Logic without metaphysics
Abstract
Standard definitions of logical consequence for formal languages are atomistic. They take as their starting point a range of possible assignments of semantic values to the extralogical atomic constituents of the language, each of which generates a unique truth value for each sentence. In modal logic, these possible assignments of semantic values are generated by Kripkestyle models involving possible worlds and an accessibility relation. In firstorder logic, they involve the standard structures of model theory, as sets of objects from which the extralogical symbols of the language receive their denotations. I argue that there is an alternative, holistic, approach to the task of defining logical consequence for a formal language. It specifies necessary and sufficient conditions for an assignment of truth values to all the sentences of the language to be compatible with the intended interpretation of its logical constants. It achieves this without invoking possible assignments of semantic values to the extralogical atomic constituents of the language, or the formal resources that are employed to generate these. I show how this approach can be successfully applied to modal propositional logic and to firstorder logic, modal as well as nonmodal. I show that the holistic definitions of logical consequence that I supply for these languages are equivalent to the standard atomistic definitions.
Keywords
Modality Necessity Logical consequence Formal semantics Substitutional quantification Possible worlds Modal semantics1 Atomism and holism in formal semantics
The subject matter of this paper is the contrast between two approaches to the task of defining the relation of logical consequence in a formal language. I am going to use for these approaches the labels atomistic and holistic. Logical consequence in a formal language is a binary relation pairing sets of sentences of the language (the premises) with sentences of the language (the conclusion). It is intended to model the relation that obtains between a set of premises and a conclusion when the forms of premises and conclusion make it impossible for the conclusion to be false if the premises are all true. The atomistic approach and the holistic approach are two alternative strategies for the task of defining this relation.
Standard semantic definitions of logical consequence for formal languages follow the atomistic approach.^{1} Their starting point is a range of possible assignments of semantic values to the extralogical atomic constituents of the sentences of the language. I am going to refer to the items that play this role in the atomistic approach as ASAs (Atomic Semantic Assignments). In propositional logic, the role of ASAs is played by atomic valuations—functions from the set of atoms to the set \(\{T,F\}\). In modal propositional logic, ASAs are possible worlds in Kripkestyle modal models. In firstorder logic, ASAs are the usual structures of model theory, consisting of a set of objects and an assignment of extensions over this set to the extralogical symbols of the language. In modal firstorder logic, ASAs are items that combine the features of modal propositional ASAs and nonmodal firstorder ASAs.
In each case, ASAs are so chosen that for every ASA and every sentence of the language, there will be a unique truth value for the sentence that’s compatible with the ASA and with the intended interpretations of the logical constants that figure in the sentence. The next step in the implementation of the atomistic approach is to specify, for each ASA and each sentence of the language, the truth value that the sentence will receive from the ASA. This is usually presented as a definition of truth for the language. Once this has been achieved, we are in a position to define logical consequence, saying that a sentence \(\phi \) of the language is a logical consequence of a set \(\varGamma \) of sentences of the language just in case every ASA is such that if it gives the value True to every element of \(\varGamma \), then it also gives the value True to \(\phi \).
The atomistic approach is almost universally employed. But is it compulsory? The main thesis of this paper is that it isn’t: for propositional and firstorder logic, modal as well as nonmodal, there is a viable alternative—the holistic approach. In the holistic approach, ASAs don’t play any role. Its starting point is the set of valuations for the language in question—functions from the set of sentences of the language (all sentences, molecular as well as atomic) to the set \(\{T,F\}\). Some valuations will be compatible with the intended interpretations of the logical constants of the language, while other valuations will be incompatible with these interpretations. This contrast is the central concept of the holistic approach. It proceeds by specifying necessary and sufficient conditions for a valuation to be compatible with the intended interpretations of the logical constants of the language. I shall refer to valuations that satisfy this condition as admissible. The holistic approach seeks to specify which valuations are admissible without invoking ASAs—by looking directly at how the intended interpretations of the logical constants rule out some combinations of truth values for the sentences of the language. Once we have specified which valuations are admissible, logical consequence can be defined directly, saying that a sentence \(\phi \) is a logical consequence of a set of sentences \(\varGamma \) just in case every admissible valuation is such that if all the elements of \(\varGamma \) receive the value True from it, then \(\phi \) also receives the value True.
My main goal in this paper is to show how the holistic approach can be successfully applied to the task of defining logical consequence for propositional and firstorder languages, modal as well as nonmodal. In Sect. 2 I shall introduce the contrast between the two approaches in the context of (nonmodal) propositional logic, where they are virtually interchangeable. In Sect. 3, I provide a holistic definition of logical consequence for modal propositional logic. In Sect. 4, I apply the holistic approach to firstorder logic. And in Sect. 5, I present a holistic treatment of modal firstorder logic, combining the ideas deployed for modal propositional logic and nonmodal firstorder logic. In each case I will start by providing a detailed presentation of the atomistic treatment of the language in question. These will be familiar to the reader, but they are offered for comparison and to facilitate the proofs of the equivalence of the holistic definitions I provide with the standard atomistic alternatives. Proving these results is the business of Sect. 7. Section provides a brief discussion of the philosophical relevance of these formal results.
2 Propositional logic
In nonmodal propositional logic, the contrast between the atomistic approach and the holistic approach is of no great consequence and can easily go unnoticed. Nevertheless, both approaches have a clear application to this case. My goal in this section is to present how both the atomistic approach and the holistic approach can be used to define logical consequence in nonmodal propositional logic.
Let the language of propositional logic be the set PL of sentences defined inductively in the usual way, with a denumerable set of atoms as the base and inductive clauses for the sentential connectives, say \(\lnot \) and \(\wedge \).
2.1 The atomistic approach
 (Atoms)

