Track-Down Operations on Bilattices

Abstract

This paper discusses a dualization of Fitting's notion of a "cut-down" operation on a bilattice, rendering a "track-down" operation, later used to represent the idea that a consistent opinion cannot arise from a set including an inconsistent opinion. The logic of track-down operations on bilattices is proved equivalent to the logic d_Sfde, dual to Deutsch's system S_fde. Furthermore, track-down operations are employed to provide an epistemic interpretation for paraconsistent weak Kleene logic. Finally, two logics of sequential combinations of cut-and track-down operations allow settling positively the question of whether bilattice-based semantics are available for subsystems of S_fde.

Full text

Track-Down Operations on Bilattices
Damian Szmuc1,2
1IIF-SADAF, National Scientific and Technical Research Council (CONICET), Argentina
2Department of Philosophy, University of Buenos Aires, Argentina
Abstract
This paper discusses a dualization of Fitting’s notion of a “cut-down” operation on
a bilattice, rendering a “track-down” operation, later used to represent the idea that a
consistent opinion cannot arise from a set including an inconsistent opinion. The logic of
track-down operations on bilattices is proved equivalent to the logic dSfde,dual to Deutsch’s
system Sfde. Furthermore, track-down operations are employed to provide an epistemic
interpretation for paraconsistent weak Kleene logic. Finally, two logics of sequential combi-
nations of cut- and track-down operations allow settling positively the question of whether
bilattice-based semantics are available for subsystems of Sfde.
1 Introduction: Aim and Definitions
This paper discusses a dualization of Fitting’s notion of a “cut-down” operation on a bilattice,
used in [9] to provide an epistemic interpretation of Kleene’s paracomplete weak three-valued
logic. The logic of such cut-down operations is equivalent, as shown in [7], to Deutsch’s four-
valued logic Sfde from [5]. Our dualization of Fitting’s notion renders a “track-down” operation
later used to represent the idea that a consistent opinion cannot arise from a set including an
inconsistent opinion.
Our first result in this paper is proving the logic of track-down operations is equivalent to the
four-valued logic dSfde, dual to Deutsch’s system. Our second contribution consists in employ-
ing such track-down operations to provide an epistemic interpretation for paraconsistent weak
Kleene logic. Our third contribution is settling positively the question, posed in [7], whether
bilattice-based semantics can be given for subsystems of Sfde. This is done by presenting two
subsystems of Sfde and dSfde corresponding to the logics of two different sequential combinations
of cut- and track-down operations.
In what follows, Lwill be the propositional language {˙¬,˙
∧,˙
∨} and F OR(L) the set of
formulae of L, defined as usual. Formulae of Lwill be denoted by ϕ, ψ, etc., while sets of
formulae will be denoted by Γ,∆, etc. For a propositional language L, a matrix Mis a structure
hV,D,Oi where hV,Oi is an algebra of the same similarity type as L, and Dis a non-empty
proper subset of V. Given M, a valuation vis a homomorphism from F OR(L) to V. A (matrix)
logic Lis a pair hF OR(L),Miwhere M⊆℘(F OR(L)) ×F O R(L) is a substitution-invariant
consequence relation defined by letting Γ Mϕiff for every valuation v, if v[Γ] ⊆ D, then
v(ϕ)∈ D. When L=hF OR(L),Miwe may denote Mby L.
1

