n, we’ll need some additional defini- tions and lemmata. In this section, we’ll introduce then-forcinginference relation - a species of paraconsistent inference relation, and the notion of coherence level of a set. From now on, we’ll assume thatmis fixed. Relative toΛ, the notion ofn-forcing and coherence level are defined as follows:
Definition 6.3.1
A non-emptyΛ-inconsistent setΓ n-forcesA,Γ `nAiff for everyn-partition,π, ofΓ,
there is a cell, C ∈ π, such thatC `Λ A. IfΓ = ∅or is Λ-consistent, thenΓ `n Aiff
Γ `Λ A.
The collection of all n-partitions of Γ will be denoted by Qn(Γ). We’ll say that a partition ofΓ is aΛ-consistent partition iff each cell of the partition isΛ-consistent.
Definition 6.3.2
The coherence level of a setΓ,`:℘(Φ)−→N∪{∞}is defined as follows:
`(Γ) = 1 ifΓ =∅orΓ 6`Λ ⊥
the cardinality of the least
Λ-consistent partition ofΓ
up to and includingω
if such partition exists
∞ otherwise
Given the usual notion of a maximal consistent set, we can state more explicitly the relationship betweenn-forcing and coherence level of a set andΛ-maximal consistent sets. We’ll use[∧C]Γ to denote {B∧C : B ∈ Γ}, ifΓ = ∅, then we let[∧C]Γ = {C}. WhereΣ= [∧C]Γ, we’ll let[∧C]∗Σ={B: B∧C∈Σ}.
Proposition 6.3.1
The following statements are equivalent 1. Γ `nA
2. `([∧¬A]Γ)> n
3. For any maximalΛ-consistent setsx1, . . . , xnsuch thatΓ ⊆S{
x1, . . . , xn},
A∈S
{x1, . . . , xn}. Proof:
(1)⇒(2): Assume that Γ `n A. If `(Γ) > n, clearly `([∧¬A]Γ) > n. So assume that `(Γ) ≤ n. Towards a contradiction, assume that `([∧¬A]Γ) ≤ n. Then there must be a consistent n-partition of [∧¬A]Γ. Let π = {C1, . . . ,Cn} be such a consistent n- partition. Then π∗ = {[∧¬A]∗C1, . . . ,[∧¬A]∗Cn} is a consistent n-partition ofΓ. But Γ `nA, so∃i(1≤i≤n):[∧¬A]∗Ci`Λ A. This contradicts our assumption that everyCiis Λ-consistent. So`([∧¬A]Γ)> nas required.
(2)⇒(3): Assume that `([∧¬A]Γ) > n. Let x1, . . . , xn be any maximalΛ-consistent sets such thatΓ ⊆ S
{x1, . . . , xn}. Clearly there must be a consistent n-partition ofΓ such that each cell, Ci, is a subset ofxi. Letπ = {C1, . . . ,Cn}be such a consistent n- partition. Thenπ0 = {[∧¬A]C1, . . . ,[∧¬A]Cn}is ann-partition of[∧¬A]Γ. But by our initial assumption`([∧¬A]Γ)> n, so∃i(1≤i≤n): Ci`Λ A. Hence,∃i(1≤i≤n) : xi`ΛA.
By deductive closure of maximal consistent sets,A∈xi. HenceA ∈S
{x1, . . . , xn}as required.
(3)⇒(1): Assume thatΓ 6`n A. Clearly, if `(Γ) > n, then Γ `n A. So`(Γ) ≤ n. Let {C1, . . . ,Cn}be a consistentn-partition such that∀i(1≤i≤n) : Ci 6`ΛA. Such a partition clearly exists, otherwiseΓ `nA. Then∀i(1≤i≤n), Ci∪{¬A}isΛ-consistent. We extend eachCi∪{¬A} to its maximalΛ-consistent extension. Hence there exist n maximal Λ-consistent setsx1, . . . , xnsuch thatΓ ⊆S{
x1, . . . , xn}, butA6∈S{
x1, . . . , xn}.
Calling`na consequence relation is well suited since it satisfies the usual proper- ties of reflexivity, monotonicity, and transitivity (we’ll leave it to the reader to verify this). And as we show later, it is also finitary. However, stepping back from the partic- ular logicΛand looking at things a bit more abstractly, proposition (6.3.1) underscores the fact that`ngeneralises the classical consequence relation`. The classical counter- part to proposition (6.3.1) is the familiar equivalence between (1)Γ `A, (2)Γ∪{¬A}is inconsistent, and (3)A∈xfor any maximal consistent extensionxofΓ.
