2.1 Independent generation

According to the first approach, the crucial feature of elementary propositions is that they are logically independent. Informally, logical independence means that for any distinct elementary propositions $e_1,\dots ,e_{n+m}$ , it is consistent for $e_1,\dots ,e_n$ to be true, and for $e_{n+1},\dots ,e_{n+m}$ to be false. That is, the conjunction of $e_1,\dots ,e_n$ with the negations of $e_{n+1},\dots ,e_{n+m}$ is non-zero (i.e., distinct from $0$ ). Algebraically, this is known as being independent. A useful way of defining this is as follows (in the context of a given Boolean algebra, left implicit).

In this definition, T and F may be empty. To see the effect this has, note that $\bigwedge \emptyset =\top $ . So, this special case of the condition requires the underlying algebra to be non-trivial, where a Boolean algebra is trivial if it has just one element. If the underlying algebra is non-trivial, this special case of the condition does not impose any further requirements.

Other regimentations of independence could be proposed, but in our setting of Boolean algebras, arguably all of the most natural options coincide. Partly, this claim is substantiated by our observations in §2.4. Correspondingly, Humberstone [Reference Humberstone15] notes that in classical logic, the most natural ways of defining the notion of formulas being independent also coincide. However, as Humberstone notes, the situation becomes more complicated when we consider non-classical logics, such as intuitionistic logic. Correspondingly, if we worked with, say, Heyting algebras rather than Boolean algebras, we would need to think more carefully about the notion of independence. Different notions of independence in wider classes of algebras have been investigated; for recent discussion with further references, see Metcalfe and Tokuda [Reference Metcalfe, Tokuda, Bezhanishvili, Iemhoff and Yang19]. See also Humberstone [Reference Humberstone15, p. 195] for discussion of independence in the context of logical atomism, including many further references. Since we restrict ourselves to working in Boolean algebras, we can restrict ourselves to the notion of independence which is standard in mathematical work on Boolean algebras.

It is important to be clear that logical independence is not a way of characterizing elementarity algebraically, since typically, there are multiple independent sets whose union is not independent. In fact, this holds for every non-trivial Boolean algebra. Up to isomorphism, there is only one trivial Boolean algebra, which we call $\mathbf {1}$ . For any other Boolean algebra, every element is distinct from its complement. Thus, for any element x, $\{x\}$ and $\{-x\}$ are distinct and independent, but $\{x,-x\}$ is not independent. Consequently, independence is only an algebraic requirement of elementarity, not a characterization of it.

Assuming an independent set E of elementary propositions, there is a straightforward way of making sense of the claim that every proposition is eventually reached from E by successive applications of conjunction and negation. We can define a sequence of sets $E_0\subseteq E_1\subseteq E_2\subseteq \dots $ , starting from $E_0=E$ , where $E_{n+1}$ contains just the negations and conjunctions of elements of $E_n$ . We can then characterize the propositions which can be obtained from elements of E using truth-functional operations as the union of $E_0,E_1,E_2,\dots $ . More formally, we can define

$$ \begin{align*} E_0&:=E\\ E_{n+1}&:=\{-x:x\in E_n\}\cup\{y\wedge z:y,z\in E_n\}\\ E_\omega&:=\bigcup_{n\in\mathbb{N}}E_n. \end{align*} $$

To say that every proposition is obtained by successive applications of conjunction and negation to elements of E is therefore naturally formalized by requiring $E_\omega $ to contain every element of A. In algebraic terminology, this requires E to generate $\mathfrak {A}$ . It will be useful to introduce an alternative way of stating this requirement, which is equivalent to the one just stated. It uses the notion of a subalgebra. A subalgebra of a Boolean algebra $\langle A,-,\wedge \rangle $ is a Boolean algebra $\mathfrak {B}$ based on a subset B of A, such that the operations of $\mathfrak {B}$ are obtained by restricting the operations of $\mathfrak {A}$ to B. A subalgebra $\mathfrak {B}$ of $\mathfrak {A}$ is a proper subalgebra of $\mathfrak {A}$ if it is distinct from $\mathfrak {A}$ . The intersection of any non-empty set of subalgebras of $\mathfrak {A}$ is itself a subalgebra of $\mathfrak {A}$ : for any such set S, the elements contained in every member of S are closed under the operations, and so form a subalgebra themselves. It follows that for a given set $E\subseteq A$ , the subalgebras which include E are closed under intersection, and so contain a smallest element. This is the intersection of all subalgebras including E. It can be shown that its underlying set is just $E_\omega $ . We can therefore define:

The first way of regimenting logical atomism therefore requires the elementary propositions E to be independent and to generate the algebra of all propositions. We introduce an obvious label for this notion.

As in the case of independence, a set $E\subseteq A$ which independently generates $\mathfrak {A}$ does not in general do so uniquely. We return to this point in more detail in §2.4 below.

2.2 Free generation

The second way of regimenting logical atomism draws on another Tractarian idea, implicit in Wittgenstein’s use of truth tables (cf. 4.442, 5.101). Here the idea that the elementary propositions are logically independent of one another is spelled out in terms of truth value assignments: any way of assigning truth values to the elementary propositions is coherent, in the sense that it extends to an assignment of truth values to all propositions that is well-behaved with respect to the truth-functional connectives; e.g., it maps a conjunction to true if and only if it maps both conjuncts to true. The idea that every proposition can be made from the elementary propositions using the truth-functional operations can be stated in similar terms. For if any given proposition is a truth-functional combination of elementary propositions, then its truth value is completely determined by the truth values of the elementary propositions. That is to say, there cannot be two different ways of coherently extending an assignment of truth values to the elementary propositions to all of the propositions.

The concepts appealed to here are reminiscent of the concept of a truth value assignment and valuation found in propositional logic, with elementary propositions playing the role of sentence letters, so we shall repurpose those terms. First, note that up to isomorphism, there is a unique two-element Boolean algebra, which we will call $\mathbf {2}$ . As a model of propositions, we can think of this as identifying propositions with truth values: the top element $\top $ is the truth value true, and the bottom element $\bot $ is the truth value false. The behaviour of $-$ and $\wedge $ conforms to the classical truth tables for negation and conjunction. For mathematical convenience, we identify $\bot $ with $0$ and $\top $ with $1$ ; by the von Neumann definition of ordinals, it follows that $2=\{0,1\}=\{\bot ,\top \}$ . A truth value assignment is then a mapping $f:E\to 2$ . A valuation is a mapping $v:A \to 2$ which respects the truth-functional operations, i.e., such that for all $x,y,z\in A$ :

1. (i) $v(-_{\mathfrak {A}}x)=-_{\mathbf {2}}v(x)$ , and

2. (ii) $v(y\wedge _{\mathfrak {A}}z)=v(y)\wedge _{\mathbf {2}}v(z)$ .

We say a valuation $v$ extends f when $v(x)=f(x)$ for all $x\in E$ .

Observe that being freely generated with respect to valuations imposes a non-trivial condition on the granularity of propositions. Propositions must be fine-grained enough to support a well-defined notion of evaluating the truth value of a proposition in terms of the truth values of its elementary components. This seems to require propositions at least be structured enough to have a well-defined notion of elementary component. (Indeed, in the next section we will make this granularity thesis precise; that all propositions have a decomposition into a unique truth-functional connective applied to the elementary propositions.) Not every Boolean algebra is like this—for instance, we will shortly see that no eight element Boolean algebra is freely generated with respect to valuations by any set of its elements.

The Tractarian picture of propositions as built up from elementary propositions suggests that a wider class of structure-sensitive operations should be well-defined. Consider, now, the operation of taking a proposition—thought of as a truth-functional combination of elementary propositions—and uniformly replacing each elementary constituent with other propositions. This is the “metaphysical” analogue of the operation of substituting the letters of a propositional formula with other formulas, so we might similarly repurpose the word “substitution.” If this notion is well-defined, then any function $f:E\to A$ mapping elementary propositions to propositions should extend uniquely to a substitution $\hat {f}:A\to A$ that informally may be thought of as taking a proposition $x\in A$ and replacing the elementary components according to f, leaving the truth-functional operations in place. It is clear, given this informal idea of leaving the truth-functional operations “in place,” that substitutions should commute with the truth-functional connectives, i.e., such that for all $x,y,z\in A$ :

1. (i) $f(-x)=-f(x)$ , and

2. (ii) $f(y\wedge z)=f(y)\wedge f(z)$ .

We take these equations as our definition of a substitution. The extendability of functions from E to A to unique substitutions seems, intuitively, to subsume the previous condition: truth value assignments can be identified with functions $f:A\to A$ that only take on two values, $\top $ and $\bot $ .

