Grounding, Entailment and Levels of Reality...Grounding, Entailment and Levels of Reality Fabrice Correia Institut de Philosophie, Universit e de Neuch^atel Groundedness in Semantics

Grounding, Entailment and Levels of Reality

Fabrice Correia

Institut de Philosophie, Universite de Neuchatel

Groundedness in Semantics and Beyond, Oslo, 24.08.2013

Intro. Proof theory I Proof theory II Grounding, truth and entailment Semantics for grounding Concl.

- Grounding = objective explanatory relation.

- Modes of expression:

The fact that q grounds the fact that p.

The fact that q explains the fact that p.

p in virtue of the fact that q.

p because q.

- Is essentially “many-one”: several facts may together or jointly ground a fact.

- Comes in various sorts: metaphysical, conceptual, logical, normative, ... (Causal?)

- Focus here: logical grounding.

Fabrice Correia Institut de Philosophie, Universite de Neuchatel

Grounding, Entailment and Levels of Reality 1/31


- Plausible view: logical grounding is strictly stronger than metaphysical grounding(compare modality).

- Another plausible view: logical grounding is strictly stronger than logicalnecessitation (entailment).

E.g. although p grounds p ∨ q, p ∧ q doesn’t ground p.

Broad aims of the presentation:

1 Build a theory of logical grounding, both proof-theoretically and semantically.

2 Determine how logical grounding and entailment are connected.

(Part of 1 and 2 already in my forthcoming “Logical Grounds”.)




Explanatory vs non-explanatory inferential links

Starting point

- Some inferential links are explanatory, others not.

Some explanatory rules of inference:

(I∧)φ ψ

φ ∧ ψ(I∨)

φ

φ ∨ ψ(I¬¬)

φ

¬¬φ

Some non-explanatory rules:

(E∧)φ ∧ ψφ

(DS)φ ∨ ψ ¬ψ

φ(E¬¬)

¬¬φφ

- Which are the explanatory rules?

To answer this question is to give a theory of logical grounding.




The basic explanatory rules

- Focus on a propositional language, with ∧, ∨ and ¬ primitive. Formulas (φ, ψ, ...)defined as usual; the literals = the atoms and the negated atoms.

- The basic explanatory rules:

(∧1)φ ψ

φ ∧ ψ(∧2)

¬φ¬(φ ∧ ψ)

(∧3)¬ψ

¬(φ ∧ ψ)

(∨1)¬φ ¬ψ¬(φ ∨ ψ)

(∨2)φ

φ ∨ ψ(∨3)

ψ

φ ∨ ψ

(¬)φ

¬¬φ

- All the other explanatory rules are derived from these rules by chaining them.




The basic explanatory rules

Remark I

No elimination rules, only introduction rules, some in positive context, some innegative contexts:

(∧1)φ ψ

φ ∧ ψ(∧2)

¬φ¬(φ ∧ ψ)

(∧3)¬ψ

¬(φ ∧ ψ)

(∨1)¬φ ¬ψ¬(φ ∨ ψ)

(∨2)φ

φ ∨ ψ(∨3)

ψ

φ ∨ ψ

(¬)φ

¬¬φ

Remark II

All the (basic and derived) explanatory rules are classically valid, and they are alsovalid in the sense of many other logics.




Grounding trees

Definition

A g-tree is a rooted tree whose nodes are occupied by formulas, and whosetransitions are given by the basic rules.

More precisely: in a g-tree, (i) no parent node is occupied by a literal and (ii) every parent node has either onechild or two children, in such a way that the principles depicted in the following table are satisfied:

Node occupied by Number of children Child(ren) occupied byφ ∧ ψ 2 φ and ψφ ∨ ψ 1 φ or ψ¬(φ ∧ ψ) 1 ¬φ or ¬ψ¬(φ ∨ ψ) 2 ¬φ and ¬ψ¬¬φ 1 φ




Grounding trees

Definitions

A g-tree for a formula φ from a set of formulas ∆ is a g-tree whose root isoccupied by φ, and such that ∆ is the set of all the formulas which occupyleaves on it.

A g-tree is degenerate iff it consists of a single node.

Examples of g-trees:

(a) p ∧ ¬q (b) ¬¬p ∧ q

¬¬p

p

q

(c) ¬¬p ∨ q

q

(d) (p ∨ q) ∧ ¬(q ∧ r)

p ∨ q ¬(q ∧ r)

¬r

(e) (p ∨ q) ∧ (p ∨ q)

p ∨ q

p

p ∨ q

q




Grounding, strict and weak

Definitions

For ∆ a set of formulas and φ a formula:

∆ strictly grounds∗ φ iff there is a non-degenerate g-tree for φ from ∆.

