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On All Strong Kleene Generalizations of Classical Logic

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Abstract

By using the notions of exact truth (‘true and not false’) and exact falsity (‘false and not true’), one can give 16 distinct definitions of classical consequence. This paper studies the class of relations that results from these definitions in settings that are paracomplete, paraconsistent or both and that are governed by the (extended) Strong Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene generalizations of classical logic also contains a host of unfamiliar logics. We first study the members of our class semantically, after which we present a uniform sequent calculus (the SK calculus) that is sound and complete with respect to all of them. Two further sequent calculi (the SKP{{\bf SK}^\mathcal{P}} and SKN{\bf SK}^{\mathcal{N}} calculus) will be considered, which serve the same purpose and which are obtained by applying general methods (due to Baaz et al.) to construct sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much simpler and shorter than those of the SKP{\bf SK}^{\mathcal{P}} and the SKN{\bf SK}^{\mathcal{N}} calculus, which is one of the reasons to prefer the SK calculus over the latter two. Besides favourably comparing the SK calculus to both the SKP{\bf SK}^{\mathcal{P}} and the SKN{\bf SK}^{\mathcal{N}} calculus, we also hint at its philosophical significance.
Stefan Wintein
On All Strong Kleene
Generalizations of Classical
Logic
Abstract. By using the notions of exact truth (‘true and not false’) and exact falsity
(‘false and not true’), one can give 16 distinct definitions of classical consequence. This
paper studies the class of relations that results from these definitions in settings that are
paracomplete, paraconsistent or both and that are governed by the (extended) Strong
Kleene schema. Besides familiar logics such as Strong Kleene logic (K3), the Logic of
Paradox (LP) and First Degree Entailment (FDE), the resulting class of all Strong Kleene
generalizations of classical logic also contains a host of unfamiliar logics. We first study the
members of our class semantically, after which we present a uniform sequent calculus (the
SK calculus) that is sound and complete with respect to all of them. Two further sequent
calculi (the SK
P
and SK
N
calculus) will be considered, which serve the same purpose
and which are obtained by applying general methods (due to Baaz et al.) to construct
sequent calculi for many-valued logics. Rules and proofs in the SK calculus are much
simpler and shorter than those of the SK
P
and the SK
N
calculus, which is one of the
reasons to prefer the SK calculus over the latter two. Besides favourably comparing the
SK calculus to both the SK
P
and the SK
N
calculus, we also hint at its philosophical
significance.
Keywords: Classical logic, Strong Kleene Logic (K3), Logic of Paradox (LP), First Degree
Entailment (FDE), Exactly True Logic, Uniform Sequent Calculus.
1. Intro duction
1.1. Strong Kleene Generalizations of Classical Logic
According to classical logic, truth and falsity are mutually exclusive and
jointly exhaustive. As a consequence, truth coincides with non-falsity. More-
over, in the classical setting truth coincides with exact truth (which we will
also denote as truth
), where a sentence is exactly true just in case it is true
and not false. Similarly, falsity coincides with exact falsity (falsity
), where
a sentence is exactly false just in case it is false and not true, and so truth
not only coincides with truth
and non-falsity, but also with non-falsity
.
Presented by Heinrich Wansing; Received April 17, 2015
Studia Logica
DOI: 10.1007/s11225-015-9649-5
c
The Author(s) 2016. This article is published with open access at Springerlink.com
S. Wintein
To sum up:
Classical setting: truth = truth
= non-falsity = non-falsity
(1)
A typical way to characterize a logical consequence relation is in terms of the
preservation of truth over a class of valuations V. According to this General
Schema (GS) a premise set Γ is said to entail a conclusion ϕ just in case, in
passing from Γ to ϕ, truth is preserved for each valuation V V, i.e.
V (γ) is true for all γ Γ = V (ϕ) is true, for all V V (GS)
When we quantify, in (GS), over the class of all classical valuations, we
obtain a characterization of the classical consequence relation. However,
(1) testifies that, when quantifying over all classical valuations, substitut-
ing ‘true
’, ‘non-false’ or ‘non-false
for any of the occurrences of ‘true’
in (GS) delivers another characterization of classical consequence. Indeed,
with x, y ∈{true, true
, non-false, non-false
}, the schema GS(x, y)that
results when we substitute x for the occurrence of ‘true’ on the left side of
the implication of (GS)andy for the occurrence of ‘true’ on its right side,
also defines—when we let V be the class of all classical valuations—classical
consequence.
In this paper, we will study the relations that are defined by our 16
schemas GS(x, y) in settings that are paracomplete, paraconsistent or both.
A setting that is both paracomplete and paraconsistent acknowledges the
possibility that a sentence is neither true nor false (hence the setting is
paracomplete) as well as the possibility that a sentence is both true and
false (hence the setting is paraconsistent). It readily follows that in such
a setting, truth, truth
, non-falsity and non-falsity
are all distinct from
one another and hence, that our 16 schemas potentially define 16 distinct
relations. Of course, whether they actually do so depends on the class of
valuations V with respect to which the schemas are evaluated, to which we
now turn.
In this paper, we will be concerned with valuations for a propositional
language
L whose BNF form is as follows (where p comes from a countably
infinite set of propositional atoms)
ϕ ::= p ϕ | ϕ ϕ | ϕ ϕ
The valuations of
L that we will consider are those associated with the
extended Strong Kleene schema (cf. Belnap [12,13] and Dunn [16]) as defined
below.
On All Strong Kleene Generalizations of Classical Logic
Definition 1. An (extended) Strong Kleene valuation is any function from
Sen(
L )to4 := {T, B, N, F} that respects the following truth tables for ,
and ¬.
TBNF
T TBNF
B
BBFF
N
NFNF
F
FFFF
TBNF
T TTTT
B
TBTB
N
TTNN
F
TBNF
¬
T F
B
B
N
N
F
T
We write V
4
to denote the set of all (extended) Strong Kleene valuations.
We use V
3n
to denote the set of all valuations in V
4
whose range is a
subset of 3n := {T, N, F}, V
3b
to denote the set of all valuations in V
4
whose range is a subset of 3b := {T, B, F},andV
2
to denote the set of all
(classical) valuations in V
4
whose range is 2 := {T, F}.
The canonical interpretation of the elements of 4 is an epistemological one:
according to Belnap’s ([12]) told-interpretation,asentenceisvaluatedasB
just in case one is told both that the sentence is true and that it is false.
In this paper however, the elements of 4 will, for sake of simplicity (and as
nothing hinges on it), be interpreted ontologically; a sentence is either T
(exactly true), F (exactly false), B (both true and false) or N (neither true
nor false). Hence, the valuations of V
4
are associated with a setting that is
both paracomplete and paraconsistent. Likewise, the valuations of V
3n
are
associated with a setting that is (only) paracomplete and the valuations of
V
3b
with a setting that is (only) paraconsistent. Finally, the valuations of
V
2
are associated with the classical setting that is neither paracomplete nor
paraconsistent.
It will be convenient to introduce a uniform notation for the Strong Kleene
Generalizations (of classical logic), i.e. for the relations
1
that are obtained
when our 16 schemas are instantiated with V
2
, V
3n
, V
3b
and V
4
respec-
tively.
2
To do so, we first introduce the following notation for subsets of 4:
1 := {T, B} 0 := {F, B} t := {T} f := {F}
ˆ
1 := {F, N}
ˆ
0 := {T, N}
ˆ
t := {F, N, B}
ˆ
f := {T, N, B}
(2)
1
As will become apparent, some Strong Kleene Generalizations will equal the empty
set (and one of the relations thus obtained will be non-transitive). It is awkward to call
the empty set a consequence relation, and hence we will not refer to the class of all Strong
Kleene Generalizations as a class of consequence relations.
2
Note that the restrictions of (the truth functions denoted by) ¬, and to 2 (3b, 3n)
are truth functions on 2 (3b, 3n). Indeed, this ensures that we can approach the Strong
Kleene Generalizations in a uniform and concise manner.
