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Abstract

I explore the logic of ground. I first develop a logic of weak ground. This logic strengthens the logic of weak ground presented by Fine in his ‘Guide to Ground.’ This logic, I argue, generates many plausible principles which Fine’s system leaves out. I then derive from this a logic of strict ground. I argue that there is a strong abductive case for adopting this logic. It’s elegant, parsimonious and explanatorily powerful. Yet, so I suggest, adopting it has important consequences. First, it means we should think of ground as a type of identity. Second, it means we should reject much of Fine’s logic of strict ground. I also show how the logic I develop connects to other systems in the literature. It is definitionally equivalent both to Angell’s logic of analytic containment and to Correia’s system G.

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Acknowledgements

Thanks to Cian Dorr, Kit Fine, Marko Malink, Chris Scambler, Alex Skiles, Trevor Teitel and a referee for this journal for their very helpful input on this paper.

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Appendix

Appendix

In this appendix, I will prove many of the claims I’ve made in the main text. The proof of some claim in the main text will fall under the section heading in which that claim was made.

1.1 A.1 What do logically complex propositions ground?

1.1.1 A.1.1 Disjunction

I first prove commutativity, associativity, supplementation and DeMorgan(1) hold for disjunction:

figure a
figure b
figure c
figure d

I then prove the distributivity principles follow from ∨-Agglomeration. Note that when I label a step ‘R’ I indicate I’m taking advantage of the repetition invariance of lists (so A is the same list as A, A…) to re-iterate the same lemma.

figure e
figure f
figure g
figure h

Note how Weak ∨-Agglomeration could be used in all but the last proof. I now use ∨∧D2 to derive ∨-Agglomeration from Weak ∨-Agglomeration. In the below proof ‘Id’ stands for ‘Identity.’

figure i

Note that we could replace Weak ∨-Agglomeration with ∨-Idempotence in this proof. So this also shows that, in the presence of Fine’s rules, ∨-Agglomeration derives from ∨-Idempotence, ∨∧D2 and ∧-Agglomeration.

1.1.2 A.1.2 Double negation

Here I just prove the ∧∨ equivalencies. We get two of these equivalencies with the agglomeration rules (and the ¬-Introduction rules) alone. However, in getting the other equivalencies ¬¬-Idempotence becomes useful. Here are the proofs:

figure j

1.2 A.2 Ground-theoretic equivalence

Proof of LL

To prove LL we show it holds in two cases: one in which C contains no ground-theoretic operators, and one in which C contains some ground-theoretic operators. This exhausts the cases. Let’s begin by considering the first case. In this case, C is a truth-functional formula. That means it is either atomic, or contains only truth-functional operators. Observe that IMP establishes that if AB, then AB. It follows that A and B are inter-substitutable in any truth-functional formula salva veritate, and so the relevant instance of LL must hold. Now let’s consider the second case, in which C contains some instance of a ground-theoretic connective. To prove that LL holds in this case, we take advantage of the following lemma:

$$ \mathbf{Substitution:} \quad A \approx B \rightarrow C \approx D$$

Where D is the result of just swapping As for Bs in C. This just says that when A and B are ground-theoretically equivalent, then any formula, C, is ground-theoretically equivalent to the result of swapping As for Bs in that formula. One can prove this by induction on the complexity of formulas. The base case –where C contains no logical operators– is trivial. For the inductive step, in the case of conjunction we use ∧-agglomeration, for disjunction we use ∨-agglomeration and for negation we use ¬-Introduction.

We then use Substitution to prove that LL holds in the ground-theoretic case. The case where C contains only instances of ≈ follows directly from Substitution, so we’re left with just cases in which C is of the form Δ ≤ E. There are two kinds of LL instances in these cases. Firstly, we might form D by replacing As for Bs in E. Secondly, we might form D by replacing As for Bs in Δ. Suppose we replace As for Bs in E. Call the resultant formula E[A/B]. By Substitution, EE[A/B]. So, by Def(≈) and transitivity, Δ ≤ E[A/B]. So this first kind of LL instances hold. Now suppose we replace As for Bs in Δ. To replace As for Bs in Δ, we need to replaces As for Bs in some member (or members), F1...Fn, of Δ. Label the formula(s) we do this to F\(^{[A/B]}_{1}\)...F\(^{[A/B]}_{n}\) and label the list of all the other formulas Γ. The resultant list can thus be written: F\(^{[A/B]}_{1}\)...F\(^{[A/B]}_{n} , {\Gamma }\). By Substitution, F\(_{k} \approx \textit {F}^{[A/B]}_{k}\) for every Fk in F1Fn. Since for every G in Γ, GG, we can therefore apply Def(≈) and CUT to derive F\(^{[A/B]}_{1}\)...F\(^{[A/B]}_{n}, {\Gamma } \leq E\). So this second kind of LL instance holds. So LL holds when C is of the form Δ ≤ E. So LL holds for every ground-theoretic C. Since LL holds for every non-ground theoretic C, it follows that LL holds generally. □

