Interpolation Methods for Dunn Logics and Their Extensions

The semantic valuations of classical logic, strong Kleene logic, the logic of paradox and the logic of first-degree entailment, all respect the Dunn conditions: we call them Dunn logics. In this paper, we study the interpolation properties of the Dunn logics and extensions of these logics to more expressive languages. We do so by relying on the Dunn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Dunn}$$\end{document}calculus, a signed tableau calculus whose rules mirror the Dunn conditions syntactically and which characterizes the Dunn logics in a uniform way. In terms of the Dunn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf{Dunn}$$\end{document} calculus, we first introduce two different interpolation methods, each of which uniformly shows that the Dunn logics have the interpolation property. One of the methods is closely related to Maehara’s method but the other method, which we call the pruned tableau method, is novel to this paper. We provide various reasons to prefer the pruned tableau method to the Maehara-style method. We then turn our attention to extensions of Dunn logics with so-called appropriate implication connectives. Although these logics have been considered at various places in the literature, a study of the interpolation properties of these logics is lacking. We use the pruned tableau method to uniformly show that these extended Dunn logics have the interpolation property and explain that the same result cannot be obtained via the Maehara-style method. Finally, we show how the pruned tableau method constructs interpolants for functionally complete extensions of the Dunn logics.


Introduction
The following conditions were laid down by Dunn [11] to equip the logic of first-degree entailment FDE [8,9,11] with an intuitive semantics based on the values T (true and not false), B (both truth and false), N (neither true nor false) and F (false and not true).i. ¬ϕ is true if and only if ϕ is false, ¬ϕ is false if and only if ϕ is true; ii. ϕ ∧ ψ is true if and only if ϕ is true and ψ is true, ϕ ∧ ψ is false if and only if ϕ is false or ψ is false; iii. ϕ ∨ ψ is true if and only if ϕ is true or ψ is true, ϕ ∨ ψ is false if and only if ϕ is false and ψ is false.
To illustrate how Dunn's conditions fix the semantics of ∧, ∨ and ¬ on 4 = {T, B, N, F}, suppose that ϕ has the value T, i.e. ϕ is true and not false.Then Dunn's condition i. tells us that ¬ϕ is false and not true, i.e. has the value F. Further reasoning along these lines leads to the following truth tables.

