) is a finite list of finitary logical connectives, a \(\Lambda \)-algebra \(\mathbb {A}\) is an underlying set A with a finite list of finitary operations on A corresponding to the symbols in \(\Lambda \). In this paper, we shall assume that
$$\begin{aligned} \{\wedge ,\mathrel {\vee },\mathbf {0},\mathbf {1}\}\subseteq \Lambda \subseteq \{\wedge ,\mathrel {\vee },\rightarrow ,\lnot ,\mathbf {0},\mathbf {1}\}\end{aligned}$$
and that \((A,\wedge ,\mathrel {\vee },\mathbf {0},\mathbf {1})\) is a bounded distributive lattice. As usual, we use the same notation for the syntactic logical connectives and the operations on \(\mathbb {A}\) interpreting them. In the rare cases where proper marking of these symbols improves readability, we attach a subscript \(\mathbb {A}\) to the algebra operations in \(\mathbb {A}\), e.g., \(\wedge _\mathbb {A}\), \(\mathrel {\vee }_\mathbb {A}\), \(\bigwedge _\mathbb {A}\), or \(\bigvee _\mathbb {A}\). We can define \(\le \) on \(\mathbb {A}\) by \(x\le y\) if and only if \(x\wedge y = x\). An element \(a\in A\) is an atom if it is \(\le \)-minimal in \(A{\setminus }\{\mathbf {0}\}\); we write \(\mathrm {At}(\mathbb {A})\) for the set of atoms in \(\mathbb {A}\). If \(\Lambda = {\{\wedge ,\mathrel {\vee },\rightarrow ,\mathbf {0},\mathbf {1}\}}\), we call \(\mathbb {A}\) an implication algebra and if \(\Lambda = { \{\wedge ,\mathrel {\vee },\rightarrow \lnot ,\mathbf {0},\mathbf {1}\}}\), we call \(\mathbb {A}\) an implication-negation algebra.
We call a \(\Lambda \)-algebra \(\mathbb {A}\) with underlying set A complete if for every \(X\subseteq A\), the \(\le \)-supremum and \(\le \)-infimum exist; in this case, we write \(\bigvee X\) and \(\bigwedge X\) for these elements of \(\mathbb {A}\). A complete \(\Lambda \)-algebra \(\mathbb {A}\) is called atomic if for every \(a\in A\), there is an \(X\subseteq \mathrm {At}(\mathbb {A})\) such that \(a = \bigvee X\).
Boolean algebras, complementation, and Heyting algebras
An algebra \(\mathbb {B}= (B,\wedge ,\mathrel {\vee },\lnot ,\mathbf {0},\mathbf {1})\) is called a Boolean algebra if for all \(b\in B\), we have that \(b\wedge \lnot b = \mathbf {0}\) and \(b\mathrel {\vee }\lnot b = \mathbf {1}\). As usual, we can define an implication by
using this definition, we can consider Boolean algebras as implication algebras or implication-negation algebras. An implication algebra \({(B,\wedge ,\mathrel {\vee },\rightarrow ,\mathbf {0},\mathbf {1})}\) is called a Boolean implication algebra if there is a Boolean algebra \({(B,\wedge ,\mathrel {\vee },\lnot ,\mathbf {0},\mathbf {1})}\) such that \(\rightarrow \) is defined by (#) from \(\mathrel {\vee }\) and \(\lnot \) or, equivalently, if the negation defined by \(\lnot _* x := x\rightarrow \mathbf {0}\) satisfies \(\lnot _* b \wedge b = \mathbf {0}\) and \(\lnot _* b\mathrel {\vee }b = \mathbf {1}\).