Another related generalisation at work is the notion of coherence level. In this framework, classically consistent sets are simply level 1 sets whereas all classically inconsistent sets are level nsets, wheren ≥ 2. Thus a classically consistent theory, in the sense of a consistent deductively closed set, is simply a level 1-theory closed under classical`. It is not difficult to see that just as the closure of a level 1 set un- der classical`yields a level 1-theory, closure of a levelnset under`nyields a level n-theory. Thus we may say that`nis a level preserving relation for any set with level
≤n. Now a theory is said to betrivialiff every formula is a deductive consequence of the theory. As is well known, closing a leveln≥2set under classical`yields a triv- ial theory. Thus as some put it colourfully, classical`is inferentially explosive with respect to inconsistent sets. Given these observations,`nprovides a possible strategy for studying inconsistent but non-trivial theories as well as paraconsistent formal sys- tems in which not everyBis a deductive consequence of{A,¬A}. More interestingly, from an information processing viewpointn-forcing provides a plausible inferential strategy to extract information from multiple sources, where n corresponds to the number of information channels or sources. More detailed discussions ofn-forcing and inconsistency-tolerant reasoning can be found in [99] and [167].
One of the crucial steps in Apostoli-Brown’s proof of the completeness of Kn is to establish the compactness of `n. But proposition (6.3.1) now makes it clear that the compactness of`nis an immediate corollary of the compactness of the coherence
levels of sets. We’ll state the problem of compactness of coherence level in terms of
trace, a kind of generalised filter base, as presented by Jennings and Schotch in [167]. The notion of trace, we may add, is in fact equivalent to the notion of non-colourability of hypergraphs.4
Definition 6.3.3
LetΣbe a collection of finite subsets of a non-empty setΓ. ThenΣis ann-trace overΓ
iff for everyn-partition,π, ofΓ, there is a cellC ∈πsuch that some element ofΣis a subset ofC.
Lemma 6.3.1
IfΣandΓ are non-empty finite sets andΣis ann-trace overΓ, thenΣis anm-trace overΓ, form < n.
Proof:
We assume thatΣandΓare non-empty, finite andΣis ann-trace overΓ. LetΣ={Ai:
1 ≤ i ≤ k}. Suppose for somem < n,Σis not anm-trace over Γ. Then there is an m-partitionπ={Cj : 1≤j≤m}ofΓ such thatAi6⊆ Cjfor allj. Let
π∗={Bl: 1≤l≤n, ∅ 6=Bl⊆ Cj, for somej}
Thenπ∗ is ann-partition such thatAi 6⊆ Blfor eachAiand eachBl. But this contra- dicts the hypothesis thatΣis ann-trace overΓ. Hence, for eachm < n,Σis anm-trace overΓ.
Lemma 6.3.2
LetΓ 6=∅andΣbe ann-trace overΓ. Then∃Σ0 ⊆finΣsuch thatΣ0 is ann-trace over Γ.
Proof:
Our strategy is to prove the contrapositive, i.e. if every finite subset of Σis not an n-trace overΓ, thenΣis not ann-trace overΓ. We proceed to construct a first order theoryT such that every finite subset ofΣis not ann-trace ofΓ iff every finite subset ofT has a model. So by the compactness of first order logic, if every finite subset ofΣ
4A hypergraph,G, is a pair,(V(G), E(G))whereV(G)is a set ofvertices, andE(G)is a collection of edges, i.e. a collection offinitesubsets ofV(G). Ann-colouring of a hypergraphGis ann-partition of V(G). A hypergraphGisn-colourable iff somen-colouringcofGis such that no edge ofGis monochro- matic underc, i.e. no edge is a subset of any cell of the partitionc. The compactness theorem for hyper- graphs states that a hypergraphGisn-colourable iff every finite sub-hypergraph ofGisn-colourable. In proving the compactness of trace, we thereby also establish the compactness of hypergraphs.
is not ann-trace ofΓ, thenT has a model and hence by the construction ofT,Σis not ann-trace overΓ. Let
Σ={Ai:i∈I} For eachi∈I, let
Ai={ai 1, . . . , a
i ki}
Make the assumption that every finite subset of Σis not ann-trace over Γ, i. e. for each finite subsetΣ0 ofΣthere is ann-partition ofΓ such that no cell in the partition contains any element ofΣ0. Towards the construction of our first order theoryT, we extend the first order language with
• nmany predicate symbols,P1, . . . , Pn, each representing a cell in then-partition
• for each i ∈ I, introduce constant symbols, ci 1, . . . , c
i
ki each naming the corre- sponding element inAi.
Let
Θ=∀x( _
1≤h≤n Phx)
For eachi∈I, let
Ωi= ^ 1≤h≤n _ 1≤j≤ki ¬Phci j ∧Θ Let Ω=[ i∈I Ωi
We obtain the first order theoryT by adding all elements ofΩ as proper axioms to standard first order logic. But our assumption is that every finite subset ofΣis not an n-trace overΓ, so every finite subset ofΩhas a model. By first order compactness,T has a model and thus Ωhas a model. Hence there must be ann-partition ofΓ such that no cell of the partition contains any element ofΣ, i.e. Σis not ann-trace overΓ.