For every atom \(\alpha , v_{a}(\alpha )=a(\alpha )\).
 (\(\lnot \))
 For every PLsentence of the form \(\lnot \phi \),$$\begin{aligned} v_{a}(\lnot \phi )= {\left\{ \begin{array}{ll} T&{} \quad \text {if } v_{a}(\phi )=F;\\ F&{} \quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$
 (\(\wedge \))
 For every PLsentence of the form \(\phi \wedge \psi \),$$\begin{aligned} v_{a}(\phi \wedge \psi )= {\left\{ \begin{array}{ll} T&{} \quad \text{ if } v_{a}(\phi )=v_{a}(\psi )=T;\\ F&{} \quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$

A PLsentence \(\phi \) is a logical consequence of a set of PLsentences \(\varGamma \) (\(\varGamma \vDash \phi \)) just in case for every atomic valuation a, if \(v_{a}(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v_{a}(\phi )=T\).
2.2 The holistic approach
 (\(\lnot \))

For every PLsentence \(\phi \), \(v(\phi ) \ne v(\lnot \phi )\).
 (\(\wedge \))

For all PLsentences \(\phi ,\psi , v(\phi \wedge \psi )=T\) iff \(v(\phi )=v(\psi )=T\).
In spite of the obvious correlation between the definition of Boolean PLvaluation and the connectiveclauses of the definition of \(v_{a}\), it is important to keep in mind the difference between the roles that they play. The connectiveclauses of the definition of \(v_{a}\) are part of a recursive definition of a function from PL to \(\{T,F\}\) (one function for each atomic valuation). For each inductive clause of the definition of PL, they specify the image of the output of the clause as a function of the image of the input of the clause. The clauses of the definition of Boolean PLvaluation play a different role. They simply rule out PLvaluations in which certain combinations of values are present.

A PLsentence \(\phi \) is a logical consequence of a set of PLsentences \(\varGamma \) (\(\varGamma \vDash \phi \)) just in case for every Boolean valuation v, if \(v(\gamma ) = T\) for every \(\gamma \in \varGamma \), then \(v(\phi ) = T\).
Proposition 1
A PLvaluation v is Boolean just in case for some atomic valuation a, \(v=v_{a}\).
Generally, there isn’t much to choose between the two definitions. By bypassing ASAs, the holistic approach has the potential for greater parsimony. However, this potential is not realised in nonmodal propositional logic, since invoking atomic valuations doesn’t seem to bring about a substantial increase in the complexity of the resulting definition of logical consequence. We shall see in the remainder that for other formal languages the situation is very different in this regard.
3 Modal propositional logic
Modal propositional logic extends propositional logic with a sentential connective whose intended interpretation models the behaviour of the expression necessarily.
The language of modal propositional logic is the set MPL of sentences defined inductively in the same way as PL, with an additional inductive clause for the necessity operator, \(\square \). An MPLvaluation is a function from MPL to \(\{ T, F \}\).
Our goal in the present section is to present the contrast between the atomistic approach and the holistic approach as it applies to modal propositional logic. Here the atomistic approach reigns supreme. Logical consequence for modal propositional languages is only ever defined along the lines of the atomistic approach. Hence it might come as a surprise that the holistic approach is also applicable in this case. My main goal will be to present a holistic definition of logical consequence for MPL which will be shown in Sect. 7 to be equivalent to the standard atomistic definition.
3.1 The atomistic approach
In order to apply the atomistic approach to MPL, our first task is to identify the items that we are going to use as ASAs in this case. Here atomic valuations won’t do. The necessity operator is not truthfunctional. Therefore there isn’t a unique MPLvaluation extending a given atomic valuation that is compatible with the intended interpretations of the logical constants of MPL.
The standard strategy for overcoming this obstacle, due to Kripke (1963), is to invoke at this point the notion of a (modal) model. A (modal) model is a triple \(M=\langle W_{M},R_{M},A_{M}\rangle \), where \(W_{M}\) is a nonempty set (the possible worlds), \(R_{M}\) is a binary relation on \(W_{M}\) (accessibility) and \(A_{M}\) is a function pairing each atom and possible world with a unique truth value.
An ASA, on this implementation of the atomistic approach, is a model and a world in this model. Each world in each model will assign a unique truth value to each MPLsentence, and the next item of business for the atomistic approach is to define, for each model, the function that pairs each world and MPLsentence with the value that the sentence receives at the world.
 (Atoms)

