An infinitely descending chain of ground without a lower bound

Abstract

Using only uncontentious principles from the logic of ground I construct an infinitely descending chain of ground without a lower bound. I then compare the construction to the constructions due to Dixon (forthcoming) and Rabin and Rabern (J Philos Log, 2015).

1 Introduction

In this note I construct an infinitely descending chain of partial ground without a lower bound. It is of particular interest that even those philosophers who think that all truths are ultimately grounded in ungrounded truths have to accept the possibility of this chain. While this is not the first such chain that has been proposed in literature, the present construction is both novel and requires less assumptions than the previous proposals.

The main part of the paper (Sects. 2–4) is taken up by the construction of the infinitely descending chain. In the final section (Sect. 5) I compare the present construction with the constructions due to Dixon (forthcoming) and Rabin and Rabern (2015).

2 Terminology

Let us use “<” as a sentential operator¹ for strict full ground.² Let us write $\Gamma <\phi$ to mean that the truths $\Gamma$ strictly fully ground the truth $\phi$.³ What we will show is that there are truths L ₀, L ₁, …, S ₀, S ₁, …, M ₁, M ₂, … such that for each n:

• (1) S _n  < L _n

• (2) L _n+1, M _n+1 < L _n

• (3) There is no set of truths $\Delta$ such that $\Delta <L_n$, for each n

The resulting structure can be depicted as follows.

Let us call a structure like this an unboundedly ever-descending tree of ground.⁴ Using $L_i \succ L_{i+1}$ to say that L _i is partially grounded in L _i+1, this tree witnesses that $L_0 \succ L_1 \succ L_2 \succ \ldots$, that is, that L ₀, L ₁, L ₂, … is an infinitely descending chain of partial ground.

3 Constructing the tree

Let S ₀, S ₁, … be countably many sentences expressing distinct ungrounded truths.⁵ For each natural number $n \ge 0$, let L _n be the sentence:

• (L _n ) Either S _n or L _n+1 is true

It is clear that L _n is true for each n. After all, S _n is true for each n, and S _n is a disjunct of L _n .

Let us write $T \ulcorner L_{n} \urcorner$ for the sentence “L _n is true”. Since L _n is true for each n, $T \ulcorner L_{n} \urcorner$ is also true for each n. Since $T \ulcorner L_{n} \urcorner$ is true for each n, and $T \ulcorner L_{n+1} \urcorner$ is a disjunct of L _n , the following holds:

• (Disjunctive-Grounding) $T \ulcorner L_{n+1} \urcorner <L_n$

This is a special case of the principle that a disjunction is grounded in its true disjuncts. While it is true that there are situations where the general principle has to be given up⁶ this is not one of those situations.⁷

For each n, let M _n be the truth that L _n means that (S _n or $T\ulcorner L_{n+1} \urcorner$). The following principle is accepted by almost all writers on ground⁸:

• (Truth-Grounding) $M_n,L_n <T\ulcorner L_n \urcorner$

• (If the sentence S means that p, then S’s being true is grounded in p’s being the case together with S’s meaning that p.)⁹ It follows that¹⁰:

$$M_{n+1}, L_{n+1} <L_n$$

This finishes the construction.

A variant of the above construction establishes that there is a dense chain of ground. Let $\{S_r :r \in {\mathbb {Q}}\}$ be some sentences expressing distinct ungrounded truths. For each rational number $r \in {\mathbb {Q}}$, let L _r be the sentence

• (L _r ) S _r or for some q less than r, L _q is true

• For each $q \in {\mathbb {Q}}$ let M _q be the truth that L _q means what it does. The same form of argument as given above establishes that ($L_r, M_{r} <L_q$ iff r is less than q).

It is intuitively obvious that the tree is unbounded, but it is instructive to make explicit the assumptions that are required to prove this: we only require some of the principles of the impure logic of ground presented in (Fine 2012a, pp. 58–67).

4 The tree is unbounded

To prove that the tree is unbounded it is convenient to introduce the notion of weak ground, writing $\Gamma \le \phi$ for the claim that the truths $\Gamma$ weakly fully ground the truth $\phi$.¹¹ Whereas strict ground is irreflexive, weak ground is not: in fact, for each truth $\phi$ we have $\phi \le \phi$. We say that $\phi$ is a weak partial ground for $\psi$ if there is some $\Gamma$ such that $\phi , \Gamma \le \psi$.

We can now dispense with the assumption that the S _i express distinct and ungrounded truths; it suffices for our purposes that the S _i , pairwise, do not have a common weak partial ground.¹²

We require two assumptions from the logic of ground. The first is that mediate ground is the closure of immediate ground under Cut.¹³ The second assumption is that the immediate grounds for a disjunction $\phi \vee \psi$ are either (1) the truth $\phi$; (2) the truth $\psi$; or (3) the truths $\phi , \psi$ taken together.

I find these assumptions from the impure logic of ground unobjectionable, but I should flag that they are not uncontroversial. Interestingly, one of the previous examples of infinitely descending chains of ground requires rejecting them; we will return to this issue in Sect. 5 below.