2 Bilattices, Cut-downs and Track-downs
Definition 1 (Ginsberg [10]).A pre-bilattice Bis a structure hB, ≤k,≤tisuch that Bis a
nonempty set and hB, ≤ki,hB, ≤tiare two complete lattices.1
Definition 2 (Ginsberg [10]).A bilattice Bis a structure of the form hB, ≤k,≤t,¬i such that
hB, ≤k,≤tiis a pre-bilattice and ¬is an involutive t-inverting function on B, i.e. ¬:B−→ B
is a function such that for all a, b ∈B: (i) ¬¬a=a, (ii) If a≤kb, then ¬a≤k¬b, (iii) If a≤tb,
then ¬b≤t¬a.
The orders ≤kand ≤tare often referred to as the “information” (sometimes, “knowledge”)
ordering, and the “truth” ordering, respectively. The lattice-theoretic operations meet and join
related to these orderings are, respectively, ⊗and ⊕, and the usual ∧and ∨.
Fitting famously offered numerous epistemic motivations to work with bilattices. For in-
stance, in [9] he proposes to consider a group Eof experts whose opinion we value and who we
are consulting on certain matters, in the form of a series of yes/no questions. When asking these
experts about a certain sentence ϕsome will say it is true, some will say it is false, some may
be willing to decline expressing an opinion and some may have reasons for calling it both true
and false.
To such a scenario corresponds, Fitting claims, the assignment of a sort of generalized truth-
value to ϕ, namely v(ϕ) = hP, Niwhere Pis the set of experts who claim that ϕis true, and
Nis the set of experts who claim that ϕis false. Given this, it is possible that P∪N6=Eand
it is also possible that P∩N6=∅. As Fitting notices
Orderings can be introduced into our people-based structure: set hP1, N1i ≤khP2, N2i
if P1⊆P2and N1⊆N2, and set hP1, N1i ≤thP2, N2iif P1⊆P2and N2⊆N1(...)
Thus, information goes up if more people express a positive or negative opinion, and truth
goes up if people drop negative opinions or add positive ones. This gives a structure of a
pre-bilattice. [9, p. 3]
In such a framework, let us additionally refer to the generalized truth-values hE ,∅i,hE,Ei,
h∅,∅i and h∅,Ei with the labels t,>,⊥,f, respectively. Focusing in the {t,>,⊥,f}-reduct of this
structure renders the famous bilattice FOU R.
t
k
⊥
ft
>
Figure 1: The bilattice F OU R
By considering situations where the generalized truth-values assigned to any sentence are
elements of FOUR (or some subset thereof) Fitting was able to provide an epistemic interpre-
tation of several Kleene logics. In fact, in [9], the way the connectives ˙¬,˙
∧,˙
∨work in the strong
1There is no need for completeness to be built into the definition of a bilattice. We assume it here, though,
to keep consistency with the literature.
2

Kleene logic K3and its four-valued generalization Efde due to Belnap and Dunn in [3] and [6]
are interpreted in terms of the bilattice operations ¬,∧,∨on FOU R.
Our target in this paper are, nevertheless, not the strong but the weak Klenee logics and
some subsystems thereof. Employing Fitting’s framework to provide an epistemic interpretation
of the operations of Kleene’s weak three-valued logic requires some subtleties. More precisely,
Fitting said in [9, pp. 66-67] that sometimes, e.g. when evaluating a conjunction ϕ˙
∧ψor a
disjunction ϕ˙
∨ψ, we may want to
‘cut this down’ by considering people who have actually expressed an opinion on both
propositions [ϕ] and [ψ]
Whence, we shall call the resulting alternative conjunctions and disjunctions—following Fergu-
son in [7]—the “cut-down variants” of these famous logical operations. Observe ¬is not altered
by this cut-down policy, as remarked in [9, p. 67] and [7, p. 24].
Definition 3 (Fitting [9]).For an element a∈B, the Kleene-Fitting cut-down of a, denoted
by [[a]] is defined as a⊕ ¬a
To this extent, the cut-down [[ϕ]] of a ϕis intended to output “those who either think ϕis
true, or think ϕis false”.
Definition 4 (Fitting [9]).For a, b ∈B, the Kleene-Fitting cut-down operations Mand Oare
defined as:
aMb= (a∧b)⊗[[a]] ⊗[[b]] aOb= (a∨b)⊗[[a]] ⊗[[b]]
Similarly, the cut-down variants of a conjunction and disjunction shall be interpreted as
cutting down the set of experts under consideration, to only those who have expressed an
opinion towards all of the propositions involved. These variants works so that no determinate
opinion on e.g. ϕ˙
∧ψor ϕ˙
∨ψcan arise from a set that includes an indeterminate opinion on
ϕor ψ.
In [9], the way the connectives ˙
∧,˙
∨work in the weak Kleene logic Kw
3was interpreted in
terms of the cut-down operations M,O, providing the target epistemic interpretation.
Definition 5. Kw
3is the three-valued logic induced by the matrix hVKw
3,DKw
3,OKw
3i, where
VKw
3={t,⊥,f},DKw
3={t},OKw
3={˙¬
Kw
3,˙
∧
Kw
3,˙
∨
Kw
3}and these truth-functions are defined by the
truth-tables in Figure 2.
˙¬
Kw
3
t f
⊥ ⊥
f t
˙
∧
Kw
3t⊥f
t t ⊥f
⊥ ⊥ ⊥ ⊥
f f ⊥f
˙
∨
Kw
3t⊥f
t t ⊥t
⊥ ⊥ ⊥ ⊥
f t ⊥f
Figure 2: Truth-tables for Kw
3
When pooling the opinion of experts, we may want to follow Fitting’s cut-down strategy or
we may want to proceed in a different, although perfectly dual, way. Sometimes, e.g. when
considering a conjunction ϕ˙
∧ψor a disjunction ϕ˙
∨ψwe may want to
‘track down’ people who have expressed an inconsistent opinion towards either propositions,
ϕor ψ
3