Clearly there is a generalization here to be made. For this, we need another algebraic concept. A homomorphism $f:A\to B$ between two Boolean algebras $\mathfrak {A}=\langle A,-_{\mathfrak {A}},\wedge _{\mathfrak {A}}\rangle $ and $\mathfrak {B}=\langle B,-_{\mathfrak {B}},\wedge _{\mathfrak {B}}\rangle $ is a function $f:A\to B$ such that for all $x,y,z\in A$ :

1. (i) $f(-_{\mathfrak {A}}x)=-_{\mathfrak {B}}f(x)$ , and

2. (ii) $f(y\wedge _{\mathfrak {A}}z)=f(y)\wedge _{\mathfrak {B}}f(z)$ .

We can now introduce a more general notion of free generation for a given Boolean algebra $\mathfrak {A}$ . The more general notion is parametric on a class $\mathsf {T}$ of algebras. This parameter delineates the algebras which may serve as targets for the functions on elementary propositions which must be uniquely extendable. (There is actually a natural further generalization in which we parameterize the notion with respect to a category, in the sense of Mac Lane [Reference Mac Lane18], of Boolean algebras and Boolean homomorphisms that contains $\mathsf {T}$ among its objects. In the generalization the homomorphism extending f must belong to and be unique with respect to the category in question. However, since we will only consider two categories in this paper—the category of Boolean algebras and homomorphisms and of complete Boolean algebras with complete homomorphisms—we won’t bother with this generalization.) Relative to this parameter, we define the following definition.

In this more general notion, free generation with respect to valuations is $\mathbf {2}$ -free generation; here, we indicate a singleton class, e.g., $\{\mathbf {2}\}$ , using its single element $\mathbf {2}$ for brevity. Similarly, $\mathfrak {A}$ is freely generated with respect to substitutions when it is $\mathfrak {A}$ -freely generated. The latter requires that any function $f:E\to A$ has a unique extension $\hat {f}$ which is a homomorphism from $\mathfrak {A}$ to $\mathfrak {A}$ ; this is also known as an endomorphism on $\mathfrak {A}$ . (An endomorphism on $\mathfrak {A}$ which is also a bijection, i.e., an isomorphism from $\mathfrak {A}$ to $\mathfrak {A}$ , is known as an automorphism of $\mathfrak {A}$ .)

We will also consider two further instances of $\mathsf {T}$ -free generation. In our terminology, the strongest notion of free generation in this context—the notion usually used in the theory of Boolean algebras—is the notion of $\mathsf {BA}$ -free generation, where $\mathsf {BA}$ is the class of Boolean algebras. Unpacking the definition, $E\subseteq A\, \mathsf {BA}$ -freely generates $\mathfrak {A}$ when any function $f:E\to B$ , where B is the underlying set of another Boolean algebra $\mathfrak {B}$ , extends to a unique homorophism $\hat {f}:\mathfrak {A}\to \mathfrak {B}$ . Intuitively, this corresponds to the idea that the operation of “replacing” elementary propositions with elements of an arbitrary Boolean algebra is well-defined. The second notion is a more modest attempt to strengthen free generation with respect to substitutions. This is $\mathsf {sub}(\mathfrak {A})$ -free generation, where $\mathsf {sub}(\mathfrak {A})$ is the class of subalgebras of $\mathfrak {A}$ . This condition takes us beyond free generation with respect to substitutions since it requires that if you have a function $f:E\to A$ , the range of its extension $\hat {f}$ must be included in the subalgebra of $\mathfrak {A}$ generated by the range of f.

2.3 Unique decomposition

The third and final regimentation of logical atomism we will explore formulates precisely the Tractarian claim that every proposition can be decomposed into a unique Boolean operation applied to the elementary propositions. In this section we will explicitly formulate versions of this Tractarian theory of propositional structure and show how it is closely related to the free generation idea by making the connection between substitutions on propositional constituents and homomorphisms more precise.

What is an n-ary Boolean connective? By a connective we shall mean a family of functions $c_{\mathfrak {A}}: A^n \to A$ on each Boolean algebra $\mathfrak {A}$ . A connective is Boolean when it is preserved by all Boolean homomorphisms. In the language of category theory, we can express this abstractly by stating that a connective is a natural transformation of the functor $-^n$ to the identity functor on the category of Boolean algebras. More concretely, we can state the condition as follows:

• For any Boolean homomorphism, $h:\mathfrak {A}\to \mathfrak {B}$ and $x_1,\ldots , x_n\in \mathfrak {A}$ , $h(c_{\mathfrak {A}}(x_1,\ldots ,x_n)) = c_{\mathfrak {B}}(h(x_1),\ldots , h(x_n))$ .

In our official definition, we will identify an element of $A^n$ with a function $f:I\to A,$ where I is an n-element index set, such as $\{1,\ldots , n\}$ . In this notation, the above condition becomes the following definition.

This notation has the benefit that it smoothly extends to infinitary connectives by letting the index set I be an infinite set.

The conjunctive connective $\wedge $ , i.e., the family $\wedge _{\mathfrak {A}}: \mathfrak {A}^2 \to \mathfrak {A}$ , is clearly a binary Boolean connective, since by the very definition of a homomorphism, $\wedge $ is preserved by every Boolean homomorphism h: $h(x_1 \wedge _{\mathfrak {A}} x_2)=h(x_1)\wedge _{\mathfrak {B}} h(x_2)$ . For similar reasons $-$ , the family $-_{\mathfrak {A}}$ for each Boolean algebra $\mathfrak {A}$ defines a unary Boolean connective. The family of operations $\vee _{\mathfrak {A}}: A^2 \to A$ , i.e., $(x,y)\mapsto -_{\mathfrak {A}}(-_{\mathfrak {A}}x \wedge _{\mathfrak {A}} -_{\mathfrak {A}}y)$ , is also easily shown to be preserved by every Boolean homomorphism, although in this case it is not immediate from the definition.

Another route into seeing why this definition is natural is that we can think of the Boolean connectives as those n-ary operations that can be “defined” from conjunction and negation, where the operative notion of definition is as follows (cf. Bacon [Reference Bacon1, sec. 3.2 and proposition 8]).

Of course, functions that preserve the operations $\wedge $ and $-$ just are Boolean homomorphisms, so that Boolean connectives are connectives defined from $\wedge $ and $-$ .

The Tractarian picture of propositions as built up uniquely from elementary propositions and Boolean connectives suggests the following condition on an algebra $\mathfrak {A}$ .

When $E = \{e_1,\ldots , e_n\}$ is finite, this simply means every element of A is of the form $c_{\mathfrak {A}}(e_1,\ldots , e_n)$ for some Boolean connective c; note that the inclusion map $\imath $ is how we represent the n-tuple of elementary propositions $\langle e_1,\ldots , e_n\rangle $ in this notation.

Like with the concept of free generation, one can introduce variants of this notion. Let $\mathsf {D}$ be a class of Boolean algebras and $\mathsf {T}$ a subclass of $\mathsf {D}$ containing a non-trivial Boolean algebra. $\mathsf {D}$ represents the domain of algebras on which our notion of a connective is to be defined, and $\mathsf {T}$ the class of algebras which are the targets of the homomorphisms that must be preserved by the type of connective in question.

Thus a Boolean connective is a $(\mathsf {BA},\mathsf {BA})$ -connective. It is immediate by definition that if c is a Boolean connective, then the family $\{c_{\mathfrak {B}} \mid \mathfrak {B}\in \mathsf {D}\}$ is a $(\mathsf {D},\mathsf {T})$ -connective for any $\mathsf {D}$ and $\mathsf {T}\subseteq \mathsf {D}$ .

Another important class of connectives is the truth-functional connectives, which, using the above terminology, can be described as the $(\mathsf {BA},\mathbf {2})$ -connectives. To see why this characterization is adequate, consider any n-ary connective c. Intuitively, c is truth-functional just in case there is some truth function $t:2\times \cdots \times 2\to 2$ such that for any arguments $x_1,\dots ,x_n$ , t maps the truth values of $x_1,\dots ,x_n$ to the truth value of $c(x_1,\dots ,x_n)$ . Of course, in a Boolean algebra $\mathfrak {B}$ , elements only have truth values relative to a valuation $v:B\to 2$ . More formally, we are thus lead to the following characterization: Let c be a family of n-ary functions $c_{\mathfrak {B}}$ on B for each algebra $\mathfrak {B}$ . c is truth-functional just in case there is a truth-function t such that for any elements $x_1,\dots ,x_n$ of any Boolean algebra $\mathfrak {B}$ and any valuation $v:B\to 2$ , $v(c_{\mathfrak {B}}(x_1,\ldots , x_n))=t(v(x_1),\ldots , v(x_n))$ . Note that in this case, it follows directly from the case of $\mathfrak {B}$ being $\mathbf {2}$ and v being the identity map that $t=c_{\mathbf {2}}$ . More generally, then, an I-ary connective c is truth-functional when for every valuation $v:B\to 2$ , and $f\in B^I$ , $v(c_{\mathfrak {B}}(f)) =c_{\mathbf {2}}(v\circ f)$ . Since the valuations of a Boolean algebra $\mathfrak {B}$ are precisely the homomorphisms from $\mathfrak {B}$ to $\mathbf {2}$ , this condition amounts to being a $(\mathsf {BA},\mathbf {2})$ -connective.