¬ T F B B N N F T
The entailment relation of FDE over the propositional language L that is based on ∧, ∨ and ¬ is then obtained by stipulating that, in passing from premisses to conclusion, truth is to be preserved over all 4-valued valuations of L's sentences that respect the above truth tables.
Observe that Dunn's conditions do not only fix the semantics of L's connectives on 4, but also on 3b = {T, B, F}, 3n = {T, N, F} and 2 = {T, F}.We may thus define, for z ∈ {2, 3b, 3n, 4}, the z-entailment relation z by stipulating that, in passing from premisses to conclusion, truth is to be preserved over all z-valued valuations of L's sentences that respect the above truth tables.Indeed, 2 comes down to classical logic CL, 3n gives us strong Kleene logic K3 [13], 3b yields the logic of paradox LP [18] and 4 gives us, as already mentioned, FDE.As Dunn's conditions thus give us a uniform semantic approach to these four familiar logics, we will refer to them as the Dunn logics.
The Dunn conditions also motivated the construction of the Dunn calculus (cf.[24]), a tableau calculus that gives us a uniform syntactic approach to the Dunn logics.The Dunn calculus is a signed tableau calculus with signs coding for truth (1), falsity (0), non-truth ( 1) and non-falsity ( 0).To explain the use of these four signs, note that the Dunn conditions may also be phrased in terms of non-truth and non-falsity.For instance, the following is equivalent to Dunn condition ii: ii .ϕ ∧ ψ is not true if and only if ϕ is not true or ψ is not true, ϕ ∧ ψ is not false if and only if ϕ is not false and ψ is not false.
The Dunn calculus translates the original Dunn conditions and their formulations in terms of non-truth and non-falsity as tableau rules.For instance, the tableau rules corresponding to Dunn conditions ii and ii are as follows: The Dunn calculus recognizes four distinct closure conditions: one for each value of z ∈ {2, 3b, 3n, 4}.These closure conditions have a straightforward rationale.For instance, {1 : ϕ, 0 : ϕ} is 2-closed and 3n-closed as no sentence can be true and false according to a 2-or 3n-valuation.On the other hand, 3b-and 4-valuations do allow for sentences that are both true and false and, accordingly, {1 : ϕ, 0 : ϕ} is neither 3b-nor 4-closed.The Dunn calculus allows us to capture the Dunn logics in a uniform manner, as we have the following result: In this paper, we will invoke the Dunn calculus to study the interpolation property of the Dunn logics.
Interpolation property If, according to a logic L, α entails β, β is not a tautology and α is not an anti-tautology, 1 we say that the L-interpolation condition for α and β is satisfied.A logic L is said to have the interpolation property if, whenever the L-interpolation condition for α and β is satisfied, there is a sentence γ, called the L-interpolant of α and β, such that every propositional variable2 that occurs in γ occurs both in α and in β and such that, according to L, α entails γ and γ entails β.
It is known that all four Dunn logics have the interpolation property.However, in the literature this information is stored as four separate facts which are proved in a highly non-uniform manner.For instance, Takeuti [23, p. 33] shows that CL has the interpolation property via what he calls Maehara's method and that proceeds via an induction on proof length in a sequent calculus for CL.On the other hand, Anderson and Belnap [1, p161] prove that FDE has the interpolation property by showing that their Hilbertstyle calculus E fde for FDE only proves so-called tautological entailments.
And both Bendova [10] and Milne [14] give a semantic proof to establish that K3 has the interpolation property. 3n the first part of this paper, we will introduce two distinct uniform interpolation methods for the Dunn logics: both methods come with a single constructive proof that establishes that all four Dunn logics have the interpolation property.Both methods construct interpolants from closed tableaux of the Dunn calculus, but they do so in rather different ways.Our first method we call the Maehara-style method.The reason for doing so is that, although there are notable differences, this method generalizes Maehara's interpolation method for CL to quite some extent.Our second method, the pruned tableau method, does not have classical (or other) ancestors, but originates in this paper.As we will see, the pruned tableau method is preferable to the Maehara-style method for, amongst others, the following two reasons: -The pruned tableau method does not, in contrast to the Maehara-style method, rely on a proof by induction but constructs the interpolant directly from a closed tableau.
-Interpolants obtained via the pruned tableau method have lower sentential complexity than those obtained with the Maehara-style method.
In a nutshell then, the pruned tableau method gives us a simpler and more direct way to obtain interpolants for the Dunn logics than the Maehara-style method does.In a recent paper, Pietz and Rivieccio [17] presented Exactly True Logic (ETL), an interesting variation upon FDE that is obtained by preserving exact truth, i.e. the value T, over all 4-valued valuations of L. Although both [17] and Wintein and Muskens [25] study ETL to quite some extent, no investigation of its interpolation property is to be found in the literature.We will first explain that the Maehara-style method cannot be invoked to construct interpolants for ETL.However, we will also show that ETL has the interpolation property by using the pruned tableau method, which is yet another reason to prefer the pruned tableau method over the Maehara-style method.
The pruned tableau method is not only interesting of itself but also has a noteworthy corollary.In a recent article, Milne [14] shows that CL's interpolation property allows for the following non-classical refinement: whenever the CL-interpolation condition for α and β is satisfied, there is a sentence γ such that (i) every propositional variable that occurs in γ occurs both in α and β (ii) α entails γ according to K3 and (iii) β entails γ according to LP.The CL-interpolant that is obtained via the pruned tableau method is readily shown to satisfy conditions (i), (ii) and (iii) and so Milne's result is an immediate corollary of the pruned tableau method.We will also show that Milne's result can be invoked to characterize CL as the transitive closure of the union of K3 and LP which, so we think, is a novel result of independent interest.The language L does not have a connective reserved for expressing implication.For CL, this is not a problem, as one may define material implication → in terms of ∨ and ¬ as usual.Material implication is an appropriate implication connective [2] for classical logic, which is to say that, according to CL, Γ ∪ {ϕ} entails ψ just in case Γ entails ϕ → ψ.However, it is well-known that no appropriate implication connective for K3, LP or FDE is definable in the language L. This motivates us to consider the language L ⊃ , which extends L with a connective ⊃ that has the following Dunn condition.iv.ϕ ⊃ ψ is true if and only if ϕ is not true or ψ is true, ϕ ⊃ ψ is false if and only if ϕ is true and ψ is false.
As this Dunn condition fixes the semantics of ⊃ on 2, 3b, 3n and 4, we may extend the Dunn logics to L ⊃ and one then readily shows that ⊃ is an appropriate implication connective for each of the extended Dunn logics thus obtained.Although quite some authors 4 have studied the Dunn logics over L ⊃ , the interpolation properties of these logics have, to the best of our knowledge, not been investigated before.We extend our Dunn calculus with tableau rules for ⊃ and then invoke the pruned tableau method to prove, in a uniform way, that the extended Dunn logics over L ⊃ all have the interpolation property.We will also explain that attempts to obtain this result via the Maehara-style method are bound to fail.Classical logic is functionally complete, which is to say that L's connectives (restricted to 2) allow us to express every truth function on 2. However, none of the other three Dunn logics, nor their extensions to L ⊃ , are functionally complete: the connectives of L ⊃ do not allow us to express all truth functions on 3b, 3n or 4. It is thus natural to consider functionally complete extensions of the Dunn logics and quite some authors5 have studied the resulting logics.Takano [22] shows that any functionally complete manyvalued logic has the interpolation property.Although it thus follows from Takano's results that the functionally complete extensions of the Dunn logics have the interpolation property, to actually construct an interpolant by the general and semantic means provided in [22] is quite cumbersome.As we will show however, the pruned tableau method constructs these interpolants in a simple and informative manner.
The paper is structured as follows.Section 2 states preliminaries.In Sections 3.1 and 3.3, we introduce our two interpolation methods and show how they can be invoked to prove, in a uniform manner, that the Dunn logics have the interpolation property.In Section 3.2, we explain in which sense the Maehara-style method resembles and in which sense it differs from Maehara's interpolation method for CL.In Section 3.4 we use the pruned tableau method to prove that ETL has the interpolation property and explain that this result cannot be obtained via the Maehara-style method.In Section 3.5 we show that Milne's result is an immediate corollary of the pruned tableau method and invoke Milne's result to characterize CL in terms of K3 and LP.In Section 4 we use the pruned tableau method to show that the Dunn logics as defined over L ⊃ have the interpolation property and explain that this result cannot be obtained via the Maehara-style method.In Section 5 we use the pruned tableau method to show that the functionally complete extensions of the Dunn logics have the interpolation property.Section 6 concludes.

