In this section I establish the results stated above, to the effect that the holistic definitions of logical consequence I have provided for modal propositional logic and nonmodal and modal first-order logic are equivalent to the atomistic alternatives.

Modal propositional logic

If M is a modal model, the set of MPL-valuations generated by M is the following set:

$$\begin{aligned} \{v:\,{\text {for some}}\,w\in W_{M},\,{\text {for every}}\,{ MPL}{\mathrm {-sentence}}\;\phi ,V_{M}(\phi ,w)=v(\phi )\} \end{aligned}$$

Lemma 1

A set of MPL-valuations is m-Boolean just in case it is generated by some modal model.

Proof

Let M be a modal model, and let V be the set of MPL-valuations it generates. We need to show that V is m-Boolean.

We can easily check that every element of V satisfies the clauses of the definition of Boolean valuation. We need to show, in addition, that for every \(v\in V\), if for every \(v'\in V\) that actualizes v, \(v'(\phi )=T\), then \(v(\square \phi ) = T\).

Let \(v\in V\) and let w be such that for every MPL-sentence \(\phi , v(\phi )=V_{M}(\phi ,w)\). Assume that for every \(v'\in V\) that actualizes v, \(v'(\phi )=T\). We need to show that \(v(\square \phi )=T\). We argue as follows:

Let V now be an m-Boolean set of MPL-valuations. Let M be the modal model defined as follows:

\(W_{M}=V\).

For all \(v,v'\in W_{M}\), \(vR_{M}v'\) just in case \(v'\) actualizes v .

For every atom \(\alpha \) and every \(v\in W_{M}\), \(A_{M}(\alpha ,v)=v(\alpha )\).

We show that M is a modal model that generates V. For this we show by induction on MPL-sentences that for every MPL-sentence \(\phi \) and every \(v\in V, V_{M}(\phi ,v)=v(\phi )\).

For the inductive clause for \(\square \) we assume (IH) that for every \(v'\in V, V_{M}(\phi ,v')=v'(\phi )\). We need to show that for every \(v\in V, V_{M}(\square \phi ,v)=v(\square \phi )\). We argue as follows:

\(\square \)

The equivalence of the atomistic and holistic definitions of K-logical consequence expresssed by Theorem 1 is a straightforward corollary of Lemma 1.

First-order logic

We show first that holistic logical consequence entails atomistic logical consequence. For this purpose we’ll need to invoke some preliminary results.

Lemma 2

If \({\mathfrak {A}}\) is an L-structure, s a variable-interpretation in \({\mathfrak {A}}\), x a variable and c an individual constant of L, then for every L-formula \(\phi \),

$$\begin{aligned}v_{{\mathfrak {A}}}(\phi , s_{(x/c_{{\mathfrak {A}}})})=v_{{\mathfrak {A}}}((\phi )[c/x], s).\end{aligned}$$

Proof

By induction on L-formulas. See, e.g., Zalabardo (2000, pp. 155–157). \(\square \)

We show next that the sentential valuations generated by certain structures are q-Boolean. If L is a first-order language, \({\mathfrak {A}}\) is an L-structure, and C is a set of individual constants not in L of the same cardinality as the universe A of \({\mathfrak {A}}\), let \(L^{+}\) be the onomastic expansion of L that we obtain by adding the elements of C to the set of individual constants of L. And let \({\mathfrak {A}}^{+}\) be the \(L^{+}\)-structure that we get from \({\mathfrak {A}}\) by adding: for every \(c\in C, c_{{\mathfrak {A}}^{+}}=f(c)\), for some one-to-one correspondence f between C and A.

Lemma 3

If L is a first-order language and \({\mathfrak {A}}\) is an L-structure, then \(vs_{{\mathfrak {A}}^{+}}\) is a q-Boolean \(L^{+}\!\)-valuation.

Proof

We need to show that \(vs_{{\mathfrak {A}}^{+}}\) satisfies the clauses of the definition of q-Boolean \(L^{+}\!\)-valuation. We provide the arguments for (\(\forall \)) and (\(\doteq b\)).

For (\(\forall \)) we argue as follows:

\(\square \)

Lemma 4

Let L be a first-order language and let \(L'\) be an onomastic expansion of L. Let \({\mathfrak {A}}'\) be an \(L'\)-structure, and let \({\mathfrak {A}}\) be the L-structure that we obtain by removing from \({\mathfrak {A}}'\) the interpretations of the new constants of of \(L'\).

