As we have already noted, premises that are inconsistent (but contradiction free) have non-trivial uncertainties. In this section we would like to continue the investigation initiated by Adams and Levine in [5] and examine how uncertainties may be trans- mitted from premises to conclusions. In [1; 2; 3], Adams extended the result of [5] to cover a language with a conditional connective. However the issue of uncertainties transmitted frominconsistentpremises to conclusions has not been addressed in any of their subsequent works. We start by identifying several candidate (uncertainty) en- tailment relations – all of which can be said to preserve the uncertainty bound of the premises under some sense:
Definition 4.6.1
For any set of formulaeΓ and formulaBwe define the following entailment between
Γ andB
Certainty Entailment: For any probability distributionPsuch thatUP(A) =0for all
Uncertainty Entailment: For any probability distributionPsuch thatUP(A) < 1for allA∈Γ, we haveUP(B)< 1. We denote this byΓ |=<1 B.
-Entailment: For any∈[0, 1], for any probability distributionPsuch thatUP(A)≤
for allA∈Γ we haveUP(B)≤. We denote this byΓ |=≤ B.
We note that an inconsistent set of formulae cannot all be certain together. Thus certainty entailment is an explosive entailment for any inconsistent premises. In fact it is just classical entailment:
Proposition 4.6.1
Certainty entailment is equivalent to classical entailment.
Proof:
We note that anyv∈ WΓ which verifies all ofΓ but falsifiesBwould also confirm the existence of aPwith allA∈Γ havingU(A) =0butU(B)6=0.
Conversely if Γ |= B, then for the uncertainty matrixA ofΓ ∪{B}, every column
[a1j, . . . a|Γ|+1j]T with all0’s in the first|Γ|entries (corresponding to members ofΓ) will havea|Γ|+1j = 0(corresponding toB). LetP = [P1. . . P2n]T be an arbitrary but fixed probability distribution such that for allAk ∈Γ,UP(Ak) =0. Clearly for eachAk∈Γ
UP(Ak) =
2n X
j=1 akjPj
is zero if eitherakj =0orPj =0for eachi≤2n. IfPj =0then obviouslya|Γ|=1j×Pj = 0. But ifPj 6= 0then akj = 0 for eachk ≤ |Γ|. But thena|Γ|=1j = 0as well and thus a|Γ|=1j×Pj=0. HenceP2j=1n a|Γ|+1jPj=0, i.e.UP(B) =0.
Turning now to uncertainty entailment, it is clearly an improvement over certainty entailment for handling inconsistencies. The basic idea of uncertainty entailment is that if each of the Ai ∈ Γ is free from complete uncertainty, then the conclusionBis also free of complete uncertainty. Since contradiction free inconsistent premises have non-trivial uncertainties, the antecedent of the conditional in our definition is never falsified in such a case. Thus we do not have A,¬A |=<1 B in general. But note that in the presence of contradictions, we do haveA∧ ¬A|=<1 B. Moreover for any classical tautology>we have|=<1 >trivially. In fact the logic which captures|=<1 completely is the discursivelogic(s) developed by Ja´skowski in [97]. For a complete sequent formulation ofdiscursivelogic, the reader can consult the systemSof Knight in [111]. But the basic idea of adiscursivelogic is to take the union of all the theorems
of an underlying logic `L together with the deductive closures (under `L) of each singleton of the premises, i.e. for anyΓ andBwe have
Γ `DBiff `LBorA`LBfor someA∈Γ
For a different choice of the underlyingLwe get a different discursive logic. Proposition 4.6.2
For anyΓ andB,Γ |=<1 Biff either`BorA`Bfor someA∈Γ where`is the usual
classical propositional logic.
Proof:
As noted before it is trivially true that ifBis a classical tautology, then for any proba- bility distributionPwe haveUP(B)≤UP(A)for anyA. So we’ll consider anyBthat is not a tautology. For the only if direction we assume thatA`Bfor someA∈Γ. Then from claim (2) of theorem (4.5.3), for every probability distribution P over Γ ∪ {B}, we have UP(B) ≤ UP(A). So in particular for any probability distribution Q with UQ(C)< 1for everyC∈Γ we haveUQ(B)< 1. This shows thatΓ |=<1B.
For the if direction, we assume that for noA∈Γ do we haveA`B. Now consider the uncertainty matrixA = (aij) forΓ ∪{B}. As usual we’ll assume thatAis a(m+
1)×2nmatrix, where the firstmrows correspond to members ofΓ and the(m+1)- th row corresponds toB. Moreover we assume thattis the number of1’s occurring in the (m+ 1)-th row of A. We note that t ≥ 1 since B is not a tautology by the initial assumption. We define the probability distribution P as follows: for every j, 1≤j≤2n, Pj = t−1 ifam+1,j =1 0 otherwise
Clearly given howPis defined,UP(B) = [Pj=12n (am+1,j×Pj)] =1. But note that since for eachA ∈ Γ,A 6` Bso we have, for eachi ≤mthere must be aji ≤ 2nsuch that aiji = 0butam+1,ji = 1. This implies that for eachA∈Γ we haveUP(A) ≤
t−1 t < 1. ThusPwitnesses the failure ofΓ |=<1B.