Uniform Notation for Dunn Logics
Throughout the paper, 2, 3b, 3n and 4 are defined as in the introduction and we will use Z to denote the set that consist of these four subsets of 4: Z = {2, 3b, 3n, 4}.In addition, the following notation for certain subsets of 4 will be in force.
And so 1, 0, 1 and 0 code for, respectively, truth, falsity, non-truth and non-falsity.The elements of {1, 0, 1, 0} will both be used semantically, as abbreviating a subset of 4 as indicated by ( 2), but also syntactically, as signs of the Dunn calculus.It will always be clear from context which usage is at stake.We consider the propositional language L that is based on {∧, ∨, ¬} and define a z-valuation for L to be to a function from the sentences of this language to z ∈ Z that respects the truth-tables of Definition 1. Also, V z will denote the set of all z-valuations for L. The L entailment relation z preserves truth over all z-valuations: Per definition, 2 , 3n , 3b and 4 are equal to, respectively, CL, K3, LP and FDE and so we may also write:

The Dunn Tableau Calculus
A signed sentence of L is an object of form x : ϕ with sign x ∈ {1, 0, 1, 0} and with ϕ a sentence of L. Tableaux in the Dunn calculus will be certain sets of branches, which are sets of signed sentences of L. The following definition specifies what it means for a valuation to satisfy a branch.Definition 2. (Satisfaction for branches) Let B be a branch and let z ∈ Z.A valuation V ∈ V z satisfies B iff every x : ϕ ∈ B is such that: The tableau rules of the Dunn calculus are displayed below.
Definition 3. (Tableau rules of the Dunn calculus) For formal considerations it will be useful to have a general form for rules, for which we choose x : ϕ/B 1 , . . ., B n , where x : ϕ is a signed sentence called the top formula of the rule and each B i is a set of signed sentences called a set of bottom formulas of the rule.For example, one instantiation of (∧ 1 ) could formally be written as 1 : ϕ ∧ ψ/{1 : ϕ, 1 : ψ} and one instantiation of (∧ 0 ) could be expressed as 0 : ϕ ∧ ψ/{0 : ϕ}, {0 : ψ}.This general form is useful, even though neither the number of sets of bottom formulas nor their cardinality ever exceeds 2.Here is our official definition of a tableau.Definition 4. (Tableaux) Let T and T be sets of branches.We say that T is a one-step expansion of T if, for some B ∈ T , x : ϕ ∈ B, and rule Let B be a finite branch.A set of branches T is a tableau with initial branch B if there is a sequence T 0 , T 1 , . . ., T n such that T 0 = {B}, T n = T , and each T i+1 is a one-step expansion of T i (0 ≤ i < n).We also say that a finite B has tableau T if T is a tableau with initial branch B.
The Dunn calculus recognizes four closure conditions, one for each value of z ∈ Z. Definition 5. (Closure conditions) Let B be a branch.We say that: A branch that is not z-closed is called z-open.When, for some propositional atom p, B contains some z-closed subset {x : p, y : p}, we say that B is atomically z-closed.A tableau is (atomically) z-closed just in case all its branches are (atomically) z-closed; if not, the tableau is (atomically) z-open.
The following theorem attests that the Dunn calculus is sound and complete with respect to z-unsatisfiable branches.
An immediate corollary of the above theorem is that the Dunn calculus allows us to capture the Dunn logics in a uniform way.For, with 1 : Γ := {1 : γ | γ ∈ Γ}, it readily follows that: The following example illustrates the convenience of the uniform approach to the Dunn logics that is provided by the Dunn calculus.
Example 1.Let α = ((p∧¬p)∧r)∨(q ∧t) and let β = ((s∨¬s)∨t)∧(q ∨r).Consider the following tableaux for {1 : α} and { 1 : β} that may be depicted as follows: Thus, the depicted tableaux of {1 : α} and { 1 : β} have branches X 1 , X 2 and Y 1 , Y 2 respectively, where: The two tableaux that are depicted in Example 1, as well as the tableau T 3 for {1 : α, 1 : β}, are fulfilled, where a tableau T is fulfilled iff, for each one-step expansion T of T , we have T = T .It is readily established 6 that a finite branch B has a unique fulfilled tableau; we will use square brackets to denote the fulfilled tableau [B] of B. If B has a z-closed tableau, then in particular [B] is z-closed.Moreover, [B] is then not only z-closed but also atomically z-closed 7 .