S. Wintein
Thus, 1 codes for truth, t codes for truth
,
ˆ
0 codes for non-falsity and
ˆ
f for
non-falsity
. Our uniform notation for the Strong Kleene Generalizations (of
classical logic) is provided by the following definition.
Definition 2. Let x, y ∈{1,
ˆ
0, t,
ˆ
f } and let z ∈{2, 3n, 3b, 4}. The relation
xy
z
between sets of sentences and sentences of L is defined as follows:
Γ
xy
z
ϕ ⇐⇒ if V (γ) x z for all γ Γ then V (ϕ)yz for all V V
z
(3)
A relation
xy
z
as defined by (3) is called a Strong Kleene Generalization
(of classical logic). When x, y ∈{t,
ˆ
f },wesaythat
xy
z
is exact . When
x, y ∈{1,
ˆ
0},wesaythat
xy
z
is regular. When
xy
z
is neither exact nor regular,
it is mixed.
Quite some of the Strong Kleene Generalizations are well-known. To be sure,
for the classical setting (z = 2) all 16 instantiations of (3) define the classical
consequence relation. But also for the considered paracomplete (z = 3n)and
paraconsistent setting (z = 3b) all instantiations of (3) are well-known. It
turns out that for z ∈{3n, 3b}, the consequence relation
xy
z
is either equal
to
K3
, the consequence relation of strong Kleene Logic K3 (cf. Kleene [19]).
LP
, the consequence relation of the Logic of Paradox LP (cf. Priest
[24]).
CL
, the consequence relation of classical logic CL.
, the empty set.
In particular, each of these 4 relations can be represented as an exact Strong
Kleene Generalization, as for z ∈{3n, 3b},wehave
3
tt
z
=
K3
ˆ
f
ˆ
f
z
=
LP
t
ˆ
f
z
=
CL
ˆ
ft
z
=
(4)
Our presentation of K3, LP, CL as in (4) naturally raises the question what
the 4 equivalents” of these familiar consequence relations look like. More
concretely, which logics are defined by
tt
4
,
ˆ
f
ˆ
f
4
and
t
ˆ
f
4
? To the best of our
knowledge, of these three relations only
tt
4
has been considered before in
the literature. Per definition, we have
3
Except for
t
ˆ
f
z
=
CL
, which we prove in Sect. 2, all equalities mentioned by (4)are
immediate consequences of the involved definitions.
On All Strong Kleene Generalizations of Classical Logic
tt
4
=
ET L
, the consequence relation of Exactly True Logic ETL which
was studied by Pietz and Rivieccio in [23].
In [23], Pietz and Rivieccio compare ETL with the logic of First Degree
Entailment (FDE), a well-known logic that is studied in [12]and[13]and
that is defined in terms of the preservation of truth or—what turns out to
be equivalent—non-falsity over V
4
valuations:
11
4
=
ˆ
0
ˆ
0
4
=
FDE
, the consequence relation of FDE.
Pietz and Rivieccio motivate their study of ETL by observing that in a 4-
valued setting, it seems natural to define one’s consequence relation in terms
of the preservation of T:
A curious feature of [. . . ] FDE is that the overdetermined value B
(both true and false) is treated as a designated value. Although there
are good theoretical reasons for this, it seems prima facie more plau-
sible to have only one of the four values designated, namely T.[23,
p. 125]
Our motivation to study ETL in this paper is rather different. The main pur-
pose of this paper is to study the Strong Kleene Generalizations in a uniform
way. As ETL is a Strong Kleene Generalization, it belongs a fortiori,tothe
object of our study. Moreover, there is some interest in the relations
tt
4
(i.e.
ETL),
ˆ
f
ˆ
f
4
and
t
ˆ
f
4
,asinlightof(4), these relations are the natural general-
izations of respectively K3, LP and CL to a 4-valued setting. We will study
the Strong Kleene Generalizations both semantically and syntactically—we
will advocate one and present three uniform sequent calculi to capture the
xy
z
relations—as explained in more detail below.
1.2. Structure of the Paper
In Sect. 2, we study the Strong Kleene Generalizations semantically. It turns
out that the different instantiations of (3) define 8 distinct non-empty Strong
Kleene Generalizations: CL, K3, LP, FDE, ETL,
ˆ
f
ˆ
f
4
,
t
ˆ
f
4
(which were already
mentioned) and
1
ˆ
f
4
. Of the 8 relations just mentioned, FDE turns out to
be the weakest, and CL the strongest one: whenever an argument is FDE
valid, it is valid according to each of the 8 relations and when an argument
is valid according to any of the 8 relations, it is CL valid. In Sect. 2.5
we present an exhaustive comparison of the strengths of all 8 relations.
However, we first show, in Sect. 2.1,thatforz ∈{2, 3b, 3n},(3) either
defines , CL, K3,orLP. Then we study the exact, regular and mixed
S. Wintein
Strong Kleene Generalizations associated with z = 4 in Sects. 2.2, 2.3 and
2.4 respectively. We will see that although quite some of the interrelations
between K3, LP and CL carry over to their 4-valued counterparts, these
counterparts have rather unusual properties at the level of meta-inferences.
For instance,
1
ˆ
f
4
turns out to be a non-transitive relation and according to
ˆ
f
ˆ
f
4
, a premise γ may entail both α and β without entailing α β.InSect.
2.6, we observe that none of the Strong Kleene Generalizations contains an
appropriate implication connective in the sense of Arieli and Avron [2]and
we show that and how such connectives can be added to our language.
In Sects. 3 and 4, we study the Strong Kleene Generalizations syntacti-
cally. In Sect. 3 we present the SK calculus (Strong Kleene calculus), which
is a uniform sequent calculus that is sound and complete with respect to
all the Strong Kleene Generalizations. The SK calculus recognizes four dis-
tinct notions of provability: showing that an argument is
xy
z
valid comes
down to showing that an appropriate sequent is z-provable. The notions of
z-provability differ only in the initial sequent rules that are allowed to occur
in a proof and in particular the operational sequent rules that can be used
in a z-proof are the same for each value of z. Although the SK calculus is a
cut-free calculus, admissible cut rules for the calculus are readily available,
as we discus in Sect. 3.4. In Sect. 3.5, we illustrate a convenient property
of the tableau calculus that is associated with the SK calculus: to check
whether an argument is valid according to, respectively, CL, K3, LP or FDE
requires the inspection of the z-closure of a single tableau.
In Sect. 4, we consider two other uniform sequent calculi for the
xy
z
relations—the SK
P
and SK
N
calculus which are based on the notions of
P(ositive) and N (egative) validity respectively—that can be obtained by
applying the general methods of Baaz et al. [6]. Although sequents in all
three calculi are sets of signed sentences, in the SK
P
and SK
N
calculus
signs are members of {t, b, n, f } that code for the corresponding elements
of 4, whereas in the SK calculus signs are elements of {1, 0,
ˆ
1,
ˆ
0} and code
for subsets of 4:theSK calculus reflects the fact that T, B, N,andF
are best thought of as combinations of (regular) truth values and its signs
capture the “underlying” values of (non-)truth and (non-)falsity. Doing so
has advantageous consequences as compared to the other two calculi, the
sequent rules and proofs of the SK calculus are much simpler and shorter.
In Sect. 5 we reflect on the results that were achieved in earlier sections.
In Sect. 5.1 we favourably compare the SK calculus to both the SK
P
and
SK
N
calculus. In Sect. 5.2 we confront the SK calculus with the fact that
On All Strong Kleene Generalizations of Classical Logic
“more standard” 2-sided sequent calculi for some of the Strong Kleene Gen-
eralizations exist. Doesn’t this deprive the SK calculus of its interest? No.
Sect. 5.3 hints at the significance of the SK calculus for (i) (a generaliza-
tion of) the bilaterlalistic account of meaning as developed by Restall [26]
and (ii) the account of logical pluralism—called intra-theoretic pluralism—as
developed by Hjortland [18].
Section 6 concludes.
2. A Taxonomy of the Strong Kleene Generalizations
2.1. Familiar Strong Kleene Generalizations: z ∈{2, 3n, 3b}
For sake of completeness, the following proposition recalls that for z = 2,
all instantiations of (3) define classical logic.