Whence, we shall call the resulting alternative conjunctions and disjunctions, the “track-down”
variants of these famous logical operations. Observe, again, that ¬is not altered by this cut-
down policy, either.
Definition 6. For an element a∈B, the track-down of a, denoted by ]]a[[ is defined as a⊗ ¬a
To this extent, the track-down ]]ϕ[[ of ϕis intended to output “those who think ϕis true,
and also think ϕis false”.
Definition 7. For a, b ∈B, the track-down operations Nand Hare defined as:
aNb= (a∧b)⊕]]a[[ ⊕]]b[[ aHb= (a∨b)⊕]]a[[ ⊕]]b[[
Similarly, the track-down variants of a conjunction and disjunction shall be interpreted
as tracking down the set of experts that expressed an inconsistent opinion towards some of
the propositions involved, and then expanding the set of experts thinking the conjunctions or
disjunctions are true, or that they are false, to account for them. These variants works so that
no consistent opinion on e.g. ϕ˙
∧ψor ϕ˙
∨ψcan arise from a set that includes an inconsistent
opinion on ϕor ψ.
In Section 5 we will establish that it is possible to interpret the way the connectives ˙
∧,˙
∨
work in the paraconsistent weak Kleene logic PWK in terms of the track-down operations N,H,
thereby providing an epistemic interpretation of PWK.
Definition 8. PWK is the three-valued logic induced by the matrix hVPWK,DPWK,OPWK i, where
VPWK ={t,>,f},DPWK ={t,>},OPWK ={˙¬
PWK,˙
∧
PWK,˙
∨
PWK}and these truth-functions are
defined by the truth-tables in Figure 3.
˙¬
PWK
t f
> >
f t
˙
∧
PWK t>f
t t >f
> > > >
f f >f
˙
∨
PWK t>f
t t >t
> > > >
f t >f
Figure 3: Truth-tables for PWK
3 Four Containment Logics
The various logics of cut- and track-down operations on bilattices we are going to study in the
sequel belong to a peculiar class of logics dubbed containment logics.
These are systems where valid inferences comply with certain set-theoretic containment
principles relating the set of propositional variables appearing in the premises and the set of
propositional variables appearing in the conclusion.
Among containment logics, the best known systems belong to the family of “Parry” logics—
so-called because their valid inferences comply with a form of Parry’s Proscriptive Principle for
entailment, discussed in [12], namely
ΓLϕonly if var(ϕ)⊆var(Γ)
where var(ϕ) is the set of propositional variables appearing in ϕ. As highlighted in [8], the
following is a Parry logic.
4