1. For every L-term t and every variable-interpretation s in \({\mathfrak {A}}\), $$\begin{aligned}den_{{\mathfrak {A}}}(t,s)=den_{{\mathfrak {A}}'}(t,s)\end{aligned}$$ 2. For every L-formula \(\phi \) and every variable-interpretation s in \({\mathfrak {A}}\), $$\begin{aligned}v_{{\mathfrak {A}}}(\phi ,s)=v_{{\mathfrak {A}}'}(\phi ,s)\end{aligned}$$

Proof

2 by induction on L-formulas. \(\square \)

We are now in a position to show that holistic logical consequence entails atomistic logical consequence.

Theorem 4

Let \(\phi \) be a sentence of a first-order language L and let \(\varGamma \) be a set of L-sentences. 1 entails 2:

1. For every onomastic expansion \(L'\) of L, for every q-Boolean \(L'\)-valuation v, if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\). 2. For every L-structure \({\mathfrak {A}}\), if \(vs_{{\mathfrak {A}}}(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(vs_{{\mathfrak {A}}}(\phi )=T\).

Proof

Assume 1, and let \({\mathfrak {A}}\) be an L-structure such that \(vs_{{\mathfrak {A}}}(\gamma )=T\) for every \(\gamma \in \varGamma \). We need to prove that \(vs_{{\mathfrak {A}}}(\phi )=T\). We argue as follows:

\(\square \)

We turn now to showing that atomistic logical consequence entails holistic logical consequence.

Let L be a first-order language with at least one individual constant, and let v be a q-Boolean L-valuation. Let E be the relation on the set of individual constants of L defined as follows: For all individual constants \(c_{1}, c_{2}, c_{1} E c_{2}\) if and only if \(v(c_{1}\doteq c_{2})=T\). It follows directly from the following result that E is an equivalence relation.

Lemma 5

If L is a first-order language and v a q-Boolean L-valuation, then, for all individual constants c, d, e the following hold:

1. \(v(c\doteq c)=T\) 2. \(v(c\doteq d)=v(d\doteq c)\) 3. If \(v(c\doteq d)=v(d\doteq e)=T\), then \(v(c\doteq e)=T\)

Proof

1 follows directly from the definition of q-Boolean valuation.

For 2 we argue as follows:

\(\square \)

Let \([c]_{E}\) denote the equivalence class generated by c with E. Now, if v is a q-Boolean L-valuation, the Henkin structure generated by v is the L-structure \({\mathfrak {A}}_{v}\) defined as follows:

The universe \(A_{v}\) of \({\mathfrak {A}}_{v}\) is the set of equivalence classes generated by E .

For every individual constant c of L , \(c_{{\mathfrak {A}}_{v}}=[c]_{E}\).

For every n-place predicate P of L, \(\langle [c_{1}]_{E},\ldots ,[c_{n}]_{E}\rangle \in P_{{\mathfrak {A}}_{v}}\) if and only if \(v(Pc_{1}\ldots c_{n})=T\).

Lemma 6

If L is a first-order language with at least one individual constant and v a q-Boolean L-valuation, then for every L-sentence \(\phi \) and every variable-interpretation s in \({\mathfrak {A}}_{v}\), \(v(\phi )=v_{{\mathfrak {A}}_{v}}(\phi ,s)\).

Proof

We define the rank of an L-formula \(\phi \), \(r(\phi )\), by the following recursion:

For every atomic L -formula \(\phi \), \(r(\phi )=1\).

For every L -formula \(\phi \), \(r(\lnot \phi )=r(\forall x\phi )=r(\phi )+1\).

For all L-formulas \(\phi ,\psi \), \(r(\phi \wedge \psi )=Max(r(\phi ),r(\psi ))+1\).

We can establish the result by strong induction on the rank of a formula in the following form: for every L-formula \(\phi \) and every variable-interpretation s in \({\mathfrak {A}}_{v}\), if \(\phi \) is an L-sentence, then \(v(\phi )=v_{{\mathfrak {A}}_{v}}(\phi ,s)\). For the inductive step, we assume (IH) that the result holds for every formula of rank no greater than n. We show that it holds for formulas of rank \(n+1\). We provide the argument for \(\forall \).

Let \(\forall x\phi \) be an L-sentence of rank \(n+1\). By the definition of rank, for every individual constant c of L, \((\phi )[c/x]\) is an L-sentence of rank n and by IH the result holds for \((\phi )[c/x]\). We argue as follows:

\(\square \)

We can now prove that atomistic logical consequence entails holistic logical consequence.