Proposition 1.
xy
2
=
CL
for any x, y ∈{1,
ˆ
0, t,
ˆ
f }
The next proposition will show that the relations
xy
z
that are induced by
letting z ∈{3b, 3n} are either equal to K3, LP, CL or . In order to prove
it, we will need the following lemma.
Lemma 1. Let Σ be a set of sentences of
L and let V V
3n
be such that
V (σ) ∈{T, F} for all σ Σ. Then there is a V
V
2
which coincides with
V on Σ.
Proof. The partial (information) order on 3n is defined by stipulating
that N T and N F. This order on 3n induces a partial order on V
3n
,
where V V
⇐⇒ V (σ) V
(σ) for all sentences σ.LetV be as indicated
above. Let atomic valuation v be the restriction of V to the propositional
atoms, define atomic valuation v
by stipulating that v
(p)=T if v(p)=N
and that v
(p)=v(p) otherwise, and let V
be the unique element of V
2
whose restriction to the propositional atoms is v
. Observe that v(p) v
(p)
for each propositional atom p and that, as the connectives of
L define truth
functions on 3n that are monotonic with respect to , this implies that
V V
, which establishes the lemma.
Proposition 2. The following relations hold.
1a.
xy
3n
=
K3
for x,y ∈{1, t} 1b.
xy
3b
=
K3
for x,y ∈{
ˆ
0, t}
2a.
xy
3n
=
LP
for x,y ∈{
ˆ
0,
ˆ
f } 2b.
xy
3b
=
LP
for x,y ∈{1,
ˆ
f }
3a.
xy
3n
=
CL
for x, y∈{1, t}×{
ˆ
0,
ˆ
f } 3b.
xy
3b
=
CL
for x, y∈{
ˆ
0, t}×{1,
ˆ
f }
4a.
xy
3n
= for x, y∈{
ˆ
0,
ˆ
f }×{1, t} 4b.
xy
3b
= for x, y∈{1,
ˆ
f }×{
ˆ
0, t}
S. Wintein
Proof. 1a, 1b, 2a and 2b all follow immediately from an inspection of the
definitions (we suppose familiarity with the definition of the consequence
relations of K3 and LP). It is easily seen that 4a and 4b hold by observing
that the valuation which assigns N to each sentence and the valuation which
assigns B to each sentence are elements of V
3n
and V
3b
respectively. In
order to prove 3a note that, as V
2
V
3n
, it easily follows that
t
ˆ
f
3n
CL
.
For suppose that an argument is not classically valid. Then there is a V V
2
that is a counter model to the classical validity of the argument: V valuates
all premisses of the argument as T and the conclusion as F.Butthen,asV is
also an element of V
3n
, V is also a counter model to the
t
ˆ
f
3n
validity of the
argument. It thus suffices to show that
CL
t
ˆ
f
3n
. To do so, we reason again
by contraposition. If an argument is not
t
ˆ
f
3n
valid, there is a counter model
V V
3n
such that V valuates all premisses of the argument as T and the
conclusion as F. According to Lemma 1,thereisaV
V
2
which valuates
the premisses and conclusion of the argument just as V does. Hence, this V
is a counter model to the classical validity of the argument, which completes
our proof of 3a. The proof of 3b is completely similar to the proof of 3a.
Propositions 1 and 2 jointly testify that for z ∈{2, 3n, 3b} the relations
xy
z
are all well-known. In the remainder of Sect. 2 we will study the Strong
Kleene Generalizations that are associated with z = 4, and the next section
starts by considering the exact ones (cf. Definition 2)amongstthem.
2.2. The Exact Strong Kleene Generalizations for z=4
This section studies the exact Strong Kleene Generalizations. These relations
are particularly interesting as in virtue of (4),
tt
4
,
ˆ
f
ˆ
f
4
and
t
ˆ
f
4
may be called
‘the 4-equivalents of K3, LP and CL’. In particular, this section investigates
to what extent this classification is justified. In order to do so, it is convenient
to have the following definition of familiar notions.
Definition 3. Let
xy
z
be a Strong Kleene Generalization. We say that ϕ is
a tautology of
xy
z
just in case, for all V V
z
, V (ϕ) y.Wesaythatϕ is
an anti-tautology of
xy
z
just in case, for all V V
z
, V (ϕ) ∈ x. We will write
Tau(
xy
z
) to denote the set of all tautologies of
xy
z
and ATau(
xy
z
)todenote
its set of anti-tautologies.
Thus, a tautology follows from any set of premisses whereas an anti-
tautology implies any conclusion. K3 has no tautologies and its anti-
tautologies coincide with those of classical logic. For LP the situation is
completely reversed: LP has no anti-tautologies and its tautologies coincide
On All Strong Kleene Generalizations of Classical Logic
with those of classical logic. Moreover a sentence is an LP tautology just
in case its negation is an anti-tautology of K3. The following proposition
attests that these well-known interrelations between K3, LP and CL carry
over to the 4-valued case.
Proposition 3. Let z ∈{3b, 3n, 4}. The following holds.
1a. Tau(
tt
z
)= 1b. ATau(
tt
z
)=ATau(
t
ˆ
f
z
)
2a. Tau(
ˆ
f
ˆ
f
z
)=Tau(
t
ˆ
f
z
)2b. ATau(
ˆ
f
ˆ
f
z
)=
3 Tau(
ˆ
f
ˆ
f
z
) ⇐⇒ ¬ ϕ ATau(
tt
z
)
Proof. 1a and 2b follow from the observation that the valuation which
assigns N to each sentence and the valuation which assigns B to each sen-
tence are elements of V
3n
V
4
and V
3b
V
4
respectively. 1b, 2a and 3
follow immediately from the definitions.
Another well-known relation between K3 and LP is that ψ implies ϕ accord-
ing to K3 just in case the negation of ϕ implies the negation of ψ according
to LP. The following proposition attests that this relation also carries over
to the 4-valued case.
Proposition 4. Let z ∈{3b, 3n, 4}. The following holds.
¬ϕ
tt
z
¬ψ ⇐⇒ ψ
ˆ
f
ˆ
f
z
ϕ
Proof. By inspection of definitions. Left to the reader.
The following observation is an immediate corollary of Proposition 3.
Corollary 1. Let z ∈{3b, 3n, 4}. The following holds.
Tau(
tt
z
) Tau(
ˆ
f
ˆ
f
z
)=Tau(
t
ˆ
f
z
) ATau(
tt
z
) ATau(
ˆ
f
ˆ
f
z
)=ATau(
t
ˆ
f
z
)
In particular, corollary 1 implies that the union of the tautologies, respec-
tively anti-tautologies, of K3 and LP gives us the tautologies, respectively
anti-tautologies, of classical logic. This observation suggests that the union
of K3 and LP just is classical logic. This suggestion however is mistaken,
4
as with α := (p ∧¬p) q and β := (r ∨¬r) q,wehave
For z ∈{3n, 3b}: α
t
ˆ
f
z
β, α
tt
z
β, α
ˆ
f
ˆ
f
z
β (5)
Observe that (5)doesnotholdforz = 4,aswedonot have that
(p ∧¬p) q
t
ˆ
f
4
(r ∨¬r) q, (6)
4
However, the mistake is not uncommon. In a recent AJP paper, P. Allo explicitly
asserts that CL is the union of K3 and LP (cf. [1, pp. 80, 83]).
S. Wintein
as revealed by a valuation V such that V (p)=V (r)=B and V (q)=N.
However, there are
t
ˆ
f
4
-valid arguments that are neither
tt
4
-nor
ˆ
f
ˆ
f
4
-valid:
with γ := (p ∧¬p) (q ∧¬q), we have
γ
t
ˆ
f
4
tt
4
ˆ
f
ˆ
f
4
q (7)
The following proposition summarizes the previous observations and states
that classical logic is a proper extension of the union of K3 and LP and that
this relation carries over to the 4-valued case.