Proof of ∨∧D2

We prove ∨∧D2 follows from Weak ∨-Agglomeration in the presence of LL, the DeMorgan(2) rules and ¬¬-Idempotence. A full natural deduction proof of this is unwieldy. But it suffices to note that the following ground-theoretic equivalences are provable:

$$\begin{array}{lrl} 1\hspace{1cm} & (A \vee B) \wedge (A \vee C)& \approx \neg \neg ((A \vee B) \wedge (A \vee C))\\ 2\hspace{1cm} & & \approx \neg (\neg(A \vee B) \vee \neg (A \vee C)) \\ 3\hspace{1cm} & &\approx \neg ((\neg A \wedge \neg B) \vee (\neg A \wedge \neg C)) \\ 4\hspace{1cm} & & \approx \neg (\neg A \wedge (\neg B \vee \neg C)) \\ 5\hspace{1cm} & & \approx \neg (\neg A \wedge \neg(B \wedge C))\\ 6\hspace{1cm} & & \approx \neg \neg (A \vee (B \wedge C) \\ 7\hspace{1cm} & & \approx A \vee (B \wedge C) \end{array}$$

Line 1 and 7 rely on ¬¬-Introduction and ¬¬-I. Line 2, 3, 5 and 6 rely on the DeMorgan rules together with LL. Line 4 relies on ∧∨D1, ∧∨D2 and LL. As I show above ∧∨D1 and ∧∨D2 are both consequence of Weak ∨-Agglomeration. So this shows that ∨∧D2 can be derived from Weak ∨-Agglomeration in the presence of the other rules. Given the proof in the Appendix A this shows that ∨-Agglomeration can, in this context, be derived from Weak ∨-Agglomeration.□

1.3 A.3 Angell’s system

I here prove AC∗∗ and LWG are equivalent. We first prove that all the basic rules of AC∗∗ are rules in LWG. E1 is trivial, E2 follows from Trans and E3 is just ¬-Introduction. E4, E5 and E6 follow from idempotence. E7 and E8 follow from commutativity. E9 and E10 follow from supplementation. E11 and E12 follow from the DeMorgan laws. E13 and E14 follow from associativity. E15 and E16 follow from distributivity. E17 follows from IMP and E18 follows from the reduction theorem (</≈).

We now prove all basic rules in LWG are rules in AC∗∗. When I use the definition of weak full ground, E18, together with classical logic I will move directly between formulas of equivalence and those of weak full ground. I will skip indicating the compression of the steps with vertical dots. Here are the proofs:Proof of disjunction and conjunction introduction.

figure k

The proof of the other disjunction rule is essentially the same. The proof of ∧-I relies just on E6 and E18.Proof of negation rules.

Here are proofs of one of the negated disjunction rules, and the negated conjunction rule:

figure l

The proof of ¬-I and ¬¬-idempotence are as follows:

figure m

Meanwhile, ¬-Introduction is just an instance of E3.Proof of agglomeration rules.

In the interests of readability, I’ll prove slightly simplified versions of the agglomeration rules. These are simplified in that they omit the arbitrary lists (Δ, Γ) following the sentences which we agglomeration into conjunctions or disjunctions. These can be easily added.

The proofs of the conjunction and disjunction agglomerations rules are as follows:

figure n

Note that in the proof of ∧-agglomeration, in the second application of E18 we take advantage of the fact that removing any number of ∧ operators from (AB) can be implemented by removing none at all.

We now prove the agglomeration rules for negated conjunction and disjunction respectively:

figure o

Proof of pure logic rules.

We have left the pure logic to last. Identity is just an instance of E13. IMP follows from E17, together with classical logic. CUT is somewhat more difficult. To prove CUT, we first prove ≤ obeys a transitivity principle:

figure p

We now prove that ∧-Supplementation is valid in AC∗∗. A full natural deduction proof of this is very unwieldy, so I begin by noting that, given \(A_{1} \wedge A_{2} \approx (A_{1} \vee \hat {\Delta }_{1}) \wedge (A_{2} \vee \hat {\Delta }_{2} )\), the following are provable in AC∗∗:

$$ \begin{array}{lrlr} 1 \hspace{1cm} & A_{1} \wedge A_{2} & \approx (A_{1} \vee \hat {\Delta}_{1}) \wedge (A_{2} \vee \hat {\Delta}_{2} ) \\ 2 \hspace{1cm}& & \approx ((A_{1}\vee \hat {\Delta}_{1}) \wedge A_{2}) \vee ((A_{1}\vee \hat {\Delta}_{1}) \wedge \hat {\Delta}_{2}) \\ 3 \hspace{1cm}& &\approx (((A_{1}\wedge A_{2}) \vee (\hat {\Delta}_{1}\wedge A_{2})) \vee ((A_{1}\wedge \hat {\Delta}_{2}) \vee (\hat {\Delta}_{1}\wedge \hat {\Delta}_{2}))) \\ 4 \hspace{1cm}& &\approx (((A_{1}\wedge A_{2}) \vee (\hat {\Delta}_{1}\wedge A_{2})) \vee (A_{1}\wedge \hat {\Delta}_{2}))) \vee ((\hat {\Delta}_{1}\wedge \hat {\Delta}_{1}) \vee (\hat {\Delta}_{1}\wedge \hat {\Delta}_{2})) & \hspace{2cm} \\ 5 \hspace{1cm}& &\approx (((A_{1}\wedge A_{2}) \vee (\hat {\Delta}_{1}\wedge A_{2})) \vee ((A_{1} \wedge \hat {\Delta}_{2} ) \vee (\hat {\Delta}_{1}\wedge \hat {\Delta}_{2}))) \vee (\hat {\Delta}_{1}\wedge \hat {\Delta}_{2})\\ 6 \hspace{1cm}&& \approx (A_{1}\wedge A_{2}) \vee (\hat {\Delta}_{1}\wedge \hat {\Delta}_{2}) \end{array} $$
(1)

Line 2 and 3 both apply distributivity (E15 and E16). Line 4 applies Idempotence (E5). Line 5 applies associativity (E13). Line 6 uses the third line in substitutions like occur in the above proofs. I’ve labelled this inference (1). Given this, the following tree is valid:

figure q

And so we have proven ∧-Supplementation. Here’s the proof of CUT in the three premise case:

figure r

Finally, let’s consider Def(≈). The left-right of this follows straightforwardly from E18 and E1. The right-left follows from E18, E1 and E8 and E2. So all the basic rules of LWG are valid in AC∗∗. So AC∗∗ and LWG are equivalent.

1.4 A.4 The logic of strict ground

Let’s start by proving Exchange. Begin by observing that if we assume BB ∨ (AC), then we can infer B ≈ ((BA) ∨ B). This is because, given BB ∨ (AC), the following are ground-theoretic equivalencies:

$$ \begin{array}{lrlr} 1 \hspace{2cm} & B & \approx B \vee (A \wedge C) \\ 2 \hspace{2cm} && \approx (B \vee A) \wedge (B \vee C) \\ 3 \hspace{2cm} && \approx ((B \vee A) \wedge (B \vee A)) \wedge (B \vee C) \\ 4 \hspace{2cm} && \approx (B \vee A) \wedge ((B \vee A) \wedge (B \vee C)) & \hspace{2cm} \\ 5 \hspace{2cm} && \approx (B \vee A) \wedge B \\ 6 \hspace{2cm} && \approx (B \vee A) \wedge (B \vee B)\\ 7 \hspace{2cm} && \approx B \vee (B \wedge A) \end{array} $$
(2)

The natural deduction proof of this is long and hard to read, so I will omit it. The important line is line five, which we obtain from the previous line via B ≈ ((BA) ∧ (BC)) and Leibniz’s law. I’ll label the inference associated with these equivalencies (2).

This allows us to prove Exchange via reasoning by cases. There are two cases. One where Δ has no members, and one where it has Δ has n members. The first case follows from Amalgamation and identity. The second case is more complex:

figure s

So, in both cases Exchange holds and so the principle is proven. Here, as above, I omit indicating the compression of steps when using (≈/≤) and the classical rules. I will do the same with the definitions below.

Let’s now prove that Subsumption(≤/≼) and Transitivity (≼/≼) hold in LWG+.

figure t

I will prove one other of Fine’s subsumption rules is valid below, and from all this it is quite easy to show the rest of Fine’s rules are valid.

1.4.1 A.4.1 Introduction rules for strict ground

Let’s now prove the introduction rules for strict ground. The ¬∧ rules are proved in the same way as the proof of ∨ rules, but use ¬∧-I1 and ¬∧-I2. Here’s the proof of the conjunction rule:

figure u

The ¬∨ rule is proved in a similar way.

1.5 A.5 Fine’s logic of strict ground

1.5.1 A.5.1 Fine’s introduction rules

I prove that Fine’s introduction rules are inconsistent with LWG+. We begin by proving subsumption:

figure v

We can now prove the ¬¬ introduction rule and the ∧ introduction rule cannot be consistently added to LWG:

figure w

Proofs for the negated conjunction and disjunction rules are the same, except rely on different idempotence principles. Any remaining proofs are available upon request.

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Lovett, A. The logic of ground. J Philos Logic 49, 13–49 (2020). https://doi.org/10.1007/s10992-019-09511-1

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