And finally, as before, we can introduce substitution theoretic notions of connectives: $(\mathfrak {A}, \mathfrak {A})$ -connectives and $(\mathsf {sub}(\mathfrak {A}), \mathsf {sub}(\mathfrak {A}))$ -connectives. The former connectives are only defined on the algebra $\mathfrak {A}$ and must commute with all endomorphisms of the algebra, and the latter are only defined on the subalgebras of $\mathfrak {A}$ and must commute with all homomorphisms between these algebras.

Unique decomposition is intuitively closely related to free generation. For the purposes of illustration, consider the notions of $\mathsf {BA}$ -free generation and unique decomposition with respect to $(\mathsf {BA},\mathsf {BA})$ -connectives. Unique decomposition tells us that propositions have a certain amount of structure, and we can use this to define a notion of substitution on propositions. At a very intuitive level, if every element of $\mathfrak {A}$ can be uniquely written as a Boolean function of elementary propositions $c_{\mathfrak {A}}(e_1,e_2,\dots )$ , then we are able to think of any function $f:E\to B$ as determining a “substitution homomorphism,” defined as replacing each elementary constituent $e_i$ of any proposition $c_{\mathfrak {A}}(e_1,e_2,\dots )$ with $f(e_i)$ . The substitution homomorphism $\hat f$ can thus be defined by mapping each element of the form $c_{\mathfrak {A}}(e_1,e_2\dots )$ to $c_{\mathfrak {B}}(f(e_1),f(e_2)\dots )$ ; by decomposition, every element does have this form, and by the uniqueness of this decomposition the definition is well-defined.

Conversely, free generation tells us that however propositions are structured, they are structured enough to admit a well-defined notion of substitution, and this turns out to uniquely determine the Boolean structure of a proposition. For if $x\in A$ could be expressed as $c_{\mathfrak {A}}(e_1,e_2,\ldots )$ for some Boolean connective c, then if we were to apply this connective to the propositions $f(e_1),f(e_2),\ldots \in B$ , the result, $c_{\mathfrak {B}}(f(e_1),f(e_2),\ldots )$ , should be a substitution instance of the original proposition x, namely: $\hat f(c_{\mathfrak {A}}(e_1,\ldots ,e_n)) = \hat f(x)$ since connectives should commute with the substitution homomorphism $\hat f$ obtained from f. Since every E-length sequence of elements of B has the form $f(e_1),f(e_2),\ldots $ for some f, the stipulation $c_{\mathfrak {B}}(f(e_1),f(e_2),\ldots ) = \hat f(x)$ totally defines a connective (which must then be shown to be a Boolean connective). This idea easily generalizes: $\mathsf {T}$ -free generation and unique decomposition with respect to $(\mathsf {T},\mathsf {T})$ -connectives are equivalent. This informal idea is made precise in Proposition 2.12 below.

2.4 Equivalence

We now have nine logical atomist principles: independent generation, $\mathsf {T}$ -free generation, for $\mathsf {T}$ being $\mathbf {2}$ , $\mathfrak {A}$ , $\mathsf {sub}(\mathfrak {A})$ , or $\mathsf {BA}$ , and unique decomposition with respect to $(\mathsf {BA},\mathsf {BA}), (\mathfrak {A},\mathfrak {A}), (\mathsf {sub}(\mathfrak {A}),\mathsf {sub}(\mathfrak {A}))$ and $(\mathsf {BA},\mathbf {2})$ -connectives. How do they relate? It turns out that they are all equivalent, at least on non-trivial algebras (i.e., algebras with at least two elements). Furthermore, we can show that independent generation is equivalent, among non-trivial algebras, to $\mathsf {T}$ -free generation, for every class of Boolean algebras $\mathsf {T}$ which contains some non-trivial Boolean algebra. And we will show that $\mathsf {T}$ -free generation is equivalent to unique decomposition into $(\mathsf {T},\mathsf {T})$ -connectives.

The proofs of these results make use of an algebraic concept which has not been introduced so far. An ultrafilter of a Boolean algebra $\langle A,-,\wedge \rangle $ is set $U\subseteq A$ such that $-x\in U$ iff $x\notin U$ , and $y\wedge z\in U$ iff $y\in U$ and $z\in U$ , for all $x,y,z\in A$ . This means that a set $U\subseteq A$ is an ultrafilter just in case its characteristic function, mapping every element of U to $\top $ and every other element of A to $\bot $ , is a homomorphism from $\mathfrak {A}$ to $\mathbf {2}$ .

We begin by connecting our unique decomposition principles to free generation.

The above proposition shows that three of the four types of unique decomposition we have discussed (namely $(\mathsf {BA},\mathsf {BA})$ , $(\mathfrak {A},\mathfrak {A})$ , and $(\textsf {sub}(\mathfrak {A}),\textsf {sub}(\mathfrak {A}))$ ) are equivalent to three of the four the sorts of free generation we have talked about: $\mathsf {BA}$ , $\mathfrak {A}$ , and $\mathsf {sub}(\mathfrak {A})$ . We have not shown such an equivalence for unique decomposition with respect to truth-functional connectives, because they aren’t defined as $(\mathsf {T},\mathsf {T})$ -connectives for some $\mathsf {T}$ . However, by using the notion of an ultrafilter defined above, we can see that all of these notions, including unique decomposition into truth-functional connectives, are equivalent.

What kinds of pairs E and $\mathfrak {A}$ satisfy independent generation (and so, equivalently, the various notions of free generation and the decomposition principles)? First, if E independently generates $\mathfrak {A}$ , then it does so uniquely, up to isomorphism, in the following sense: if E and $E'$ independently generate $\mathfrak {A}$ and $\mathfrak {A}'$ , respectively, and E and $E'$ have the same cardinality, then any bijection between E and $E'$ has a unique extension to an isomorphism between $\mathfrak {A}$ and $\mathfrak {A}'$ . Second, any set E independently generates some Boolean algebra $\mathfrak {A}$ . Up to isomorphism, we may therefore speak of the Boolean algebra which is independently generated by a set of a given cardinality $\kappa $ . (For a more detailed discussion of these and the following observations, see Givant and Halmos [Reference Givant and Halmos13, chap. 28].)

To describe independently generated Boolean algebras in more detail, it is useful to present one way of generating them using a technique commonly employed in logic. Take the language of classical propositional logic, based on a set of proposition letters of some finite or infinite cardinality $\kappa $ . Quotient this set under provable equivalence: let A be the set of equivalence classes $[\varphi ]$ of formulas $\varphi $ under the relation of provable equivalence in classical propositional logic. This relation of provable equivalence is not only an equivalence relation, but a congruence with respect to the connectives: if $\varphi ,\psi ,\chi $ are provably equivalent to $\varphi ',\psi ',\chi '$ , respectively, then $\neg \varphi $ and $\psi \wedge \chi $ are also provably equivalent to $\neg \varphi '$ and $\psi '\wedge \chi '$ , respectively. Consequently, there is a function $-$ which maps $[\varphi ]$ to $[\neg \varphi ]$ , and a function $\wedge $ which maps $[\psi ]$ and $[\chi ]$ to $[\psi \wedge \chi ]$ , for all formulas $\varphi ,\psi ,\chi $ . $\mathfrak {A}_\kappa =\langle A,-,\wedge \rangle $ is a Boolean algebra, known as the Lindenbaum–Tarski algebra (of cardinality $\kappa $ ). $\mathfrak {A}_\kappa $ can be shown to be independently generated by the set E of equivalence classes of proposition letters. No two proposition letters are provably equivalent, so $|E|=\kappa $ . As noted above, the Boolean algebra independently generated by a set of a given cardinality is unique up to isomorphism. Thus, a Boolean algebra is independently generated just in case it is isomorphic to a Lindenbaum–Tarski algebra. One particular consequence of this is worth noting: It is well-known that the Lindenbaum–Tarski algebra based on a countably infinite language is, up to isomorphism, the only countably infinite atomless Boolean algebra. So up to isomorphism, there is a unique countably infinite independently generated Boolean algebra, which is the countably infinite atomless Boolean algebra.

In the finite case, an alternative presentation is helpful as well. The powerset $\mathcal {P}(W)$ is the set of subsets of a given set W. This forms a Boolean algebra when we add the operations of relative complement and intersection (mapping any $X\subseteq W$ to $W\backslash X$ and any $Y,Z\subseteq W$ to $Y\cap Z$ , respectively). We write $\mathfrak {P}(W)$ for this algebra. W itself can be taken to be a powerset $\mathcal {P}(S)$ , which gives us the double powerset algebra $\mathfrak {P}\mathcal {P}(S)$ , whose members are sets of subsets of S. For any $s\in S$ , let $s^*=\{w\subseteq S:s\in w\}$ . It can be shown that $\Gamma _S=\{s^*:s\in S\}$ independently generates $\mathfrak {P}\mathcal {P}(S)$ . (We will see an intuitive explanation of this observation in §3.3.) We call $\Gamma _S$ the set of canonical generators of $\mathfrak {P}\mathcal {P}(S)$ . Further, the function mapping any s to $s^*$ is a bijection from S to $\Gamma _S$ , whence S and $\Gamma _S$ have the same cardinality. Thus, any finite set E independently generates a Boolean algebra which is isomorphic to $\mathfrak {P}\mathcal {P}(E)$ . So, a finite Boolean algebra is independently generated just in case it is isomorphic to a double powerset algebra. This does not extend to the infinite case: it follows from the characterization in terms of Lindenbaum–Tarski algebras that no infinite independently generated algebra is isomorphic to a powerset algebra.