Proposition 5. For z ∈{3b, 3n, 4}:
tt
z
ˆ
f
ˆ
f
z
t
ˆ
f
z
Proof. Observe that any counter model V to the
t
ˆ
f
z
-validity of an argument
is both a counter model to its
tt
z
-validity as well as to its
ˆ
f
ˆ
f
z
-validity. Hence
tt
z
ˆ
f
ˆ
f
z
t
ˆ
f
z
for any z ∈{3b, 3n, 4}. That the inclusion is proper follows
for z ∈{3b, 3n} from observation (5)andforz = 4 from (7).
Remember that
tt
4
is called Exactly True Logic (ETL) by Pietz and Riv-
ieccio [23]. ETL is explosive in the sense that a contradiction, i.e. a sentence
of form ϕ ∧¬ϕ, implies any sentence whatsoever. Indeed, it is easily verified
that a contradiction is never valuated as T which is to say that ETL has
the contradictions amongst its anti-tautologies. As every ETL anti-tautology
clearly is a classical anti-tautology and as contradictions are classical anti-
tautologies, one wonders whether the anti-tautologies of ETL coincide with
those of classical logic. They do not and in particular the disjunction of two
distinct contradictions such as (p ∧¬p) (q ∧¬q) is not an anti-tautology
of ETL , as a valuation according to which V (p)=B and V (q)=N testi-
fies. It is interesting to note that these observations imply that the conse-
quence relation of ETL has certain unusual properties on the level of meta-
inferences. In particular the following meta-inference is not valid according
to ETL.
α |= γ, β |= γ = α β |= γ (8)
Indeed, taking α = p∧¬p, β = q ∧¬q and γ = r testifies that meta-inference
(8) fails for ETL
. And although (8) is valid according to
ˆ
f
ˆ
f
4
, Proposition 4
readily implies that a dual meta-inference must fail for
ˆ
f
ˆ
f
4
:
γ |= α, γ |= β = γ |= α β (9)
Taking α = p∨¬p, β = q∨¬q and γ = r shows that (9) fails for
ˆ
f
ˆ
f
4
. Moreover,
ˆ
f
ˆ
f
4
does not only have unusual properties at the level of meta-inferences but
also at the level of inferences, as for instance it does not validate (10)
p, q |= p q (10)
On All Strong Kleene Generalizations of Classical Logic
In a multiple conclusion setting,
5
we see that the failure of (10)for
ˆ
f
ˆ
f
4
is
mirrored by the failure of (11)forETL:
p q |= p, q (11)
Interestingly,
t
ˆ
f
4
satisfies both (10) and (in a multiple conclusion setting)
(11) as is easily verified. On the other hand, it violates both (8)and(9),
which is testified by the same examples that were considered above. In addi-
tion,
t
ˆ
f
4
is non-transitive. That is, it invalidates the following structural
meta-inference (a meta-inference is structural when it is expressible without
referring to logical connectives):
α |= β, β |= γ = α |= γ (12)
To see that
t
ˆ
f
4
is non-transitive, one takes α =(p ∧¬p) (q ∧¬q), β = q ∧¬q
and γ = r. Together with reflexivity and monotonicity, transitivity is often
taken to be an essential property of logical consequence. For a proponent
of this view,
t
ˆ
f
4
does not define a “genuine” consequence relation. Such a
proponent may hold that although
t
ˆ
f
4
results from a certain generalization
of classical logic, its non-transitivity testifies that it does not result from a
proper such generalization. Proper generalizations of classical logic, one may
hold, are defined in terms of the Preservation of Designated Value, i.e. they
are instantiations of the (PDV) schema for some set of designated values
D
and class of valuations V
Γ |= ϕ V (γ)
D for all γ Γ = V (ϕ) D , for all V V
(PDV)
It readily follows that any relation that is defined in terms of the preserva-
tion of designated values is transitive. However, not the other way around:
t
ˆ
f
3n
is not defined in terms of the preservation of designated value but coin-
cides with the (transitive) consequence relation of classical logic. This paper
is not the place to argue that transitivity is / is not central to logical conse-
quence. It should be noted though that recently, non-transitive consequence
relations have attracted quite some attention. Although their motivations
differ widely, Weir [33], Zardini [35], Cobreros et al. [14] and Ripley [27]all
advocate a non-transitive consequence relation.
Let us now turn to the regular Strong Kleene Generalizations associated
with z = 4.
5
In Sect. 2.5 we discuss the consequence of going multiple conclusion in some more
detail.
S. Wintein
2.3. The Regular Strong Kleene Generalizations for z=4
As the next proposition attests, the regular Strong Kleene Generalizations
are also familiar. To establish the proposition, we first have the following
lemma.
Lemma 2. Let V V
4
. Then there is a V
V
4
, called the BN-swap of
V , such that
V
(ϕ)=T V (ϕ)=T V
(ϕ)=F V (ϕ)=F
V
(ϕ)=B V (ϕ)=N V
(ϕ)=N V (ϕ)=B
Proof. Define the atomic valuation v
by letting v
(p)=B if V (p)=
N, v
(p)=N if V (p)=B and v
(p)=V (p) otherwise.
6
Let V
be the
recursive extension of v
in accordance with the truth tables of Definition
1. By induction on sentential complexity one shows that V
has the desired
properties. This can safely be left to the reader.
Proposition 6. The following relations hold.
1a.
11
4
=
FDE
1b.
ˆ
0
ˆ
0
4
=
FDE
2a.
1
ˆ
0
4
= 2b.
ˆ
01
4
=
Proof. 1a. is just the definition of FDE. 1b immediately follows from
Lemma 2. 2a and 2b follow from the observations that the valuations which
assign respectively B to each sentence and N to each sentence are elements
of V
4
.
The study of the mixed Strong Kleene Generalizations is taken up in the
next section.
2.4. The Mixed Strong Kleene Generalizations for z=4
Quite some of the mixed Strong Kleene Generalizations equal the empty set,
as shown by the following proposition.
Proposition 7. We have:
ˆ
f 1
4
=
ˆ
f
ˆ
0
4
=
1t
4
=
ˆ
0t
4
=
Proof. From the observation that the valuations which assign, respectively,
N to each sentence and B to each sentence, are elements of V
4
.
Besides contributing to our taxonomy of the mixed Strong Kleene Gen-
eralizations, the following proposition is interesting as it provides two alter-
native characterizations of ETL.
6
Using Fitting’s [17] conflation operator , this can be expressed more concisely as
v
(p)= V (p).
On All Strong Kleene Generalizations of Classical Logic
Proposition 8. We have:
ET L
=
tt
4
=
t1
4
=
t
ˆ
0
4
Proof. Per definition, we have
tt
4
t1
4
. To show the reverse inclusion,
suppose that an argument is not valid according to
tt
4
. Then there is a
counter model V according to which all premisses are valuated as T and
according to which the conclusion ϕ is valuated either as B, N or F.If
V (ϕ)=N or if V (ϕ)=F,thenV is also a counter model to the
t1
4
validity
of the argument. If V (ϕ)=B, then the BN-swap of V (cf. Lemma 2)is
a counter model to the
t1
4
validity of the argument. Hence
t1
4
tt
4
and so
t1
4
=
tt
4
. Similarly, one shows that
tt
4
=
t
ˆ
0
4
.
In light of Proposition 8 it is natural to hypothesize that the remaining
two mixed consequence relations coincide with (one another and with)
ˆ
f
ˆ
f
4
.
That hypothesis is almost correct, as the following proposition attests.
Proposition 9. The following relations hold.
1.
1
ˆ
f
4
=
ˆ
0
ˆ
f
4
2
ˆ
f
ˆ
f
4
ψ ⇐⇒ ϕ
1
ˆ
f
4
ψ 3.
ˆ
f
ˆ
f
4
1
ˆ
f
4
Proof. Both 1 and 2 follow by a by now familiar recipe: consider counter
models and BN-swaps. 3 follows per definition.
Thus, for single premise arguments,
1
ˆ
f
4
and
ˆ
0
ˆ
f
4
coincide with (one another
and with)
ˆ
f
ˆ
f
4
. This result cannot be strengthened to cover arbitrary argu-
ments, as:
p, q
1
ˆ
f
4
p qp,q
ˆ
f
ˆ
f
4
p q
Dually, Proposition 8 crucially relies on the fact that we are working in a
single conclusion setting, as p q |= p, q is valid according to (the multi-
ple conclusion version of)
t1
4
but not according to (the multiple conclusion
version of) ETL.