˙¬
Sfde
t f
> >
⊥ ⊥
f t
˙
∧
Sfde t> ⊥ f
t t > ⊥ f
> > > ⊥ f
⊥ ⊥⊥⊥⊥
f f f ⊥f
˙
∨
Sfde t> ⊥ f
t t t ⊥t
>t>⊥>
⊥ ⊥⊥⊥⊥
f t > ⊥ f
Figure 4: Truth-tables for the logic Sfde
Definition 9 (Deutsch [5]).Sfde is the four-valued logic induced by the matrix hVSfde,DSfde ,OSfde i,
where VSfde ={t,>,⊥,f},DSfde ={t,>},OSfde ={˙¬
Sfde ,˙
∧
Sfde ,˙
∨
Sfde }and these truth-functions are
defined by the truth-tables in Figure 4.
The logic Sw
fde, semantics for which were first given in [13], is also a Parry logic. This can be
established by noting, as we do below, that Sw
fde is a fragment of Sfde.
Definition 10. Sw
fde is the four-valued logic induced by the matrix hVSw
fde ,DSw
fde ,OSw
fde i, where
VSw
fde ={t,>,⊥,f},DSw
fde ={t,>},OSw
fde ={˙¬
Sw
fde ,˙
∧
Sw
fde ,˙
∨
Sw
fde }and these truth-functions are defined
by the truth-tables in Figure 5.
˙¬
Sw
fde
t f
> >
⊥ ⊥
f t
˙
∧
Sw
fde t> ⊥ f
t t > ⊥ f
> >>⊥>
⊥ ⊥⊥⊥⊥
f f > ⊥ f
˙
∨
Sw
fde t> ⊥ f
t t > ⊥ t
> >>⊥>
⊥ ⊥⊥⊥⊥
f t > ⊥ f
Figure 5: Truth-tables for the logic Sw
fde
That Sw
fde is a fragment of Sfde can be observed by showing the definability of the Sw
fde truth-
functions in terms of Sfde truth-functions.2Letting the trivial case of negation aside, it can be
easily checked, by looking at the corresponding truth-tables, that ˙
∧
Sw
fde (x, y) is definable in Sfde
as
˙
∨
Sfde ˙
∧
Sfde (x, y),˙
∨
Sfde ˙
∧
Sfde x, ˙¬
Sfde (x),˙
∧
Sfde y, ˙¬
Sfde (y)!
later defining ˙
∨
Sw
fde in terms of ˙
∧
Sw
fde and ˙¬
Sw
fde , as is customary.
Now, besides Parry logics, there is yet another important and not so widely discussed family
of containment logics, which we may naturally refer to as “Dual Parry” logics—for all of their
valid inferences comply with a dual of Parry’s Proscriptive Principle for entailment, namely
ΓLϕonly if ∃Γ0⊆Γ,Γ06=∅, var(Γ0)⊆var(ϕ)
It can be proved (as we do in Observation 1) that the logic defined below, semantics for
which were given for the first time in [13], is indeed a Dual Parry logic.3
2This follows the strategy of [11, §5.1] to show that Kleene’s weak three-valued logics (e.g. Kw
3) are fragments
of their strong counterparts (e.g. K3)—whence, the addition of the superscript wto these systems.
3Notice that the clause Γ06=∅would render enjoying the dual Parry Principle incompatible with having
theorems. Thus, a reviewer suggests replacing it with the condition that Γ0Lϕ. Although this is an interesting
alternative, the incompatibility induced by the current clause is dual—as expected—to the fact that enjoying
the Parry Principle is incompatible with having anti-theorems. that is, with there being formulae ψsuch that
ψL∅.
5