With the relationships of immediate ground made explicit the tree looks as follows:

In this figure the single arrows indicate relationships of strict full ground; the double arrows indicate that M _n , L _n together strictly fully ground $T\ulcorner L_n \urcorner$. (The truths M _i and S _j might themselves be (weakly) grounded, but we (harmlessly) omit displaying their grounds.)

We require two non-logical assumptions. The first assumption is that M ₀, M ₁, … do not together ground any L _i . This is reasonable: that the sentences L _i mean what they do is not enough to account for the truth of any L _j . The second assumption is that no grounds for the truths S _i (weakly) partially ground any of the truths M _j (for j > i). This assumption, too, is reasonable. The truth M _j is the truth that L _j means what it does: we can surely choose the truths S _i such that no grounds for S _i grounds M _j , for j > i.

From these assumptions it follows that the above tree is unboundedly ever-descending.

Observe first that if a truth $\psi$ is a partial ground of L ₀, then any occurrence of $\psi$ in the above tree occurs some finite number of steps below L ₀. (This follows from mediate ground’s being the closure of immediate ground.)¹⁴

Suppose for contradiction that there are some truths $\Delta$ such that $\Delta <L_n$ for each n. Suppose first that $T\ulcorner L_n \urcorner$ is in $\Delta$ for some n. Let us write $\Delta = \Delta ' \cup \{T\ulcorner L_n \urcorner \}$. Then, since $\Delta ', T\ulcorner L_n \urcorner <L_{n}$ (by assumption) and $L_n, M_n <T\ulcorner L_n \urcorner$, a strict partial ground for $T\ulcorner L_n \urcorner$ is $T\ulcorner L_n \urcorner$ itself, contradicting the irreflexivity of partial ground. The same argument establishes that $L_n \notin \Delta$ for each n.

Suppose next that $\Delta ' \le S_n$ for no $\Delta ' \subset \Delta$ and no n. Since each $\delta \in \Delta$ is such that each occurrence of $\delta$ is only finitely many steps away from L ₀ it follows that each occurrence of a $\delta \in \Delta$ has to be below some M _i . Since the truths M ₀, M ₁, … do not ground L ₀, the truths $\Delta$ do not ground L ₀ either.

It follows that there is an m such that $\Delta = \Delta _m \cup \Delta _m'$ and $\Delta _m \le S_m$. If $i \ne j$, S _i and S _j have no common (weak partial) ground. It follows that $\Delta _m$ does not weakly partially ground any S _n , $n \ne m$. By our second non-logical assumption $\Delta _m$ does not weakly partially ground any $M_n, n > k$. It follows that $\Delta _m \cup \Delta _m' = \Delta$ does not strictly ground L _n for n > m.

Since $\Delta$ was arbitrary this shows that there is no $\Delta$ such that $\Delta <L_n$ for each n. The tree is unbounded.¹⁵

A notable feature of the construction is that all the principles we have relied on should be acceptable to those who think that, necessarily, all truths are grounded in ungrounded truths. Even if it is necessary that all truths are grounded in ungrounded truths it is possible that there are infinitely descending chains of ground without a lower bound.

5 Comparison with other unbounded chains

This is not the first purported unbounded infinitely descending chain of ground in the literature. Dixon (forthcoming) gives two examples of infinitely descending chains without a lower bound. It must be admitted that in one respect his chains are more striking: they are examples of unbounded infinitely descending chains of full ground. Unfortunately, there are problems. One of his example does not work; and while the other might work, it requires infinitary resources and unusually problematic ones at that.¹⁶

In one of his examples Dixon considers a situation where we have countably many mereologically simple objects $o_0, o_1, \ldots$. They have the following masses. $o_0$ has mass exactly 1 kg; $o_1$ has mass exactly 1/2 kg; and so on. He then considers the following two sequences of truths $a_0, a_1, \ldots , b_0, b_1, \ldots$

a 0	$\exists z$ (z has non-zero mass at most 1 kg)
a 1	$\exists z$ (z has non-zero mass at most 1/2 kg)
a 2	$\exists z$ (z has non-zero mass at most 1/4 kg)
$\vdots$
b 0	$\exists z$ (z has non-zero mass exactly 1 kg)
b 1	$\exists z$ (z has non-zero mass exactly 1/2 kg)
b 2	$\exists z$ (z has non-zero mass exactly 1/4 kg)
$\vdots$

Dixon then claims that the following is an unboundedly ever-descending tree of ground.

Dixon here relies on the principle labeled (Determinables) below.

(Determinables):

For any positive real numbers x and y, if (1) something with a non-zero mass of at most y kg exists and (2) $y = 0.5x$, then $\exists z\,(z\,{\text {has}}\,{\text{non-zero}}\,{\text{mass}}\,{\text{of}}\,{\text{at}}\,{\text{most}}\,y) <\exists z\,(z\,{\text {has}}\,{\text{non-zero}}\,{\text{mass}}\,{\text{of}}\,{\text{at}}\,{\text{most}}\,x)$

(Determinables) is false; we can see this as follows. Since $a_0$ is an existentially quantified truth, it follows by the elimination rules for $\exists$ (Fine 2012a, pp. 65–66) that if $a_1$ strictly fully grounds $a_0$, then $a_1$ weakly fully grounds that some $o_i$ has non-zero mass at most $1/2^i$ kg.¹⁷ Since $a_1$ is existential it will be grounded in its instances. In particular, for each $j>i$ it will be grounded in the truth that $o_j$ has non-zero mass at most $1/2^j$. But since $o_i, o_j$ are distinct mereologically simple objects the truths about the mass of one does not ground the truths about the mass of the other.