So, the finite and infinite cases of independently generated Boolean algebras differ sharply in their structural properties. We will shortly make this more concrete, in the form of the properties of atomicity and completeness, introduced below, which finite independently generated Boolean algebras possess but infinite independently generated Boolean algebras lack. A more straightforward observation concerns the relative cardinalities of E and $\mathfrak {A}$ : assuming E independently generates $\mathfrak {A}$ , if E is finite, then $|A|=2^{2^{|E|}}>|E|$ , whereas if E is infinite, then $|A|=|E|$ .

Finally, it is worth illustrating that a set which independently generates a Boolean algebra does not do so uniquely (even though it does so uniquely up to isomorphism). Consider first the finite case of a double powerset algebra $\mathfrak {P}\mathcal {P}(S)$ based on a finite set S. Any permutation f of $\mathcal {P}(S)$ induces an automorphism $\bar {f}$ of $\mathfrak {P}\mathcal {P}(S)$ , where $\bar {f}(x)=\{f(w):w\in x\}$ . Thus, for any such permutation f, the set $\{\bar {f}(x):x\in \Gamma _S\}$ independently generates $\mathfrak {P}\mathcal {P}(S)$ . To make this more concrete, let S be the two-element set $2=\{0,1\}$ . Then the canonical generators form the following set $\Gamma _2=\{0^*,1^*\}$ :

$$\begin{align*}\{\{\{0\},\{0,1\}\},\{\{1\},\{0,1\}\}\}.\end{align*}$$

As our permutation f of $\mathcal {P}(2)=\{\emptyset ,\{0\},\{1\},\{0,1\}\}$ , let us choose the transposition of $\emptyset $ and $\{0\}$ , which maps these two elements to each other, and the other two elements to themselves. Then $\bar {f}$ maps the elements of $\Gamma _2$ to the elements of the following set, which therefore also independently generates $\mathfrak {P}\mathcal {P}(S)$ :

$$\begin{align*}\{\{\emptyset,\{0,1\}\},\{\{1\},\{0,1\}\}\}.\end{align*}$$

Similarly, in the case of a Lindenbaum–Tarski algebra, the equivalence classes of proposition letters are not unique as independent generators. A simple example of a distinct set of independent generators is the set of equivalence classes of negations of proposition letters.

3.1 Quantification

Logical atomists have to find a way of accommodating quantification. If there are finitely many individuals, this is easily done using conjunction and disjunction. For example, the proposition that everything is F can be identified with the conjunction of propositions $Fx$ , for every individual x (cf. 5.52). Existential quantification can be treated analogously, using disjunction. However, if there are infinitely many individuals, this idea will not obviously work, since Boolean algebras only provide finitary truth-functional operations.

One way of accommodating quantification is therefore to provide infinitary analogs of conjunction and disjunction. This can be done by assuming the algebra of propositions to be complete. A Boolean algebra is complete if every set X of elements has a greatest lower bound $\bigwedge X$ and a least upper bound $\bigvee X$ . Just as the greatest lower bound of two elements $x\wedge y$ can be understood as their conjunction, the greatest lower bound of a set of elements $\bigwedge X$ can be understood as their (possibly infinite) conjunction. The same applies to disjunction and least upper bounds.

The need to accommodate quantifiers therefore motivates us to only consider complete Boolean algebras in modeling logical atomism. Further, the availability of potentially infinitary truth-functional operations suggests adapting the nine conditions discussed above, taking into account not just the binary operation $\wedge $ , but the more general operation $\bigwedge $ on sets. ( $\bigvee $ can be defined in terms of $-$ and $\bigwedge $ , just as $\vee $ can be defined in terms of $-$ and $\wedge $ .)

Although moving to complete Boolean algebras is a natural way of accommodating quantification, it is not the only way. Using the tools of algebraic logic, one could also work with Boolean-valued models of first-order logic, and require only those greatest lower bounds and least upper bounds to exist which need to exist in order to evaluate every first-order formula (see, e.g., Rasiowa and Sikorski [Reference Rasiowa and Sikorski25]). Which greatest lower bounds and least upper bounds then need to exist depends on the choice of the interpretation of non-logical constants. On the level of Boolean algebras, this will not impose any further constraints, since every Boolean algebra can be expanded to an algebraic model of first-order logic by choosing a suitable interpretation of non-logical constants. Along these lines, one could retain the focus on all Boolean algebras, but still adapt the nine conditions discussed above, taking into account now precisely the existing greatest lower bounds and least upper bounds. We leave exploring this interesting variant for another occasion.

3.2 Regimentations, revised

Let us therefore revisit the regimentations of logical atomism discussed above in the setting of complete Boolean algebras. First, we adjust the definition of finite descriptions to take into account the possibility of conjoining arbitrary sets of propositions. This allows us to consider complete descriptions, which decide, for each elementary proposition, whether to include it or its negation. This gives rise to a notion of a set being completely independent.

In the definition of complete independence, it would also have been natural to use a variant of the notion of a complete description which omits the requirement that $T\cup F=E$ ; we could call this an arbitrary description. It is easy to see that the definition of complete independence is not affected by this. We have chosen to work with complete rather than arbitrary descriptions since the notion of a complete description will be very useful in establishing a number of our results.

We adapt the notion of a subalgebra as follows: $\langle B,-',\wedge '\rangle $ is a complete subalgebra of a complete Boolean algebra $\langle A,-,\wedge \rangle $ if it is a subalgebra of $\mathfrak {A}$ which preserves greatest lower bounds (and therefore least upper bounds), in the following sense: for any $X\subseteq B$ , the greatest lower bound of X in $\mathfrak {B}$ is the greatest lower bound in $\mathfrak {A}$ . This gives us a complete notion of generation.

With this, we can adapt the notion of independent generation by defining, for any complete Boolean algebra $\mathfrak {A}$ :

Moving on to the notions of free generation, let a complete homomorphism from $\langle A,-,\wedge \rangle $ to $\langle B,-',\wedge '\rangle $ (both complete Boolean algebras) be a homomorphism f which preserves greatest lower bounds (and therefore least upper bounds), in the following sense: for any $X\subseteq A$ , f maps the greatest lower bound of X in $\mathfrak {A}$ to the greatest lower bound of $\{f(x):x\in X\}$ in $\mathfrak {B}$ . (Correspondingly, a complete endomorphism on an algebra $\mathfrak {A}$ is a complete homomorphism from $\mathfrak {A}$ to $\mathfrak {A}$ .) With this, let $\mathsf {CBA}$ be the class of complete Boolean algebras along with the class of complete homomorphisms among complete Boolean algebras.

We can now consider a variant of $\mathsf {T}$ -free generation, complete $\mathsf {T}$ -free generation, for various classes of complete Boolean algebras $\mathsf {T}$ .

Corresponding to the four cases of free generation considered above, we will be interested in complete $\mathbf {2}$ -free, complete $\mathfrak {A}$ -free, complete $\mathsf {csub}(\mathfrak {A})$ -free, and complete $\mathsf {CBA}$ -free generation, where $\mathsf {csub}(\mathfrak {A})$ is the class of complete subalgebras of $\mathfrak {A}$ .

Let us finally weaken the definition of a Boolean connective so as to include connectives like infinitary conjunction. By a complete Boolean connective we will mean a family of operations defined on complete Boolean algebras that are preserved by all complete homomorphisms. More generally, we define this concept for any class $\mathsf {D}$ of complete Boolean algebras and $\mathsf {T}$ a subclass of $\mathsf {D}$ containing a non-trivial algebra.

We will call a complete $(\mathsf {CBA},\mathsf {CBA})$ -connective a complete Boolean connective. Corresponding to the four classes of connectives we discussed, (namely $(\mathsf {BA},\mathsf {BA})$ , $(\mathfrak {A},\mathfrak {A})$ , $(\mathsf {sub}(\mathfrak {A}),\mathsf {sub}(\mathfrak {A}))$ , and $(\mathsf {BA},\mathbf {2})$ ), there are four corresponding classes of complete connectives that we will consider: complete $(\mathsf {CBA},\mathsf {CBA})$ , $(\mathfrak {A},\mathfrak {A})$ , $(\mathsf {csub}(\mathfrak {A}),\mathsf {csub}(\mathfrak {A}))$ , and $(\mathsf {CBA},\mathbf {2})$ -connectives.