The next section exploits the results obtained in Sects. 2.1, 2.2, 2.3 and
2.4 to exhaustively compare the strength of all the Strong Kleene General-
izations.
2.5. Comparing the Strength of All Strong Kleene Generalizations
In this section, we compare the strength of all the Strong Kleene General-
izations. For sake of completeness, we will also compare the single premise
and multiple conclusion versions of these relations, for which we introduce
the following notation.
S. Wintein
FDE
ET L
ˆ
f
ˆ
f
4
K3
t
ˆ
f
4
LP
CL
FDE
ET L
ˆ
f
ˆ
f
4
1
ˆ
f
4
K3
t
ˆ
f
4
LP
CL
FDE
+
ET L
+
ˆ
f
ˆ
f
4+
1
ˆ
f
4+
t1
4+
K3
+
t
ˆ
f
4+
LP
+
CL
+
Figure 1. Comparing the Strong Kleene Generalizations in three set-
tings: single premise, normal and multiple conclusion
Definition 4. Let x, y ∈{1,
ˆ
0, t ,
ˆ
f } and let z ∈{2, 3n, 3b, 4}. The single
premise relation
xy
z
is the restriction of
xy
z
to singleton premise sets. The
multiple conclusion relation
xy
z+
between sets of sentences of L and sets of
sentences of
L is defined as follows:
Γ
xy
z+
Δ ⇐⇒ if V (γ) x for all γ Γ then V (δ) y
for some δ Δ for all V V
z
We will also write
ET L
for the single premise version of ETL,
FDE
+
for the
multiple conclusion version of FDE etc. Figure 1 contains Hasse diagrams of
the partial orders on the three types (single premise, normal, multiple con-
clusion) of Strong Kleene Generalizations that are induced by . For sake of
brevity, Figure 1 picks a single representative definition for each consequence
relation that is involved in the comparison. How the other definitions map
onto the ones used in Figure 1 is, for z = 4 displayed by Table 1 below.
7
Figure 1 and Table 1 jointly deliver an exhaustive comparison of all the
Strong Kleene Generalizations. Besides the results that were established in
earlier parts of Sect. 2,Figure1 also exploits the following, easily established,
proposition.
Proposition 10. Let x, y ∈{1,
ˆ
0, t,
ˆ
f } and z ∈{3b, 3n}. The following
holds.
xy
4
xy
z
xy
2
xy
4
xy
z
xy
2
xy
4+
xy
z+
xy
2+
7
For z ∈{2, 3b, 3n} this information is provided by propositions 1 and 2.
On All Strong Kleene Generalizations of Classical Logic
Table 1. The 4-valued Strong Kleene Generalizations and their name in Figure 1
Single premise Normal Multiple conclusion
Fig. 1 Equals Fig. 1 Equals Fig. 1 Equals
FDE
11
4
,
ˆ
0
ˆ
0
4
FDE
11
4
,
ˆ
0
ˆ
0
4
FDE
+
11
4+
,
ˆ
0
ˆ
0
4+
ET L
tt
4
,
t1
4
,
t
ˆ
0
4
ET L
tt
4
,
t1
4
,
t
ˆ
0
4
ET L
+
tt
4+
ˆ
f
ˆ
f
4
ˆ
f
ˆ
f
4
,
1
ˆ
f
4
,
ˆ
0
ˆ
f
4
ˆ
f
ˆ
f
4
ˆ
f
ˆ
f
4
ˆ
f
ˆ
f
4+
ˆ
f
ˆ
f
4+
t
ˆ
f
4
t
ˆ
f
4
t
ˆ
f
4
t
ˆ
f
4
t
ˆ
f
4+
t
ˆ
f
4+
1
ˆ
f
4
1
ˆ
f
4
,
ˆ
0
ˆ
f
4
1
ˆ
f
4+
1
ˆ
f
4+
,
ˆ
0
ˆ
f
4+
t1
4+
t1
4+
,
t
ˆ
0
4+
the 7 other
xy
4
the 7 other
xy
4
the 7 other
xy
4+
Proof. We show that
xy
4
xy
3b
. All other cases are similar and are left
to the reader. Suppose that it is not the case that Γ
xy
3b
ϕ. Then, there is
a valuation V V
3b
such that V (γ) x z for all γ Γ andsuchthat
V (ϕ) ∈ y z.NotethatV V
4
as V
3b
V
4
and hence it is not the case
that Γ
xy
4
ϕ.
This ends our taxonomy of the Strong Kleene Generalizations as they are
defined over the (classical propositional) language
L . In the next subsec-
tion, we briefly consider what becomes of the Strong Kleene Generalizations
when they are defined over a propositional language that has more (truth-
functional) expressive power than
L .
2.6. Implication Connectives, Expressivity and Functional Completeness
The vocabulary of our language
L is quite restricted as it does not contain
an implication connective. With respect to classical logic CL,thisisnot
a genuine restriction: the implication , defined by stipulating that ϕ
ψ := ¬ϕ ψ, is what Arieli and Avron [2]callanappropriate implication
connective for CL as it corresponds to CL entailment in the sense of (13).
Γ, ϕ
CL
ψ ⇐⇒ Γ
CL
ϕ ψ (13)
The definition of in terms of ¬ and determines its truth-functional
behaviour with respect to each V V
4
and hence, one may study the infer-
ential behaviour of with respect to all Strong Kleene Generalizations.
By doing so, one readily observes that fails to be an appropriate impli-
cation connective for any (non-empty) Strong Kleene Generalization other
than CL. For some Strong Kleene Generalizations, such as K3, only the left-
to-right direction (expressing a deduction theorem) of the definition of an
S. Wintein
appropriate implication connective fails:
Γ, ϕ
K3
ψ ⇒ Γ
K3
ϕ ψΓ,ϕ
K3
ψ Γ
K3
ϕ ψ
For other Strong Kleene Generalizations, such as LP, only the right-to-left
direction (expressing a resolution theorem) fails:
Γ, ϕ
LP
ψ Γ
LP
ϕ ψΓ,ϕ
LP
ψ ⇐ Γ
LP
ϕ ψ
And yet other Strong Kleene Generalizations such as FDE neither enjoy
a deduction nor a resolution theorem (in terms of or any
L definable
connective whatsoever).
The fact that
L does not allow the Strong Kleene Generalizations to
enjoy appropriate implication connectives provides a motivation to consider
extensions of
L . Extensions of L with appropriate implication connectives
for K3, LP and FDE haven been considered at various places in the literature
(for example in Avron [4], Batens and de Clerq [8] or in Arieli and Avron
[3]). It is in the spirit of this paper to consider appropriate implication
connectives for all the Strong Kleene Generalizations and to add them to
our language in one fell swoop. In order to do so, we will extend
L with four
implication connectives
x
,wherex ∈{1,
ˆ
0, t,
ˆ
f }, and consider language L
whose BNF form is as follows:
ϕ ::= p ϕ | ϕ ϕ | ϕ ϕ | ϕ
1
ϕ | ϕ
ˆ
0
ϕ | ϕ
t
ϕ | ϕ
ˆ
f
ϕ
The semantics of the connectives, as well as the notion of an
L
valuation
andthatofaStrong Kleene
Generalization, are given by the following
definition.
Definition 5. An
L
valuation is a function V from the sentences of L
to 4 that respects the truth tables for , and ¬ as given by Definition 1
and which is such that, for each x ∈{1,
ˆ
0, t,
ˆ
f }:
V (ϕ
x
ψ)
V (ψ)ifV (ϕ) x
T if V (ϕ) ∈ x
We will use V
4
to denote the set of all L
valuations and we will use V
2
,
V
3b
andV
3n
to denote the sets of L
valuations whose range is, respectively,
a subset of 2, 3b and 3n.Withx ∈{1,
ˆ
0, t ,
ˆ
f } and z ∈{2, 3n, 3b, 4},the
relation
xy
z
is defined as expected:
Γ
xy
z
ϕ ⇐⇒ if V (γ) x z for all γ Γ then V (ϕ) y z for all V V
z
(14)
The relations
xy
z
as defined by (14)wecallStrong Kleene
Generalizations.
