Grounding Generalizations

Abstract

Some propositions are true, and it is true that some propositions are true. Each of these facts looks like an impeccable ground of the other. But they cannot both ground each other, since grounding is asymmetric. This paper explores two new diagnoses of this much discussed puzzle. The tools of higher-order logic are used to show how both diagnoses can be fleshed out into strong and consistent theories of grounding. These theories of grounding in turn demand new theories of the granularity of propositions, properties, and relations. Even those who are uninterested in grounding should take seriously these pictures of reality’s logical structure, which are in many ways reminiscent of Russell’s and Wittgenstein’s logical atomism.

Appendices

Appendix A: Truth Condition Models

Here we give a precise characterization of the models described in Section 2. We there described two possible treatments of the connectives ∧ and ∨ and two possible treatments of λ-terms; these are distinguished by subscripts below.

Let W be some non-empty set and c≠d be arbitrary urelements.

Definition 1 (Conjunction and disjunction)

$\bigwedge X = \{c\}\cup X$ if X≠∅ and W otherwise. $\bigvee X = \{d\}\cup X$ if X≠∅ and ∅ otherwise.

Definition 2 (Propositions)

$P_{0} = \mathcal {P}(W)$ $P_{n+1} = P_{0} \cup \{\bigwedge X: X\subseteq P_{n}\}\cup \{\bigvee X: X\subseteq P_{n}\}$

Definition 3 (Ranks)

κ(τ) = 0 if τ is not monadic; κ(〈τ〉) = 1 + κ(τ).

Definition 4 (Domains)

D_e is some set $D_{\langle \rangle } = \bigcup _{n\in \mathbb {N}} P_{n}$ $D_{\langle \tau \rangle }= {P_{\kappa (\tau )}}^{D_{\tau }}$ $D_{\langle \tau _{1},\dots ,\tau _{n}\rangle }= {D_{\langle \rangle }}^{D_{\tau _{1}}\times \dots \times D_{\tau _{n}}}$ for n > 1

Definition 5 (Truth conditions)

$tc: D_{\langle \rangle }\to \mathcal {P}(W)$ such that tc(p) = p for $p\in \mathcal {P}(W)$ $tc(p)=\bigcap \{tc(x): x\in p\}$ if c ∈ p $tc(p)=\bigcup \{tc(x): x\in p\}$ if d ∈ p

Definition 6 (Interpretation)

⟦x⟧^g = g(v) for variables v $\llbracket Fa_{1}{\dots } a_{n}\rrbracket ^{g} = \llbracket F\rrbracket ^{g}(\llbracket a_{1}\rrbracket ^{g},\dots , \llbracket a_{n}\rrbracket ^{g})$⟦¬⟧^g(p) = W∖tc(p) $\llbracket \wedge _{1}\rrbracket ^{g}(p,q) = \bigwedge \{p,q\}$ $\llbracket \vee _{1}\rrbracket ^{g}(p), = \bigvee \{p,q\}$⟦∧₂⟧^g(p,q) = p if $\{q,c\}\subseteq p$; q if $\{p,c\}\subseteq q$; and $\bigwedge \{p,q\}$ otherwise ⟦∨₂⟧^g(p,q) = p if $\{q,d\}\subseteq p$; q if $\{p,d\}\subseteq q$; and $\bigvee \{p,q\}$ otherwise $\llbracket \prec \rrbracket ^{g}(p)(q) = \bigvee \{\bigwedge \{r_{1},\dots ,r_{n}\}:p= r_{1}\in \dots \in r_{n}= q\neq p\}$ $\llbracket \forall _{\tau }\rrbracket ^{g}(f) = \bigwedge \{f(x): x\in D_{\tau }\}$ $\llbracket \exists _{\tau }\rrbracket ^{g}(f) = \bigvee \{f(x): x\in D_{\tau }\}$ $\llbracket (\lambda _{1} x_{1}{\dots } x_{n}.\varphi )\rrbracket ^{g}(y_{1},\dots ,y_{n}) = tc(\llbracket \varphi \rrbracket ^{g[x_{i}\to y_{i}]})$ $\llbracket (\lambda _{2} x_{1}^{\tau _{1}}{\dots } x_{n}^{\tau _{n}}.\varphi )\rrbracket ^{g}(y_{1},\dots ,y_{n})=$

(i)$\llbracket \varphi \rrbracket ^{g[x_{i}\to y_{i}]}$ if $\exists f\in D_{\langle \tau _{1},\dots ,\tau _{n}\rangle }\forall y_{i}\in D_{\tau _{i}}:f(y_{1},\dots ,y_{n})=\llbracket \varphi \rrbracket ^{g[x_{i}\to y_{i}]}$,

(ii) $tc(\llbracket \varphi \rrbracket ^{g[x_{i}\to y_{i}]})$ otherwise.