On an atomic bounded distributive lattice \(\mathbb {A}= (A,\wedge ,\mathrel {\vee },\mathbf {0},\mathbf {1})\), we have a canonical definition for a negation operation, the complementation negation: since \(\mathbb {A}\) is atomic, every element \(a\in A\) is uniquely represented by a set \(X\subseteq \mathrm {At}(\mathbb {A})\) such that \(a = \bigvee X\). Then we define the complementation negation by
$$\begin{aligned} \lnot _\mathrm {c}(\bigvee X) := \bigvee \{t\in \mathrm {At}(\mathbb {A})\,;\,t\notin X\}.\end{aligned}$$
In this situation, \((A,\wedge ,\mathrel {\vee },\lnot _\mathrm {c},\mathbf {0},\mathbf {1})\) is an atomic Boolean algebra. Moreover, if \((A,\wedge ,\mathrel {\vee },\lnot ,\mathbf {0},\mathbf {1})\) is an atomic Boolean algebra and \(\lnot _\mathrm {c}\) is the complementation negation of the atomic bounded distributive lattice \((A,\wedge ,\mathrel {\vee },\mathbf {0},\mathbf {1})\), then \(\lnot = \lnot _\mathrm {c}\). Of course, for every set X, the power set algebra \((\wp (X),\cap ,\cup ,\varnothing ,X)\) forms an atomic bounded distributive lattice and, with the set complementation operator, a Boolean algebra.
If \((H,\wedge ,\mathrel {\vee },\mathbf {0},\mathbf {1})\) is a bounded distributive lattice, then an implication algebra \(\mathbb {H}= (H,\wedge ,\mathrel {\vee },\rightarrow ,\mathbf {0},\mathbf {1})\) is called a Heyting algebra if and only if the Law of Residuation holds, i.e., for all \(a,b,c\in H\), we have that
$$\begin{aligned} c\wedge a\le b \text{ if } \text{ and } \text{ only } \text{ if } c\le a\rightarrow b.\end{aligned}$$
If \(\mathbb {H}\) is a complete lattice, then this is equivalent to
and we say that \(\mathbb {H}\) is a complete Heyting algebra. In a Heyting algebra \(\mathbb {H}\), we can define a negation \(\lnot _\mathbb {H}\) by \(\lnot _\mathbb {H}x := x\rightarrow \mathbf {0}\). Note that Boolean implication algebras are Heyting algebras.
It is well known that the class of Heyting algebras forms a variety [13, p. 8] and that not every complete bounded distributive lattice can be turned into a Heyting algebra (e.g., the dual of the Heyting algebra of open subsets of \(\mathbb {R}\); cf. [2, Proposition 51.2]).
A Heyting algebra is called linear if \((H,\le )\) is a linear order; the formula \((p\rightarrow q)\mathrel {\vee }(q\rightarrow p)\) characterises the variety of Heyting algebras generated by the linear Heyting algebras [7, 11, 20] (cf. also [19] for a discussion of Skolem’s 1913 results).
We shall later use the following linear three element complete Heyting algebra \(\mathbf {3} := (\{\mathbf {0},\mathbf {\nicefrac {1}{2}},\mathbf {1}\},\wedge ,\mathrel {\vee },\rightarrow ,\mathbf {0},\mathbf {1})\) with \(\mathbf {0}\le \mathbf {\nicefrac {1}{2}}\le \mathbf {1}\). Then \(\rightarrow \) is uniquely determined by (\(\dag \)):
$$\begin{aligned} \begin{array}{c|cccl} \rightarrow &{}\quad \mathbf {0}&{}\quad \mathbf {\nicefrac {1}{2}}&{}\quad \mathbf {1}\\ \mathbf {0}&{}\quad \mathbf {1}&{}\quad \mathbf {1}&{}\quad \mathbf {1}\\ \mathbf {\nicefrac {1}{2}}&{}\quad \mathbf {0}&{}\quad \mathbf {1}&{}\quad \mathbf {1}\\ \mathbf {1}&{}\quad \mathbf {0}&{}\quad \mathbf {\nicefrac {1}{2}}&{}\quad \mathbf {1}&{} . \end{array} \end{aligned}$$
Languages
Fix a set S of non-logical symbols, a countable set P of propositional variables, and a countable set V of first-order variables. We denote the set of well-formed propositional formulas with connectives \(\Lambda \) and propositional variables P by \(\mathcal {L}_\Lambda \) and the set of well-formed first-order formulas with connectives \(\Lambda \), variables in V and constant, relation and function symbols in S by \(\mathcal {L}_{\Lambda ,S}\). The subset of sentences of \(\mathcal {L}_{\Lambda ,S}\) will be denoted by \(\mathrm {Sent}_{\Lambda ,S}\). Note that both \(\mathcal {L}_\Lambda \) and \(\mathrm {Sent}_{\Lambda ,S}\) have the structure of a \(\Lambda \)-algebra and that the \(\Lambda \)-algebra \(\mathcal {L}_\Lambda \) is generated by closure under the connectives in \(\Lambda \) from the set P.
