Interpolation in 16-Valued Trilattice Logics

In a recent paper we have defined an analytic tableau calculus PL16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbf {\mathsf{{PL}}}}}_{\mathbf {16}}$$\end{document} for a functionally complete extension of Shramko and Wansing’s logic based on the trilattice SIXTEEN3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${SIXTEEN}_3$$\end{document}. This calculus makes it possible to define syntactic entailment relations that capture central semantic relations of the logic—such as the relations , , and that each correspond to a lattice order in SIXTEEN3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${SIXTEEN}_3$$\end{document}; and , the intersection of and . It turns out that our method of characterising these semantic relations—as intersections of auxiliary relations that can be captured with the help of a single calculus—lends itself well to proving interpolation. All entailment relations just mentioned have the interpolation property, not only when they are defined with respect to a functionally complete language, but also in a range of cases where less expressive languages are considered. For example, we will show that , when restricted to Ltf\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}_{tf}$$\end{document}, the language originally considered by Shramko and Wansing, enjoys interpolation. This answers a question that was recently posed by M. Takano.


Introduction
In Muskens and Wintein [4] we have presented an analytic tableau calculus PL 16 for a functionally complete extension of the logic considered in Shramko and Wansing [8]. Both Shramko and Wansing's original logic and our extension are based on the trilattice SIXTEEN 3 and PL 16 can capture three semantic entailment relations, |= t , |= f , and |= i , that each correspond to one of SIXTEEN 3 's three lattice orderings. 1 The calculus has a relatively simple formulation-only one rule scheme is needed for each of the three negations present in the logic, while each of the three conjunctions and each of the three disjunctions comes with two rule schemes.
In this paper we build upon [4] and study interpolation in Shramko and Wansing's trilattice logics. Using what is essentially Maehara's method we will prove a variant of his lemma for PL 16 . Interpolation theorems for |= t , |= f , |= i , and the intersection |= of |= t and |= f readily follow if these notions are interpreted as relations between sentences of the functionally complete language L tfi . We will also consider restrictions of these relations to those sublanguages of L tfi that have the property that if one of the conjunctions or disjunctions of the language is present then so is its dual. All these restrictions enjoy interpolation. In particular, |= is shown to have the (perfect) interpolation property on Shramko and Wansing's original language L tf , which answers a question by Takano [11].
The rest of the paper will be set up as follows. We will first give concise definitions of SIXTEEN 3 , of the functionally complete language L tfi and its semantics, and of the tableau system PL 16 . Once the stage is set in this way we will state and prove our interpolation results-first for logics based on L tfi and then for the restrictions. A short conclusion will end the paper.

The Trilattice SIXTEEN 3
The introduction of SIXTEEN 3 in Shramko and Wansing [8] was motivated by a wish to generalise the well-known four-valued Belnap-Dunn logic (Belnap [1,2], Dunn [3]). The latter is based on the values T = {1} (true and not false), F = {0} (false and not true), N = ∅ (neither true nor false), and B = {0, 1} (both true and false) and can be viewed as a generalisation of classical logic-a move from {0, 1} with its usual ordering to P({0, 1}) with two lattice orders. Shramko and Wansing in fact repeat this move, going from the set of truth-values P({0, 1}) = {T, F, N, B} to its power set P(P({0, 1})), now with three lattices. While the four-valued logic is meant to model the reasoning of a computer that is fed potentially incomplete or conflicting information, the 16-valued logic that results models networks of such computers (for more complete information, see the papers cited above, Wansing [12], or Shramko and Wansing [9], for example).
While the logic is thus based on P({T, F, N, B}) and can have a direct formulation on the basis of this set of truth-values, it is in fact slightly more convenient to follow Odintsov [5], who represents subsets of {T, F, N, B} with the help of matrices of the following form.