For every atom \(\alpha \) and \(w\in W_{M}\), \(V_{M}(\alpha ,w) = A_{M}(\alpha ,w)\).
 (\(\square \))
 For every MPLsentence \(\phi \) and \(w\in W_{M}\),$$\begin{aligned} V_{M}(\square \phi ,w) = {\left\{ \begin{array}{ll} T &{} \quad \text {if for every }w'\in W_{M}\text { such that }wR_{M}w', V_{M}(\phi ,w') = T;\\ F &{} \quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

An MPLsentence \(\phi \) is a Klogical consequence of a set of MPLsentences \(\varGamma \) (\(\varGamma \vDash _{K}\phi \)) just in case for every model M and every \(w\in W_{M}\), if \(V_{M}(\gamma ,w) = T\) for every \(\gamma \in \varGamma \), then \(V_{M}(\phi ,w) = T\).
3.2 The holistic approach
 (K)

For every MPLsentence \(\phi \) and every set \(\varGamma \) of MPLsentences, if \(\varGamma \vDash \phi \), then \(\{\square \gamma :\gamma \in \varGamma \}\vDash \square \phi \).
Here I’m going to present a different strategy for overcoming the obstacle.^{2} A minor adjustment to the holistic template will suffice to obtain the intended result. The proposal is to impose necessary and sufficient conditions, not directly on individual MPLvaluations, but on sets of MPLvaluations. Then we’ll be able to say that an MPLvaluation is compatible with the intended interpretations of the logical constants of MPL just in case it is an element of a set that satisfies these conditions.

An MPLvaluation \(v'\)actualizes an MPLvaluation v just in case, for every MPLsentence \(\phi \), if \(v(\square \phi )=T\) then \(v'(\phi )=T\).

For every MPLsentence \(\phi \) and every \(v\in V\), if for every \(v'\in V\) such that \(v'\) actualizes v, \(v'(\phi )=T\), then \(v(\square \phi )=T\).

An MPLsentence \(\phi \) is a Klogical consequence of a set of MPLsentences \(\varGamma \) (\(\varGamma \vDash _{K}\phi \)) just in case for every mBoolean set of MPLvaluations V and every \(v\in V\), if \(v(\gamma ) = T\) for every \(\gamma \in \varGamma \), then \(v(\phi ) = T\).
 (T)

For every MPLsentence \(\phi \), if \(v(\square \phi )=T\) then \(v(\phi )= T\).
 (4)

For every MPLsentence \(\phi \), if \(v(\square \phi )=T\) then \(v(\square \square \phi )= T\).
The holistic definition of Klogical consequence presented in this section is equivalent to the atomstic definition. This is expressed by the following result:
Theorem 1
 1.
For every modal model M and every \(w\in W_{M}\), if \(V_{M}(\gamma ,w)=T\) for every \(\gamma \in \varGamma \), then \(V_{M}(\phi ,w)=T\).
 2.
For every mBoolean set V of MPLvaluations and every \(v\in V\), if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\).
In Sect. 7.1, below, I offer a proof of this result.
4 Firstorder logic
In this section I present the contrast between the atomistic and the holistic approach for (nonmodal) firstorder logic. We consider firstorder languages whose logical vocabulary consists of the connectives and brackets of PL, the universal quantifier \(\forall \), the identity sign, \(\doteq \), and denumerably many variables. The extralogical vocabulary of a firstorder language may contain nplace predicates, for any positive integer n, and individual constants. For any extralogical vocabulary the terms of the language are the variables and the individual constants of the vocabulary. The set of formulas of the language is defined by induction in the usual way. Then the set of sentences of the language is defined as the set of formulas with no free variables.
Our goal in this section is to present the application of the atomistic approach and the holistic approach to the task of defining logical consequence as a relation between sets of sentences of a firstorder language L and sentences of L. The holistic definition of logical consequence presented in this section will be shown in Sect. 7.2 to be equivalent to the standard atomistic definition.
4.1 The atomistic approach

For every individual constant c of L, \(c_{{\mathfrak {A}}}\in A\).