For arbitrary sets \(\Lambda \) of logical connectives and S of non-logical symbols, we define \(\mathrm {NFF}_\Lambda \) to be the closure of P under the logical connectives other than \(\lnot \) and \(\mathrm {NFF}_{\Lambda ,S}\) to be the closure of the atomic formulae of \(\mathcal {L}_{\Lambda ,S}\) under the logical connectives other than \(\lnot \). These formulas are called the negation-free \(\Lambda \)-formulas. Clearly, if \(\lnot \notin \Lambda \), then \(\mathcal {L}_\Lambda = \mathrm {NFF}_\Lambda \) and \(\mathcal {L}_{\Lambda ,S} = \mathrm {NFF}_{\Lambda ,S}\).
Homomorphisms, assignments, and translations
For any two \(\Lambda \)-algebras \(\mathbb {A}\) and \(\mathbb {B}\), a map \(f:A\rightarrow B\) is called a \(\Lambda \)-homomorphism if it preserves all operations in \(\Lambda \); it is called a \(\Lambda \)-isomorphism if it is a bijective \(\Lambda \)-homomorphism; isomorphisms from \(\mathbb {A}\) to \(\mathbb {A}\) are called \(\Lambda \)-automorphisms.
If \(\mathbb {A}\) and \(\mathbb {B}\) are two complete \(\Lambda \)-algebras and \(f{:}A{\rightarrow } B\) is a \(\Lambda \)-homomorphism, we call it complete if it preserves the operations \(\bigvee \) and \(\bigwedge \), i.e.,
$$\begin{aligned}\textstyle f({\textstyle \bigvee _\mathbb {A}} X)&= {\textstyle \bigvee _\mathbb {B}}(\{f(x)\,;\,x\in X\}) \text{ and }\\ f({\textstyle \bigwedge _\mathbb {A}} X)&= {\textstyle \bigwedge _\mathbb {B}}(\{f(x)\,;\,x\in X\}) \end{aligned}$$
for \(X\subseteq A\).
Since \(\mathcal {L}_\Lambda \) is generated from P, we can think of any \(\Lambda \)-homomorphism defined on \(\mathcal {L}_\Lambda \) as a function on P, homomorphically extended to all of \(\mathcal {L}_\Lambda \). If \(\mathbb {A}\) is a \(\Lambda \)-algebra with underlying set A, we say that \(\Lambda \)-homomorphisms \(\iota :\mathcal {L}_\Lambda \rightarrow A\) are \(\mathbb {A}\)-assignments; if S is a set of non-logical symbols, we say that \(\Lambda \)-homomorphisms \(T:\mathcal {L}_\Lambda \rightarrow \mathrm {Sent}_{\Lambda ,S}\) are S-translations.
The propositional logic of an algebra
A set \(D\subseteq A\) is called a designated set or filter if the following four conditions hold: (i) \(\mathbf {1}\in D\), (ii) \(\mathbf {0}\notin D\), (iii) if \(x\in D\) and \(x\le y\), then \(y\in D\), and (iv) for \(x,y\in D\), we have \(x\wedge y\in D\). For any designated set D, the propositional logic of \((\mathbb {A},D)\) is defined as
$$\begin{aligned} \mathbf {L}(\mathbb {A},D) := \{\varphi \in \mathcal {L}_\Lambda \,;\,\iota (\varphi )\in D \text{ for } \text{ all } \mathbb {A}\text{-assignments } \iota \}.\end{aligned}$$
Since the classical propositional calculus \(\mathbf {CPC}\) is maximally consistent, we obtain that if \(\mathbb {B}\) is a Boolean algebra and D is any designated set, then \(\mathbf {L}(\mathbb {B},D) = \mathbf {CPC}\) [3, Theorem 5.11].