We will aim to characterize these conditions in more direct, structural terms, and to delineate how they relate to each other. The unique decomposition principles turn out to be equivalent to free generation principles, apart from unique decomposition with respect to $(\mathsf {CBA},\mathbf {2})$ -connectives, which is unsatisfiable, as we show in §4. This leaves us with only the remaining five conditions to consider. We have been able to provide structural characterizations of three of these conditions: complete independent generation, complete $\mathbf {2}$ -free generation, and complete $\mathsf {CBA}$ -free generation. These will be presented in §5. The remaining two conditions of complete $\mathfrak {A}$ -free generation and complete $\mathsf {csub}(\mathfrak {A})$ -free generation are more difficult to understand, and we have only been able to describe partially how they relate to each other and the other three notions. The relevant results will be discussed in §6. Before delving into these mathematical questions, it is worth mentioning a closely related way of regimenting logical atomist ideas, which also uses certain complete Boolean algebras.

3.3 Possible worlds

In the literature on logical atomism, some authors have also adopted (complete) Boolean algebras, but imposed a condition which is at least apparently different from those considered above. In particular, Suszko [Reference Suszko30], Moss [Reference Moss21], and Button [Reference Button7] effectively suggest modeling the logical atomist account of propositions using double powerset algebras, as defined above. Somewhat less explicitly, one may also find this understanding of logical atomism in Ramsey [Reference Ramsey22, Reference Ramsey23].

Why double powerset algebras? First, powerset algebras provide very natural models of propositions, using the ideas of possible world semantics. Think of a set W as the set of possible worlds, and assume that the propositions are the sets of possible worlds, identifying any proposition with the set of possible worlds in which it is true (cf. 4.2, 4.4, 4.431). Since a proposition is true in a world just in case its negation is not true there, relative complement serves as the operation of negation. Similarly, a conjunctive proposition is true in a world just in case its conjuncts are true in this world; consequently, intersection serves as the operation of conjunction. Thus, on this picture, we can identify the propositional algebra with the powerset algebra $\mathfrak {P}(W)$ .

Second, from this perspective of possible world semantics, it is natural to think that the logical atomist would take the elementary propositions to determine the possible worlds. That is, taking elementary propositions again to be logically independent, it is natural to think that for any set of elementary propositions X, there is a possible world in which every member of X is true, and every elementary proposition not in X is false. Further, since all propositions are truth-functional combinations of elementary propositions, the truth of every proposition should be determined by the truth and falsity of elementary propositions. Thus, there should only be one world in which every member of X is true and every elementary proposition not in X is false. Consequently, we should be able to identify every possible world with the set of elementary propositions true in it; more generally, we should be able to identify the possible worlds W with the sets of elementary propositions (cf. 4.27, 4.28, 4.45).

It’s important to be clear that on this picture, one world may be a subset of states of another world, but this doesn’t make the latter world more specific. Naturally, all worlds are fully specific, in the sense that they determine which states obtain and which ones don’t obtain. The matter is perhaps easier to see if we work not with worlds as sets of states, but with worlds as characteristic functions of such sets. Then, we can think of a world as an assignment of truth values to states, telling us which states obtain and which ones don’t obtain. On this presentation, it is clear that all worlds are fully specific.

Putting these ideas together, we arrive at double powerset algebras. To be precise, if we literally think of propositions as sets of worlds, we have to be a bit careful about the status of elementary propositions. (If we only think of the possible worlds account as constraining the structure of the algebra of propositions, the following issues do not arise.) By way of example, consider an elementary proposition $e\in E$ . The informal ideas sketched above suggested that the set of worlds in which e is true is $e^*=\{w\subseteq E:e\in w\}$ . Since e is a proposition, it has to be the set of worlds in which it is true, and so $e=e^*$ . But $e\in \{e\}\in e^*$ , whence $e\neq e^*$ . Therefore, we should strictly speaking start from a set of, say, elementary states S, with the set of worlds $W=\mathcal {P}(S)$ . Each state s then corresponding uniquely to an elementary proposition $s^*=\{w\subseteq S:s\in w\}$ , the proposition that state s obtains. We thus arrive at the elementary propositions forming the set $E=\{s^*:s\in S\}$ of elements of the double powerset algebra of propositions $\mathfrak {P}\mathcal {P}(S)$ .

Among finite algebras, we have seen above that this regimentation of logical atomism (relying on the possible worlds account of propositions) fits exactly the condition of independent generation and its equivalent formulations in terms of different kinds of free generation and unique decomposition, at least up to isomorphism. However, among infinite algebras, it diverges sharply from these conditions, since no infinite powerset algebra is independently generated. This follows from the fact that infinite independently generated Boolean algebras are incomplete, while all powerset algebras are complete, with intersections and disjunctions serving as greatest lower bounds and least upper bounds.

The divergence can also be shown by considering the atoms of algebras. An atom of a Boolean algebra $\mathfrak {A}$ is a non-zero element a for which there is no x such that $\bot <x<a$ . $\mathfrak {A}$ is atomless if it has no atoms, and atomic if for every non-zero element x, there is some atom $a\leq x$ . Every powerset algebra is atomic, with the singleton sets serving as atoms. Indeed, a Boolean algebra is isomorphic to a powerset algebra if and only if it is both complete and atomic. In contrast, every infinite free Boolean algebra is not only incomplete but also atomless.

Intuitively, it is not that surprising that the double powerset condition comes apart so decisively from the conditions formulated above. The reason is that in the double powerset condition, possible worlds are essentially generated from elementary propositions by taking conjunctions of elementary propositions and their negations. If there are infinitely many elementary propositions, then these conjunctions have to be infinite. As noted, Boolean algebras which are not complete will not in general provide such an infinitary operation of conjunction. Even if the algebra under consideration is complete, the conditions formulated above do not take into account any infinitary truth-functional operations.

These observations suggest an intriguing possibility: Could it be that independent generation and its variant definitions in terms of free generation and unique decomposition come to match the double powerset condition when they are adjusted to accommodate infinitary operations? We will see below that this is not the case, although some of the conditions come close. In general, we will see that the mathematical situation becomes much more complicated once we move to complete Boolean algebras in the way sketched here. Starting to explore this territory will be the main task for the rest of this paper. In the following, results concerning E and $\mathfrak {A}$ assume implicitly that E is a set of elements of a non-trivial complete Boolean algebra $\mathfrak {A}$ .

5.1 Complete $\mathsf {CBA}$ -free generation

We begin with complete $\mathsf {CBA}$ -free generation, since it can be characterized using a very simple condition. The complete Boolean algebras which are completely $\mathsf {CBA}$ -freely generated are just the finite double powerset algebras. To state this more carefully, we write again $e^*$ for $\{w\subseteq E:e\in w\}$ , where e is an element of a contextually salient set E, and we continue to use this notation in the following.

Complete $\mathsf {CBA}$ -free generation is thus inconsistent with the existence of an infinity of propositions, and so is implausible insofar as it is plausible that there are infinitely many propositions (cf. 4.2211). The Gaifman–Hales theorem also sheds some light on the notion of complete generation: If $\mathfrak {A}$ is generated by E, then every element of A can be obtained from E by a finite sequence of applications of $-$ and $\wedge $ to the members of E. If $\mathfrak {A}$ is completely generated by E, then to construct an arbitrary member of A, we may have to use applications of $\bigwedge $ to infinite sets. However, one might naturally wonder whether we may also need to iterate the applications of $-$ and $\bigwedge $ an infinite number of times. That this is the case follows from Gaifman–Hales. To make this precise, we define, analogous to §2.1:

$$ \begin{align*} E^\infty_0&:=E\\ E^\infty_{n+1}&:=\{-x:x\in E^\infty_n\}\cup\{\bigwedge Y:Y\subseteq E^\infty_n\}\\ E^\infty_\omega&:=\bigcup_{n\in\mathbb{N}}E^\infty_n. \end{align*} $$

Assume for the sake of the example that E is countably infinite. Then by construction, the cardinality of $E^\infty _\omega $ cannot be greater than $\beth _\omega =\mathrm {sup}\{\aleph _0,2^{\aleph _0},2^{2^{\aleph _0}},\dots \}$ . By Gaifman–Hales, $\mathfrak {A}$ may be generated by a countable set G, but be of larger cardinality than $\beth _\omega $ . (Similar considerations apply if we start from an uncountable set E, adjusting the cardinality bound on $E^\infty _\omega $ .) The construction of generated elements may therefore have to be iterated into the transfinite, as we will do in the proof of Lemma 5.5. Of course, this is a consequence of the fact that only $-$ and $\bigwedge $ are treated as primitive here. If in constructing elements of complete Boolean algebras we allowed also $\bigvee $ and all the other possible operations in complete Boolean algebras, every element of A could trivially be constructed from G in just one step (cf. 5.3, 5.32).