Turning now to -entailment, the basic requirement is that the uncertainty of a conclusionBshould never exceed the maximum value of the uncertainty of any given A∈Γ, i.e. for any probability distributionP,U(B)≤ max{U(A) : A∈Γ}. As it turns
out-entailment is in fact equivalent to uncertainty entailment: Proposition 4.6.3
For anyΓandB,Γ |=≤BiffΓ |=<1 B. Proof:
The only if direction is trivial since|=<1is a special case of|=≤when < 1.
For the if direction, consider Bwhere Bis a tautology. Then for any Γ we have Γ |=≤ BsinceU(B) =0for any probability distribution. Suppose then thatBis not a tautology but for someA∈ Γ,A `Bholds. Then from claim (2) of theorem (4.5.3) again, for every probability distributionP overΓ ∪{B}, we haveU(B) ≤ U(A). Thus we haveU(B)≤max{U(A) : A∈Γ}as required.
To put the matter in terms of preservation, discursive logic is exactly the logic which preserves the uncertainty bounds of premises. Note however that discursive logic does not allow for fullaggregationof premises. In general we haveU(Vm
i=1Ai)≤ Pm
i=1U(Ai), but notU(
Vm
i=1Ai) ≤ max{U(Ai)| 1 ≤ i ≤ m}. In light of this, discur- sive logic is a very extreme approach to bounding the uncertainty of the conclusion. When the value of max{U(A) : A∈Γ}is close to1, it is of course desirable to ensure that the conclusion’s uncertainty should not exceed this bound. But when the value of max{U(A) : A ∈ Γ} is small, a slightly riskier inference with a higher conclusion uncertainty may be acceptable. More importantly, aggregation is particularly useful for fusing information from multiple sources. We’ll introduce a kind of entailment relation which permits a limited form of aggregation by bounding the size of the ag- gregating set. Our entailment relation also provides a partial solution to a problem stated in Knight [111] (page 360). But first we need to fix some terminologies and definitions.
Definition 4.6.2
Letk∈Z+be arbitrary but fixed. LetΓ be a finite set of formulae innvariables. The
set of all subsets ofΓ of size≤kis denoted by℘k(Γ).
IfP = [P1, . . . , P2n]T is a probability distribution overΓ, we say thatPisi-positive if Pi> 0.
If∆⊆Γ, we say thatPverifies∆if there exists ani≤2nsuch thatPisi-positive and theith term ofUP(A)is0for eachA∈ ∆, i.e. whereAis the uncertainty matrix forΓ
andj(1), . . . , j(t)are the respective enumeration of members of∆, we haveaj(1)i×Pi=
. . .=aj(t)i×Pi=0underP.
Note that if∆∈℘k(Γ)is inconsistent, then noPwill verify∆. Intuitively,Pverifies a∆only if P distributes non-zero probability into at least one model of∆. We now introduce a generalised version of-entailment with an additional parameterkas a bound on the size of the aggregating set.
Definition 4.6.3
Letk ∈ Z+ be arbitrary but fixed. LetΓ be any finite set of formulae innvariables.
For any formulaB, we say thatΓ k-entailsB,Γ |=kB, iffU(B)< 1on every probability
distributionPwhich verifies every∆∈℘k(Γ).
Now for different choices ofkwe can regain different degrees of aggregation. So for instance ifk≥2andA1, A2 ∈ΓthenΓ |=kA1∧A2. Againkis the absolute upper bound on the number of (independent) members ofΓ that can be conjoined. Note also that any tautology >isk-entailed by anyΓ since U(>) = 0 < 1 holds trivially. Moreover if the size of the smallest minimal inconsistent subset ofΓ ismandm < k, then noP will verify every∆ ∈ ℘k(Γ)and thus Γ |=k Bfor anyBholds trivially, i.e. |=k explodes when m < k. We summarise the properties of |=k in theorem (4.6.1). The content of our theorem is self-explanatory. Part (1) shows that|=kis an extension of |=≤. Part (2) shows that|=k is a kind of substructural logic. Part (3) shows that |=k is monotonically increasing with respect to k. Part (4) is a generalised version of proposition (4.6.2) and therein shows that|=k can be viewed as a kind of generalised discursive logic (and thus is decidable). Part (5) shows that|=k, like|=≤, preserves the uncertainty bound of the premises in a certain sense.
Theorem 4.6.1
1. For anyk∈Z+,|=≤⊆|=k.
2. |=kis reflexive and monotonic but transitivity fails.
3. Ifk0, k∈Z+andk0< k, then|=k0⊆|=k.
4. Letk∈Z+be fixed. For anyΓ andB,Γ |=kBiffB∈S{Cn(∆)|∆∈℘k(Γ)}.
5. Let∈[0, 1]such that < 1. LetΓ |=k B. Then for any probability distribution P, ifPA∈∆UP(A)≤holds for each∆∈℘k(Γ)thenUP(B)≤.