∆ strictly grounds φ iff there is a covering of ∆ (i.e. a family of sets whoseunion is ∆) such that for each ∆′ in this covering, ∆′ strictly grounds∗ φ.

∆ weakly grounds φ iff for some ∆′ ⊆ ∆, there is a g-tree for φ from ∆′.

Corollary

∆ weakly grounds φ iff either φ ∈ ∆, or for some ∆′ ⊆ ∆, ∆′ strictlygrounds∗ φ.

∆ weakly grounds φ iff either φ ∈ ∆, or for some ∆′ ⊆ ∆, ∆′ strictlygrounds φ.

strict grounding∗ ⇒ strict grounding ⇒ weak grounding




Grounding, strict and weak

Properties of strict grounding∗ (�∗):

1. If ∆ �∗ φ, then ∆ 6= ∅ and is finite

2. If ∆ �∗ φ, then φ is not a literal

3. If ∆ �∗ φ, then φ’s degree of complexity > the degree of complexity of any ψ ∈ ∆

4. Not: ∆, φ �∗ φ Generalised Irreflexivity

5. If ∆ �∗ φ and φ,∆′ �∗ ψ, then provided that φ /∈ ∆′, ∆,∆′ �∗ ψ Restricted Cut

Properties of strict grounding (�):

1. If ∆ � φ, then ∆ 6= ∅ and is finite

2. If ∆ �∗ φ, then φ is not a literal

3. If ∆ �∗ φ, then φ’s degree of complexity > the degree of complexity of any ψ ∈ ∆

4. Not: ∆, φ � φ Generalised Irreflexivity

5. If ∆ � φ and φ,∆′ � ψ, then ∆,∆′ � ψ Cut

6. If ∆ � φ and ∆′ � φ, then ∆,∆′ � φ Amalgamation

Properties of weak grounding (�):

1. If ∆ � φ, then ∆ 6= ∅2. If ∆ � φ and φ /∈ ∆, then φ is not a literal

3. ∆, φ � φ Generalised Reflexivity

4. If ∆ � φ and φ,∆′ � ψ, then ∆,∆′ � ψ Cut

5. If ∆ � φ, then ∆,∆′ � φ Weakening




Grounding and entailment

- Already visible: ∆ weakly grounds φ ⇒ ∆ classically entails φ.

- Also already visible: ∆ weakly grounds φ ⇒ ∆ entails, in several other senses of“entails”, φ.

- In particular, in the senses captured by the following logics:

First-Degree Entailment (FDE)

Logic of Paradox (LP)

Strong Kleene 3-Valued Logic (K3)

- We focus on them, and try to deepen our understanding of how logical grounding

connects to the corresponding notions of entailment.




Sequent calculi for entailment

Multiple conclusion sequent calculus for FDE:

Introduction axioms:

i1. φ, ψ ` φ ∧ ψi2. φ ` φ ∨ ψi3. ψ ` ψ ∨ φi4. ¬φ,¬ψ ` ¬(φ∨ψ)

i5. ¬φ ` ¬(φ ∧ ψ)

i6. ¬ψ ` ¬(φ ∧ ψ)

i7. φ ` ¬¬φ

Elimination axioms:

e1. φ ∧ ψ ` φe2. φ ∧ ψ ` ψe3. φ ∨ ψ ` φ, ψe4. ¬(φ ∨ ψ) ` ¬φe5. ¬(φ ∨ ψ) ` ¬ψe6. ¬(φ∧ψ) ` ¬φ,¬ψe7. ¬¬φ ` φ

Structural rules:

s1.∆ ` Γ, φ φ,∆′ ` Γ′

∆,∆′ ` Γ, Γ′Cut

s2.∆ ` Γ

∆,∆′ ` Γ, Γ′Weakening

A sequent ∆ ` Γ is FDE-provable—in symbols: ∆ FDE Γ—iff it is provable from a set of sequents whosemembers are all among the introduction and the elimination axioms.

For LP:

Add the “excluded middle” axiom ` φ,¬φ.

For K3:

Add the “non-contradiction” axiom φ,¬φ `.

For PC:

Add both axioms.




Sequent calculi for grounding

- Take the calculus for FDE, drop the elimination axioms and add the reflexivity axiomφ ` φ (which is provable in FDE). Call the X the resulting relation.

- Then X is essentially many-one: for all ∆ and Γ, ∆ X Γ iff ∆ X φ for someφ ∈ Γ.

- And it turns out that for all ∆ and φ, ∆ X φ iff ∆ � φ .