Interpolation: Notation and a Useful Lemma
In this section we first define the interpolation property, and some associated notions, for an arbitrary propositional language L. Then we state a lemma pertaining to the Dunn logics that will be useful for showing that these logics have the interpolation property.
Let L be an arbitrary propositional language and let Sen(L) be its set of sentences.We write Voc(ϕ) (the vocabulary of ϕ) to denote the set of propositional variables that occur in ϕ ∈ Sen(L).With Σ a set of sentences of L, and with Θ a signed set of such sentences, Voc(Σ) = ϕ∈Σ Voc(ϕ) and Voc(Θ) = x:ϕ∈Θ Voc(ϕ).
Let |= be any relation between sets of sentences and sentences of L. A sentence ϕ of L is said to be a tautology of |= when ∅ |= ϕ and ϕ is called an anti-tautology of |= when ϕ |= ψ for any ψ ∈ Sen(L).With α and β sentences of L, we say that the |=-interpolation condition for α and β is satisfied just in case α |= β, α is not an anti-tautology of |= and β is not a tautology of |=.Also, we say that γ is a |=-interpolant for α and β iff α |= γ, γ |= β and γ is a sentence in the joint vocabulary of α and β: Voc(γ) ⊆ Voc(α) ∩ Voc(β).The interpolation property for |= is then defined as follows.
Definition 6. (The Interpolation Property) |= has the interpolation property iff whenever the |=-interpolation condition for α and β is satisfied, there is a |=-interpolant for α and β.
The following lemma, pertaining to the Dunn logics, will turn out to be useful.
Proof.Suppose, for a reductio ad absurdum, that the z -interpolation condition for α and β is satisfied whereas Voc(α) ∩ Voc(β) = ∅.As α is not an anti-tautology and as β is not a tautology, there have to be valuations be the (unique) valuation that valuates the propositional atoms of L as follows.
Hence, V testifies that α z β so that the z -interpolation condition for α and β is not satisfied.

The Maehara-Style Interpolation Method
In this section we first present our Maehara-style interpolation method which is then used to show, in one fell swoop, that all four Dunn logics have the interpolation property.
By the bifurcation of a set of signed sentences Θ, we mean the ordered pair Θ L , Θ R such that {Θ L , Θ R } is the partition of Θ in which all elements of Θ L have their sign in {1, 0} and all elements of Θ R have their sign in { 1, 0}.To show that the Dunn logics have the interpolation property via our Maehara-style method, the following lemma is crucial.
Proof.We will prove the Bifurcation lemma via induction on the minimal number k of one-step expansions that are needed to convert T 0 = {Θ} into a closed tableau of Θ.
Induction base If k = 0, Θ is z-closed.It suffices to consider the following four cases, which are easily established.Below, p is a propositional atom that is contained in Voc(Θ L ) ∩ Voc(Θ R ).
Induction step Suppose that Θ has a z-closed tableau T k+1 that results from T 0 = {Θ} in k + 1 one-step expansions.Let (• x ) be the tableau rule that is used in the one-step expansion from T 0 to T 1 .
For tableau rules other than (∧ 1 ) and (∧ 0 ), the proof is completely similar.For the actual construction of interpolants however, it will be convenient to explicitly write down the remaining cases.To do so, let us abbreviate the statement 'Θ has a z-closed tableau and γ is a sentence of L that satisfies conditions (i) and (ii) with respect to the bifurcation Θ L , Θ R of Θ' as Θ z γ.
Using this notation the results from above, together with the remaining cases, can be stated as follows: Here is our uniform Maehara-style proof which establishes that all the Dunn logics have the interpolation property.
It readily follows from Theorem 2 that FDE has the perfect interpolation property (cf.[1]) which is to say that an FDE-interpolant of α and β exists whenever α entails β according to FDE, as recorded by the following corollary.
Corollary 1.If α F DE β then there exists an FDE-interpolant of α and β.
Proof.An inspection of the 4-closure conditions and the tableau rules of the Dunn calculus reveals that neither {1 : α} nor { 1 : β} can have a 4closed tableau: FDE has no (anti-)tautologies.Hence, the result follows from Theorem 2.
The example below, which continues Example 1, illustrates how one constructs interpolants for the Dunn logics via our Maehara-style interpolation method.
Example 2. (Interpolants via the Maehara-style method) Let α, β, X 1 , X 2 , Y 1 , Y 2 and T 3 = {B 1 , B 2 , B 3 , B 4 } all be as in Example 1.For z ∈ {2, 3b, 3n} the z -interpolation condition for α and β is satisfied and we will show how the Maehara-style method constructs a z -interpolant for α and β for these three values of z.
As T 3 as defined in Example 1 is a tableau of {1 : α, 1 : β} that is zclosed for all z ∈ {2, 3b, 3n} one may also use the Maehara-style method to define z -interpolants based on this tableau.However, doing so results in interpolants with higher sentential complexity.As the reader may care to verify, by applying the Maehara-style method to T 3 we get that: γ :=((r ∧ ¬r) ∧ r) ∨ (q ∧ t) is a z -interpolant for α and β for z ∈ {2, 3n} δ :=((r ∨ ¬r) ∧ r) ∨ (q ∧ t) is a z -interpolant for α and β for z ∈ {2, 3b}