n f t b
Here each element of the matrix is a 0 or a 1 and signals the presence or the absence of an element of {T, F, N, B}. Rivieccio [7] linearises this notation, obtaining the more manageable b, f, t, n . We shall follow him in this and define 16 as {0, 1} 4 . Any A ⊆ {T, F, N, B} will be represented by a quadruple S B , S F , S T , S N ∈ 16 such that S X = 1 iff X ∈ A, for X ∈ {T, F, N, B}. With this representation in place, the three lattice orderings of the trilattice can be defined as follows (we let ≤ f be the inverse of the relation originally defined in [8], so that it becomes a nonfalsity ordering, not a falsity ordering, see also [5,6]).  Figure 1 depicts the orderings ≤ t and ≤ f on 16, while Figure 2 shows ≤ i and the intersection ≤ t ∩ ≤ f . The node names employed in these pictures belong to the object language defined in Table 1 below (with tb denoting 1, 0, 1, 0 , for example). While the definition above provides lattice orderings, the next definition gives the lattices via their meet and join operations. The official definition of SIXTEEN 3 is based upon these operations.
It is worthwhile to observe that, for each pairwise distinct x, y ∈ {t, f, i}, the following contraposition, monotonicity, and involution properties hold.

The Language L tfi and its Semantics
The language L tfi is defined by the following BNF form (where p comes from some countably infinite set of propositional constants).
This language receives an interpretation as follows.
Definition 4. A valuation function is a function V from the sentences of L tfi to 16 such that L tfi sentences ϕ and ψ are logically equivalent if V (ϕ) = V (ψ), for all V .
Muskens and Wintein [4] show that L tfi is functionally complete. Indeed, it is possible to denote each of the elements of 16 with the help of an L tfi sentence, as in the following definition.
Definition 5. Let p 0 be some fixed propositional constant. The formulas in the first column of Table 1 will be defined by the corresponding entries in the second column. For any of these abbreviations ξ and any p, we will write ξ p for the result of replacing each p 0 in ξ by p.
It is not difficult to verify that, for any valuation V , any ξ in the first column of Table 1, and any p, V (ξ p ) equals the corresponding entry in the third column.
We now come to the definition of the semantic consequence relations. As was already announced in the introduction, the relations |= t , |= f , and |= i are directly based upon ≤ t , ≤ f , and ≤ i respectively, while |= is the intersection of |= t and |= f . Definition 6. Let the relations |= t , |= f , |= i , and |= be defined as follows.
Further decomposition of these relations is in fact possible and useful. This decomposition will be in terms of the relations |= B , |= F , |= T , and |= N , defined below. We follow the convention that, for any V and ϕ, V B (ϕ) refers to the first element of V (ϕ), V F (ϕ) to its second element, V T (ϕ) to its third, and V N (ϕ) to its fourth (so that V (ϕ) = V B (ϕ), V F (ϕ), V T (ϕ), V N (ϕ) ).
Definition 7. For each x ∈ {T, B, F, N}, define the auxiliary entailment relation |= x by letting, for each two L tfi sentences ϕ and ψ, ϕ |= x ψ iff for It is not difficult to see, on the basis of these definitions and the ones in Definition 1, that the equivalences in the following proposition hold.

The Calculus PL 16 and Satisfiability
In order to capture these semantic entailment relations, Muskens and Wintein [4] define the calculus PL 16 . Entries in this calculus are signed formulas x : ϕ, where ϕ is an L tfi formula and x is one of the signs b, f, t, n, b, f, t, and n. While the role of these signed sentences in the calculus is a purely formal one, they also have an intuitive meaning. b : ϕ, for example, can be read as saying that the first (i.e. B) component of the value of ϕ is 1; that of b : ϕ is that it is 0. The other signs can be interpreted similarly.
Definition 8. The following are expansion rules of the calculus PL 16 . The general form of these rules is ϑ/B 1 , . . . , B n , where ϑ 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, using this general form one instantiation of the (∧ 1 i ) rule can be expressed as On the basis of these rules tableaux can be obtained in the usual way (see [4] for a precise definition). A tableau branch will be closed if it contains signed sentences x : ϕ and x : ϕ for x ∈ {n, f, t, b}, while a tableau is closed if all its branches are closed.
As we shall see shortly there is an intimate connection between the PL 16 rules just given and the following notion of satisfiability.
Definition 9. Let Θ be a set of signed L tfi sentences and let V be an L tfi valuation. V satisfies Θ iff the following statements hold.
A set of signed sentences will be called satisfiable if some V satisfies it, unsatisfiable otherwise.
It is shown in [4] that a finite set of sentences is unsatisfiable if and only if it has a closed tableau. In this paper we will stay entirely on the semantic side of this equation, but will make use of the following relation between the PL 16 rules and satisfiability. It follows from an easy inspection of the relevant definitions.
Hence if Θ is a set of signed L tfi sentences and ϑ/B 1 , . . . , B n is a PL 16 rule, We also note that the following connection between unsatisfiability and our auxiliary entailment relations obtains.