Definition 11. The logic dSfde is induced by the matrix hVdSfde ,DdSfde ,OdSfde i, where VdSfde =
{t,>,⊥,f},DdSfde ={t,>},OdSfde ={˙¬
dSfde ,˙
∧
dSfde ,˙
∨
dSfde }and these truth-functions are defined by
the truth-tables in Figure 6.
˙¬
dSfde
t f
> >
⊥ ⊥
f t
˙
∧
dSfde t> ⊥ f
t t > ⊥ f
> >>>>
⊥ ⊥ > ⊥ f
f f >f f
˙
∨
dSfde t> ⊥ f
t t >t t
> >>>>
⊥t>⊥⊥
f t > ⊥ f
Figure 6: Truth-tables for the logic dSfde
Observation 1. dSfde is a Dual Parry logic.
Proof. Suppose, for reductio, that there is an inference Γ dSfde ϕsuch that it is not true that
∃Γ0⊆Γ,Γ06=∅, var(Γ0)⊆var(ϕ). Notice, furthermore, that for all ψthere is a valuation v
such that for all p∈var(ψ), v(p) = ⊥, whence v(ψ) = ⊥/∈ DdSfde . Consider a valuation vof
this sort, where v(ϕ) = ⊥/∈ DdSfde, to construct the valuation v∗
v∗(p) = (>if p∈var(Γ) \var(ϕ)
v(p) otherwise
By the above, we know for all γ∈Γ, var(γ)*var(ϕ), which implies for all γ∈Γ, there is a
q∈var(γ)\var(ϕ) such that v∗(q) = >. Whence, for all γ∈Γ, v∗(γ) = >, implying v∗(Γ) ⊆
DdSfde . Simultaneously, by our assumptions, v∗(ϕ)/∈ DdSfde. Then, v∗witnesses Γ 2dSfde ϕ,
contradicting our initial assumption. Therefore, dSfde is a Dual Parry logic.
Observation 2. Letting Σ¬={¬ψ|ψ∈Σ}, the following readily verifiable fact establishes the
duality between dSfde and Deutsch’s logic Sfde
ΓSfde ∆⇐⇒ ∆¬dSfde Γ¬
Moreover, the logic dSw
fde, presented first in [13], is also a Dual Parry logic. This can be
established—once again—by noting, as we do below, that dSw
fde is a fragment of dSfde.
Definition 12. The logic dSw
fde is induced by the matrix hVdSw
fde ,DdSw
fde ,OdSw
fde i, where VdSw
fde =
{t,>,⊥,f},DdSw
fde ={t,>},OdSw
fde ={˙¬
dSw
fde ,˙
∧
dSw
fde ,˙
∨
dSw
fde }and these truth-functions are defined by
the truth-tables in Figure 7.
˙¬
dSw
fde
t f
> >
⊥ ⊥
f t
˙
∧
dSw
fde t> ⊥ f
t t > ⊥ f
> >>>>
⊥ ⊥>⊥⊥
f f > ⊥ f
˙
∨
dSw
fde t> ⊥ f
t t > ⊥ t
> >>>>
⊥ ⊥>⊥⊥
f t > ⊥ f
Figure 7: Truth-tables for the logic dSw
fde
That dSw
fde is a fragment of dSfde can be proved in the same way in which it was proved that
Sw
fde is a fragment of Sfde. For this purpose, the main argument carries over to this case, and so
6

in fact we can check, by looking at the corresponding truth-tables, that ˙
∧
dSw
fde (x, y) is definable
in dSfde as
˙
∨
dSfde ˙
∧
dSfde (x, y),˙
∨
dSfde ˙
∧
dSfde x, ˙¬
dSfde (x),˙
∧
dSfde y, ˙¬
dSfde (y)!
Next, we will focus on reviewing and showing some connections these containment systems
have with bilattices, similar to those enjoyed by Belnap-Dunn logic Efde and bilattices.
4 Logical Bilattices
We now turn to logical systems based on bilattices.
Definition 13 (Arieli and Avron [2]).Abifilter on a bilattice Bis a nonempty and proper
subset F ⊂ Bsuch that for all a, b ∈B:a∧b∈ F iff a∈ F and b∈ F iff a⊗b∈ F . A bifilter
Fis prime if for all a, b ∈B:a∨b∈ F iff either a∈ F or b∈ F iff a⊕b∈ F.
Definition 14 (Arieli and Avron [2]).Alogical bilattice is a pair hB,Fi where Bis a bilattice
and Fis a prime bifilter on B.
Observation 3 (Arieli and Avron [2]).The set {t,>} is the only prime bifilter on the bilattice
FOU R. Thus, the only logical bilattice definable on FOUR is hF OU R,{t,>}i.
The key to defining different consequence relations on logical bilattices is the notion of a
valuation based on a bilattice. Arieli and Avron, e.g. opted for the following.
Definition 15 (Arieli and Avron [1]).An Arieli-Avron valuation (AA-valuation) on a bilattice
Bis a function v:F OR(L)−→ Binterpreting the connectives in {˙¬,˙
∧,˙
∨} with the operations
in {¬,∧,∨}, respectively.
Definition 16 (Arieli and Avron [1]).With regard to a logical bilattice hB,Fi, an inference
from Γ to ϕis AA-valid (symbolized Γ hB,F i
AA ϕ) if for all AA-valuations vsuch that v(Γ) ⊆ F,
then also v(ϕ)∈ F.
Thus, Arieli and Avron proved the following, interpreted by them as showing that Efde is
the logic of logical bilattices—a fortiori establishing that Efde has a similar relation to logical
bilattices than Boolean algebras have with Classical Logic.
Fact 4 (Arieli and Avron [1]).For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
AA ϕ⇐⇒ ΓEfde ϕ
Generalizing Fitting’s work, Ferguson focused in [7] on different definitions inspired by the
“cut-down” interpretation, proving similar results about the relation of Sfde and bilattices.
Definition 17 (Ferguson [7]).A Kleene-Fitting valuation on a bilattice Bis a function v:
F OR(L)−→ Binterpreting the connectives in {˙¬,˙
∧,˙
∨} with the operations in {¬,M,O}, re-
spectively.
Definition 18 (Ferguson [7]).With regard to a logical bilattice hB,Fi, an inference from Γ to
ϕis KF -valid (symbolized Γ hB,F i
KF ϕ) if for all Kleene-Fitting valuations vsuch that v(Γ) ⊆ F,
then also v(ϕ)∈ F.
This allowed him to prove the next result.
7