Definition 7 (Validity)

φ is valid := tc(⟦φ⟧^g) = W for all g in any model

Appendix B: Plural Quantification and Full Ground

For every type τ we add a corresponding type τ^∗ for pluralities of entities of type τ. (Compare the ‘extensional types’ of [13].) Formally, we enrich our type system as follows: e is a type; for any type τ, τ^∗ is a type; for any types $\tau _{1},\dots ,\tau _{n}$, $\langle \tau _{1},\dots ,\tau _{n}\rangle$ is a type; nothing else is a type. Domains for non-plural types are defined as before. $D_{\tau ^{*}}=\mathcal {P}(D_{\tau })$. For every type τ, we enrich our language with an ‘is one of’ connective ∈_τ of type 〈τ,τ^∗〉 such that ⟦∈_τ⟧^g(x,Y ) = W if x ∈ Y and = ∅ otherwise.

Now to full grounding. Intuitively, a way for Γ to fully ground p is given by a collection of propositions that can be organized in such a way that any such proposition not in Γ is assigned some such propositions that immediately fully ground it (by being either all of its conjuncts or some of its disjuncts), every maximal chain of these immediate full grounding relationships begins in Γ and ends at p, and every member of Γ is used in the process.

Definition 8

X is a way for Γ to fully ground p if and only if, for some non-trivial rooted tree G = 〈V,E,v〉 and surjective function f : V → X,

• 1. f(v) = p

• 2. {f(x) : x has no children} = Γ

• 3. if x is a child of y, then f(x) is either a conjunct or a disjunct or f(y)

• 4. if f(x)∉Γ and q is a conjunct of f(x), then q = f(y) for some child y of x.

Finally, we enrich our language with a full grounding connective < of type 〈〈〉^∗,〈〉〉, subject to the interpretation:

$$ \llbracket<\rrbracket^{g}({\Gamma},p)=\bigvee\left\{\bigwedge X: X ~ \text{is a way for}~ {\Gamma} ~ \text{to fully ground} p\right\}. $$

(Parallel strategies for introducing a full grounding connective in the context of the other models of grounding discussed below are straightforward.)

This clause validates all principles of the pure logic of (strict) ground in [16] (isolated in [11]), as well as the ‘internality’ of full grounding: $({\Gamma }<p)\to \Box (\forall q(q\in _{\langle \rangle }{\Gamma } \to q)\to ({\Gamma }<p))$. Partial grounds are all and only the parts of full grounds: $(p\prec q)\leftrightarrow \exists {\Gamma }(p\in _{\langle \rangle }{\Gamma }\wedge {\Gamma } < q)$. But not every full ground together with a partial ground is a full ground: p ∨ (q ∧ r) may be fully grounded in p alone and partially grounded in q without being fully grounded in p together with q.

The clause also validates the following controversial but popular principle about iterated grounding: (Γ < p) → (Γ < (Γ < p)).¹ The main reservation about that principle in the literature is that Γ < p and Γ < q can have different full grounds when p and q are different propositions. But that is validated too: $(({\Gamma }<p)\wedge ({\Gamma }<q)) \to (p\neq q\to \neg \forall {\Delta }(({\Delta }<({\Gamma }<p))\leftrightarrow ({\Delta }<({\Gamma }<q))))$. There is no incompatibility here because < is a many-one relation: true grounding claims typically have many full grounds.

Appendix C: Atomist Models

Here we give a precise characterization of the modified model construction described in Section 3. This involves (i) changing the definition of propositional domains, (ii) introducing a normalization operation that replaces the truth condition operation in the clause for λ-abstraction, and (iii) introducing parallel changes in the clause for negation.

P₀ is now a set of basic atomic propositions, imbued with a negation operation ν : P₀ → P₀ such that ν(p)≠ν(ν(p)) = p for all p ∈ P₀. Domains are defined as in Appendix A, with two changes:

$$ \begin{array}{@{}rcl@{}} P_{n+1} & = & P_{0} \cup \{\{c\}\cup X: X\subseteq P_{n}\}\cup \{\{d\}\cup X: X\subseteq P_{n}\}\\ D_{\langle\tau \rangle} & = & {P_{\kappa(\tau)+2}}^{D_{\tau}} \end{array} $$

The first change is to allow for a trivial tautology {c} and contradiction {d}. We make a parallel adjustment in the interpretation of ≺:

$$ \llbracket \prec\rrbracket^{g}(p,q) =\{d\}\cup\{\{c,r_{1},\dots,r_{n}\}:p= r_{1}\in\dots\in r_{n}= q\neq p\} $$

We now define an operation for turning level n + 3 propositions into Boolean-equivalent level n + 2 propositions. The intuitive idea is that every conjunction is replaced with its conjunctive normal form and every disjunction with its disjunctive normal form. For example, suppose we have a level-3 conjunction. We first turn it into a level-3 conjunction of disjunctions, by replacing every level-0 conjunct with its singleton disjunction and replacing every conjunctive conjunct with the singleton disjunctions of its members. Next, we do the same to this conjunctions’ disjuncts, yielding a level-3 conjunction of disjunctions of conjunctions. Then, we invert each of its conjuncts from a disjunction of conjunctions to a conjunction of disjunctions, yielding a level-3 conjunction of conjunctions of disjunctions. We then merge the conjuncts together, yielding a level-2 conjunction of disjunctions equivalent to our original proposition; for an example see footnote (12). By iterating this procedure n times any proposition of level n + 2 can be reduced to a level-2 normal form. More precisely:

Definition 9 (Sign)

σ(p) = c if c ∈ p and σ(p) = d if d ∈ p.