It is worth noting that while (Determinables) is false there is a correct principle in its vicinity, namely:

(Determinables′):

If $y= .5x$ then the truth that z has non-zero mass at most y grounds the truth that z has non-zero mass at most x

Dixon’s second example—an example also given by Rabin and Rabern (2015)—relies on infinitary disjunction. Let $p_0, p_1, \ldots$ be countably many distinct fundamental truths. Consider now the infinite disjunction:

$$p_0 \vee \left( p_1 \vee \left( p_2 \vee \ldots \right) \right) )$$

They claim that we now get the following infinitely descending chain of ground:

One advantage of the construction given in this paper is that it requires no infinitary resources at all. But even if one has no problem with infinitary disjunction as such—and I do not—Dixon and Rabin and Rabern have to make some problematic assumptions about how infinitary disjunctions work.¹⁸

Let us first set aside a superficial worry. From the notation it might appear that the infinite disjunction is formed by infinitely many applications of the binary operation of disjunction. Dixon, however, is clear that he does not understand infinite disjunction in this way; rather, he thinks that we form infinite disjunctions by applying the disjunction operator once to a collection of infinitely many sentences $\{p_0, p_1, \ldots \}$, in a single operation forming the complex sentence: $\bigvee \{p_0, p_1, \ldots \}$.

I have no objection to being able to form infinite disjunctions in this way. What I am worried about is the claim that $\bigvee \{p_0, p_1, \ldots \}$ is strictly fully grounded in $\bigvee \{p_1, p_2, p_3 \ldots \}$. Let us use $\approx$ as an operator for mutual weak full ground (or factual equivalence in the sense of (Correia 2010)). The crux of the issue is whether one accepts the following associativity thesis:

• (Associativity)

$$\bigvee \{p_0, p_1, p_2, \ldots \} \approx \bigvee \{p_0, \bigvee \{p_1, p_2, \ldots \}\}$$

(Dixon explicitly accepts this principle.¹⁹)

I do not accept (Associativity).

First, it seems we should distinguish between $\bigvee \{p_0, p_1, \ldots \}$ and $\bigvee \{p_0, \bigvee \{p_1, p_2, \ldots \}\}$. In forming the first truth we apply the disjunction operator once; in forming the second truth we apply it twice. The immediate reply is that this is merely a difference in the expression of the truth and not a difference in the truths expressed.

Secondly, however, (Associativity) is inconsistent with the assumption that the immediate full grounds for a truth $\phi \vee \psi$ are all and only $\phi$, $\psi$, and $\phi , \psi$ taken together. (We here assume that $\phi , \psi$ are both true.) To see this let $p , q , r$ be three sentences expressing three truths that, pairwise, have no common ground. It follows by (Associativity) that $(p \vee (q \vee r ))\approx ((p \vee q ) \vee r )$. Since $(p \vee q)$ is an immediate full ground for $(p \vee q) \vee r$, if $(p \vee (q \vee r)) \approx ((p \vee q) \vee r)$ then $p \vee q$ has to be an immediate full ground for $p \vee (q \vee r)$ as well. But $p \vee q$ is identical to neither p nor $q \vee r$.

Third, and most importantly, while one might respond by giving up the claim about the immediate grounds of disjunctions, holding on to (Associativity) is subject to a serious but hitherto unnoticed difficulty.

Let S, T be two sentences expressing distinct, fundamental truths. Consider now the sentences (Good) and (Bad).

• (Good) (The unique sentence with label (Bad) is true $\vee S) \vee T$

• (Bad) The unique sentence with label (Bad) is true $\vee (S \vee T)$

These sentences are of the forms: $(\phi \vee \psi ) \vee \theta$ and $\phi \vee (\psi \vee \theta )$, respectively. However, (Good) and (Bad) do not mutually weakly fully ground each other.²⁰ For (the truth expressed by) “The unique sentence with label (Bad) is true” is a ground for (the truth expressed by) (Good), but it is not a ground for (the truth expressed by) (Bad): the latter would contradict the irreflexivity of partial ground.²¹

Admittedly, this only shows that (Associativity) is not valid; it does not show that there is anything wrong with the particular application Dixon and Rabin and Rabern make of (Associativity). Recently, however, it has become clear that grounding is treacherous when account is taken of self-reference and impredicativity [see e.g., Fine (2010), Litland (2015), Correia (2014) and Krämer (2013)]. Until we have a general theory of ground telling us which instances of (Associativity) are safe and which are not, some skepticism about the chain due to Dixon and Rabin and Rabern is warranted.

I conclude that the herein presented chain is the most secure unbounded infinitely descending chain yet presented.