A Maehara Style Theorem and Interpolation in L tfi
Interpolation theorems usually come in two flavours, depending on whether the logical language that was defined is capable of naming truth-values with the help of zero-place connectives or not. Classical propositional logic, for example, has the property that whenever ϕ |= 2 ψ (with |= 2 the classical entailment relation), there is an interpolant χ such that ϕ |= 2 χ, χ |= 2 ψ, and all propositional letters occurring in χ also occur in both ϕ and ψ. If the language that was defined contains ⊥ or as zero-place connectives, that is, otherwise a condition is needed that excludes cases where ϕ and ψ have no propositional letters in common. The usual condition is that ϕ is not a contradiction and that ψ is not a tautology.
A similar condition will not always work here. Consider the relation |= t and let p and q be two (distinct) propositional letters. Then f p |= t ftb q clearly holds, f p is not a contradiction in any sense (f p |= t nf for example), ftb q is not a tautology (tb |= t ftb q ), but since there are no formulas that do not contain any propositional letters there cannot be an interpolant. One obvious way to get rid of this somewhat artificial conundrum would be to reintroduce, say, tb as a zero-place connective, but here we will stick to our earlier set-up of the language in [4] and will state conditions on interpolation where necessary. These conditions will be stated in terms of the existence of shared vocabulary.
We will prove a general Maehara-style theorem in this section, but will first prepare the ground and start with laying down conventions with respect to signs.
b} then x is the opposite of x, and x is the opposite of x. The opposite of any sign x ∈ {n, f, t, b, n, f, t, b} will be denoted by x . If S is any set of signs, then {x | x ∈ S} will be denoted as S and will also be called the opposite of S. A signed sentence x : ϕ will be called S-signed or signed in S if x ∈ S and a set of signed sentences Θ will be said to be S-signed or signed in S if each of its elements is signed in S. We will formulate our theorem not just for the functionally complete language, but also for (virtually) all sublanguages of L tfi . Languages will be identified with their basic set of connectives, as usual.
Note that the only rules in PL 16 that change the signs of signed formulas are the negation rules (∼ t ), (∼ f ), and (∼ i ). In Figure 3 we have summarised them. The eight signs of the calculus form the nodes of a labelled graph that is arranged in such a way that whenever x and y are vertices connected with an edge labeled ∼ k , any signed sentence x : ϕ can be obtained from y : ∼ k ϕ with the help of rule (∼ k )-and vice versa, the graph is undirected. We see at a glance, for example, that b : ϕ can be obtained from t : There clearly are an infinite number of paths between any two nodes x and y, but we find it expedient to define canonical short paths between them and canonical strings of negations labelling these paths.
Definition 11. We denote the empty string with . Define C to be the following set of strings of negations.
If τ ∈ C then τ is called a canonical string of negations. Consider Figure 3 and let x and y be signs. There is a unique σ ∈ C labelling a path in Figure  3 from x to y. σ is called the (canonical) x, y-string. Figure 3 there is a path labelled ∼ k ∼ from x to y (k, ∈ {t, f, i}), there is also a path labelled ∼ ∼ k from x to y. Also, if there is a path labelled ∼ k ∼ k from x to y, then x = y. It follows that if there is any string of negations from a language L ⊆ L tfi , labelling a path from x to y, there