Sequent calculus for weak grounding:

Introduction axioms:

i1. φ, ψ ` φ ∧ ψi2. φ ` φ ∨ ψi3. ψ ` ψ ∨ φi4. ¬φ,¬ψ ` ¬(φ∨ψ)

i5. ¬φ ` ¬(φ ∧ ψ)

i6. ¬ψ ` ¬(φ ∧ ψ)

i7. φ ` ¬¬φ

Structural rules:

s3.∆ ` φ φ,∆′ ` ψ

∆,∆′ ` ψCut

s4.∆ ` φ

∆,∆′ ` φWeakening

Structural axiom:

s5. φ ` φ Reflexivity





Sequent calculus for strict grounding:

The introduction axioms, plus Cut (s3) and the extra rule:

s6.∆ ` φ ∆′ ` φ

∆,∆′ ` φAmalgamation

Sequent calculus for strict grounding∗:

The introduction axioms, plus the rules:

s7.∆ ` φ φ,∆′ ` ψ

∆,∆′ ` ψφ /∈ ∆′ Restricted Cut i8.

∆ ` φ ∆′ ` ψ∆,∆′ ` φ ∧ ψ

i9.∆ ` ¬φ ∆′ ` ¬ψ∆,∆′ ` ¬(φ ∨ ψ)

i10.∆ ` φ

∆, φ ` φ ∧ φ

i11.∆ ` ¬φ

∆,¬φ ` ¬(φ ∨ φ)




Truth: a general framework

Definition (Valuation)

A valuation is a distribution of truth-values (T and F) over the atoms.

Gaps and gluts are not excluded: a valuation can be maximal, coherent, neither orboth (= classical).

Definition (Truth and Falsity)

For v a valuation:

v |= p iff p is T according to v

v =| p iff p is F according to v

v |= φ ∧ ψ iff v |= φ and v |= ψ

v =| φ ∧ ψ iff v =| φ or v =| ψv |= φ ∨ ψ iff v |= φ or v |= ψ

v =| φ ∨ ψ iff v =| φ and v =| ψv |= ¬φ iff v =| φv =| ¬φ iff v |= φ.




Semantic characterisations of entailment

Definitions (Semantic consequence relations)

∆ �FDE Γ iff for every valuation v : (v � φ for all φ ∈ ∆)⇒ (v � ψ for some ψ ∈ Γ).

∆ �LP Γ iff for every maximal valuation v : (v � φ for all φ ∈ ∆)⇒ (v � ψ for some ψ ∈ Γ).

∆ �K3 Γ iff for every coherent valuation v : (v � φ for all φ ∈ ∆)⇒ (v � ψ for some ψ ∈ Γ).

∆ �PC Γ iff for every maximal coherent valuation v : (v � φ for all φ ∈ ∆)⇒ (v � ψ for some ψ ∈ Γ).

Theorem (Adequacy – where S is FDE, LP, K3 or PC)

For all ∆ and Γ: ∆ S Γ iff ∆ �S Γ.

(See e.g. Priest 2008.)




Grounding and truth

Definitions

A situation is a set of literals.

The situation determined by valuation v is the set of all literals true in v .

FUNDAMENTAL CONNECTION

For all formulas φ and all valuations v :

φ is true in v ⇔ φ is weakly grounded in the situation determined by v .




Characterisation of entailment in ground-theoretic terms

Definition

A situation Λ is maximal iff for every atom p, either p ∈ Λ or ¬p ∈ Λ, and coherentiff for every atom p, not both p ∈ Λ and ¬p ∈ Λ.

Thanks to FUNDAMENTAL CONNECTION we immediately get:

Theorem (Characterisation of entailment in terms of weak grounding)

∆ �FDE Γ iff for every situation Λ: (Λ � φ for all φ ∈ ∆)⇒ (Λ � ψ for some ψ ∈ Γ).

∆ �LP Γ iff for every maximal situation Λ: (Λ � φ for all φ ∈ ∆)⇒ (Λ � ψ for some ψ ∈ Γ).

∆ �K3 Γ iff for every coherent situation Λ: (Λ � φ for all φ ∈ ∆)⇒ (Λ � ψ for some ψ ∈ Γ).

∆ �PC Γ iff for every maximal coherent situation Λ: (Λ � φ for all φ ∈ ∆)⇒ (Λ � ψ for some ψ ∈ Γ).




Collapse

Corollary (Collapse)

For ∆ a situation and φ a formula:

∆ � φ iff ∆ �FDE φ.

∆ � φ iff ∆ �LP φ provided that ∆ is maximal.