Maehara-Style Interpolation Versus Maehara's Method
In this section we sketch Maehara's interpolation method for CL.Doing so is instructive as it allows us to point out an important difference between our Maehara-style method and Maehara's method that, as we will see later on, has some interesting consequences.
Although Maehara's method (cf.[23, p 33]) is presented in terms of a sequent calculus for CL, the method is readily translated in terms of a tableau calculus.To do so, let us observe that the Dunn calculus contains a traditional signed tableau calculus for CL, which we'll call the CL calculus, as a subcalculus.Tableaux of the CL calculus are sets of positive L-branches, i.e. sets of sentences of L that are signed with 1 or 0. The tableau rules of the CL calculus are the positive rule of the Dunn calculus, i.e. the rules (• x ) with • ∈ {¬, ∧, ∨} and x ∈ {1, 0}.The closure conditions of the CL calculus are as follows: a positive branch B is closed iff {1 : ϕ, 0 : ϕ} ⊆ B for some sentence ϕ.The CL calculus is a notational variant of the calculus presented by Smullyan [21] and so it readily follows that Γ CL ϕ ⇐⇒ 1 : Γ ∪ {0 : ϕ} has a closed tableau (in the CL calculus) To apply Maehara's classical interpolation method, one extends the language L with the propositional constant symbol ⊥ to obtain the language L ⊥ and one then defines a positive L ⊥ -branch B to be closed ⊥ just in case B is closed or 1 : ⊥ ∈ B. The essential ingredient that is needed to prove that CL has the interpolation property by Maehara's method is the following lemma.Proof.If the CL-interpolation conditions are satisfied, 1 : α, 0 : β has a closed tableau so that there is a sentence γ of L ⊥ that satisfies condition (i) and (ii) of the Partition lemma.By replacing all occurrences of ⊥ in γ with p ∧ ¬p for some p ∈ Voc(α) ∩ Voc(β) (which exists in virtue of Lemma 1) one obtains a sentence of L that is a CL-interpolant for α and β.
The Partition lemma of Maehara's method plays a similar role as the Bifurcation lemma of our Maehara-style method.A notable difference between the two methods is that our Maehara-style method does not, in contrast to Maehara's method, involve a detour through the extended language L ⊥ .The reason that our Maehara-style method does not need such a detour may be accounted for by the fact that the Bifurcation lemma is formulated in terms of the unique bifurcation of a branch whereas the Partition lemma is formulated in terms of all the partitions of a branch.The latter difference is an essential one, as the following two remarks purport to illustrate.
First, observe that the Bifurcation lemma cannot be phrased in terms of arbitrary partitions.To see this, note that Θ = {1 : p, 1 : p, } has a 4closed tableau so that, when phrased in terms of arbitrary partitions, the "Bifurcation lemma" would require the existence of a sentence γ of L such that Voc(γ) ⊆ {p} and such that both { 1 : p, 1 : γ} and {1 : p, 1 : γ} have a 4-closed tableau.One readily shows that such a γ cannot exist.Hence, we may say that bifurcations are an essential ingredient of our Maehara style method.
Second, one may consider getting rid of the arbitrary partitions of the Partition lemma by phrasing that lemma in terms of separated partitions, where the separated partition Θ L , Θ R of a positive branch Θ is such that Θ L contains all elements of Θ with sign 1 and Θ R contains all elements of Θ with sign 0. Call the lemma that results from rephrasing the Partition lemma in terms of separated partitions the Separated partition lemma.The truth of the Separated partition lemma immediately follows from the truth of the Partition lemma.However, the inductive proof that is underlying the Partition lemma breaks down for the Separated partition lemma, as the reader may care to verify by trying to prove the inductive step associated with a tableau rule for negation.Without such an inductive proof, the Separated partition lemma does not tell us how to construct a classical interpolant.Hence, we may say that arbitrary partitions are an essential ingredient of Maehara's method.
In Sections 3.4 and 4, we will see some interesting consequences of the fact that the Maehara-style method essentially relies on bifurcations.