On All Strong Kleene Generalizations of Classical Logic
With z ∈{2, 3b, 3n, 4} we will write
z
x
to denote the restriction of
x
to z. As is readily verified,
z
x
defines a truth function on z. The following
proposition explains how the restricted implications are related.
Proposition 11. The following relations hold.
2
1
=
2
ˆ
0
=
2
t
=
2
ˆ
f
(the material conditional of classical logic)
3b
1
=
3b
ˆ
f
,
3b
t
=
3b
ˆ
0
,
3n
1
=
3n
t
,
3n
ˆ
f
=
3n
ˆ
0
Proof. It readily follows from the definition of the implication connectives
that whenever x z = y z we have that
z
x
=
z
y
.
In Avron [4], the extension of K3 to the language L augmented with appro-
priate implication connective
3n
t
is studied. Likewise, [4], studies the exten-
sion of LP to the language
L augmented with
3b
1
and the same logic
is also studied by, for instance Batens [7] and Batens and de Clerq [8].
Finally, the extension of FDE to the language
L augmented with
4
1
is
studied in, amongst others, Arieli and Avron [3]. In our uniform framework
all these logics are available as Strong Kleene
Generalizations (restricted
to the appropriate fragment of
L
). The following proposition attest that
the introduction of the four connectives
x
ensures that in fact all Strong
Kleene
Generalizations have an appropriate implication connective.
Proposition 12. For all x, y ∈{1,
ˆ
0, t ,
ˆ
f }: Γ, ϕ
xy
z
ψ ⇐⇒ Γ
xy
z
ϕ
x
ψ
Proof. Assume that Γ, ϕ
xy
z
ψ.LetV V
z
and suppose that V (γ) x
for all γ Γ . We need to show that V (ϕ
x
ψ) x. When V (ϕ) x,
it follows that V (ϕ
x
ψ)=V (ψ) y as Γ, ϕ
xy
z
ψ. When V (ϕ) ∈ x,it
follows that V (ϕ
x
ψ)=T y as y ∈{1,
ˆ
0, t ,
ˆ
f }.
Reason by contraposition and assume that Γ,ϕ
xy
z
ψ.Thus,forsome
V V
z
,wehavethatV (γ) x for all γ Γ , V (ϕ) x and V (ψ) ∈ y.It
thus follows that V (ϕ
x
ψ)=V (ψ) ∈ y and that Γ
xy
z
ϕ
x
ψ.
Thus, it is relatively straightforward to extend our basic language L
with appropriate implication connectives, and to do so in a uniform way.
It should be noted though, that most of the results that we established in
Sect. 2 for the Strong Kleene Generalizations do not carry over to the Strong
Kleene
Generalizations. As an example, whereas we have that
1
ˆ
0
3n
=
1
ˆ
0
2
(cf.
3a of Proposition 2)wedonot have that
1
ˆ
0
3n
=
1
ˆ
0
2
as, for instance we have
1
ˆ
0
2
p
1
q ∨¬q but not
1
ˆ
0
3n
p
1
q ∨¬q. More generally, the reader may
observe that Lemma 2 is no longer valid when V
4
is replaced with V
4
and
that this lemma is used in the proof of Propositions 8 and 9. Likewise, the
S. Wintein
valuations which assign B, respectively N to every sentence are elements
of V
4
but not of V
4
and this fact ensures that (the counterparts of) some
propositions of Sect. 2 do not hold for the Strong Kleene
Generalizations. To
provide a detailed taxonomy of the Strong Kleene
Generalizations, however,
is far beyond the scope of this paper.
Although the (extended) language
L
contains appropriate implication
connectives, it lacks, for instance, a connective expressing strong negation
and so one may also consider adding such a connective, together with var-
ious others. Or, in order to ensure that all possible truth functions on 4
are expressible in one’s language, one may consider a language that is func-
tionally complete with respect to 4. In the literature, various authors have
studied the relation of FDE consequence on functional complete extensions
of
L (see e.g. Muskens [20], Arieli and Avron [3], Ruet [29], Pynko [25],
or Omori and Sano [22]). In this paper, we will not consider functionally
complete extensions of
L . The reason we do not, however, has not only to
do with restrictions of scope and length. For, as the reader will have already
observed (see also footnote 2) our uniform treatment of the Strong Kleene
(and Strong Kleene
) Generalizations relies on the fact that the restric-
tions of the (truth functions denoted by the) connectives of
L
and L
to z ∈{2, 3b, 3n} are truth functions on z. When we consider a language
that is functionally complete with respect to 4, all truth functions on 4 are
expressible and so in particular those whose restriction to z ∈{2, 3b, 3n}
is not a truth function on z. As a consequence, there are no valuations of
a functionally complete language whose range is a proper subset of 4 and
hence, to study the “Strong Kleene Generalizations” for z ∈{2, 3b, 3n} is
nonsensical in this case.
3. A Uniform Sequent Calculus for the Strong Kleene
Generalizations
3.1. Sequent Calculi for Many-Valued Logics
The main goal of Sect. 3 is to present a uniform signed sequent calculus
(which we call the SK calculus) for all the Strong Kleene Generalizations:
a calculus that can be used to characterize each and every Strong Kleene
Generalization. In Sect. 4, we then present two other such calculi (the SK
P
and SK
N
calculus) than can be obtained by applying the general methods of
Baaz et al. [6] to construct sequent calculi for many-valued logics. We think
that it is instructive to first briefly sketch the rationale of these general
On All Strong Kleene Generalizations of Classical Logic
methods and to indicate in which sense our SK calculus differs from the
calculi obtained by these general methods.
In a standard sequent calculus for classical logic, sequents are 2-sided
objects of form Γ Δ, where Γ and Δ can be taken to be sets of sentences.
Equivalently though, a 2-sided sequent Γ Δ can be presented as a set
{l : γ | γ Γ }∪{r : δ | δ Δ} consisting of signed sentences, where the
sign of a sentence indicates whether it occurs on the left side or on the right
side of . But one may also choose other signs than l and r that are more
informative in the sense that they hint at the semantic interpretation of a
provable sequent. Interestingly, there are two (equivalent but) distinct ways
to interpret provable sequents, associated with the following two translations
of a 2-sided sequent to a 2-signed one:
i Translate Γ Δ as {F : γ | γ Γ }∪{T : δ | δ Δ}. Provable sequents
correspond to
P(ostive) valid ones: when {F : γ | γ Γ }∪{T : δ | δ
Δ} is provable, in every V V
2
, V (γ)=F for some γ Γ or V (δ)=T
for some δ Δ.
ii Translate Γ Δ as {T : γ | γ Γ }∪{F : δ | δ Δ}. Provable sequents
correspond to
N (egative) valid ones: when {T : γ | γ Γ }∪{F : δ |
δ Δ} is provable, in every V V
2
, V (γ) = T for some γ Γ or
V (δ) = F for some δ Δ.
Of course, the proof systems for classical logic that result from transla-
tion i and ii are, modulo an insignificant difference in signs, exactly alike.
Interestingly though, Baaz et al [6] show how, for any n-valued logic, two
dual n-signed sequent calculi can be given that correspond to the (general-
ized) notions of
P- validity and N -validity respectively. The dual calculi
that are obtained as such may differ substantially when n>2(asthennot
being valuated as T is not the same as being valuated as F) which is vividly
illustrated by the SK
P
and SK
N
calculus that we obtain by applying those
methods in Sect. 4.
The signs exploited by the SK
P
and SK
N
calculus are members of
{t, b, n, f } and code for the corresponding elements of 4. In contrast, the
signs of the SK calculus are elements of {1, 0 ,
ˆ
1,
ˆ
0} and code for the corre-
sponding subsets of 4 as indicated by (2). The SK calculus (which is based
on a generalized notion of
N -(in)validity as will become apparent) reflects
the fact that T, B, N,andF are best thought of as combinations of (reg-
ular) truth values and its signs capture the “underlying” values of (non-)
truth and (non-) falsity. Doing so has formal and philosophical advantageous
consequences as we point out in Sect. 5.