For every nplace predicate P of L, \(P_{{\mathfrak {A}}}\in {\mathscr {P}}(A^{n})\).
 (IC)

For every individual constant c of L, \(den_{{\mathfrak {A}}}(c, s)=c_{{\mathfrak {A}}}\).
 (Var)

For every variable x, \(den_{{\mathfrak {A}}}(x, s)=s(x)\).
 (Pred)
 For every Lformula of the form \(Pt_{1}\ldots t_{n}\),$$\begin{aligned} v_{{\mathfrak {A}}}(Pt_{1}\ldots t_{n}, s)= {\left\{ \begin{array}{ll} T&{} \quad \text{ if } \langle den_{{\mathfrak {A}}}(t_{1}, s),\ldots , den_{{\mathfrak {A}}}(t_{n}, s)\rangle \in P_{{\mathfrak {A}}};\\ F&{} \quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$
 (\(\doteq \))
 For every Lformula of the form \(t\doteq u\),$$\begin{aligned} v_{{\mathfrak {A}}}(t\doteq u, s)= {\left\{ \begin{array}{ll} T&{} \quad \text{ if } den_{{\mathfrak {A}}}(t, s)=den_{{\mathfrak {A}}}(u, s);\\ F&{} \quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$
 (\(\forall \))
 For every Lformula of the form \(\forall x\phi \),$$\begin{aligned} v_{{\mathfrak {A}}}(\forall x\phi , s)= {\left\{ \begin{array}{ll} T&{} \quad \text{ if } \text{ for } \text{ every } a\in A, v_{{\mathfrak {A}}}(\phi , s_{(x/a)})=T;\\ F&{} \quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$

An Lsentence \(\phi \) is a logical consequence of a set of Lsentences \(\varGamma \) (\(\varGamma \vDash \phi \)) just in case for every Lstructure \({\mathfrak {A}}\), if \(vs_{{\mathfrak {A}}}(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(vs_{{\mathfrak {A}}}(\phi )=T\).^{4}
4.2 The holistic approach

For every Lformula \(\phi \), individual constant c of L and variable x, the c/xsubstitution of \(\phi \), \((\phi )[c/x]\), is the formula that we obtain by replacing x with c wherever it occurs in free \(\phi \).
We are going to use for this purpose the notion of a qBooleanLvaluation, defined, as with Boolean PLvaluations, with a list of individually necessary and jointly sufficient conditions, specifying which Lvaluations are ruled out by the intended interpretation of each logical constant. For the connectives we use the same conditions as in propositional logic. We just need to add to these one for the universal quantifier and two for the identity sign.
 (\(\lnot \))

For every Lsentence \(\phi \), \(v(\phi ) \ne v(\lnot \phi )\).
 (\(\wedge \))

For all Lsentences \(\phi ,\psi , v(\phi \wedge \psi )=T\) iff \(v(\phi )=v(\psi )=T\).
 (\(\forall \))

For every Lformula \(\phi \) in which no variable other than x is free, \(v(\forall x\phi )=T\) iff for every individual constant c of L, \(v((\phi )[c/x])=T\).
 (\(\doteq a\))

For every individual constant c of L, \(v(c\doteq c)=T\).
 (\(\doteq b\))

For all individual constants \(c, c'\) of L, if \(v(c\doteq c')=T\), then for every Lformula \(\phi \) in which no variable other than x is free, \(v((\phi )[c/x])=v((\phi )[c'/x])\).

An Lsentence \(\phi \) is an Llogical consequence of a set \(\varGamma \) of Lsentences (\(\varGamma \vDash ^{L}\phi \)) just in case for every qBoolean Lvaluation v, if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\).

An Lsentence \(\phi \) is a logical consequence of a set \(\varGamma \) of Lsentences (\(\varGamma \vDash \phi \)) just in case for every onomastic expansion \(L'\) of L, \(\varGamma \vDash ^{L'}\phi \).
Theorem 2
 1.
For every onomastic expansion \(L'\) of L, for every qBoolean \(L'\)valuation v, if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\).
 2.
For every Lstructure \({\mathfrak {A}}\), if \(vs_{{\mathfrak {A}}}(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(vs_{{\mathfrak {A}}}(\phi )=T\).
In Sect. 7.2, below, I provide a proof of this result.
5 Modal firstorder logic
We consider now modal firstorder languages whose logical vocabulary contains the connectives and brackets of PL, the necessity operator \(\square \), the universal quantifier \(\forall \), the identity sign \(\doteq \), and denumerably many variables. The extralogical vocabulary of a modal firstorder language may contain nplace predicates, for any positive integer n, and individual constants.
As in nonmodal firstorder languages, the terms of a modal firstorder language are the variables and the individual constants. The set of formulas is also defined with the usual induction, including this time inductive clauses for both \(\forall \) and \(\square \). As with nonmodal firstorder logic, the sentences of a modal firstorder language are the formulas with no free variables.
5.1 The atomistic approach
Standard definitions of logical consequence for modal firstorder languages follow the atomistic approach. The role of ASAs is played by items that combine the ideas of firstorder structures and modal models. There are several important decisions one needs to make in order to effect this combination. Here we are going to present a particularly simple version of the idea.^{8}

\(W_{M}\) is a nonempty set (the possible worlds).