∆ � φ iff ∆ �K3 φ provided that ∆ is coherent.

∆ � φ iff ∆ �PC φ provided that ∆ is maximal coherent.




Gentzenian thought

Characterisation theorem Why not define entailment in ground-theoretic terms?

This could fit well with the Gentzenian thought that the introduction rules forthe logical connectives are basic and the elimination rules are derivative.




On a proposal by B. Schnieder

Definitions

For S a situation:

∆ ≺S φ iffdf S � ψ for all ψ ∈ ∆, and (either φ ∈ ∆, or ∆ � φ).

ψ 2S φ iffdf for some ∆, ψ,∆ ≺S φ.

φ is groundedS iffdf for some ∆, ∆ ≺S φ.

- Fact: φ is groundedS iff S � φ.

- Hence my characterisation of �FDE can be reformulated as follows:

∆ �FDE φ iff given any situation S such that all the members of ∆ are groundedS , φ is also groundedS .

- We also have:

Theorem (Characterisation of FDE-consequence in terms of grounding)

∆ �FDE φ iff given any situation S such that all the members of ∆ are groundedS , there is a set of formulas ∆′

such that:

∆′ ≺S φ, and

For every δ ∈ ∆′, there’s a ψ ∈ ∆ such that δ 2S ψ.

This corresponds to a recent suggestion to define entailment made by B. Schnieder in “On Ground and

Consequence” (manuscript). (To be accurate, he doesn’t use the very relations ≺S and 2S , but invokes similarrelations.)




What we already have

- Remember the collapse result:

For ∆ a situation and φ a formula:

∆ � φ iff ∆ �FDE φ, i.e. iff every valuation which verifies each ψ ∈ ∆ verifies φ.

Provides a semantic characterisation of weak grounding in a restricted case.

- For ∆ arbitrary, we can “cheat”: redefine valuations so that any type of formula canbe treated as atomic.

Can we do better?




Kit Fine’s truth-maker semantics?

- Verifying = making true, falsifying = making false.

- Verifiers and falsifiers are facts: parts of worlds rather than whole worlds.

- Facts fuse; the fusion f · g of facts f and g is conjunctive.

- Closure principle: f · g verifies φ if both f and g do.

- Truth-clauses for Kit Fine’s notions of grounding (see Fine 2012a, 2012b):

φ1, φ2, ... weakly ground ψ iff:

For all fact f1, f2, ...: (f1 verifies φ1, f2 verifies φ2, ...) ⇒ (f1 · f2 · ... verifies ψ).

φ helps weakly ground ψ iff (roughly):

For some φ1, φ2, ...: φ, φ1, φ2, ... weakly ground ψ.

φ1, φ2, ... strictly ground ψ iff (roughly):

φ1, φ2, ... weakly ground ψ and ψ doesn’t help weakly ground any of φ1, φ2, ...





- Plausible semantic clauses for the connectives (see Fine 2012b, manuscript):

f |= ¬φ iff f =| φ; f =| ¬φ iff f |= φ.

f |= φ ∧ ψ iff for some f1 and f2 such that f = f1 · f2, f1 |= φ and f2 |= ψ.

f |= φ ∨ ψ iff either f |= φ, or f |= ψ, or for some f1 and f2 such thatf = f1 · f2, f1 |= φ and f2 |= ψ.

- Consequences:

¬¬φ and φ have the same verifiers.

Hence they weakly ground each other.

Hence it’s not the case that φ strictly grounds ¬¬φ.

Any verifier for φ ∧ φ is a verifier for φ (closure is at work here).

Hence φ ∧ φ weakly grounds φ.

Hence it’s not the case that φ strictly grounds φ ∧ φ.

φ ∨ φ and φ have the same verifiers (closure is also at work here).

Hence they weakly ground each other.

Hence it’s not the case that φ strictly grounds φ ∨ φ.

=⇒ We must give up these semantic clauses...





... Unless we work with a “worldly” conception of grounding, as opposed to a“conceptual” conception.

- In “Grounding and Truth-Functions” (2010) I distinguish between these twoconceptions and formulate a logic of “worldly” grounding.

- Worldly = coarse-grained. E.g. p doesn’t ground p ∧ p, because p and p ∧ p are thesame fact / proposition.

- Conceptual = fine-grained. E.g. p grounds p ∧ p, p and p ∧ p are not the same fact/ proposition.

- Fine’s truth-maker semantics turns out to be adequate for the system I put forwardin the paper!





- The system extends a logic of “analytic equivalence” proposed by R. B. Angell(1977, 1989). I understand A ≈ B as saying that A and B are the same (wordly) fact.