S. Wintein
3.2. The SK Calculus
First some notational conventions. We introduce the complement operation
com on the set {1,
ˆ
1, 0,
ˆ
0} of signs of the SK calculus by stipulating that
com(1)=
ˆ
1, com(
ˆ
1)=1, com(0)=
ˆ
0 and com(
ˆ
0)=0.Further,withx
a sign and, with Γ a set of sentences, we write x : Γ as shorthand for
{x : γ | γ Γ }. Finally, it will be convenient to let Z := {2, 3b, 3n, 4}.
Sequents of the SK calculus will be sets of signed sentences of
L ,as
indicated above. Provable sequents of our calculus will correspond to unsat-
isfiable sets of signed sentences. In fact, we will distinguish four kinds of
unsatisfiability, as explained by the following definition.
Definition 6. Let Θ be a sequent let V V
4
be a valuation. We say that
V satisfies Θ iff every x : ϕ Θ is such that:
x = 1 = V (ϕ) ∈{T, B} x = 0 = V (ϕ) ∈{F, B}
x =
ˆ
1 = V (ϕ) ∈{F, N} x =
ˆ
0 = V (ϕ) ∈{T, N}
With z Z,wesaythatΘ is z-unsatisfiable just in case no V V
z
satisfies
Θ.
Observe that all regular Strong Kleene Generalizations have a natural
alternative definition in terms of z-unsatisfiability. For, if x, y ∈{1,
ˆ
0} we
have that
Γ
xy
z
ϕ ⇐⇒ x : Γ ∪{com(y):ϕ} is z-unsatisfiable (15)
Corresponding to the four notions of z-unsatisfiability, the SK calculus will
distinguish four notions of z-provability. The latter notions will differ only
with respect to the initial sequent rules (or axioms) that may be used in
a proof. To define the four sets of initial sequent rules associated with the
notions of z-provability, we consider the (initial) sequent rules of form (R
xy
),
(R
xy
)
x : ϕ, y : ϕ
in terms of which we define four sets of initial sequent rules. 4-provability
will only allow initial sequent rules that occur in R
4
:
R
4
= {(R
xy
) |x, y∈{1,
ˆ
1, 0,
ˆ
0}}
More generally, z-provability will only allow initial sequent rules that occur
in R
z
, where:
R
3b
= R
4
∪{(R
ˆ
1
ˆ
0
)} R
3n
= R
4
∪{(R
10
)} R
2
= R
3b
R
3n
On All Strong Kleene Generalizations of Classical Logic
Indeed, for all z Z,wehavethatR
z
R
2
. The structural rules of the SK
calculus will consist of the initial sequent rules that are contained in R
2
,
together with (W )eakening:
Σ
(W )
Σ
where Σ Σ
The SK calculus augments these structural rules with operational rules for
the logical connectives of
L that are provided by the following definition.
Definition 7. (The SK calculus) The structural rules of the SK calculus
are the initial sequent rules of R
2
, together with (W ). The operational rules
of the SK calculus are as follows:
Σ, y : ϕ
(¬
x
)
Σ, x : ¬ϕ
if x, y∈{1, 0,
ˆ
1,
ˆ
0, 0, 1,
ˆ
0,
ˆ
1}
Σ, x : ϕ, x : ψ
(
x
)
Σ, x : ϕ ψ
Σ, x : ϕΣ,x: ψ
(
x
)
Σ, x : ϕ ψ
if x ∈{1,
ˆ
0} if x ∈{
ˆ
1, 0}
Σ, x : ϕΣ,x: ψ
(
x
)
Σ, x : ϕ ψ
Σ, x : ϕ, x : ψ
(
x
)
Σ, x : ϕ ψ
if x ∈{1,
ˆ
0} if x ∈{
ˆ
1, 0}
With z Z, the rules of the SK
z
calculus are (W ), the initial sequent rules of
R
z
and all operational sequent rules. A sequent Θ is said to be z-provable if
some finite Θ
0
Θ has a proof tree respecting the rules of the SK
z
calculus.
It will turn out to be convenient to introduce a general form for our
sequent rules. For this, we pick T
1
,...,T
n
/B,whereeachT
i
is a set of
signed sentences, called a top set of the rule and where B is a set of signed
sentences called the bottom set of the rule. For instance one instantiation of
(R
10
) could formally be written as ∪{1 : ϕ, 0 : ϕ} and one instantiation
of (
0
)asΣ ∪{0 : ϕ}∪{0 : ψ} ∪{0 : ϕ ψ}. The following proposition
exploits the general form a sequent rule and explains how the rules of the
SK calculus can be interpreted in terms of satisfaction.
Proposition 13. The following claims hold.
1. The bottom set of each initial sequent rule in R
z
is z-unsatisfiable.
S. Wintein
2. A valuation V V
4
satisfies the bottom set of an operational rule of the
SK calculus iff it satisfies some top set of that rule.
3. A valuation V V
4
dissatisfies (i.e. does not satisfy) the bottom set of
an operational rule of the SK calculus iff it dissatisfies all top sets of
that rule.
Proof. By inspection.
Claims 1 and 3 of Proposition 13 hint at Theorem 1, which states that the
SK calculus is sound and complete with respect to z-unsatisfiability.
Theorem 1. With z Z,asequentΘ is z-provable if and only if Θ is
z-unsatisfiable.
Proof. First, it should be noted that it follows from the general results
on the compactness of propositional (finitely) many-valued logics due to
Woodruff [34] that a sequent Θ is z-unsatisfiable iff some finite Θ
0
Θ is
z-unsatisfiable. This, together with the definition of z-provability, ensures
that it suffices to establish the theorem for finite sequents.
Let z = 4. The direction follows easily by an induction on proof depth
(of 4-proofs) plus observation 3 of Proposition 13.Forthe direction, let
Θ be a finite sequent that is not 4-provable. We use induction on the total
number n of connectives occurring in Θ.Ifn =0,Θ is a set of signed
propositional constants that is not a conclusion of the 2 rules in R
4
.This
means that Θ does not contain a pair 1 : p,
ˆ
1 : p,orapair0 : p,
ˆ
0 : p.
Consider the valuation V V
4
which valuates the propositional atoms of
L as follows:
V (p)=
T 1 : p Θ, 0 : p ∈ Θ
B 1 : p Θ, 0 : p Θ
N 1 : p ∈ Θ, 0 : p ∈ Θ
F 1 : p ∈ Θ, 0 : p Θ
Observe that this definition of V , together with the fact that Θ is 4-provable
implies that if Θ contains 1 : p then V (p) ∈{T, B} ;ifΘ contains 0 : p
then V (p) ∈{F, B};ifΘ contains
ˆ
1 : p then Θ does not contain 1 : p and
so V (p) ∈{F, N};andifΘ contains
ˆ
0 : p then Θ does not contain 0 : p and
so V (p) ∈{T, N}. Indeed, we just established that V satisfies Θ.
If n>0, Θ can be written as a sequent Σ,θ, where θ is some signed
sentence containing at least one connective and θ/ Σ. Inspection of the
rules shows that in this case Θ follows from a sequent Θ
1
or from a pair of
sequents Θ
1
and Θ
2
, each containing fewer than n connectives. One of these
On All Strong Kleene Generalizations of Classical Logic
top sequents must be 4-unprovable and hence, by induction, satisfied by
some valuation V . Observation 2 of Proposition 13 gives that Θ is satisfied
by the same V . Thus, from the fact that Θ is not 4-provable, it follows that
there is a V V
4
which satisfies Θ.
For z ∈{2, 3b, 3n} the proof is entirely similar, albeit that the definition
of the valuation V V
z
that is needed to establish the induction base is
slightly different.
3.3. Capturing the Strong Kleene Generalizations Via the SK Calculus
With Theorem 1 at our disposal, we can now explain how the SK calculus
captures the Strong Kleene Generalizations. In light of observation (15),
our calculus straightforwardly captures the regular consequence relations,
as attested by the following proposition.