Definition 10 (Members)

m(p) = p∖{σ(p)}

Definition 11 (Involution)

Let ^∗ be the operation on D_〈〉∪{c,d} such that p^∗ = ν(p) for p ∈ P₀ c^∗ = d d^∗ = c p^∗ = {x^∗ : x ∈ p} for p ∈ D_〈〉∖P₀

Definition 12 (Polarization)

p^‡ = {σ(p)}∪{q : q ∈ p and σ(q) = σ(p)^∗}∪{{σ(p)^∗,q} : q ∈ p∩P₀}∪{{σ(p)^∗,q} : ∃r ∈ p s.t. q ∈ m(r) and σ(r) = σ(p)}

Definition 13 (Inversion)

If m(p) = ∅, then ι(p) = {σ(p)^∗,p}. Otherwise, ι(p) = {σ(p)^∗}∪{{σ(p)}∪image(f) : f a choice function on m(p)}.

Definition 14 (Flattening)

$f(p) = \bigcup \{\iota (q^{\dagger }): q\in m(p^{\dagger })\}$

Definition 15 (Levels)

$\mathfrak {l}(p)=\min \limits \{n: p\in P_{n}\}$

Definition 16 (Normal form)

Let norm : D_〈〉→ P₂ s.t. $norm(p)=f^{\mathfrak {l}(p)-2}(p)$ if $\mathfrak {l}(p)>2$, and norm(p) = p otherwise.

We interpret our language as in Appendix A with three exceptions: (i) we modify the clause for ≺ as described above, (ii) we replace tc with norm in the clause for λ-abstraction, and (iii) we modify the clauses for ¬ as follows:

$$ \llbracket \neg\rrbracket^{g}(p) = norm(p^{*}) $$

Finally, we define a notion of truth conditions as follows:

Definition 17 (Worlds)

$W= \{w\subseteq P_{0}: \forall p\in P_{0},p\in w\leftrightarrow \nu (p)\not \in w\}$

Definition 18 (Truth conditions)

$tc: D_{\langle \rangle }\cup \{c,d\}\to \mathcal {P}(W)$ such that tc(p) = {w : p ∈ w} for p ∈ P₀ tc(c) = W tc(d) = ∅ $tc(p)=\bigcap \{tc(x): x\in p\}$ if c ∈ p $tc(p)=\bigcup \{tc(x): x\in p\}$ if d ∈ p

As before, a formula is valid if and only if it has trivial truth conditions in every model relative to every assignment.

Appendix D: Syncategoremtic Models

This appendix explains how to modify the above construction to model the theory sketched in Section 5.3, which endorses (30) and (31) (formulated in terms of new quantifiers which take relations as arguments), and in which conjunction and disjunction are formalized not as connectives but as syntactic operations for forming new formulas from finite sets of formulas.

We first modify the definition of higher-order domains to give a uniform treatment of monadic and polyadic types. Ranks and domains are now defined as follows: κ(e) = κ(〈〉) = 0; $\kappa (\langle \tau _{1},\dots ,\tau _{n}\rangle )=\max \limits (\kappa (\tau _{1}),\dots ,\kappa (\tau _{n}))+\nobreak 1$; $D_{\langle \tau _{1},\dots ,\tau _{n}\rangle }={P_{\kappa (\langle \tau _{1},\dots ,\tau _{n}\rangle )+1}}^{D_{\tau _{1}}\times \dots \times D_{\tau _{n}}}$ (for n > 0).

We then interpret set conjunction and disjunction in the obvious way: $\llbracket \bigwedge {\Gamma }\rrbracket ^{g}=\{c\}\cup \{\llbracket \varphi \rrbracket ^{g}: \varphi \in {\Gamma }\}$, and likewise for disjunction. Similarly for universal and existential quantifiers applying to polyadic predicates: $\llbracket \forall _{\tau _{1},\dots ,\tau _{n}}\rrbracket ^{g}(f)=\{c\}\cup \{f(x_{1},\dots ,x_{n}):x_{i}\in D_{\tau _{i}}\}$, and likewise for existential quantifiers.

The rest of the construction is as before with the obvious adjustments.²

Appendix E: Schematic Instance Structure

In this appendix we establish the consistency of the schemas:

$$ \begin{array}{@{}rcl@{}} \forall p(inst(p,\exists x\varphi)& \leftrightarrow & \exists x(p=\varphi))\\ \forall p(inst(p,\forall x\varphi) & \leftrightarrow & \exists x(p=\varphi)) \end{array} $$

For convenience, we will operate in a higher-order language of the sort discussed in Section 1, where quantifier prefixes are treated as syncategorematic variable-binding sentential operators. That is, ∃x is a single expression that combines with a formula to yield a formula; however, unlike other expressions with this syntactic behavior (i.e., of type 〈〈〉〉), we do not assign a semantic value to ∃x, but instead assign semantic values directly to formulas ∃xφ, as is standard in Tarskian model-theory for first-order languages.