The Pruned Tableau Method
The next theorem shows how to construct interpolants for the Dunn logics via the pruned tableau method.Proof.Let z ∈ Z and suppose that the z -interpolation condition for α and β is satisfied.It suffices to show that there is a sentence γ ∈ Sen(L) with Voc(γ) ⊆ Voc(α) ∩ Voc(β) and such that (i) {1 : α, 1 : γ} has a z-closed tableau and (ii) {1 : γ, 1 : β} has a z-closed tableau.
Let [1 : α] and [ 1 : β] be the fulfilled tableaux of {1 : α} and { 1 : β} respectively and note that these tableau are z-open as α is not an antitautology and β is not a tautology of z .Also, note that for any A ∈ [1 : α] and B ∈ [ 1 : β], A ∪ B is z-closed as α z β.Let A be a z-open branch of [1 : α] and define the sets A 1 and A 0 as follows, where p is a propositional variable: As A is z-open, and as A ∪ B is z-closed for any B ∈ [ 1 : β], it follows that A 1 ∪ A 0 is non-empty and so the following sentence is well-defined: In terms of the sentences γ(A), we define the sentence γ as follows.
Then, from the definition of γ and the tableau rule (∨ 1), it follows that 1 : γ(A) is an element of every branch of [ 1 : γ] and so in particular of X. Further, from the definition of γ(A) and the tableau rule (∧ 1) it follows that there is some atomic p for which: (1 : p ∈ A and 1 : p ∈ X) or (0 : p ∈ A and 1 : ¬p ∈ X) from which it readily follows that A ∪ X is 4-closed and hence z-closed.Suppose, without loss of generality, that 1 : p ∈ A and 1 : p ∈ B. Then p is one of the conjuncts of γ(A) and hence, as 1 : γ(A) ∈ X, it follows that 1 : p ∈ X.But then X ∪ B is 4-closed and hence z-closed.
Table 1.z -interpolants via the pruned tableau and the Maehra-style method Thus, in contrast to the Maehara-style method, constructing interpolants via the pruned tableau method does not rely on a construction by induction but reads off the interpolant directly from a closed tableau.The pruned tableau method is not only simpler and more direct in this sense, but also the obtained interpolants typically have lower sentential complexity than those obtained with the Maeharae-style method, as illustrated by the following example.
The pruned tableau method delivers a unique z -interpolant for α and β whenever their z -interpolation condition is satisfied.In contrast, {1 : α, 1 : β} may have several z-closed tableaux, which ensures that the z -interpolants that are obtained via the Maehara-style method are not unique, as witnessed by (the discussion following) Example 2. With α and β as defined in Example 2 and Example 3, the below table displays the z-interpolants-for z = 2, 3n and 3n-for α and β that we obtained in those examples by the pruned tableau method and by applying the Maehara-style method to the tableaux T 3 , T 3n and T 3b as defined in Example 2.

The Pruned Tableau Method and ETL
In a recent paper, Pietz and Rivieccio introduce and study exactly true logic (ETL), whose semantic definition is as follows.
In order to capture ET L via the Dunn calculus, we define ET L as follows.
Although perhaps a bit surprising at first sight, ET L coincides with ET L , as the proof of the following proposition explains.
However, an inspection of the tableau rules readily verifies that a branch B has a 4-closed tableau iff its counterpart has a 4-closed tableau, where the counterpart of B is defined as follows: To establish that ETL has the interpolation property in terms of the Dunn calculus, we thus need to show the following: whenever the ETLinterpolation condition for α and β is satisfied, there is a sentence γ in the joint vocabulary of α and β such that both {1 : α, 0 : α, 1 : γ} and {1 : γ, 0 : γ, 1 : β} have a 4-closed tableau.Observe that, as {1 : α}, { 0 : α, 1 : β} is the bifurcation of {1 : α, 0 : α, 1 : β}, the Bifurcation lemma and hence the Maehara-style method cannot be used to establish this.Moreover, given the problems with rephrasing the Bifurcation lemma in terms of arbitrary partitions as discussed in Section 3.2, to show that ETL has the interpolation property via a Maehara-inspired method seems to be problematic in the current setting.However, the pruned tableau method can be readily used to construct ETL-interpolants, as attested by the following proposition.The pruned tableau method thus readily establishes that ETL has the interpolation property whereas the Maehara-style method cannot be used to establish this fact.We take it that this is an additional (to the reasons mentioned in Section 3.3) reason to prefer the pruned tableau method over the Maehara-style method.