5.2 Complete $\mathbf {2}$ -free generation

For much of the following, we need two more basic notions for Boolean algebras, namely, relativization and product. First, let $\langle A, -,\wedge \rangle $ be a Boolean algebra, and $r\in A$ . The relativization of $\mathfrak {A}$ to r is the Boolean algebra $\langle B,-',\wedge '\rangle ,$ where $B=\{y\in A:y\leq r\}$ , $-'x=r\wedge -x$ , and $y\wedge 'z=y\wedge z$ , for all $x,y,z\in B$ . Second, if $\langle A,-_{\mathfrak {A}},\wedge _{\mathfrak {A}}\rangle $ and $\langle B,-_{\mathfrak {B}},\wedge _{\mathfrak {B}}\rangle $ are Boolean algebras, then $\mathfrak {A}\times \mathfrak {B}$ , the product of $\mathfrak {A}$ and $\mathfrak {B}$ , is the Boolean algebra $\langle A\times B,-_{\mathfrak {A\times B}},\wedge _{\mathfrak {A\times B}}\rangle $ such that, for all $x,y,z\in A$ and $x',y',z'\in B$ ,

• $-_{\mathfrak {A\times B}}\langle x,x'\rangle =\langle -_{\mathfrak {A}}x,-_{\mathfrak {B}}x'\rangle $

• $\langle y,y'\rangle \wedge _{\mathfrak {A\times B}}\langle z,z'\rangle =\langle y\wedge _{\mathfrak {A}}z,y'\wedge _{\mathfrak {B}}z'\rangle .$

For brevity, we will often use $-$ and $\wedge $ for the two operations of any Boolean algebra, letting context disambiguate to which algebra the relevant operation belongs.

In the following, we will rely on a basic observation about products of complete Boolean algebras, namely, that any complete Boolean algebra $\mathfrak {A}$ is isomorphic to a product $\mathfrak {B}\times \mathfrak {C}$ of a complete atomless Boolean algebra $\mathfrak {B}$ and a complete atomic Boolean algebra $\mathfrak {C}$ . More specifically, let x be the least upper bound of the set of atoms in $\mathfrak {A}$ . Let $\mathfrak {B}$ be the relativization of $\mathfrak {A}$ to $-x$ , and $\mathfrak {C}$ the relativization of $\mathfrak {A}$ to x. Then $\mathfrak {B}$ is complete and atomless, $\mathfrak {C}$ is complete and atomic, and $\mathfrak {A}$ is isomorphic to $\mathfrak {B}\times \mathfrak {C}$ . (See Givant and Halmos [Reference Givant and Halmos13, p. 227, corollary 2]. To see why this observation applies in particular to atomic and atomless Boolean algebras, note that the trivial Boolean algebra $\mathbf {1}$ is both atomic and atomless.)

We first note that double powerset algebras are completely $\mathbf {2}$ -freely generated by their canonical generators, in the finite as well as in the infinite case. Indeed, we can expand this observation to any product of a complete atomless Boolean algebra with a double powerset algebra, turning each canonical generator of the double powerset algebra into a generator of the product algebra by adjoining an arbitrary element of the atomless algebra. The proof of this observation appeals to another basic notion and observation which will remain important in the following: An ultrafilter U is principal if it has a least element, i.e., if $\bigwedge U\in U$ . Equivalently, a set $U\subseteq A$ is a principal ultrafilter just in case its characteristic function (mapping every element of U to $\top $ and every other element of A to $\bot $ ) is a complete homomorphism from $\mathfrak {A}$ to $\mathbf {2}$ . Furthermore, the principal ultrafilters of a Boolean algebra correspond uniquely to the atoms of the algebra; in particular, if U is a principal ultrafilter, then $\bigwedge U$ is the corresponding atom, and if a is an atom, then $\{x:x\geq a\}$ is the corresponding principal ultrafilter.

Proof. Consider any function $f:E\to 2$ . Let $\Sigma =\{s\in S:f(\langle \beta (s),s^*\rangle )=1\}$ . Define $\hat {f}:A\to 2$ such that

$$\begin{align*}\hat{f}(\langle b,c\rangle)=1\text{ iff }\Sigma\in c.\end{align*}$$

It is straightforward to verify that $\hat {f}$ is a complete homomorphism extending f.

For uniqueness, consider any complete homomorphism g extending f. Since g is a complete homomorphism, the set $U\subseteq A$ of elements mapped by g to $1$ forms a principal ultrafilter. Thus, g maps the atom $a=\bigwedge U$ to $1$ . By construction of $\mathfrak {A}$ , a must be identical to $\langle \bot _{\mathfrak {B}},\{w\}\rangle $ , for some $w\subseteq S$ . We show that $w=\Sigma $ . Consider any $s\in S$ : $f(\langle \beta (s),s^*\rangle )=1$ iff $\langle \beta (s),s^*\rangle \in U$ iff $\langle \bot _{\mathfrak {B}},\{w\}\rangle \leq _{\mathfrak {A}}\langle \beta (s),s^*\rangle $ iff $\{w\}\subseteq s^*$ iff $w\in s^*$ iff $s\in w$ . Thus $w=\Sigma $ . With this, it follows that $\hat {f}(\langle b,c\rangle )=1$ iff $w\in c$ iff $\langle \bot _{\mathfrak {B}},\{w\}\rangle \leq \langle b,c\rangle $ iff $\langle b,c\rangle \in U$ iff $g(\langle b,c\rangle )=1$ . So $\hat {f}=g$ .

Next, we note that up to isomorphism, these are the only examples of complete $\mathbf {2}$ -free generation. To state this result, we assume the convention of writing $\pi _i$ for the ith projection function, which maps any n-tuple with $n\geq i$ to its ith coordinate. For example, $\pi _2(\langle x,y\rangle )=y$ .

Proof. Assume E completely $\mathbf {2}$ -freely generates $\mathfrak {A}$ . Let c be the least upper bound of the set of atoms of $\mathfrak {A}$ , $b=-c$ , and $\mathfrak {B}$ the relativization of $\mathfrak {A}$ to b. As noted above, $\mathfrak {B}$ is complete and atomless. For every $w\subseteq E$ , let $D_w=w\cup \{-e:e\in E\backslash w\}$ . Define a function f on A such that for all $x\in A$ ,

$$\begin{align*}f(x)=\langle x\wedge b,\{w\subseteq E:\bigwedge D_w\wedge c\leq x\}\rangle.\end{align*}$$

We show that $\mathfrak {B}$ and f witness the claim.

First, we show that for every $w\subseteq E$ , $\bigwedge D_w\wedge c$ is an atom of $\mathfrak {A}$ . Let g be the characteristic function of w, mapping any $e\in w$ to $1$ and $e\in E\backslash w$ to $0$ . By complete $\mathbf {2}$ -free generation, g has a unique extension $\hat {g}$ to a complete homomorphism from $\mathfrak {A}$ to $\mathbf {2}$ . Let U be the corresponding ultrafilter, the set of elements mapped by $\hat {g}$ to $1$ . Since $\hat {g}$ is complete, U is principal, whence $a=\bigwedge U$ is an atom. Further, $g(x)=1$ for all $x\in D_w$ , and so $\hat {g}(\bigwedge D_w)=1$ . Thus, $a\leq \bigwedge D_w$ , and so $a\leq \bigwedge D_w\wedge c$ . We show that $a=\bigwedge D_w\wedge c$ . Assume otherwise for contradiction. Then there is another atom $a'\leq \bigwedge D_w\wedge c$ , and so a corresponding principal ultrafilter $U'=\{x\in A:a'\leq x\}$ . Let h be the characteristic function of $U'$ , mapping every $x\in U'$ to $1$ and $x\in A\backslash U'$ to $0$ . Then h is a complete homomorphism from $\mathfrak {A}$ to $\mathbf {2}$ . Further, since $a'\leq \bigwedge D_w$ , $D_w\subseteq U'$ , so h extends g. Thus h contradicts the uniqueness of $\hat {g}$ .

With this observation, it is straightforward to verify that f is a homomorphism. To establish that f is surjective, it suffices to show that any $\langle x,y\rangle \in B\times \mathcal {PP}(E)$ is the image of $x\vee \bigvee _{w\in y}\bigwedge D_w$ under f, which is routine. To establish that f is injective, consider any distinct members x and y of A. Then either $x\wedge b\neq y\wedge b$ or $x\wedge c\neq y\wedge c$ . In the former case, it is immediate that $\pi _1f(x)\neq \pi _1f(y)$ . In the latter case, we may assume without loss of generality that there is an atom $a\leq x$ such that $a\nleq y$ . Let $w=\{e\in E:a\leq e\}$ . As established above, $\bigwedge D_w\wedge c$ is an atom, and so $\bigwedge D_w\wedge c=a$ . Thus $\bigwedge D_w\wedge c\nleq y$ . Thus, w witnesses $\pi _2f(x)\neq \pi _2f(y)$ . So, in either case, $f(x)\neq f(y)$ .