The model construction has some high-level similarities to the ones discussed in Appendices A and C. In both cases, since we can recover the instances of a generalization from that generalization, Cantor’s theorem implies that not all sets of propositions can be all and only the instances of some generalization. The main difference from the previous constructions is that generalizations can be instances of themselves. Another difference is a reversal in what is possible regarding conjunction and negation: the present construction is compatible with distinct propositions always having distinct negations (although for simplicity we will start off with models where this fails), but not with distinct pairs of propositions always having distinct conjunctions (given the inconsistency of instance structure and conjunctive structure).

I’ll begin with an informal discription of the models. There are two distinguished ‘boring’ propositions, one true and one false. Each has itself and only itself as an instance. Every categorematic predication (i.e., every formula that is neither a free variable nor a generalization) denotes one of these two propositions. Generalizations are individuated by their instances. So the boring truth is ∀p(p ∨¬p) (equivalently, ∃p(p ∨¬p)), since its instances are true categorematic predications (i.e., the boring truth), and the only proposition with the boring truth as its only instance is the boring truth. Likewise, the boring falsehood is ∀p(p ∧¬p) (equivalently, ∃p(p ∧¬p)).

Now consider ∃p¬p and ∀p¬p. These two propositions have the same instances: namely, both boring propositions. So they cannot be either of the two boring propositions. Now, if we didn’t have sentences like ∃q∃p¬p and ∃pp, we could get by with just these four propositions. This is because, for any formula φ and assignment g, it will turn out that ⟦φ⟧^g will be one of these four propositions as long as any occurrences of vacuous propositional quantifiers or of propositional variables in φ are in the scope of some categorematic operator. But the possibility of vacuous propositional quantification and of propositional variables occurring only under quantifiers complicates things.

First, notice that, for any proposition p, ∀xp has p as its only instance. Since, as we have seen, some propositions have more than one instance, it follows that there must be infinitely many propositions – and not only infinitely many propositions with a single instance. For example, ∀p∀x∀q(p → q) has not only the boring truth and falsehood as instances, but also the proposition whose only instance is the proposition that has the boring truth and boring falsehood as its only instances. ∀p∀x∀q∀y∀r(p → (q → r)) has more instances still, etc.

Next, consider bound propositional variables occurring only under quantifiers. For example, both ∃pp and ∀pp have every proposition as an instance. In fact, there is an infinite sequence of propositions with infinitely many instances: the instances of ∃p∀xp are all propositions that have a single instance; the instances of ∀p∃x∃yp are all the propositions that have a single instance and whose instance has a single instance; etc. Fortunately, it can be show by induction on the complexity of formulas that these are the only propositions with infinitely many instances expressed by any closed sentence. It is this fact that allows us to build models of instance structure.

We now turn to a formal description of the models. Let $D_{\langle \rangle }=\mathbb {Z}\backslash \{0\}$. Let u be a partial function from $\mathcal {P}(D_{\langle \rangle })\to D_{\langle \rangle }$ satisfying the following conditions:

• u is injective

• u is defined on all finite $X\subseteq D_{\langle \rangle }$

• u is defined on {fⁿ(p) : p ∈ D_〈〉}, for all $n\in \mathbb {N}$, where f(p) := u({p})

• − u(X) = u({−p : p ∈ X}) if u is defined on X

• u({1}) = 1

• u(X) > 0 if and only if p > 0 for all p ∈ X

While it is possible (but tedious) to explicitly specify such a function, the existence of such a function is easily seen by cardinality considerations. Intuitively, u(X) is the universal generalization whose instances are all and only the members of X, and − u({−p : p ∈ X}) is the corresponding existential generalization.

We let D_e be an arbitrary set and $D_{\langle \tau _{1}\dots \tau _{n}\rangle }=\{1,-1\}^{D_{\tau _{1}}\times \dots \times D_{\tau _{n}}}$. We then interpret our language as follows:

• ⟦x⟧^g = g(x)

• $\llbracket Fa_{1}{\dots } a_{n}\rrbracket ^{g}=\llbracket F\rrbracket ^{g}(\llbracket a_{1}\rrbracket ^{g},\dots ,\llbracket a_{n}\rrbracket ^{g})$

• ⟦¬⟧^g(p) = 1 iff p < 0

• ⟦∧⟧^g(p,q) = 1 iff p > 0 and q > 0

• $\llbracket \forall x\varphi \rrbracket ^{g}=u\{\llbracket \varphi \rrbracket ^{g^{\prime }}: g^{\prime }$ an x-variant of g}

• $\llbracket \exists x\varphi \rrbracket ^{g}=-u\{-\llbracket \varphi \rrbracket ^{g^{\prime }}: g^{\prime }$ an x-variant of g}

• ⟦inst⟧^g(p,q) = 1 iff ∃X((p ∈ X ∧ q = u(X)) ∨ (−p ∈ X ∧ q = −u(X)))

It can be shown by induction on the complexity of formulas that the clauses for quantifiers are well-defined given the conditions on u.