\(D_{M}\) is a nonempty set (the universe).

\(R_{M}\) is a binary relation on \(W_{M}\) (accessibility).
 \(I_{M}\) is an interpretation of each extralogical symbol of L, as follows:

For every individual constant c of L, \(c_{M}\in D_{M}\). I.e. c is interpreted with an object in the universe.

For every nplace predicate P of L, \(P_{M}\in {}^{W_{M}}\! {\mathscr {P}} (D^{n}_{M})\). I.e. P is interpreted with a function pairing each element of \(W_{M}\) with a set of ntuples of elements of \(D_{M}\)—its extension at that world. We shall refer to the set of ntuples of elements of \(D_{M}\) that \(P_{M}\) pairs with w as \(P^{w}_{M}\).

 (IC)

For every individual constant c of L, \(den_{M}(c,w,s)=c_{M}\).
 (Var)

For every variable x, \(den_{M}(x,w,s)=s(x)\).
 (Pred)
 For every Lformula of the form \(Pt_{1}\ldots t_{n}\),$$\begin{aligned} V_{M}(Pt_{1}\ldots t_{n},w,s)= {\left\{ \begin{array}{ll} T&{}\quad \text{ if } \langle den_{M}(t_{1},w, s),\ldots , den_{M}(t_{n},w, s)\rangle \in P^{w}_{M};\\ F&{}\quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$
 (\(\doteq \))
 For every Lformula of the form \(t\doteq u\),$$\begin{aligned} V_{M}(t\doteq u, w,s)= {\left\{ \begin{array}{ll} T&{}\quad \text{ if } den_{M}(t, w,s)=den_{M}(u, w,s);\\ F&{}\quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$
 (\(\forall \))
 For every Lformula of the form \(\forall x\phi \),$$\begin{aligned} V_{M}(\forall x\phi , w,s)= {\left\{ \begin{array}{ll} T&{}\quad \text{ if } \text{ for } \text{ every } a\in D_{M}, V_{M}(\phi , w,s_{(x/a)})=T;\\ F&{}\quad \text{ otherwise. }\end{array}\right. } \end{aligned}$$
 (\(\square \))
 For every Lformula of the form \(\square \phi \),$$\begin{aligned} V_{M}(\square \phi ,w,s) = {\left\{ \begin{array}{ll} T&{}\quad \text {if for every }w'\in W_{M}\text { such that }wR_{M}w',\\ &{}\quad V_{M}(\phi ,w',s) = T;\\ F&{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$

An Lsentence \(\phi \) is a Klogical consequence of a set of Lsentences \(\varGamma \) (\(\varGamma \vDash _{K}\phi \)) just in case for every Lmodel M and every \(w\in W_{M}\), if \(V\!S_{M}(\gamma ,w)=T\) for every \(\gamma \in \varGamma \), then \(V\!S_{M}(\phi ,w)=T\).
5.2 The holistic approach
The application of the holistic approach to modal firstorder logic draws on the ideas that we developed to adapt the holistic template to nonmodal firstorder logic and to modal propositional logic. As in nonmodal firstorder logic, we start with a languagerelative definition of logical consequence and then define the relation that will do the job in terms of the languagerelative relations on the onomastic expansions of the target language. As in modal propositional logic, we don’t formulate necessary and sufficient conditions for a valuation to be admissible. We formulate necessary and sufficient conditions on sets of valuations. Admissible valuations are elements of sets that satisfy these conditions.
 (\(\square \))

For every Lsentence \(\phi \) and every \(v\in V\), \(v(\square \phi )=T\) if for every \(v'\in V\) such that \(v'\) actualizes v, \(v'(\phi )=T\).
 (\(\doteq \))

For all individual constants \(c, c'\) of L and all \(v, v'\in V\), \(v(c\doteq c')=v'(c\doteq c')\).

An Lsentence \(\phi \) is an LKlogical consequence of a set of Lsentences \(\varGamma \) (\(\varGamma \vDash _{K}^{L}\phi \)) just in case for every mqBoolean set of Lvaluations V and every \(v\in V\), if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\).