Algebra-valued structures and their propositional logic
If \(\mathbb {A}\) is a \(\Lambda \)-algebra and S is a set of non-logical symbols, then any \(\Lambda \)-homomor-phism \(\llbracket \cdot \rrbracket :\mathrm {Sent}_{\Lambda ,S}\rightarrow A\) will be called an \(\mathbb {A}\)-valued S-structure. Note that if \(S'\subseteq S\), \(\Lambda '\subseteq \Lambda \), \(\mathbb {A}\) is a \(\Lambda \)-algebra and \(\mathbb {A}'\) its \(\Lambda '\)-reduct, and \(\llbracket \cdot \rrbracket \) is an \(\mathbb {A}\)-valued S-structure, then \(\llbracket \cdot \rrbracket {\upharpoonright }\mathrm {Sent}_{\Lambda ,S'}\) is an \(\mathbb {A}\)-valued \(S'\)-structure and \(\llbracket \cdot \rrbracket {\upharpoonright }\mathrm {Sent}_{\Lambda ^*,S}\) is an \(\mathbb {A}^*\)-valued S-structure.
We define the propositional logic of \((\llbracket \cdot \rrbracket ,D)\) as
$$\begin{aligned} \mathbf {L}(\llbracket \cdot \rrbracket ,D) := \{\varphi \in \mathcal {L}_\Lambda \,;\,\llbracket T(\varphi )\rrbracket \in D \text{ for } \text{ all } S\text{-translations }\,\, T\}.\end{aligned}$$
Note that if T is an S-translation and \(\llbracket \cdot \rrbracket \) is an \(\mathbb {A}\)-valued S-structure, then \(\varphi \mapsto \llbracket T(\varphi )\rrbracket \) is an \(\mathbb {A}\)-assignment, so
Clearly, \(\mathrm {ran}(\llbracket \cdot \rrbracket ) \subseteq A\) is closed under all operations in \(\Lambda \) (since \(\llbracket \cdot \rrbracket \) is a homomorphism) and thus defines a sub-\(\Lambda \)-algebra \(\mathbb {A}_{\llbracket \cdot \rrbracket }\) of \(\mathbb {A}\). The \(\mathbb {A}\)-assignments that are of the form \(\varphi \mapsto \llbracket T(\varphi )\rrbracket \) are exactly the \(\mathbb {A}_{\llbracket \cdot \rrbracket }\)-assignments, so we obtain
$$\begin{aligned} \mathbf {L}(\llbracket \cdot \rrbracket ,D) = \mathbf {L}(\mathbb {A}_{\llbracket \cdot \rrbracket },D \cap \mathbb {A}_{\llbracket \cdot \rrbracket }).\end{aligned}$$
We should like to point out that the propositional logic of the structure \((\llbracket \cdot \rrbracket ,D)\) as defined above treats all \(\Lambda ,S\)-sentences as propositional atoms and thus cannot take their internal construction into account; this is in line with the usual definitions of propositional logics of first-order theories (cf., e.g., [5]). Note that ignoring the internal structure of sentences can result in a situation where a structure \((\llbracket \cdot \rrbracket ,D)\) is non-classical, but satisfies \(\mathbf {L}(\llbracket \cdot \rrbracket ,D) = \mathbf {CPC}\). E.g., consider the Heyting algebra \(\mathbb {H}\) with \(H = \mathbb {Z}\cup \{\mathbf {0},\mathbf {1}\}\) from Proposition 4.7 where we prove that \(\mathbf {L}(\llbracket \cdot \rrbracket _\mathbb {H},\{\mathbf {1}\}) = \mathbf {CPC}\). It is easy to see that the sentence \(\varphi := {\forall x\forall y (x\in y \vee x\notin y)}\) (cf. the proof of Proposition 3.3) evaluates to \(\mathbf {0}\) in \(\mathbb {H}\), so \(\mathbb {H}\) is non-classical. (This was pointed out by one of the referees.)