I’ll now describe two natural ways in which the construction might be modified or extended. First, in order to validate double-negation equivalence (and hence the injectivity of negation), we could replace the above interpretation of negation with the syncategorematic clause:

$$ \llbracket\neg\varphi\rrbracket^{g}=-\llbracket\varphi\rrbracket^{g} $$

Second, we can prune down the model in a natural way to eliminate arbitrary structure and superfluous propositions like u(∅). To do this, start with a model of the kind just described, and let the propositional domain $D^{*}_{\langle \rangle }$ of the new model be all and only the propositions denoted by closed sentences in the old model. $D^{*}_{\langle \rangle }$ is clearly a subset of the image of u. Moreover, it can be shown that, for any $p,q\in D^{*}_{\langle \rangle }$, if $u^{-1}(p)\cap D^{*}_{\langle \rangle }=u^{-1}(q)\cap D^{*}_{\langle \rangle }$, then u^− 1(p) = u^− 1(q). So we can generate a new model by replacing u with the function $u^{*}: u^{-1}(p)\cap D^{*}_{\langle \rangle }\mapsto p$. Models generated in this way are unique up to isomorphism.

Appendix F: Predicational Models

The model construction here repurposes many of the ideas from Appendices A and C. We first characterize the recoverable types, entities of which are constituents of atomic propositions, as follows:

Definition 19 (Recoverable types)

Let R be the smallest set of types containing e that is closed under the formation of non-empty sequences.

Next, we define our domains. Propositional domains are constructed as in Appendix C, except with the level-0 propositions P₀ now treated as having additional structure, viz predications involving recoverable entities. Recoverable properties and relations are identified with functions from entities of the relevant types to truth conditions. Non-recoverable properties and relations are identified with arbitrary functions from entities of the relevant types to propositions. The distinction here between recoverable and non-recoverable types is similar to that between monadic and polyadic types in Appendix A.

Definition 20 (Domains)

For some non-empty W and urelements c≠d: D_e = some set $D_{\langle \tau _{1},\dots ,\tau _{n}\rangle }= \mathcal {P}(W)^{D_{\tau _{1}}\times \dots \times D_{\tau _{n}}}$ for $\langle \tau _{1},\dots ,\tau _{n}\rangle \in R$ $D_{\langle \tau _{1},\dots ,\tau _{n}\rangle } = D_{\langle \rangle }^{D_{\tau _{1}}\times \dots \times D_{\tau _{n}}}$ for $\langle \tau _{1},\dots ,\tau _{n}\rangle \not \in R$ $D_{\langle \rangle }= \bigcup _{n\in \mathbb {N}} P_{n}$ $P_{0} = \{\langle f,\langle x_{1},\dots ,x_{n}\rangle \rangle : f\in D_{\langle \tau _{1},\dots ,\tau _{n}\rangle },x_{i}\in D_{\tau _{i}},\langle \tau _{1},\dots ,\tau _{n}\rangle \in R\}$ $P_{n+1} = P_{0} \cup \{\{c\}\cup X: X\subseteq P_{n}\}\cup \{\{d\}\cup X: X\subseteq P_{n}\}$ (same as in Appendix C)

Definition 21 (Truth conditions)

$tc: \langle f,\langle x_{1},\dots ,x_{n}\rangle \rangle \in P_{0}\mapsto f(x_{1},\dots ,x_{n})$; tc is defined as in Appendix C for p ∈ D_〈〉∖P₀.

Definition 22 (Property negation)

$': \bigcup _{\tau \in R\backslash \{e\}}D_{\tau }\to \bigcup _{\tau \in R\backslash \{e\}}D_{\tau }$ such that $f^{\prime }(x_{1},\dots ,x_{n})=W\backslash f(x_{1},\dots ,x_{n})$ for all $\langle x_{1},\dots ,x_{n}\rangle \in \text {domain}(f)$.

Definition 23 (Involution)

$^{*}: \langle f,\langle x_{1},\dots ,x_{n}\rangle \rangle \in P_{0}\mapsto \langle f^{\prime },\langle x_{1},\dots ,x_{n}\rangle \rangle$; ^∗ is defined as in Appendix C on D_〈〉∖P₀.

Since instance structure is no longer a species of conjunctive/disjunctive structure, we need to add it to the list of kinds of immediate non-factive grounding relations in terms of which we interpret ≺, which we do as follows.

Definition 24 (Projection functions)

$\pi _{i}(\langle x_{1},\dots ,x_{n}\rangle )=x_{i}$

Definition 25 (Immediate non-factive grounds)

For p,q ∈ D_〈〉,p ∝ q := either p ∈ q, or $\pi _{1}(q) \in \bigcup _{\tau _{i}\in R}\{\llbracket \forall _{\tau _{1},\dots ,\tau _{n}}\rrbracket ,\llbracket \exists _{\tau _{1},\dots ,\tau _{n}}\rrbracket \}$ (see below) and π₂(q) = 〈π₁(p)〉.