Proof:
is not a tautology and for an arbitraryΓ, we haveΓ |=≤ B. From propositions (4.6.2) and (4.6.3), it follows that for some A ∈ Γ, A ` B. Again from claim (2) of theo- rem (4.5.3) it follows that for any probability distributionQ, we haveUQ(B)≤UQ(A). Clearly{A}∈℘k(Γ)for anyk∈Z+. So ifPverifies every∆∈℘
k(Γ), it must also verify {A}. This implies the existence of someisuch thatPi> 0and theith-term ofUP(A)is0. SinceA`B, theithterm ofUP(B)must be0as well. ThusUP(B)≤[(Pj2nPj)−Pi]< 1.
(2): For reflexivity, clearly ifA ∈ Γ then{A} ∈ ℘k(Γ) for anyk ∈ Z+. So ifP verifies every∆∈℘k(Γ), it must also verify{A}as well. This implies that for somei,U(A)≤
[(P2jnPj) −Pi]< 1as required.
For monotonicity, we note that℘k(Γ)⊆℘k(Γ, Σ)so ifPverifies every member℘k(Γ, Σ), it must also verify every member of℘k(Γ). So on the assumption thatΓ |=k Aholds Γ, Σ|=kAmust hold as well.
To see the failure of transitivity, considerΓ ={p, ¬p∨r, ¬r}. We have
Γ |=2p∧(¬p∨r)andΓ, p∧(¬p∨r)|=2q
But note thatΓ 6|=2 q.
(3) We note ifk0 < kthen℘k0(Γ)⊆℘k(Γ)for anyΓ. Thus ifPverifies every member of ℘k(Γ)it must also verify every member of℘k0(Γ). So on the assumption thatΓ |=k0 A, Γ |=kAmust hold as well.
(4) For the if direction letP be any probability distribution which verifies every∆ ∈
℘k(Γ). We want to show that U
P(B) < 1on the assumption thatB ∈
S{
Cn(∆)| ∆ ∈
℘k(Γ)}. So we assume that for some∆0 ∈ ℘k(Γ), ∆0 ` B. By the initial assumption howeverPmust verify∆0, so there exists someisuch thatPi > 0and theith term of UP(A)is0for everyA∈∆0. But∆0`Bso theithterm ofUP(B)is0as well. As in (1) and (2), this suffices to show thatUP(B)≤[(Pj=12n Pj) −Pi]< 1.
For the only if direction, we assume that B 6∈ S
{Cn(∆)| ∆ ∈ ℘k(Γ)}, i.e. for every ∆∈℘k(Γ),∆6`B. We’ll show the existence of aPwhich verifies every∆∈℘k(Γ)but onPwe haveUP(B) =1.
Consider the uncertainty matrixA = (aij)forΓ ∪{B}. As usual we’ll assume that
(m+1)-th row corresponds toB. Moreover we assume thatt is the number of1’s occurring in the(m+1)-th row ofA. We note thatt≥1sinceBcannot be a tautology by the initial assumption. We define the probability distribution P as follows: for everyj,1≤j≤2n, Pj = t−1 ifam+1,j =1 0 otherwise
Clearly given howPis defined,UP(B) = [Pj=12n (am+1,j×Pj)] =1.
Claim:Pverifies every∆∈℘k(Γ).
Proof of claim: Let∆∈℘k(Γ)be arbitrary. By the initial assumption∆6`Bso there must be a column inAwhich witnesses this. Let the witnessing column be thesth column in A. We note that P must bes-positive sincePs = t−1 > 0. Moreover the sth term of U(A) must be0for everyA ∈ ∆. HenceP verifies∆. Since∆was arbitrary, this suffices to show thatPverifies every member of℘k(Γ).
(5) We assume thatΓ |=k Band thatP= [P1. . . P2n]T is an arbitrary probability distri- bution such thatPA∈∆UP(A) ≤ < 1 holds for each∆ ∈ ℘k(Γ). From (4) above it follows thatB∈S{
Cn(∆)|∆∈℘k(Γ)}. This implies that for some∆0∈℘k(Γ)we have ∆0 ` B. But by the initial assumption
P
A∈∆0UP(A) ≤ < 1. By theorem (4.3.2)), it follows that∆0must be consistent. Let|Γ|=mwithnvariables and let|∆0| =t. With- out loss of generality we may assume that the firsttrows of the uncertainty matrixA
correspond to members of∆0and the(m+1)-th row ofAcorresponds toB. Using the usual column rotation,Acan be reconfigured into the following sub-matrices:
A=
" B C D E #
B is a t×s submatrix with each column containing at least one entry of 1; C is a t×(2n−s)zero submatrix. By the consistency of∆0,Ccannot be empty. We note that since∆0 ` B, the last row of Emust be 0’s. This gives the following absolute upper bound onUP(B):
However we note that since each column ofBcontains at least one entry of1, we have the following absolute lower bound onPA∈∆
0UP(A): P1+. . .+Pt ≤ X A∈∆0 UP(A) HenceUP(B)≤PA∈∆ 0UP(A)≤ < 1as required.