An Lsentence \(\phi \) is a Klogical consequence of a set of Lsentences \(\varGamma \) (\(\varGamma \vDash _{K}\phi \)) just in case for every onomastic expansion \(L'\) of L, \(\varGamma \vDash _{K}^{L'}\phi \).
In Sect. 7.3, below, we show that this holistic definition of logical consequence for modal firstorder languages is equivalent to the atomistic definition provided above. The claim can be expressed as follows:
Theorem 3
 1.
For every onomastic expansion \(L'\) of L, for every mqBoolean 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 Lmodel M and every \(w\in W_{M}\), if \(V\!S_{M}(\gamma ,w)=T\) for every \(\gamma \in \varGamma \), then \(V\!S_{M}(\phi ,w)=T\).
6 Atomism or holism?
The atomistic approach and the holistic approach correspond to two different conceptions of the source of logical properties and relations, such as logical consequence. It is widely accepted that language makes contact with reality at the level of its atomic constituents. Compound sentences obtain their semantic properties from their components, and ultimately from the atomic constituents that figure in them. The definitions of truth for formal languages provide us with the tools for investigating this phenomenon.
By using these same tools for explicating logical consequence, the atomistic approach presents the concept as arising from the mechanisms by which language comes to represent the world. A sentence \(\phi \) follows from a set of sentences \(\varGamma \), on this approach, when the possible ways in which \(\phi \) and the elements of \(\varGamma \) might represent the world don’t include any in which \(\phi \) comes out false and all the elements of \(\varGamma \) come out true.
On the holistic approach, by contrast, logical properties and relations are not connected with the mechanisms through which language makes contact with the world. They arise instead from factors internal to language—from the fact that some combinations of truth values for sentences are incompatible with structural features of these sentences. Atomic components play no special role in the account, and the ways in which these make contact with the world play no role whatsoever.
This is not the place to adjudicate the contest between these conceptions of the source of logical properties and relations. My goal is to show that the second conception can produce workable definitions of logical consequence for standard formal languages. This result, established in the next section, will block what some may have seen as a powerful argument for the atomistic approach. The premise of this argument is the thought that what makes \(\phi \) a logical consequence of \(\varGamma \) is the fact that structural features of \(\phi \) and the elements of \(\varGamma \) make it impossible for \(\phi \) to be false if all the elements of \(\varGamma \) are true.^{10} This premise would render the atomistic approach unavoidable if the explanation of how the structures of propositions rule out certain truthvalue combinations needed to invoke how their truth values are determined by the semantic values of their atomic components. The availability of the holistic approach shows that this isn’t the case. We can explain how the structures of propositions rule out certain truthvalue combinations without considering how their truth values are determined by the semantic values of their atomic components. Hence the holistic approach deprives the atomistic approach of this line of support.
Notice, in addition, that the resources deployed by the atomistic approach for explaining logical consequence go well beyond what is required for the task. I’m going to illustrate the point with the case of nonmodal firstorder logic, although it applies in the same way to modal firstorder logic. In order to define logical consequence for a firstorder language L, all we need to do is specify which Lvaluations are compatible with the intended interpretations of the logical constants. Hence the value of Lstructures for the task of defining logical consequence is restricted to the fact that each Lstructure singles out a unique Lvaluation. But Lstructures do much more than this, as can be seen by the fact that (infinitely) many different nonisomorphic Lstructures single out the same Lvaluation. This is a direct consequence of what is usually presented as the expressive limitations of firstorder logic. Thus it follows from the Löwenheim–Skolem results that if \({\mathfrak {A}}\) is an Lstructure with an infinite universe, then for every infinite cardinality \(\kappa \) greater than or equal to the cardinality of L there is an Lstructure \({\mathfrak {B}}\) with a universe of cardinality \(\kappa \) that singles out the same Lvaluation as \({\mathfrak {A}}\).^{11} I.e., in the symbolism introduced in Sect. 4, \(vs_{{\mathfrak {A}}} = vs_{{\mathfrak {B}}}\).^{12} The ways in which \({\mathfrak {A}}\) might differ from \({\mathfrak {B}}\) are of the greatest importance in the study of the representational properties of L, but they are completely irrelevant for the task of defining logical consequence.^{13} From the point of view of parsimony, the holistic approach has a clear advantage over the atomistic approach.
7 Equivalence results
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 firstorder logic are equivalent to the atomistic alternatives.
7.1 Modal propositional logic
Lemma 1
A set of MPLvaluations is mBoolean just in case it is generated by some modal model.
Proof
Let M be a modal model, and let V be the set of MPLvaluations it generates. We need to show that V is mBoolean.
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 \(v(\square \phi ) = T\).

\(W_{M}=V\).