Fact 5 (Ferguson [7]).For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
KF ϕ⇐⇒ ΓSfde ϕ
Ferguson took this fact to establish that Sfde is the logic of cut-down operations on logical
bilattices and, hence, that Sfde has the same relation with bilattices equipped with cut-down
operations than Classical Logic has with Boolean algebras.
5 The Logic of Track-Down Operations
In this section we prove a result similar to the previously discussed about Efde and Sfde, but we
now focus on dSfde.
Definition 19. A track-down valuation on a bilattice Bis a function v:FO R(L)−→ B
interpreting the connectives in {˙¬,˙
∧,˙
∨} with the operations in {¬,N,H}, respectively.
Definition 20. With regard to a logical bilattice hB,Fi, an inference from Γ to ϕis T D-
valid (symbolized Γ hB,Fi
TD ϕ) if for all track-down valuations vsuch that v(Γ) ⊆ F , then also
v(ϕ)∈ F.
Observation 6. For all sets of formulae Γ∪ {ϕ},
ΓhFO UR,{t,>}i
TD ϕ⇐⇒ ΓdSfde ϕ
Proof. This follows straightforwardly, by identifying, one the one hand, the sets VdSfde and
F OU R ={t,>,⊥,f}and, on the other hand, the truth-functions of dSfde with the corresponding
track-down operations on FOUR.
Definition 21 (Arieli and Avron [2]).Let hB,F i be a logical bilattice, we define the following
exclusive and exhaustive subsets of B:
ThB,F i
t={a∈B|a∈ F,¬a /∈ F}
ThB,F i
>={a∈B|a∈ F,¬a∈ F}
ThB,F i
⊥={a∈B|a /∈ F,¬a /∈ F}
ThB,F i
f={a∈B|a /∈ F,¬a∈ F }
Definition 22 (Arieli and Avron [2]).Let hB0,F0iand hB1,F1ibe logical bilattices and let
a0∈B0and a1∈B1be elements of each. Then a0∈B0and a1∈B1are similar (a0'a1) if:
(i) a0∈ F0iff a1∈ F1, and (ii) ¬a0∈ F0iff ¬a1∈ F1.
Definition 23. Two track-down valuations v0and v1for logical bilattices hB0,F0iand hB1,F1i,
respectively, are similar (v0'v1) if for all atomic p∈F OR(L): v0(p)'v1(p).
Proposition 7. Let v0and v1be two track-down valuations for logical bilattices hB0,F0iand
hB1,F1i, respectively. If v0'v1, then for all formulae ϕ∈F O R(L):v0(ϕ)'v1(ϕ).
Proof. By induction on the complexity of ϕ.
Definition 24 (Ferguson [7]).Let ghB,Fi :B−→ F OU R be a function such that ghB,F i(x) is
the unique yfor which x∈ T hB,Fi
y.
8