Definition 26 (Surrogates for truth conditions)

$s: \mathcal {P}(W)\to P_{0}$ s.t. $s(X)=\langle \{X\}^{D_{\langle e\rangle }},\langle \{W\}^{D_{e}}\rangle \rangle$ (gloss: self-identity is such that an X-world obtains)

Definition 27 (Interpretation)

We interpret ∧ and ∨ as in Appendix A. $\llbracket F^{\tau } a_{1}{\dots } a_{n}\rrbracket ^{g} = \langle \llbracket F\rrbracket ^{g},\langle \llbracket a_{1}\rrbracket ^{g},\dots , \llbracket a_{n}\rrbracket ^{g}\rangle \rangle$ if τ ∈ R and $\llbracket F\rrbracket ^{g}(\llbracket a_{1}\rrbracket ^{g},\dots , \llbracket a_{n}\rrbracket ^{g})$ if τ∉R ⟦T⟧^g(p) = {c,p} (alternatively {d,p}) ⟦¬⟧^g(p) = p^∗ (alternatively ⟦T⟧^g(p)^∗; see note 26) $\llbracket \prec \rrbracket ^{g}(p,q) =\{d\}\cup \{\{c,r_{1},\dots ,r_{n}\}:p= r_{1}\propto \dots \propto r_{n}= q\neq p\}$ $\llbracket \forall _{\tau }\rrbracket ^{g}(f) = \bigcap \{f(x): x\in D_{\tau }\}$ for τ ∈ R and $s(\bigcap \{tc(f(x)): x\in D_{\tau }\})$ for τ∉R; similarly for $\forall _{\tau _{1},\dots ,\tau _{n}}$ when n > 1 $\llbracket \exists _{\tau }\rrbracket ^{g}(f) = \bigcup \{f(x): x\in D_{\tau }\}$ for τ ∈ R and $s(\bigcup \{tc(f(x)): x\in D_{\tau }\})$ for τ∉R; similarly for $\exists _{\tau _{1},\dots ,\tau _{n}}$ when n > 1 $\llbracket (\lambda x_{1}^{\tau _{1}}{\dots } x_{n}^{\tau _{n}}.\varphi )\rrbracket ^{g}(y_{1},\dots ,y_{n}) = tc(\llbracket \varphi \rrbracket ^{g[x_{i}\to y_{i}]})$ for $\langle \tau _{1},\dots ,\tau _{n}\rangle \in R$ and $\llbracket \varphi \rrbracket ^{g[x_{i}\to y_{i}]}$ for $\langle \tau _{1},\dots ,\tau _{n}\rangle \not \in R$

Appendix G: Atomism with Predicational Structure

This appendix shows how the ideas of Appendices C and F can be combined to model a version of logical atomism where atomic propositions decompose into predications of fundamental properties and relations.

We model fundamental entities (logical atoms) as members of a set A of variables of our language such that, if $a_{0}^{\langle \tau _{1},\dots ,\tau _{n}\rangle } \in A$, then (i) $\langle \tau _{1},\dots ,\tau _{n}\rangle \in R$ and (ii) for all τ_i, there is some $a_{i}^{\tau _{i}}\in A$. Atomic propositions correspond to predications $\ulcorner a_{0}a_{1}{\dots } a_{n}\urcorner$ involving these variables (although, for reasons that will emerge, they are not themselves members of D_〈〉). We will now define a set F the members of which correspond to Boolean combinations of logical atoms:

Definition 28 (Fundamentally based propositions)

$F_{0} = \{\ulcorner a_{0}^{\langle \tau _{1},\dots ,\tau _{n}\rangle }a_{1}^{\tau _{1}}{\dots } a_{n}^{\tau _{n}}\urcorner : a_{i}\in A\}$ $F_{m+1} = F_{0} \cup \{\{n,p\}:p\in F_{m}\} \cup \{\{c\}\cup X: X\subseteq F_{m}\}\cup \{\{d\}\cup X: X\subseteq F_{m}\}$ $F = \bigcup _{n} F_{n}$

Note the new structure building operation: {n,p} is the negation of p.

Definition 29 (Domains)

Drawing on Appendices C and F: D_e = {a^e : a ∈ A} $D_{\langle \tau _{1},\dots ,\tau _{n}\rangle } = F^{D_{\tau _{1}}\times \dots \times D_{\tau _{n}}}$ for τ_i ∈ R $P_{0} = \{\langle f,\langle x_{1},\dots ,x_{n}\rangle \rangle : f\in D_{\langle \tau _{1},\dots ,\tau _{n}\rangle },x_{i}\in D_{\tau _{i}}, f(x_{1},\dots ,x_{n})\in F_{0}\}$ $P_{m+1} = \{\{n,p\}:p\in P_{m}\}\cup \{\{c\}\cup X: X\subseteq P_{m}\}\cup \{\{d\}\cup X: X\subseteq P_{m}\}\cup \phantom {}$ $\{\langle f,\langle x_{1},\dots ,x_{n}\rangle \rangle : f\in D_{\langle \tau _{1},\dots ,\tau _{n}\rangle },x_{i}\in D_{\tau _{i}}, f(x_{1},\dots ,x_{n})\in F_{m+1}\}$ $D_{\langle \rangle }= \bigcup _{n\in \mathbb {N}} P_{n}$ $D_{\langle \tau _{1},\dots ,\tau _{n}\rangle }=D_{\langle \rangle }^{D_{\tau _{1}}\times \dots \times D_{\tau _{n}}}$ if any τ_i∉R $D_{\langle \tau \rangle }= {P_{\kappa (\tau )+3}}^{D_{\tau }}$ for τ∉R

We now define three functions: $\mathfrak {a}$ maps every member of A to the corresponding entity in D; ⋅^↑ maps every member of D_〈〉 to the member of F that results from “unpacking” the content of its predicational constituents; ⋅^↓ maps every member of F to the corresponding member of D_〈〉.