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

For every atom \(\alpha \) and every \(v\in W_{M}\), \(A_{M}(\alpha ,v)=v(\alpha )\).
The equivalence of the atomistic and holistic definitions of Klogical consequence expresssed by Theorem 1 is a straightforward corollary of Lemma 1.
7.2 Firstorder 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
Proof
By induction on Lformulas. See, e.g., Zalabardo (2000, pp. 155–157). \(\square \)
We show next that the sentential valuations generated by certain structures are qBoolean. If L is a firstorder language, \({\mathfrak {A}}\) is an Lstructure, 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 onetoone correspondence f between C and A.
Lemma 3
If L is a firstorder language and \({\mathfrak {A}}\) is an Lstructure, then \(vs_{{\mathfrak {A}}^{+}}\) is a qBoolean \(L^{+}\!\)valuation.
Proof
We need to show that \(vs_{{\mathfrak {A}}^{+}}\) satisfies the clauses of the definition of qBoolean \(L^{+}\!\)valuation. We provide the arguments for (\(\forall \)) and (\(\doteq b\)).
Lemma 4
 1.For every Lterm t and every variableinterpretation s in \({\mathfrak {A}}\),$$\begin{aligned}den_{{\mathfrak {A}}}(t,s)=den_{{\mathfrak {A}}'}(t,s)\end{aligned}$$
 2.For every Lformula \(\phi \) and every variableinterpretation s in \({\mathfrak {A}}\),$$\begin{aligned}v_{{\mathfrak {A}}}(\phi ,s)=v_{{\mathfrak {A}}'}(\phi ,s)\end{aligned}$$
Proof
2 by induction on Lformulas. \(\square \)
We are now in a position to show that holistic logical consequence entails atomistic logical consequence.
Theorem 4
 1.
For every onomastic expansion \(L'\) of L, for every qBoolean \(L'\)valuation v, if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\).
 2.
For every Lstructure \({\mathfrak {A}}\), if \(vs_{{\mathfrak {A}}}(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(vs_{{\mathfrak {A}}}(\phi )=T\).
Proof
We turn now to showing that atomistic logical consequence entails holistic logical consequence.
Let L be a firstorder language with at least one individual constant, and let v be a qBoolean Lvaluation. 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
 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 qBoolean valuation.

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 nplace 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 firstorder language with at least one individual constant and v a qBoolean Lvaluation, then for every Lsentence \(\phi \) and every variableinterpretation s in \({\mathfrak {A}}_{v}\), \(v(\phi )=v_{{\mathfrak {A}}_{v}}(\phi ,s)\).
Proof

For every atomic Lformula \(\phi \), \(r(\phi )=1\).

For every Lformula \(\phi \), \(r(\lnot \phi )=r(\forall x\phi )=r(\phi )+1\).

For all Lformulas \(\phi ,\psi \), \(r(\phi \wedge \psi )=Max(r(\phi ),r(\psi ))+1\).
We can now prove that atomistic logical consequence entails holistic logical consequence.
Theorem 5
 1.
For every onomastic expansion \(L'\) of L, for every qBoolean \(L'\)valuation v, if \(v(\gamma )=T\) for every \(\gamma \in \varGamma \), then \(v(\phi )=T\).
 2.
For every Lstructure \({\mathfrak {A}}\), 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 qBoolean \(L'\)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 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.
7.3 Modal firstorder logic
Our first goal is to show that holistic logical consequence entails atomistic logical consequence. We proceed in the same way as with nonmodal firstorder logic.
Lemma 7
Proof
By induction on Lformulas. 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 \(\square \).
We show next that the sets of sentential valuations generated by certain structures are mqBoolean. If L is a modal firstorder language, M is an Lstructure, 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 onetoone correspondence f between C and \(D_{M}\).
Lemma 8
If L is a modal firstorder language and M is an Lstructure, then \(\{vs^w_{M^{+}}:w\in W_{M^{+}}\}\) is a mqBoolean set of \(L^{+}\)valuations.
Proof
We first need to show that, for every \(w\in W_{M^{+}}, vs^w_{M^{+}}\) is qBoolean. Let \(w\in W_{M^{+}}\). We provide the arguments for clauses (\(\forall \)) and (\(\doteq b\)) of the definition.
Lemma 9
 1.
For every Lterm t, every \(w\in W_{M}\) and every variableinterpretation s in M, \(den_{M}(t,w,s)=den_{M'}(t,w,s)\).
 2.
For every Lformula \(\phi \), every \(w\in W_{M}\) and every variableinterpretation s in M, \(v_{M}(\phi ,w,s)=v_{M'}(\phi ,w,s)\).
Proof
2 by induction on Lformulas. \(\square \)
We can now establish that holistic Klogical consequence entails atomistic Klogical consequence.
Theorem 6
 1.
For every onomastic expansion \(L'\) of L, for every mqBoolean 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 Lmodel 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
We turn now to our final task of establishing that atomistic Klogical consequence entails holistic Klogical consequence.
Let L be a modal firstorder language with at least one individual constant, and let V be an mqBoolean set of Lvaluations. 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 firstorder logic, that E is an equivalence relation. Let \([c]_{E}\) denote the equivalence class generated by c with E.