What gessentially does is to find the unique member of the partition provided in Defini-
tion 21, to which xbelongs, and then maps it to the corresponding element of F OU R.
Proposition 8. If vis a track-down valuation on a logical bilattice hB,Fi, then ghB ,Fi ◦vis a
track-down valuation on hFOUR,{t,>}i such that v'ghB,F i ◦v.
Proof. Immediate from the definition of ghB,Fi .
Lemma 9. For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
TD ϕ⇐⇒ ΓhFOU R,{t,>}i
TD ϕ
Proof. RTL: suppose Γ 2hB,F i
TD ϕand let vbe a track-down valuation on hB,Fi witnessing this
fact, i.e. v(Γ) ⊆ F and v(ϕ)/∈ F. By Proposition 8, ghB,F i ◦vis a track-down valuation on
hFOU R,{t,>}i behaving in the following way: ghB,F i ◦v(Γ) ⊆ {t,>} and ghB,Fi ◦v(ϕ)/∈ {t,>}.
This implies Γ 2hFOUR,{t,>}i
TD ϕ.
LTR: suppose Γ 2hF OU R,{t,>}i
TD ϕand let vbe a track-down valuation on hFOU R,{t,>}i
witnessing this fact, i.e. v(Γ) ⊆ {t,>} and v(ϕ)/∈ {t,>}. Given F OU R ⊆B, for all bilattices
hB,F i, we know that vis a valuation on hB,Fi too. Moreover, von hB,Fi is similar to von
hFOU R,{t,>}i, since >,t∈F and ⊥,f/∈F. Thus, vwitnesses Γ 2hB,F i
TD ϕ.
Theorem 10. For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
TD ϕ⇐⇒ ΓdSfde ϕ
Proof. From Observation 6 and Lemma 9.
This establishes the first contribution of this paper, namely that dSfde is the logic of track-
down operations on bilattices. This allows us to conclude—following Avron, Arieli and Ferguson—
that the relation between dSfde and bilattices endowed with track-down operations is the same
that Boolean algebras have with Classical Logic.
Furthermore, this also facilitates the second contribution of this paper, consisting in an
epistemic interpretation for Paraconsistent Weak Kleene. In fact, let us consider a situation in
which we apply the track-down policy, but all experts consulted have determinate opinions on
absolutely all propositions ϕ. This will amount, formally, to restricting the valuations of dSfde
to the values in {t,>,f}. The three-valued logic induced by this restriction is no other than
PWK, whence its operations can be interpreted in terms of the cut-down operations N,H.
6 Combined Cut-downs and Track-downs
Finally, peculiar combinations of the cut-down and track-down approaches can, and perhaps
should be conceived, for matters of exhaustivity. The combinations we are going to study next
rely on applying the cut- and track-down approaches sequentially. By this we mean, defining
cut-down operations which take the output of the track-down operations as an input, and
viceversa.
To be more specific, concerning the cut-down approach, we could alternatively cut down
experts who have expressed an opinion towards both ϕand ψ, taking into consideration the
expert’s opinion on ϕand ψonly after the track-down process. That is, after tracking down
whether their expressed an inconsistent opinion towards either ϕor ψ. This renders what we
call the weak cut-down operations.
9

Definition 25. For elements a, b ∈B, the weak cut-down operations Mwand Oware defined
as:
aMwb= (aNb)⊗[[a]] ⊗[[b]] aOwb= (aHb)⊗[[a]] ⊗[[b]]
Analogously, concerning the track-down approach, we could track down experts who have
expressed an inconsistent opinion towards either ϕand ψ, taking into consideration the expert’s
opinion on ϕand ψonly after the cut-down process. That is, after cutting down those experts
which have not expressed an opinion towards either ϕor ψ. This renders what we call the weak
track-down operations.
Definition 26. For elements a, b ∈B, the weak track-down operations Nwand Hware defined
as:
aNwb= (aMb)⊕]]a[[ ⊕]]b[[ aHwb= (aOb)⊕]]a[[ ⊕]]b[[
We show, next, some results concerning the relation between bilattices, Sw
fde and dSw
fde, which
are equally conclusive than those previously discussed regarding Efde,Sfde and dSfde.
Definition 27. A weak cut-down valuation on a bilattice Bis a function v:F OR(L)−→ B
interpreting the connectives in {˙¬,˙
∧,˙
∨} with the operations in {¬,Mw,Ow}, respectively.
Definition 28. With regard to a logical bilattice hB,Fi, an inference from Γ to ϕis CDw-valid
(symbolized, Γ hB,F i
CDwϕ) if for all weak cut-down valuations vsuch that v(Γ) ⊆ F , then also
v(ϕ)∈ F.
Definition 29. A weak track-down valuation on a bilattice Bis a function v:F OR(L)−→ B
interpreting the connectives in {˙¬,˙
∧,˙
∨} with the operations in {¬,Nw,Hw}, respectively.
Definition 30. With regard to a logical bilattice hB,F i, an inference from Γ to ϕis T D w-valid
(symbolized, Γ hB,F i
TDwϕ) if for all weak track-down valuations vsuch that v(Γ) ⊆ F, then also
v(ϕ)∈ F.
We can adapt the definitions and proofs of Section 5 to show that Sw
fde and dSw
fde are,
respectively, the logic of weak cut-down and weak track-down operations on bilattices.
Observation 11. For all sets of formulae Γ∪ {ϕ},
ΓhFO UR,{t,>}i
CDwϕ⇐⇒ ΓSw
fde ϕ
Proof. We establish this by identifying the sets VSw
fde and F OU R ={t,>,⊥,f}, and the truth-
functions of Sw
fde with the corresponding weak cut-down operations on FOU R.
Observation 12. For all sets of formulae Γ∪ {ϕ},
ΓhFO UR,{t,>}i
TDwϕ⇐⇒ ΓdSw
fde ϕ
Proof. For this, we identify the sets VdSw
fde and F OU R ={t,>,⊥,f}, and the truth-functions of
dSw
fde with the corresponding weak track-down operations on FOUR.
Definition 31. Two weak cut-down (alternatively, weak track-down) valuations v0and v1
for logical bilattices hB0,F0iand hB1,F1i, respectively, are similar (v0'v1) if for all atomic
p∈F OR(L): v0(p)'v1(p).
10