Milne's Result and a Novel Characterization of CL
In a recent paper, Peter Milne [14] establishes what he calls 'a non-classical refinement of the interpolation property for classical logic'.Milne's refinement tells us that, whenever the CL -interpolation condition for α and β is satisfied, there is a sentence γ in the joint vocabulary of α and β such that: Milne presents a semantic proof of (10) but his result also follows readily from our proof of Theorem 4. This is attested by the following corollary, which shows that the CL-interpolant that is obtained via the pruned tableau method satisfies (10).
Corollary 2. Suppose that the CL -interpolation condition for α and β is satisfied.Let γ be the CL -interpolant for α and β as defined in the proof of Theorem 4. Then α K3 γ and γ LP β.
Proof.Suppose that the CL -interpolation condition for α and β is satisfied and let γ be as indicated above.Observe that in the proof of Theorem Proof.For the left-to-right direction, we distinguish 3 cases.First, if α is an anti-tautology of CL, setting χ = β establishes the claim.Second, if β is a tautology of CL, setting χ = α establishes the claim.Third, if α is not an anti-tautology of CL and β is not a tautology of CL, the claim is established by Corollary 2. The right-to-left direction follows from the fact that CL extends both K3 and LP and the transitivity of classical consequence.

Interpolation and Appropriate Implication
As was discussed in the introduction, K3, LP and FDE as defined over L do not enjoy an appropriate implication connective, which motivates an extension of L with ⊃.The Dunn conditions of ⊃, which were given in the introduction, determine the following truth table.

⊃ T B N F T T B N F B T B N F N T T T T F T T T T
We consider the propositional language L ⊃ that is based on {∧, ∨, ¬, ⊃} and define a z-valuation for L ⊃ to be to a function from the sentences of this language to z ∈ Z that respects the truth tables of Definition 1 and the above truth table of ⊃.Also, V ⊃ z will denote the set of all z-valuation for L ⊃ .We define the L ⊃ entailment relations z ⊃ which preserve truth over all z-valuations of L ⊃ : is just classical logic with ⊃ denoting material implication, the other z ⊃ logics are genuine extensions of K3, LP and FDE for which ⊃ is an appropriate implication connective, as the following proposition attests.
Proof.By inspection.
The Dunn conditions for ⊃ do no only determine its truth table, but they also give rise to the following tableau rules.
The tableau rules of the Dunn ⊃ calculus are obtained by adding the above rules for ⊃ to those of the Dunn calculus.The Dunn ⊃ calculus has the same z-closure conditions as the Dunn calculus and the following proposition will not come as a surprise.Proof.See [24].
In Section 3.4, we explained that the Maehara-style interpolation method cannot be used to show that ETL has the interpolation property.It turns out that showing that the z ⊃ -logics have the interpolation property via the Maehara-style method is also problematic.To see the problem, note that all the rules of the Dunn calculus are bifurcation-neutral, which means that: whenever the sign x the top formula x : ϕ of a rule x : ϕ/B 1 , . . ., B n is positive (i.e.x ∈ {1, 0}) so are the signs of the formulas that occur in B 1 , . . ., B n , and whenever x is negative (i.e.x ∈ { 1, 0}), so are the signs of the formulas that occur in B 1 , . . ., B n .In contrast to the rules of the Dunn calculus, (⊃ 1 ) and (⊃ 1) are not bifurcation-neutral.As a consequence, the proof of the Bifurcation lemma breaks down when it is phrased in terms of the Dunn ⊃ calculus, as the reader may care to verify.
However, by invoking the pruned tableau method, we will show, in a uniform way, that the logics z ⊃ all have the interpolation property.In order to present the proof in a neat way, we first have the following definition.
(1) χ = q ∧ ¬q.Then, every positive branch of [1 : α] is z-closed.But then, as [1 : α] has at least one positive branch it follows that z is either 2 or 3n.But if z is 2 or 3n, any branch which contains 1 : q ∧ ¬q is z-closed and so X ∪ B in particular.
(2) χ = (A 1 ∪ A 0 ) for some positive z-open branch A of [1 : α].As A is z-open and as A∪B is z-closed it follows that there is some propositional atom r such that (1 : r ∈ A and 1 : r ∈ B) or (0 : r ∈ A and 0 : r ∈ B).
If 1 : r ∈ A it follows that r is one of the conjuncts of χ, that 1 : r ∈ X and hence that X ∪ B is z-closed.If 0 : r ∈ A it follows that ¬r is one of the conjuncts of χ, that 0 : r ∈ X and hence that X ∪ B is z-closed.
If x = 0 it follows that ¬p ⊃ χ is one of the conjuncts of γ(A) and hence that 1 : ¬p ∈ X or 1 : χ ∈ X.If 1 : ¬p ∈ X then 0 : p ∈ X and hence X ∪ B is z-closed.If 1 : χ ∈ X, an argument entirely similar to that given above reveals that X ∪ B is z-closed.