Lemma 6.3.3
LetΣbe finite,Γ 6=∅. IfΣis ann-trace overΓ, then∃Γ0⊆finΓ such thatΣis ann-trace overΓ0.
Proof:
Assume that Σis finite and Γ 6= ∅. We’ll show that if Σ is not ann-trace over Γ0,
lemma (6.3.2). We proceed to construct a first order theoryT such thatΣis not an n- trace overΓ0,∀Γ0 ⊆finΓ, iff every finite subset ofThas a model. So by the compactness of first order logic, ifΣis not ann-trace overΓ0, ∀Γ0 ⊆fin Γ, thenT has a model and hence by the construction ofT,Σis not ann-trace overΓ. LetIbe an index set. For each i∈I, letΓi⊆fin Γ. LetΣ= {A1, . . . ,Am}. For1 ≤j≤m, letAj ={aj1, . . . , a
j
kj}. Make the assumption thatΣis not ann-trace over any finite subset ofΓ, i. e. for eachi∈I, there is ann-partition ofΓisuch that no cell in the partition includesaj,1 ≤ j ≤ m. Towards the construction of our first order theoryT, extend the first order language with
• for eachi∈I,nmany predicate symbols,Pi 1, . . . , P
i
n, each represents a cell in the n-partition ofΓi
• for 1 ≤ j ≤ m, introduce constant symbols, cj1, . . . , c j
kj, each name the corre- sponding element ofAj
For eachi∈I, let
Θi=∀x( _ 1≤h≤n Pi hx) and Ωi= ^ 1≤h≤n _ 1≤j≤k1 ¬Pi hc 1 j ∧. . .∧ ^ 1≤h≤n _ 1≤j≤km ¬Pi hc m j ∧Θi Let Ω=[ i∈I Ωi
We obtain our first order theoryT by adding all elements ofΩ as proper axioms to first order logic. But our assumption is thatΣis not ann-trace overΓi, for eachi∈I, so every finite subset ofΩhas a model. By first order compactness,Thas a model and thusΩhas a model. Hence there must be ann-partition ofΓ such that no cell of the partition contains any element ofΣ, i. e.Σis not ann-trace overΓ.
Theorem 6.3.1
Trace Compactness: letΓ be a non-empty set andΣbe a collection of finite subsets of
Γ. ThenΣis ann-trace overΓ iff there is aΓ0 ⊆finΓ and aΣ0⊆finΣsuch thatΣ0is an n-trace overΓ0.
Proof:
that Σis an n-trace over Γ, it suffices to show that Σ0 is an n-trace overΓ. Clearly eachn-partition ofΓ must also partitionΓ0intonor fewer cells. If ann-partition ofΓ partitionsΓ0intoncells, then by our initial hypothesis, some element ofΣ0is a subset of some cell in the partition. And if ann-partition ofΓpartitionsΓ0intomcells, where m < n, then by lemma (6.3.1),Σ0 is anm-trace overΓ0, so some element ofΣ0 must also be a subset of some cell of the partition. So either way, some element ofΣ0 is a subset of some cell in eachn-partition ofΓ. HenceΣ0is ann-trace overΓ.
(⇒): Assume thatΣis ann-trace overΓ. By lemma (6.3.2),∃Σ0 ⊆fin Γ such thatΣ0is ann-trace overΓ. By lemma (6.3.3),∃Γ0 ⊆finΓ such thatΣ0is ann-trace overΓ0. Having now established the compactness of traces, we are now in a position to prove the compactness of coherence level of sets.
Theorem 6.3.2
Level Compactness: ForΓ ⊆ Φ, if`(Γ) > n, then there is a finite subsetΓ0 ofΓ, such that`(Γ0)> n.
Proof:
LetΓ ⊆Φsuch that`(Γ)> n. Let
Σ={A: A ⊆finΓ andA `Λ⊥}
By our assumption that`(Γ)> n,Σis ann-trace overΓ. By theorem (6.3.1),∃Γ0⊆finΓ,
∃Σ0 ⊆fin Σsuch thatΣ0 is an n-trace overΓ0. But then everyn-partition ofΓ0 must contain a cell which includes some element ofΣ0. ButΣ0 ⊆Σ. Hence everyn-partition ofΓ0contains an inconsistent cell, i.e.`(Γ0)> nas required.
Theorem 6.3.3
Compactness of`n: ForΓ ⊆Φ,A∈Φ, ifΓ `nA, then∃Γ0 ⊆finΓ such thatΓ0`nA.
Proof:
Let Γ ⊆ Φ, A ∈ Φ, such thatΓ `n A. Then by proposition (6.3.1), `([∧¬A]Γ) > n. By theorem (6.3.2), there is Γ0 ⊆fin Γ such that `([∧¬A]Γ0) > n. Hence by proposi- tion (6.3.1) again,Γ0 `nA.