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

\(D_{M_{V}}\) 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 nplace 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 firstorder language with at least one individual constant and V is an mqBoolean set of Lvaluations, then for every Lsentence \(\phi \) and every \(v\in V\), \(v(\phi )=V_{M_{V}}(\phi , v,s)\), for any variableinterpretation 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 Lformula \(\phi \), every \(v\in V\) and every variableinterpretation 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 \(\square \).
We can now establish that atomistic Klogical consequence entails holistic Klogical consequence.
Theorem 7
 1.
For every onomastic expansion \(L'\) of L, for every mqBoolean 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 Lmodel 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 mqBoolean 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.
It follows from Theorems 6 and 7 that the holistic definition of logical consequence for modal firstorder logic is equivalent to the atomistic definition, as expressed by Theorem 3.
8 Conclusion
I have shown that the holistic approach can be successfully applied to the task of defining logical consequence in propositional and firstorder logic, modal as well as nonmodal. I have shown that the resulting definitions are equivalent to the standard definitions based on the atomistic template. One salient feature of the holistic definitions I’ve provided is that they make no use of the technical apparatus of firstorder, modal and firstorder modal models, employed by the atomistic approach for this purpose. This technical apparatus is a fascinating subject of study in its own right and nothing I’ve said detracts from the interest of this study. What does follow from the results I’ve presented is that modal, firstorder and modal firstorder models are not required for defining logical consequence. In general, we can maintain that logical properties and relations arise from the fact that some combinations of truth values are incompatible with formal features of sentences, while rejecting the link between logical properties and relations and the mechanisms by which sentences come to represent the world.
Footnotes
 1.
Semantic definitions are those that take as their starting point the thought that \(\phi \) is a logical consequence of \(\varGamma \) just in case structural features of \(\phi \) and of the elements of \(\varGamma \) are incompatible with assigning to \(\phi \) the value False if all the elements of \(\varGamma \) receive the value True. In the present paper I’m restricting my attention to semantic definitions. The modeltheoretic approach to logical consequence produces atomistic semantic definitions. The holistic definitions that I’m going to provide are also semantic. Prooftheoretic approaches produce nonsemantic definitions of logical consequence, and fall outside the scope of this paper. It could be argued, although I won’t do it here, that the holistic semantic approach that I’m going to present has some of the advantages claimed for the prooftheoretic approach.
 2.
I sketch this strategy in Zalabardo (2018).
 3.
My proposal differs in this respect from Hughes Leblanc’s truthfunctional semantics for modal logic (Leblanc 1976), which invokes a set of (atomic) valuations and a binary relation on this set.
 4.
In standard applications of the atomistic approach, logical consequence is defined for all Lformulas. Here I’ve restricted the definition to Lsentences in order to facilitate comparisons with the holistic approach.
 5.
Dunn and Belnap’s (1968) presentation of substitutional quantification follows this atomistic template (p. 179). The same goes for Leblanc (1976). Leblanc (1983, pp. 213–214) briefly mentions a holistic version of substitutional semantics that is essentially the one developed here. Holistic ideas can also be found in the work of Smullyan (1968, p. 47) and Hintikka (1955).
 6.
In the limiting case, for firstorder languages with no individual constants, a qBoolean valuation will give the value T to every universal sentence.
 7.
 8.
 9.
Instead of (\(\doteq \)) we could require that for every valuation v in an mqBoolean set, if \(v(\lnot c\doteq c')=T\), then \(v(\square \lnot c\doteq c')=T\). However, having (\(\doteq \)) will facilitate the proof of the equivalence of holistic and atomistic logical consequence.
In systems at least as strong as B, (\(\doteq \)) is not needed to establish that if \(v(\lnot c\doteq c')=T\), then \(v(\square \lnot c\doteq c')=T\) either. In these systems, every valuation v in an mqBoolean set V is such that if \(v(\lnot \phi )=T\), then \(v(\square \lnot \square \phi )=T\). Hence, if \(v(\lnot c\doteq c')=T\), then for every valuation \(v'\in V\) that actualizes v we have that \(v'(\square c\doteq c')=F\). But we have seen that if \(v'(c\doteq c')=T\), then \(v'(\square c\doteq c')=T\). Therefore, for every \(v'\in V\) that actualizes v we have that \(v'(c\doteq c')=F\), and \(v(\square \lnot c\doteq c')=T\).
 10.
This premise will be rejected by those who endorse prooftheoretic accounts of logical consequence.
 11.
See Zalabardo (2000, pp. 263–271).
 12.
Of course there are also cases in which \(vs_{{\mathfrak {A}}} = vs_{{\mathfrak {B}}}\) and \({\mathfrak {A}}\) is not isomorphic to \({\mathfrak {B}}\) although their universes are of the same cardinality. This is the situation, for example, with nonstandard models of firstorder arithmetic. See Zalabardo (2000, pp. 272–282).
 13.
In Zalabardo (2018) I argue that we encounter a similar phenomenon in the atomistic treatment of modal propositional logic—different nonisomorphic models that generate the same mBoolean set of MPLvaluations.
Notes
Acknowledgements
For their comments on this material, I’m grateful to Kit Fine and Antti Hautam\(\ddot{\text {a}}\)ki, to my colleagues Lavinia Picollo and Daniel Rothschild, and to two anonymous referees for this journal.
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