Interpolation and Functional Completeness
In order to define functionally complete extensions of the Dunn logics, we consider the propositional constant symbols f, b and n, which we interpret as constant functions denoting the values F, B and N respectively.Also, we consider the unary connective − that will denote Fitting's [12] conflation operator, i.e. the following truth function on 4: Let L 4 be the language that is obtained by extending L ⊃ with f, b, n and −.Proposition 6. L 4 (with its logical vocabulary interpreted as indicated above) is functionally complete with respect to 4.
By an L 4 -valuation, we mean a function from the sentences of L 4 to 4 that respects the above indicated interpretation of ¬, ∧, ∨ ⊃, −, f, b and n.We will use V # 4 to denote the set of all L 4 -valuations and define the L 4 entailment relation 4 # as follows: Takano [22] shows that any functionally complete many-valued logic, and so in particular 4 # , has the interpolation property.However, to actually By an argument similar to the one used in the proof of Theorem 4, it follows that γ is a 4 # -interpolant for α and β.
Indeed, the proof of proposition 8 is a straightforward generalization of the proof of Theorem 4, which ensures that the pruned tableau construction of 4 # -interpolants is just as simple as the construction of interpolants for the Dunn logics over L. The proof of proposition 8 is also informative, as it shows that 4 # -interpolants can always be found L − , the sublanguage of L 4 that is based on {¬, −, ∧, ∨}.That is, we have the following corollary to proposition 8. Let us, for sake of completeness, also briefly discuss interpolation for functionally complete extensions of 3b and 3n .Avron [5] shows that the language L 3b , obtained by extending L ⊃ with f and b, is functionally complete with respect to 3b.From this result, it is not hard to show that L 3n , obtained by extending L ⊃ with f and n, is functionally complete with respect to 3n.Completely similar to the above, one defines the notion of an L 3b -and L 3n -valuation and by preserving truth over these valuations one defines the L 3b -and L 3n -entailment relations 3b

Lemma 2 .
(Bifurcation lemma) Let Θ be a finite set of signed sentences of L and let Θ L , Θ R be the bifurcation of Θ.If Θ has a z-closed tableau and if Voc

Lemma 3 .
(Partition lemma) Let Θ be a finite positive L-branch and let {Θ L , Θ R } be an arbitrary partition of Θ.If Θ has a closed tableau there is a γ ∈ Sen(L ⊥ ) such that: (i) Voc(γ) ⊆ Voc(Θ L ) ∩ Voc(Θ R ) and such that (ii) both Θ L ∪ {0 : γ} and {1 : γ} ∪ Θ R have a closed ⊥ tableau.Proof.By induction on the number of one-step expansions needed to convert {Θ} into a closed tableau of Θ.It then readily follows from the Partition lemma that CL has the interpolation property,as attested by the following theorem.Theorem 3. CL has the interpolation property.

Theorem 4 .
The Dunn logics have the interpolation property.
we are done.So suppose B is z-open.Per definition of γ and the tableau rule (∨ 1 ) it follows that there is a z-open branch A of [1 : α] such that 1 : γ(A) ∈ X.As A and B are atomically z-open and as A ∪ B is atomically z-closed, it follows that there is an atomic p such that (1 : p ∈ A and 1 : p ∈ B) or (0 : p ∈ A and 0 : p ∈ B)

Corollary 3 .
If the 4 # -interpolation condition for α and β is satisfied, then there is a γ ∈ Sen(L − ) that is a 4 # -interpolant for α and β.Proof.By inspecting the proof of Proposition 8.

#
have the interpolation property by invoking the Dunn ⊃ calculus in combination with the 3b # -and 3n # closure conditions, which are defined as follows.B is 3b # -closed ⇔ B is 3b-closed or {1 : f, 0 : f, 1 : b, 0 : b} ∩ B = ∅ B is 3n # -closed ⇔ B is 3n-closed or {1 : f, 0 : f, 1 : n, 0 : n} ∩ B = ∅One may show that, for z ∈ {3b, 3n}:Γ 3b # ϕ ⇐⇒ 1 : Γ ∪ { 1 : ϕ} has a z # -closed tableauBy invoking the pruned tableau method one readily shows that 3b # and 3n#have the interpolation property and that their interpolants can always be found in L ⊃ : 4 it is shown that for any 2-open branch A of [1 : α] and for any X ∈ [ 1 : γ], A ∪ X is 4-closed.Now a branch A of [1 : α] is 2-open if and only if A is 3n-open and so it follows that for any 3n-open branch A of [1 : α], A ∪ X is 4-closed and hence 3n-closed.This establishes that α K3 γ.Similarly, one shows that γ LP β.Although Milne's result is interesting in itself, we also feel that his has an even more interesting consequence: CL can be characterized in terms of K3 and LP, as the following propositions attests.Proposition 3. (Characterizing CL in terms of K3 and LP) α CL β ⇐⇒ there is a sentence χ such that α K3 χ and χ LP β.