- A key principle: strict grounding is closed under ≈.

- Definitions:

A ≤c B (A is a conjunctive part of B) as: ∃x(A ∧ x ≈ B).

A ≤d B (A is a disjunctive part of B) as: ∃x(A ∨ x ≈ B).

A ≤cd B (A is a conjunctive part of a disjunctive part of B) as: ∃y(A∧ y ≤d B).

- Another key principle: A, B strictly ground C iff:

(i) A ∧ B ≤d C

(ii) Both C �cd A and C �cd B.

- Fine’s “Angellic Content” (manuscript): A ≈ B iff A and B have the same verifiers.

- As Fine pointed out to me: mix his semantics for ≈ and his proposed semanticclause for strict grounding, and you get my second key principle.

- In more details: A, B weakly ground C iff A ∧ B ≤d C , and A helps weakly groundB iff A ≤cd B.




Levels of reality

- Focus on statements of type ∆ � φ, where ∆ is a (finite) situation.

- The intuitive idea:

Each world is endowed with a structure of “levels of reality”, which is a linearordering.

At any world, each truth is assigned such a level.

φ1, φ2, ... strictly ground ψ iff in every world where φ1, φ2, ... hold, ψ alsoholds, and is a assigned a level that is strictly higher than the level assigned toany of φ1, φ2, ...




Levels of reality

- We may use the structure of the natural numbers to represent levels of reality.

- A model is then defined as a partial function m from the literals to N.

- Given such a function, we define a partial function lr (levels of reality) from theformulas to N as follows (clauses for negated conjunctions and negated disjunctionsomitted or simplicity):

Conditions under which lr is defined:

For φ a literal: lr(φ) is defined iff m(φ) is defined.lr(¬¬φ) is defined iff lr(φ) is defined.lr(φ ∧ ψ) is defined iff lr(φ) and lr(ψ) are defined.lr(φ ∨ ψ) is defined iff lr(φ) or lr(ψ) is defined.

Values of lr when defined:

For φ a literal: lr(φ) = m(φ).lr(¬¬φ) = lr(φ) + 1.lr(φ ∧ ψ) = max{lr(φ), lr(ψ)} + 1.

lr(φ ∨ ψ) =

max{lr(φ), lr(ψ)} + 1, if both lr(φ) and lr(ψ) are defined;lr(φ) + 1, if only lr(φ) is defined;lr(ψ) + 1, if only lr(ψ) is defined.

- A formula φ is true in a model iff φ has a level of reality according to that model,and false iff ¬φ is true.

⇒ Truth and falsity behave as above, and gaps and gluts are likewise allowed.




Levels of reality

Definition (Semantic consequence for strict ground)

For ∆ any set of formulas and φ a formula:

∆ �SG φ iff in every model where the members of ∆ are true, φ is also true, andits level of reality is strictly greater than the level of any of the members of ∆.

We have soundness for the unrestricted system (i.e. in which the grounders are notrequired to be literals). But not completeness: for instance, ¬¬φ �SG ¬¬(φ ∨ ψ) butnot ¬¬φ� ¬¬(φ ∨ ψ).

We thus have soundness for the restricted system (i.e. in which only sets ofliterals may ground other formulas), and it turns out that we also havecompleteness here.




Further work that suggests itself:

- More on the semantics for logical grounding.

- Logical grounding and other concepts of entailment.

- Quantified languages (cf. “Logical Grounds”).

- Other notions of grounding, in particular metaphysical grounding.




References:

Angell, R. B. 1977. “Three Systems of First Degree Entailment”, Journal of Symbolic Logic, 42:147.

— 1989. “Deducibility, Entailment and Analytic Containement”, in J. Norman and R. Sylvan (eds.), Directions inRelevant Logic, Dordrecht: Kluwer Academic Publishers.

Correia, F. 2010. Grounding and Truth-Functions, Logique et Analyse, 53(211), 251-279.

— forthcoming. Logical Grounds, The Review of Symbolic Logic.

Fine, K. 2012a. The Pure Logic of Ground, The Review of Symbolic Logic, 5(1), 1-25.

— 2012b. Guide to Ground, in F. Correia, & B. Schnieder (eds.), Metaphysical Grounding: Understanding theStructure of Reality, CUP.

— manuscript. Angellic Content.

Priest, G. 2008. An Introduction to Non-Classical Logic: From If to Is, CUP.

Schnieder, B. manuscript. On Ground and Consequence.



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Grounding, Entailment and Levels of Reality...Grounding, Entailment and Levels of Reality Fabrice Correia Institut de Philosophie, Universit e de Neuch^atel Groundedness in Semantics