Theorem 5

Let \(\phi \) be a sentence of a first-order language L and let \(\varGamma \) be a set of sentences of L. 2 entails 1:

1. For every onomastic expansion \(L'\) of L, for every q-Boolean \(L'\)-valuation v, if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\). 2. For every L-structure \({\mathfrak {A}}\), if \(vs_{{\mathfrak {A}}}(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(vs_{{\mathfrak {A}}}(\phi )=T\).

Proof

Assume 2. Let \(L'\) be an onomastic expansion of L, let v be a q-Boolean \(L'\)-valuation such that \(v(\gamma )=T\) for every \(\gamma \in \varGamma \). We need to prove that \(v(\phi )=T\). Let \({\mathfrak {A}}^{L}_{v}\) be the restriction to L of the Henkin structure generated by v, \({\mathfrak {A}}_{v}\).

We argue as follows:

\(\square \)

We have now attained our goal for the present section. It follows from Theorems 4 and 5 that atomistic logical consequence and holistic logical consequence are one and the same relation, as expressed by Theorem 2.

Modal first-order logic

Our first goal is to show that holistic logical consequence entails atomistic logical consequence. We proceed in the same way as with nonmodal first-order logic.

Lemma 7

Let L be a modal first-order language. If M is an L-model , \(w\in W_{M}\), s a variable-interpretation in M and c an individual constant of L, then for every L-formula \(\phi \),

$$\begin{aligned} V_{M}(\phi ,w, s_{(x/c_{M})})=V_{M}((\phi )[c/x], w, s). \end{aligned}$$

Proof

By induction on L-formulas. The base and the inductive clauses for \(\lnot ,\wedge \) and \(\forall \) are handled in the same way as in the proof of Lemma 2. We provide the clause for \(\square \).

Let \(\phi \) be an L-formula. Assume (IH) that for every \(w\in W_{M}\) and every variable-interpretation s in M, \(V_{M}(\phi ,w, s_{(x/c_{M})})=V_{M}((\phi )[c/x], w, s)\). We need to show that for every \(w\in W_{M}\) and every variable-interpretation s in M, \(V_{M}(\square \phi ,w, s_{(x/c_{M})})=V_{M}((\square \phi )[c/x], w, s)\). We argue as follows:

\(\square \)

We show next that the sets of sentential valuations generated by certain structures are mq-Boolean. If L is a modal first-order language, M is an L-structure, and C is a set of individual constants not in L of the same cardinality as the universe \(D_{M}\) of M, let \(L^{+}\) be the the onomastic expansion of L that we obtain by adding the elements of C to the set of individual constants of L. And let \(M^{+}\) be the \(L^{+}\)-structure that we get from M by adding, for every \(c\in C, c_{M^{+}}=f(c)\), for some one-to-one correspondence f between C and \(D_{M}\).

Lemma 8

If L is a modal first-order language and M is an L-structure, then \(\{vs^w_{M^{+}}:w\in W_{M^{+}}\}\) is a mq-Boolean set of \(L^{+}\)-valuations.

Proof

We first need to show that, for every \(w\in W_{M^{+}}, vs^w_{M^{+}}\) is q-Boolean. Let \(w\in W_{M^{+}}\). We provide the arguments for clauses (\(\forall \)) and (\(\doteq b\)) of the definition.

For (\(\forall \)) we argue as follows:

\(\square \)

Lemma 9

Let L be a modal first-order language and let \(L'\) be an onomastic expansion of L. Let \(M'\) be an \(L'\) structure, and let M be the L-structure that we obtain by removing from \(M'\) the interpretations of the symbols of \(L'\) not in L.

1. For every L-term t, every \(w\in W_{M}\) and every variable-interpretation s in M, \(den_{M}(t,w,s)=den_{M'}(t,w,s)\). 2. For every L-formula \(\phi \), every \(w\in W_{M}\) and every variable-interpretation s in M, \(v_{M}(\phi ,w,s)=v_{M'}(\phi ,w,s)\).

Proof

2 by induction on L-formulas. \(\square \)

We can now establish that holistic K-logical consequence entails atomistic K-logical consequence.