Loyalty and faithfulness
An \(\mathbb {A}\)-valued S-structure \(\llbracket \cdot \rrbracket \) is called loyal to \((\mathbb {A},D)\) if the converse of (\(\ddag \)) holds as well, i.e., if \(\mathbf {L}(\mathbb {A},D) = \mathbf {L}(\llbracket \cdot \rrbracket ,D)\); it is called faithful to \(\mathbb {A}\) if for every \(a\in A\), there is a \(\varphi \in \mathrm {Sent}_{\Lambda ,S}\) such that \(\llbracket \varphi \rrbracket = a\); equivalently, if \(\mathbb {A}_{\llbracket \cdot \rrbracket } = \mathbb {A}\). (Cf. the paragraph on Related Work in Section 1 for the genesis of these notions.)
Lemma 2.1
Let \(\Lambda \) be any set of propositional connectives, S be any set of non-logical symbols, \(\mathbb {A}\) be a \(\Lambda \)-algebra, and \(\llbracket \cdot \rrbracket \) be an \(\mathbb {A}\)-valued S-structure. Then, if \(\llbracket \cdot \rrbracket \) is faithful to \(\mathbb {A}\), then it is loyal to \((\mathbb {A},D)\) for any designated set D.
Proof
By (\(\ddag \)), we only need to prove one inclusion; if \(\varphi \notin \mathbf {L}(\mathbb {A},D)\), then let \(p_1,\ldots ,p_n\) be the propositional variables occurring in \(\varphi \) and let \(\iota \) be an assignment such that \(\iota (\varphi )\notin D\). By faithfulness, find sentences \(\sigma _i\in \mathrm {Sent}_{\Lambda ,S}\) such that \(\llbracket \sigma _i\rrbracket = \iota (p_i)\) for \(1\le i\le n\). Let T be any translation such that \(T(p_i) = \sigma _i\) for \(1\le i \le n\). Then \(\llbracket T(\varphi )\rrbracket = \iota (\varphi )\notin D\), and hence T witnesses that \(\varphi \notin \mathbf {L}(\llbracket \cdot \rrbracket ,D)\). \(\square \)
A proof of Lemma 2.1 in the more general setting for classes of Heyting structures can be found in [17, Proposition 2.50].
Note that faithfulness and loyalty depend on the choice of S. As mentioned above, if \(S^*\subseteq S\) and \(\Lambda ^*\subseteq \Lambda \) then \(\mathrm {Sent}_{\Lambda ^*,S^*} \subseteq \mathrm {Sent}_{\Lambda ,S}\) and thus we can easily see the following:
Observation 2.2
Let \(\mathbb {A}\) be a \(\Lambda \)-algebra, \(\mathbb {A}^*\) its \(\Lambda ^*\)-reduct, and \(\llbracket \cdot \rrbracket \) be an \(\mathbb {A}\)-valued S-structure. If \(\llbracket \cdot \rrbracket {\upharpoonright }\mathrm {Sent}_{\Lambda ^*,S^*}\) is faithful to \(\mathbb {A}^*\), then \(\llbracket \cdot \rrbracket \) is faithful to \(\mathbb {A}\).
However, the converse is not true in general: faithfulness cannot hold if the algebra \(\mathbb {A}\) is bigger than the set \(\mathrm {Sent}_{\Lambda ,S}\), so for countable languages, no \(\mathbb {A}\)-valued S-structure can be faithful to an uncountable algebra \(\mathbb {A}\). Thus, if \(\mathbb {A}\) is an uncountable algebra, S an uncountable set of non-logical symbols, \(\llbracket \cdot \rrbracket \) is an \(\mathbb {A}\)-valued S-structure that is faithful to \(\mathbb {A}\), and \(S'\) is a countable subset of S, then \(\llbracket \cdot \rrbracket {\upharpoonright }\mathcal {L}_{\Lambda ,S'}\) cannot be faithful to \(\mathbb {A}\). The constructions in this paper will give another example that does not use a cardinality argument (cf. the remark after Theorem 5.10 at the end of this paper).