Proposition 14. Let z Z and let x, y ∈{1,
ˆ
0}. We have:
Γ
xy
z
ϕ ⇐⇒ x : Γ ∪{com(y):ϕ} is z-provable
Proof. Immediate from Theorem 1 and observation (15).
According to Propositions 1, 2 and 6 the (non-empty) regular Strong Kleene
Generalizations are either equal to CL, K3, LP,orFDE and so Proposition
14 attests that the SK calculus is tailor made to capture these four familiar
consequence relations.
Indeed, the very signs of the SK calculus reflect that it is tailor made
to capture the regular Strong Kleene Generalizations. As for any z = 4,
every (exact or mixed) relation
xy
z
can (in virtue of propositions 1 and 2)
be expressed as a regular consequence relation, our calculus can capture all
these relations. However, as not all of the considered 4-valued consequence
relations can be expressed as regular consequence relations, the question
arises whether our calculus can also be invoked to capture those. More con-
cretely, this question comes down (cf. Table 1) to asking whether the SK
calculus can capture
ET L
,
ˆ
f
ˆ
f
4
,
t
ˆ
f
4
and
1
ˆ
f
4
. Let us first consider
ET L
in terms
of its canonical definition
tt
4
and observe that
Γ
tt
4
ϕ ⇐⇒
1 : Γ
ˆ
0 : Γ ∪{0 : ϕ} is 4-unsatisfiable and
1 : Γ
ˆ
0 : Γ ∪{
ˆ
1 : ϕ} is 4-unsatisfiable
This observation naturally suggests to define the syntactic correlate of
ET L
in terms of two provable sequents of our calculus. However, as
ET L
can also
be defined as
t1
4
(cf. Proposition 8), the following proposition attests that a
single proof tree suffices to establish
ET L
consequence.
S. Wintein
Proposition 15. The following relations hold.
Γ
t1
4
ϕ ⇐⇒ 1 : Γ
ˆ
0 : Γ ∪{
ˆ
1 : ϕ} is 4-provable
Γ
t
ˆ
f
4
ϕ ⇐⇒ 1 : Γ
ˆ
0 : Γ ∪{
ˆ
1 : ϕ, 0 : ϕ } is 4-provable
Γ
1
ˆ
f
4
ϕ ⇐⇒ 1 : Γ ∪{
ˆ
1 : ϕ, 0 : ϕ } is 4-provable
Proof. By inspection of definitions, using Theorem 1.
The remaining relation on our list,
ˆ
f
ˆ
f
4
, can also be captured by the SK
calculus, albeit in the rather artificial manner of the following proposition.
Proposition 16. The following relation holds.
Γ
ˆ
f
ˆ
f
4
ϕ ⇐⇒ f ∈{1,
ˆ
0}
Γ
{f(γ):γ | γ Γ }∪{0 : ϕ,
ˆ
1 : ϕ} is 4-provable,
where {1,
ˆ
0}
Γ
is the set of all functions from Γ to {1,
ˆ
0}.
Proof. In light of Theorem 1, it suffices to show that Γ
ˆ
f
ˆ
f
4
ϕ iff for every
f ∈{1,
ˆ
0}
Γ
the sequent Θ
f
:= {f (γ):γ | γ Γ }∪{0 : ϕ,
ˆ
1 : ϕ} is
unsatisfiable. For the left-to-right direction, suppose that for some f,some
V V
4
satisfies Θ
f
. Then V is such that V (γ) ∈{T, B, N} for all γ Γ
while V (ϕ)=F. Hence it is not the case that Γ
ˆ
f
ˆ
f
4
ϕ. For the right-to-left
direction, suppose that there is a V V
4
such that V (γ) ∈{T, B, N} for
all γ Γ and V (ϕ)=F.LetΓ
1
:= {γ Γ | V (γ) ∈{T, B}} and let
f ∈{1,
ˆ
0}
Γ
be such that f(γ)=1 iff γ Γ
1
.ThenV satisifies Θ
f
and hence
Θ
f
is not unsatisfiable.
And so although the SK calculus captures the
ˆ
f
ˆ
f
4
relation, the way it
does so is rather cumbersome: when Γ is a finite set of premises, a proof of
Γ
ˆ
f
ˆ
f
4
ϕ consists of 2
|Γ |
distinct proof trees.
Together, Propositions 14, 15 and 16 testify that the SK calculus allows
us to capture all the Strong Kleene Generalizations. The SK calculus is a
cut-free calculus; in the next section we explain that and which cut rules
can be added to the SK calculus.
3.4. A Note on Adding Cut-Rules to the SK Calculus
Although the structural rules of the SK calculus merely consist of Weaken-
ing and the initial sequent rules of R
2
, the usual structural rules of Contrac-
tion and Permutation are implicitly built into our calculus as our sequents
are sets of signed sentences. Further, although SK is a cut-free sequent cal-
culus, several (admissible) cut rules are readily available. To introduce them
On All Strong Kleene Generalizations of Classical Logic
in a convenient manner, we introduce the following four auxiliary sets of
signed sentences.
S
ϕ
T
= {1 : ϕ,
ˆ
0 : ϕ},S
ϕ
B
= {1 : ϕ, 0 : ϕ},S
ϕ
N
= {
ˆ
1 : ϕ,
ˆ
0 : ϕ},
S
ϕ
F
= {
ˆ
1 : ϕ, 0 : ϕ}
An auxiliary set S
ϕ
x
is satisfied by a valuation V iff V (ϕ)=x.Withz Z,
therule(Cut
z
) has, as its top sets, the sets Σ S
ϕ
x
for x z and has, as its
bottom set, Σ. As an example, consider the rule (Cut
4
):
Σ,1 : ϕ,
ˆ
0 : ϕΣ,1 : ϕ, 0 : ϕΣ,
ˆ
1 : ϕ,
ˆ
0 : ϕΣ,
ˆ
1 : ϕ, 0 : ϕ
(Cut
4
)
Σ
The following proposition explain the sense in which adding a (Cut
z
)rule
to the SK calculus is admissible.
Proposition 17. AsequentΘ is z-provable if and only if it is provable
according to the rules of the SK
z
calculus with (Cut
z
) added to it.
Proof. A valuation V V
z
dissatisfies (cf. Proposition 13.3) all top sets of
(Cut
z
)justincaseV dissatisfies its bottom set. And so, as the SK
z
calculus
is complete with respect to z-unsatisfiable sets (cf. Theorem 1), the result
follows.
Although adding the (Cut
z
) rules to the SK calculus is admissible in the
sense of Proposition 17, adding those cut rules is unnatural in the following
sense. Basically, (Cut
z
) states that a sentence always takes one of the truth
values in z. This insistence on values is in conflict with the spirit of the SK
calculus, whose four signs code for sets of values rather than for values as
such. More natural cut rules are available though. To define them, we first
consider all cut rules of form (Cut
xy
),
Σ,x : ϕΣ,y: ϕ
(Cut
xy
)
Σ
in terms of which we define, for each z Z, the set of cut rules CUT
z
, where:
CUT
4
= {(Cut
1
ˆ
1
), (Cut
ˆ
00
)} CUT
3b
= CUT
4
∪{(Cut
10
)}
CUT
3n
= CUT
4
∪{(Cut
ˆ
1
ˆ
0
)} CUT
2
= CUT
3b
CUT
3n
Observe that, when (Cut
xy
) CUT
z
, a valuation V V
z
dissatisfies the two
top sets of (Cut
xy
) just in case it dissatisfies its bottom set. Intuitively then,
(Cut
xy
) CUT
z
may be interpreted as specifying that, for each valuation
V V
z
and sentence ϕ,wehavethatV (ϕ) x or V (ϕ) y. Although we
feel that it is more natural to add the rules of CUT
z
to the SK calculus
S. Wintein
rather than (Cut
z
), the rules in CUT
z
turn out to be interderivable with
(Cut
z
), as follows from the following proposition.
Proposition 18. Let z Z. In the pr