Theorem 6

Let \(\phi \) be a sentence of a modal first-order language L and let \(\varGamma \) be a set of sentences of L. 1 entails 2:

1. For every onomastic expansion \(L'\) of L, for every mq-Boolean set V of \(L'\)-valuations and every \(v\in V\), if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\). 2. For every L-model M and every \(w\in W_{M}\), if \(V\!S_{M}(\gamma ,w)=T\) for every \(\gamma \in \varGamma \), then \(V\!S_{M}(\phi ,w)=T\).

Proof

Assume 1, and let M be an L-model and \(w\in W_{M}\) such that \(V\!S_{M}(\gamma , w)=T\) for every \(\gamma \in \varGamma \). We need to prove that \(V\!S_{M}(\phi ,w)=T\). We argue as follows:

\(\square \)

We turn now to our final task of establishing that atomistic K-logical consequence entails holistic K-logical consequence.

Let L be a modal first-order language with at least one individual constant, and let V be an mq-Boolean set of L-valuations. Let E be the relation on the set of individual constants of L defined as follows: For all individual constants \(c_{1}, c_{2}, c_{1}E c_{2}\) if and only if \(v(c_{1}\doteq c_{2})=T\), for any v in V. We can easily prove, as we did for nonmodal first-order logic, that E is an equivalence relation. Let \([c]_{E}\) denote the equivalence class generated by c with E.

Now, if V is an mq-Boolean set of L-valuations, the Henkin model generated by V is the L-model \(M_{V}\) defined as follows:

\(W_{M_{V}}=V\).

\(D_{M_{V}}\) is the set of equivalence classes generated by E .

\(R_{M_{V}}\) is the actualization relation on V .

For every individual constant c of L , \(c_{M_{V}}=[c]_{E}\).

For every n-place predicate P of L and every \(v\in V\), \(\langle [c_{1}]_{E},\ldots ,[c_{n}]_{E}\rangle \in P^{v}_{M_{V}}\) if and only if \(v(Pc_{1}\ldots c_{n})=T\).

Lemma 10

If L is a modal first-order language with at least one individual constant and V is an mq-Boolean set of L-valuations, then for every L-sentence \(\phi \) and every \(v\in V\), \(v(\phi )=V_{M_{V}}(\phi , v,s)\), for any variable-interpretation s in \(M_{V}\).

Proof

By strong induction on the rank of a formula (see proof of Lemma 6; add: \(r(\square \phi )=r(\phi )+1\)), in the following form: for every L-formula \(\phi \), every \(v\in V\) and every variable-interpretation s in \(M_{V}\), if \(\phi \) is a sentence, then \(v(\phi )=v_{M_{V}}(\phi ,v,s)\). The argument is the same as in the proof of Lemma 6. We provide the inductive clauses for \(\forall \) and \(\square \).

Let \(\forall x\phi \) be an L-sentence of rank \(n+1\). By the definition of rank, for every individual constant c, \((\phi )[c/x]\) is an L-sentence of rank n and by IH the result holds for \((\phi )[c/x]\). We argue as follows:

Let \(\square \phi \) be an L-sentence of rank \(n+1\). By the definition of rank, \(\phi \) is an L-sentence of rank n and by IH the result holds for \(\phi \): for every \(v\in V\) and every variable-interpretation s in \(M_{V}\), \(v(\phi )=V_{M_{V}}(\phi ,v,s)\). We argue as follows:

\(\square \)

We can now establish that atomistic K-logical consequence entails holistic K-logical consequence.

Theorem 7

Let \(\phi \) be a sentence of a modal first-order language L and let \(\varGamma \) be a set of sentences of L. 2 entails 1:

1. For every onomastic expansion \(L'\) of L, for every mq-Boolean set V of \(L'\)-valuations and every \(v\in V\), if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\). 2. For every L-model M and every \(w\in W_{M}\), if \(V\!S_{M}(\gamma ,w)=T\) for every \(\gamma \in \varGamma \), then \(V\!S_{M}(\phi ,w)=T\).

Proof

Assume 2. Let \(L'\) be an onomastic expansion of L, let V be an mq-Boolean set of \(L'\)-valuations, and let v be an \(L'\)-valuation in V such that \(v(\gamma )=T\) for every \(\gamma \in \varGamma \). We need to prove that \(v(\phi )=T\). Let \(M^{L}_{V}\) be the restriction to L of \(M_{V}\), the Henkin model generated by V.

We argue as follows:

\(\square \)

It follows from Theorems 6 and 7 that the holistic definition of logical consequence for modal first-order logic is equivalent to the atomistic definition, as expressed by Theorem 3.