Finally, consider any $e\in E$ , with a view to establishing $\pi _2f(e)=e^*$ . Let $w\subseteq E$ ; it suffices to show that $\bigwedge D_w\wedge c\leq w$ iff $w\in e^*$ , i.e., iff $e\in w$ . If $e\in w$ , the claim is immediate. Otherwise, $-e\in D_w$ . As shown above, $\bigwedge D_w\wedge c$ is an atom, and so non-zero; thus, $\bigwedge D_w\wedge c\nleq e$ , as required.

Putting these two lemmas together, we obtain the following characterization of complete $\mathbf {2}$ -free generation.

5.3 Complete independent generation

To characterize complete independent generation, we begin with a basic observation on complete descriptions based on complete generators of complete algebras. The observation is based on the fact that as in the finite case, we can characterize the smallest complete subalgebra of a given algebra containing a starting set of elements as the result of successively closing the starting set under complementation and greatest lower bounds, now admitting arbitrary greatest lower bounds and iterating the construction into the transfinite.

Proof. For every ordinal $\alpha $ , define a set $G_\alpha \subseteq A$ as follows:

$$ \begin{align*} G_0&:=G\\ G_{\alpha+1}&:=\{-x:x\in G_\alpha\}\cup\{\bigwedge X:X\subseteq G_\alpha\}\\ G_\lambda&:=\bigcup_{\alpha<\lambda}G_\alpha \quad (\lambda \text{ a limit ordinal}). \end{align*} $$

By cardinality considerations, there is some ordinal $\alpha $ at which $G_\alpha $ reaches a fixed point. This set is closed under the two operations, and so a complete subalgebra, which means that $A=G_\alpha $ . It thus suffices to show by a transfinite induction that for every ordinal $\alpha $ and $x\in G_\alpha $ , $\bigwedge D\leq x$ or $\bigwedge D\leq -x$ , which is routine.

We can now characterize complete independent generation in very similar terms to those just used to characterize complete $\mathbf {2}$ -free generation. First, we can again show that a product of a complete atomless Boolean algebra $\mathfrak {B}$ with a double powerset algebra $\mathfrak {P}\mathcal {P}(S)$ is completely independently generated, with one additional cardinality constraint: the atomless factor $\mathfrak {B}$ must be generated by a set G which is no larger than the base set S on which the double powerset algebra is constructed. To produce the witnessing set E of completely independent complete generators, we just need to adjoin to every canonical generator $s^*$ of the double powerset algebra an element $\beta (s)$ of G, ensuring that the image of S under $\beta $ contains each element of G. In other words:

Proof. We begin by showing that the atoms of $\mathfrak {A}$ can be described as the set $\{\bigwedge D:D\vartriangleleft E\}$ . Note that $\{D:D\vartriangleleft E\}=\{D_w:w\subseteq S\}$ , where $D_w$ is defined as follows:

$$\begin{align*}D_w:=\{\langle\beta(s),s^*\rangle:s\in w\}\cup\{-\langle\beta(s),s^*\rangle:s\in S\backslash w\}.\end{align*}$$

It thus suffices to show that $\bigwedge D_w=\langle \bot ,\{w\}\rangle $ , for all $w\subseteq S$ . We consider the two coordinates in turn.

First, working in the atomless algebra $\mathfrak {B}$ : Let $G:=\{\beta (s):s\in S\}$ . If $w\subseteq S$ , then $\{\pi _1(x):x\in D_w\}$ includes some $D\vartriangleleft G$ . By Lemma 5.5, for all $x\in B$ , $\bigwedge D\leq x$ or $\bigwedge D\leq -x$ . Since $\mathfrak {B}$ is atomless, it follows that $\bigwedge D=\bot $ , and so $\bigwedge \{\pi _1(x):x\in D_w\} =\bot $ , as required.

Second, working in the double powerset algebra $\mathfrak {P}\mathcal {P}(S)$ : If $w\subseteq S$ , then $D:=\{\pi _2(x):x\in D_w\}\vartriangleleft \Gamma _S$ . In particular,

$$\begin{align*}D=\{s^*:s\in w\}\cup\{-s^*:s\in S\backslash w\}.\end{align*}$$

It is routine to verify that for all $v\subseteq S$ , $v\in \bigcap D$ iff $v=w$ . Thus $\bigcap D=\{w\}$ , and so $\bigwedge \{\pi _2(x):x\in D_w\}=\{w\}$ , as required.

From this initial observation, we can immediately conclude that E is completely independent: for every $D\vartriangleleft E$ , $\bigwedge D$ is an atom, and so non-zero.

It is also easy to conclude from this observation that E completely generates $\mathfrak {A}$ : Let $\mathfrak {A}'$ be the smallest complete subalgebra of $\mathfrak {A}$ including E. By the initial observation, $\mathfrak {A}'$ contains every atom of $\mathfrak {A}$ . Since every member of a powerset algebra is a union (least upper bound) of some of its atoms, it follows that $\langle \bot ,y\rangle \in \mathfrak {A}'$ , for all $y\in \mathcal {PP}(S)$ . In particular, $\mathfrak {A}'$ contains $\langle \bot ,\top \rangle $ , and so also $\langle \beta (s),\bot \rangle $ , for all $s\in S$ . Since G completely generates $\mathfrak {B}$ , it follows that $\mathfrak {A}'$ also contains $\langle x,\bot \rangle $ , for all $x\in B$ . Thus, $\mathfrak {A}'$ contains any $\langle x,y\rangle \in \mathfrak {A}$ , as required.

As in the case of complete $\mathbf {2}$ -free generation, we can show that up to isomorphism, these are the only examples of complete independent generation.

Proof. The proof follows the pattern of Lemma 5.3. Assume E completely independently generates $\mathfrak {A}$ . Let c be the least upper bound of the set of atoms of $\mathfrak {A}$ , $b=-c$ , and $\mathfrak {B}$ the relativization of $\mathfrak {A}$ to b. As noted above, $\mathfrak {B}$ is complete and atomless. For every $w\subseteq E$ , let $D_w=w\cup \{-e:e\in E\backslash w\}$ . Define a function f on A such that for all $x\in A$ ,

$$\begin{align*}f(x)=\langle x\wedge b,\{w\subseteq E:\bigwedge D_w\leq x\}\rangle.\end{align*}$$

We show that $\mathfrak {B}$ and f witness the claim.

First, we show that for every $w\subseteq E$ , $\bigwedge D_w$ is an atom of $\mathfrak {A}$ . That $\bigwedge D_w>\bot $ is guaranteed by the complete independence of E. As E completely generates $\mathfrak {A}$ , it therefore follows with Lemma 5.5 that $D_w$ is an atom. The remainder of the proof proceeds along the lines of the proof of Lemma 5.3.

Putting the last two lemmas together, we obtain the following characterization of complete independent generation.

One obvious consequence of Theorems 5.4 and 5.8 is that by moving from arbitrary Boolean algebras to complete Boolean algebras, we include not only the finite double powerset algebras, but also infinite double powerset algebras. It is worth noting that in this case, the relevant generators need not be determined uniquely up to isomorphism. Recall how we noted that for any cardinality $\kappa $ , there is a Boolean algebra $\mathfrak {A}$ which is independently generated by a set E of cardinality $\kappa $ . Furthermore, in this case, $\mathfrak {A}$ and E determine each other uniquely, up to isomorphism: $\mathfrak {A}$ is uniquely determined, up to isomorphism, by the cardinality of E, and the cardinality of E is uniquely determined by the structure—indeed, by the cardinality—of $\mathfrak {A}$ . In the case of double powerset algebras, it is obvious that the cardinality of the starting set S, which is identical to the cardinality of the set of canonical generators $\Gamma _S$ , determines $\mathfrak {P}\mathcal {P}(S)$ uniquely, up to isomorphism. If S is finite, its cardinality is also determined by the cardinality of $\mathfrak {P}\mathcal {P}(S)$ , since there is a unique number n such that $2^{2^n}=|\mathfrak {P}\mathcal {P}(S)|$ . This need not hold if S is infinite, since it is consistent with ZFC that there are distinct cardinals $\kappa ,\lambda $ such that $2^\kappa =2^\lambda $ . (This consistency claim can be obtained using forcing, the standard technique for independence proofs in set theory. For example, it follows as a corollary of Easton’s theorem; see Jech [Reference Jech16, p. 232, theorem 15.18].) In this case, $\mathfrak {P}\mathcal {P}(\kappa )$ is isomorphic to $\mathfrak {P}\mathcal {P}(\lambda )$ while $\Gamma _\kappa =\kappa \neq \lambda =\Gamma _\lambda $ . In such a case, there is a double powerset algebra which is, e.g., completely independently generated by two sets of distinct cardinalities. Thus, the relevant regimentations of logical atomism admitting all double powerset algebras need not even pin down the elementary propositions up to isomorphism.

