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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

ΓLϕonly 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

ΓLϕonly 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 Γ0Lϕ. 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

ψL∅.

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

ΓSfde ∆⇐⇒ ∆¬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},

ΓhB,F i

AA ϕ⇐⇒ ΓEfde 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},

ΓhB,F i

KF ϕ⇐⇒ ΓSfde 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},

ΓhFO UR,{t,>}i

TD ϕ⇐⇒ ΓdSfde 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},

ΓhB,F i

TD ϕ⇐⇒ ΓhFOU 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},

ΓhB,F i

TD ϕ⇐⇒ ΓdSfde 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},

ΓhFO UR,{t,>}i

CDwϕ⇐⇒ ΓSw

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},

ΓhFO UR,{t,>}i

TDwϕ⇐⇒ ΓdSw

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},

ΓhB,F i

CDwϕ⇐⇒ ΓhFOU 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},

ΓhB,F i

TDwϕ⇐⇒ ΓhFOU 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},

ΓhB,F i

CDwϕ⇐⇒ ΓSw

fde ϕ

Proof. From Observation 11 and Lemma 15.

Theorem 18. For all logical bilattices hB,Fi and all sets of formulae Γ∪ {ϕ},

ΓhB,F i

TDwϕ⇐⇒ ΓdSw

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

References

[1] O. Arieli and A. Avron. Reasoning with logical bilattices. Journal of Logic, Language and

Information, 5(1):25–63, 1996.

[2] O. Arieli and A. Avron. The logical role of the four-valued bilattice. In Proc. 13th Sym-

posium on Logic in Computer Science, pages 118–126, Los Alamitos, CA, 1998. IEEE

Computer Society.

[3] N. Belnap. A useful four-valued logic. In M. Dunn and G. Epstein, editors, Modern uses

of Multiple-Valued Logic, pages 8–37. Reidel, Dordrecht, 1977.

[4] M. Coniglio and M. I. Corbal´an. Sequent calculi for the classical fragment of Bochvar and

Halld´en’s Nonsense Logics. In Proc. 7th Workshop on Logical and Semantic Frameworks

with Applications (LSFA), pages 125–136. EPTCS 113, 2012.

[5] H. Deutsch. Relevant analytic entailment. The Relevance Logic Newsletter, 2(1):26–44,

1977.

[6] M. Dunn. Intuitive semantics for ﬁrst-degree entailments and ‘coupled trees’. Philosophical

Studies, 29(3):149–168, 1976.

[7] T. M. Ferguson. Cut-Down Operations on Bilattices. In Proc. 45th International Symposium

on Multiple-Valued Logic (ISMVL), pages 24–29, Los Alamitos, CA, 2015. IEEE Computer

Society.

[8] T. M. Ferguson. Logics of nonsense and Parry systems. Journal of Philosophical Logic,

44(1):65–80, 2015.

[9] M. Fitting. Bilattices are nice things. In T. Bolander, V. Hendricks, and S. Pedersen,

editors, Self-Reference, pages 53–78. CSLI Publications, Stanford, 2006.

[10] M. Ginsberg. Multivalued logics: A uniform approach to reasoning in artiﬁcial intelligence.

Computational Intelligence, 4(3):265–316, 1988.

[11] S. Gottwald. Many-valued logics. In D. Jacquette, editor, Philosophy of Logic, pages

675–722. North Holland, Amsterdam, 2007.

[12] W. T. Parry. Ein Axiomensystem f¨ur eine neue Art von Implikation (analytische Implika-

tion). Ergebnisse eines mathematischen Kolloquiums, 4:5–6, 1933.

[13] D. Szmuc. Deﬁning LFIs and LFUs in extensions of infectious logics. Journal of Applied

Non-Classical Logics, 26(4):286–314, 2017.

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