The following are proved exactly like in Section 5.
Proposition 13. Let v0and v1be two weak cut-down (alternatively, weak track-down) valua-
tions for logical bilattices hB0,F0iand hB1,F1i, respectively. If v0'v1, then for all formulae
ϕ∈F OR(L):v0(ϕ)'v1(ϕ).
Proposition 14. If vis a weak cut-down (alternatively, weak track-down) valuation on a
logical bilattice hB,Fi, then ghB ,Fi ◦vis a weak cut-down (respectively, weak track-down) on
hFOU R,{t,>}i, such that v'ghB,F i ◦v.
Lemma 15. For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
CDwϕ⇐⇒ ΓhFOU R,{t,>}i
CDwϕ
Proof. RTL: suppose Γ 2hB,Fi
CDwϕand let vbe a weak cut-down valuation on hB,Fi witnessing this
fact, i.e. v(Γ) ⊆ F and v(ϕ)/∈ F. By Proposition 14, ghB,F i ◦vis a weak cut-down valuation on
hFOU R,{t,>}i behaving in the following way: ghB,F i ◦v(Γ) ⊆ {t,>} and ghB,Fi ◦v(ϕ)/∈ {t,>}.
This implies Γ 2hFOUR,{t,>}i
CDwϕ.
LTR: suppose Γ 2hF OU R,{t,>}i
CDwϕand let vbe a weak cut-down valuation on hFOU R,{t,>}i
witnessing this fact, i.e. v(Γ) ⊆ {t,>} and v(ϕ)/∈ {t,>}. Given F OU R ⊆B, for all hB,F i, we
know vis a valuation on hB,Fi too. Moreover, von hB,Fi is similar to von hFOU R,{t,>}i,
since >,t∈ F and ⊥,f/∈ F. Thus, vwitnesses Γ 2hB,F i
CDwϕ.
Lemma 16. For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
TDwϕ⇐⇒ ΓhFOU R,{t,>}i
TDwϕ
Proof. Similar to the proof of Lemma 15, replacing “weak cut-down valuation” for “weak track-
down valuation”.
Theorem 17. For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
CDwϕ⇐⇒ ΓSw
fde ϕ
Proof. From Observation 11 and Lemma 15.
Theorem 18. For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},
ΓhB,F i
TDwϕ⇐⇒ ΓdSw
fde ϕ
Proof. From Observation 12 and Lemma 16.
7 Further Research
A proof-theoretic exploration of the logics of cut-down and track-down operations is pending. It
appears straightforward to build Gentzen-style sequent calculi for dSfde and Sfde, using the tech-
niques described in [4], by imposing variable inclusion restrictions on the rules—respectively—of
the sequent calculi for K3and its paraconsistent dual LP. However, we conjecture special care
needs to be taken in trying to reflect, in potential calculi for Sw
fde and dSw
fde, the sequential nature
of the combinations of cut- and track-downs behind these systems.
11

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