Definition 30 (Logical atoms)

$\mathfrak {a}: a^{\tau }\to D_{\tau }$ s.t. $\mathfrak {a}(a^{e})=a$ $\mathfrak {a}(a_{0}^{\langle \tau _{1},\dots ,\tau _{n}\rangle })(\mathfrak {a}(a_{1}^{\tau _{1}}),\dots ,\mathfrak {a}(a_{n}^{\tau _{1}}))=\ulcorner a_{0}a_{1}{\dots } a_{n}\urcorner$ $\mathfrak {a}(a_{0}^{\langle \tau _{1},\dots ,\tau _{n}\rangle })(x_{1},\dots ,x_{n})=\{d\}$ if any $x_{i}\not \in \text {image}(\mathfrak {a})$

Remark 31

The third clause in the definition of $\mathfrak {a}$ says that applying an atomic relation to a non-atomic entity yields the trivial contradiction. This is somewhat arbitrary, but there appears to be no more principled option. Note that this possibility does not arise if A contains no higher-order predicates. This is a count in favor of the view that all atomic/fundamental properties and relations apply to individuals, held by Lewis [30] against Armstrong [1] (who held that laws of nature are a matter of a fundamental relation of nomic necessitation holding between properties of individuals). Note also that no denotations of logical connectives will be in the image of $\mathfrak {a}$ – logicality is distinguished from fundamentality (as Dorr [12] recommends against Sider [41]). This is a natural separation once we grant that only entities of recoverable types are fundamental, and also when conjunctive/disjunctive structure is modeled as an operation on sets of propositions rather than on pairs of propositions.

Definition 32 (Unpacking)

Let ⋅^↑ : D_〈〉→ F s.t. (i) p^↑ = π₁(p)(π₂(p)) if p is an ordered pair, and (ii) ({∗}∪ X)^↑ = {∗}∪{p^↑ : p ∈ X} for ∗∈{n,c,d}.

Definition 33 (Correspondence)

Let ⋅^↓ : F → D_〈〉 s.t. (i) $\ulcorner a_{0}a_{1}{\dots } a_{n}\urcorner ^{\downarrow }=\langle \mathfrak {a}(a_{0}),\langle \mathfrak {a}(a_{1}),\dots ,\mathfrak {a}(a_{n})\rangle \rangle$ for $\ulcorner a_{0}a_{1}{\dots } a_{n}\urcorner \in F_{0}$, and (ii) ({∗}∪ X)^↓ = {∗}∪{p^↓ : p ∈ X} for ∗∈{n,c,d}.

Note that the image of ⋅^↑ is F and that p^↑↓↑ = p^↑ for all p ∈ D_〈〉.

As in Appendix C, the interpretations of ¬ and of λ-abstraction appeal to an operation norm mapping every member of D_〈〉 to a Boolean-equivalent normal form of degree ≤ 3 (not 2, since now we have negation structure to account for). There are many natural such operations, so we omit an explicit characterization. The clauses for negation and λ-abstraction are then:

• ⟦¬⟧^g(p) = norm({n,p^↑↓})

• $\llbracket (\lambda x_{1}^{\tau _{1}}{\dots } x_{n}^{\tau _{n}}.\varphi )\rrbracket ^{g}(y_{1},\dots ,y_{n})=(\llbracket \varphi \rrbracket ^{g[x_{i}\mapsto y_{i}]})^{\uparrow }$ for τ_i ∈ R

• $\llbracket (\lambda x_{1}^{\tau _{1}}{\dots } x_{n}^{\tau _{n}}.\varphi )\rrbracket ^{g}(y_{1},\dots ,y_{n})=\llbracket \varphi \rrbracket ^{g[x_{i}\mapsto y_{i}]}$ if some τ_i∉R,n > 1

• $\llbracket (\lambda x^{\tau }.\varphi )\rrbracket ^{g}(y)=norm((\llbracket \varphi \rrbracket ^{g[x_{i}\mapsto y_{i}]})^{\uparrow \downarrow })$ for τ∉R

We now turn to the interpretation of quantifiers. As in Appendices A and C, if f is a monadic property of a non-recoverable type, then its universal generalization is the conjunction of its instances. (There is no primitive notion of universal generalization applying to non-recoverable polyadic relations.) The situation for recoverable types is more complicated. The denotation of a recoverable-type universal quantifier $\forall _{\tau _{1},\dots ,\tau _{n}}$ cannot always map $f\in D_{\langle \tau _{1},\dots ,\tau _{n}\rangle }$ to the conjunction of the instances of the resulting generalization, since the image of f may be unbounded in level and so have no conjunction; nor can it map f to appropriate truth conditions, as in Appendix F, since worlds are absent from the present model construction. Instead, we interpret the universal quantifier as the function mapping f to the conjunction of normalizations of propositions its image (where norm is defined on F as in D_〈〉). This is inelegant but harmless, since application at recoverable types corresponds to ordered-pair formation rather than to function application, so the clauses below won’t threaten the claim that the only immediate grounds of a generalization are it instances.