Another obvious consequence of Theorems 5.4 and 5.8 is that neither of the relevant constraints requires the generated algebra to be a double powerset algebra. In the case of complete $\mathbf {2}$ -free generation, it is perhaps not all that surprising that some complete Boolean algebra which is not isomorphic to a double powerset algebra is completely $\mathbf {2}$ -freely generated by one of its sets. After all, complete $\mathbf {2}$ -free generation only captures the idea that the truth values of all propositions are uniquely determined by the truth values of the elementary propositions. The matter is perhaps more surprising in the case of complete independent generation. Along lines discussed in the proof of Lemma 5.7, if E completely independently generates $\mathfrak {A}$ , then the atoms of $\mathfrak {A}$ correspond uniquely to the subsets of E, via the greatest lower bounds of complete descriptions in terms of E. This is as one might expect. The (arbitrary) disjunctions of atoms naturally form a double powerset structure. However, we have seen that it is consistent with complete independent generation that further elements are generated: $\mathfrak {A}$ might be the product of an atomless algebra $\mathfrak {B}$ with a double powerset algebra $\mathfrak {P}\mathcal {P}(S)$ . In this case, every generating element consists of a component of the atomless algebra and a component of the double powerset algebra. When we form the atoms as greatest lower bounds of complete descriptions in terms of E, the atomless components disappear. Thus, once we have formed the disjunction of all atoms, we can use this to access the atomless component of the generating elements, and so generate the rest of the algebra. Thus, aside from the expected double powerset structure containing $2^{2^{|E|}}$ elements, complete independent generation allows the generation of further, maybe unexpected, propositions, based on the atomless factor $\mathfrak {B}$ . It is worth noting that with the Gaifman–Hales theorem, it follows immediately that this atomless factor may be arbitrarily large: If E is countably infinite, the corresponding double powerset algebra will have cardinality $2^{2^{|E|}}=2^{2^{\aleph _0}}=\beth _2$ . But since there are arbitrarily large complete Boolean algebras completely generated by a countable set, the atomless factor may be arbitrarily large. The unexpected atomless factor may therefore dwarf the expected double powerset factor by an arbitrarily large degree.

So, in the case of complete $\mathbf {2}$ -free and completely independent generation at least, the rapprochement with the double powerset picture is incomplete. Naturally, one can close the gap by making further assumptions, such as atomicity: it follows immediately from Theorem 5.8 that a complete atomic Boolean algebra $\mathfrak {A}$ is, e.g., completely independently generated by a set E just in case there is an isomorphism from $\mathfrak {A}$ to $\mathfrak {P}\mathcal {P}(E)$ which maps every $e\in E$ to $e^*$ . Thus, to arrive at the double powerset picture, it also suffices to assume, in addition to completely independently generated, that the disjunction of the conjunctions of complete descriptions in terms of E is the top element, i.e.,

$$\begin{align*}\bigvee_{D\vartriangleleft E}\bigwedge D=\top.\end{align*}$$

Such additional assumptions may even be taken to be part of the more general logical atomist picture (cf. 4.46, 4.461).

6.1 Basic observations

We begin with the relationship between complete $\mathsf {CBA}$ -free generation and complete $\mathsf {csub}(\mathfrak {A})$ -free generation. That the former entails the latter is immediate. That the entailment cannot be reversed follows by Theorem 5.1 and the following result.

Proof. Consider any complete subalgebra $\mathfrak {B}$ of $\mathfrak {A}=\mathfrak {P}\mathcal {P}(S)$ , and function $f:\Gamma _S\to B$ . Let $\hat {f}:A\to A$ such that for all $x\in A$ ,

$$\begin{align*}\hat{f}(x)=\{w\subseteq S:\{s\in S:w\in f(s^*)\}\in x\}.\end{align*}$$

It is straightforward to show that $\hat {f}$ is a complete homomorphism from $\mathfrak {A}$ to $\mathfrak {A}$ which extends f.

Recall that $\Gamma _S$ completely generates $\mathfrak {A}$ . By inductions on the generation of $\mathfrak {A}$ (as in the proof of Lemma 5.5), it follows that $\mathrm {im}(\hat {f})\subseteq B$ whence $\hat {f}$ is a complete homomorphism from $\mathfrak {A}$ to $\mathfrak {B}$ . It also follows that $\hat {f}$ is the only such homomorphism extending f: Let g be a complete homomorphism from $\mathfrak {A}$ to $\mathfrak {B}$ extending f. By an induction, we can show that at every stage of the generation of $\mathfrak {A}$ from $\Gamma _S$ , $\hat {f}$ and g agree on the elements generated up to this stage. The base case follows from the fact that both $\hat {f}$ and g extend f. Successor stages follow from the fact that both $\hat {f}$ and g are complete homomorphisms, and union stages are immediate. Thus g is identical to $\hat {f}$ , meaning that $\hat {f}$ is unique.

It is immediate that complete $\mathsf {csub}(\mathfrak {A})$ -free generation entails complete $\mathfrak {A}$ -free generation. That complete $\mathsf {csub}(\mathfrak {A})$ -free generation entails complete independent generation is shown in the next result.

It follows immediately from Theorems 5.8 and 5.4 that complete independent generation entails complete $\mathbf {2}$ -free generation, but not vice versa. Moreover, there are complete Boolean algebras completely $\mathbf {2}$ -freely generated by some set, and not completely independently generated by any set.

It only remains to show that complete independent generation does not entail complete $\mathfrak {A}$ -free generation. Since we have seen that complete $\mathsf {csub}(\mathfrak {A})$ -free generation entails both complete independent generation and complete $\mathfrak {A}$ -free generation, it immediately follows that complete independent generation does not entail complete $\mathsf {csub}(\mathfrak {A})$ -free generation. Moreover, we can show not only that some set which completely independently generates an algebra $\mathfrak {A}$ fails to completely $\mathfrak {A}$ -freely generate it, but more specifically that there is a completely independently generated algebra $\mathfrak {A}$ such that no set which completely independently generates $\mathfrak {A}$ also completely $\mathfrak {A}$ -freely generates $\mathfrak {A}$ . There is a further question which we leave open here, which is whether there is a completely independently generated algebra which is not completely $\mathfrak {A}$ -freely generated at all. Establishing our results on the relationship between complete independent generation and complete $\mathfrak {A}$ -free generation is somewhat complicated. We therefore devote a separate section to it.

6.2 Complete independent generation without complete $\mathfrak {A}$ -free generation

For the result of this section, we need a number of further notions for a Boolean algebra $\mathfrak {A}$ . First, elements $x,y\in A$ are disjoint if $x\wedge y=\bot $ . An antichain of a $\mathfrak {A}$ is a set $X\subseteq A$ which is pairwise disjoint, i.e., such that any two distinct elements of X are disjoint. $\mathfrak {A}$ satisfies the countable chain condition if every antichain of $\mathfrak {A}$ is countable. Finally, $\mathfrak {A}$ is homogeneous if for every non-zero $x\in A$ , the relativization of $\mathfrak {A}$ to x is isomorphic to $\mathfrak {A}$ itself. To these standard notions, we add two non-standard notions which will be helpful as well.

Using these definitions, we aim to show the following result.

We split up the proof of this result into a number of lemmas. Theorem 5.8 tells us that to construct an algebra which is completely independently generated, we can start, without loss of generality, with a product $\mathfrak {A}=\mathfrak {B}\times \mathfrak {P}\mathcal {P}(S)$ , with S of at least the size of some set of complete generators for $\mathfrak {B}$ . We can easily satisfy this cardinality constraint by letting $S=B$ . The challenge is now to ensure that complete $\mathfrak {A}$ -free generation fails, for any completely independently generating set $E\subseteq A$ . It turns out that this can be done by choosing $\mathfrak {B}$ to be infinite, homogeneous, and satisfying the countable chain condition.

That there are algebras $\mathfrak {B}$ satisfying these conditions is well-known. For example, for any infinite set I, consider the partial order of finite partial functions from I to $2$ , ordered by reverse subset inclusion. The corresponding partial order topology determines a complete Boolean algebra $\mathfrak {B}$ . This algebra is infinite and homogeneous, and so atomless, and satisfies the countable chain condition. The construction is commonly used, so we omit the details (see Koppelberg [Reference Koppelberg, Monk and Bonnet17, p. 181 and p. 64, example 6] and Bell [Reference Bell5, p. 50]).

So, for the remainder of this section, let $\mathfrak {B}$ be infinite, homogeneous, and satisfying the countable chain condition. Let $\mathfrak {A}:=\mathfrak {B}\times \mathfrak {P}\mathcal {P}(B)$ . Let $E\subseteq A$ be any set which completely independently generates $\mathfrak {A}$ , and let $G:=\{\pi _1(e):e\in E\}$ . Note that by Lemma 5.7, G completely generates $\mathfrak {B}$ .

Next, we note that if E completely $\mathfrak {A}$ -freely generates $\mathfrak {A}$ , then its complete descriptions conjoin to the atoms of $\mathfrak {A}$ .

The next lemma is central; it shows that in the envisaged algebra $\mathfrak {A}$ , the first coordinates of any completely $\mathfrak {A}$ -freely generating set must be divisible.