• ⟦∀_τ⟧^g(f) = {c}∪{f(x) : x ∈ D_τ} for τ∉R

• ⟦∃_τ⟧^g(f) = {d}∪{f(x) : x ∈ D_τ} for τ∉R

• $\llbracket \forall _{\tau _{1},\dots ,\tau _{n}}\rrbracket ^{g}(f)=\{c\}\cup \{norm(f(x_{1},\dots ,x_{n})): x_{i}\in D_{\tau _{i}}\}$ for τ_i ∈ R

• $\llbracket \exists _{\tau _{1},\dots ,\tau _{n}}\rrbracket ^{g}(f)=\{d\}\cup \{norm(f(x_{1},\dots ,x_{n})): x_{i}\in D_{\tau _{i}}\}$ for τ_i ∈ R

We now define immediate non-factive grounds. There are three new cases to consider: negated conjunctions and disjunctions, negated negations, and predications of non-fundamental recoverable properties and relations; unlike all of the previous constructions, this third case is an example of grounding not attributable to Boolean or quantificational structure.

Definition 34 (Immediate non-factive grounds)

p ∝ q := p,q ∈ D_〈〉 and one of the following conditions holds:

• 1. $\pi _{1}(q) \in \bigcup _{\tau _{i}\in R}\{\llbracket \forall _{\tau _{1},\dots ,\tau _{n}}\rrbracket ,\llbracket \exists _{\tau _{1},\dots ,\tau _{n}}\rrbracket \}$ and π₂(q) = 〈π₁(p)〉

• 2. $\pi _{1}(q)\not \in \bigcup _{\tau _{i}\in R}\{\llbracket \forall _{\tau _{1},\dots ,\tau _{n}}\rrbracket ,\llbracket \exists _{\tau _{1},\dots ,\tau _{n}}\rrbracket \}, \pi _{1}(q)(\pi _{2}(q))^{\downarrow }=p$, and p≠q

• 3. n∉q and p ∈ q

• 4. q = {n,r},n∉r,s ∈ r and p = {n,s}, for some r,s

• 5. q = {n,{n,p}}

We interpret ≺ using ∝ as in Appendix F. To model full grounding, we modify the definition of ways of fully grounding a proposition as below, and then give the same interpretation of < as in Appendix B.

Definition 35

X is a way for Γ to fully ground p if and only if, for some non-trivial rooted tree G = 〈V,E,v〉 and surjective function f : V → X,

• 1. f(v) = p

• 2. {f(x) : x has no children} = Γ

• 3. if x is a child of y, then f(x) ∝ f(y)

• 4. if q ∈ X∖Γ and either (i) c ∈ q, (ii) for some r, q = {n,r} and d ∈ r, or (iii) $\pi _{1}(q)\in \bigcup _{\tau _{i}\in R}\{\llbracket \forall _{\tau _{1},\dots ,\tau _{n}}\rrbracket \}$, then $\{r: r\propto q\}\subseteq X$.

We interpret predication as in Appendix F: as ordered-pair formation at recoverable types and as function application at non-recoverable types. While both options for ∧ and ∨ described in Appendix A remain available, the second seems even more natural here, given the precedent of ¬ failing to express the model-theoretic negation operation.

One attraction of this theory is that the notions of being a fundamental entity (i.e., in the image of $\mathfrak {a}$) and of being an atomic proposition (i.e., being a predication involving only such entities) are definable in the object language:

• Atomic := (λp.(λq.q ≺ p) = (λq.q ≺ q))

• Fundamental_e:= (λx^e.x = x)

• Fundamental_τ:= (λx^τ.x≠x) for τ∉R

• Fundamental$_{\langle \tau _{1},\dots ,\tau _{n}\rangle }:= (\lambda F.\exists x_{1}^{\tau _{1}}{\dots } \exists x_{n}^{\tau _{n}}(\text {Atomic}(Fx_{1}{\dots } x_{n}$))) for τ_i ∈ R

Note, finally, that the constructions here can be modified in the manner of Appendix D to validate the kind of view discussed in Section 5.3. On this view, all generalizations – even of polyadic relations at non-recoverable types – carry a record of their instances, which will be all and only their immediate grounds. At recoverable types, they do so by having predicational structure which records the kind of generalization they are and what property or relation is being generalized. Such structure is inconsistent at non-recoverable types: there, generalizations are identified with conjunctions/disjunctions of their instances. On this view conjunction and disjunction are not relations between propositions but operations for forming new propositions from sets of propositions. It is then natural to regiment negation in a similarly syncategorematic way. We can then have negated formulas express the model-theoretic negation of the proposition expressed by their unnegated counterparts, without restricting classical quantification theory, as discussed in Section 5.3. Doing so validates the standard principles about the grounds of negated propositions from Fine [15].