If in
Note that the first partition corresponds to the cube in Figure 3 as a whole, the next three partitions each correspond to opposing faces of that cube, the following three to sets of edges, and the last to its set of vertices is also a canonical x, y-string of L negations. Another observation is that, for any x and y, the x, y-string is identical to the x , y -string. The following proposition is easily seen to be true. Of course, if one or more negations are not present in L ⊆ L tfi , there may be no x, y-string of L negations (and hence no path labelled with negations from L at all) between two given nodes. We introduce the notion of Lreachability.
Definition 12. Let L ⊆ L tfi , and let x and y be signs. x and y are in the L-reachability relation if the x, y-string contains only negations from L. L-reachability clearly is an equivalence relation. For each L and each sign x, let [x] L be the set {y | y is L-reachable from x}. For ease of reference, Table 2 gives an overview of the various partitions We define a general notion of interpolant. In the following, as in the rest of the paper, Voc(ϕ) will be used for the set of propositional letters occurring in ϕ and Voc(Θ) will be the set of propositional letters occurring in signed sentences in Θ.
Definition 13. Let L ⊆ L tfi , let Θ 1 and Θ 2 be sets of signed L-sentences, let z be any sign, and let p be a proposition letter. An L-sentence χ is called We now state and prove a general theorem for the calculus. The proof is in fact an adaptation of Maehara's method-most often used in the context of Gentzen sequent calculi-to the present setting.
x is a sign} and let Θ 1 be a set of S-signed sentences, while Θ 2 is a set of S -signed sentences, and Θ 1 ∪ Θ 2 is unsatisfiable. Let z ∈ S and let p be a proposition letter. Then there is a z, p-interpolant of Θ 1 and Proof. We will proceed by induction on the number of connectives occurring in signed sentences in Θ 1 ∪ Θ 2 . For the base step, assume that Θ 1 ∪ Θ 2 only contains signed propositional letters.
In general, if a set of signed sentences Ξ has only elements of the form y : q, with q a propositional variable, and, for no q and y, {y : q, y : q} ⊆ Ξ, then Ξ is easily shown to be satisfiable. By contraposition we find that {x : r, x : r} ⊆ Θ 1 ∪ Θ 2 , for some x and r.
We consider two main subcases and in each define a z, p-interpolant χ.
I. x : r ∈ Θ 1 and x : r ∈ Θ 2 , for some x and r. In this case we can let χ = σr, where σ is the x, z-string. Since x and z are both elements of S, σ only contains negation symbols from L. Note that in this case χ is in fact a z-interpolant of Θ 1 and Θ 2 in L.
for some x and r. Then L must contain at least one conjunction or disjunction and so either tb, or nf, or nt, or fb, or nftb, or ∅ is definable in L (compare Table 1). In the first case (in which ∨ t ∈ L) we can consider the following further subcases. In each of these subcases Voc(χ) ⊆ (Voc(Θ 1 )∩Voc(Θ 2 ))∪{p}, while Θ 1 ∪ {z : χ} and Θ 2 ∪ {z : χ} are unsatisfiable. The cases where conjunctions or disjunctions other than ∨ t are present in L are entirely similar and left to the reader.
For the induction step, assume that Θ 1 and Θ 2 satisfy the constraints mentioned in the theorem, while the unsatisfiable Θ 1 ∪ Θ 2 contains n + 1 connectives and the theorem holds for all Θ 1 and Θ 2 such that Θ 1 ∪ Θ 2 contains at most n connectives. Let ϑ ∈ Θ 1 ∪ Θ 2 be a signed sentence containing at least one connective. There is a unique tableau rule ρ such that ϑ is an instantiation of its top formula. We prove the induction step by cases, taking into account 1) which rule ρ matches ϑ and 2) whether ϑ ∈ Θ 1 or ϑ ∈ Θ 2 . This gives 30 cases, but they cluster in two similarity groups. Note that all rules have the property that if their top formula is an L sentence signed in S, their bottom formulas will also be signed in S.
Since a. and c. are unsatisfiable, e. and f. below are too, and from this we deduce that g. is unsatisfiable.
There are now two possibilities. The first is that z is one of the signs mentioned in the side condition of (∧ 2 t ), i.e. z ∈ {n, f, t, b}. Then z ∈ {n, f, t, b}, i.e. z is one of the signs mentioned in the side condition of (∧ 1 t ). Using (∧ 1 t ) we see that h. is unsatisfiable since g. is and using (∧ 2 t ) it follows that i. is unsatisfiable because b. and d. are. We conclude that χ 1 ∧ t χ 2 is a z, p-interpolant of Θ 1 and Θ 2 in this case.
If, on the other hand, z ∈ {n, f, t, b}, we reason as follows. Since x ∈ {n, f, t, b}, while x ∈ S and z ∈ S, it must be the case that ∼ t ∈ L. This means that ∼ t (∼ t χ 1 ∧ t ∼ t χ 2 ), a sentence equivalent to χ 1 ∨ t χ 2 (note that we have not assumed that ∨ t ∈ L), is an L sentence. Using (∨ 1 t ) we conclude from g. that Θ 1 ∪ {z : χ 1 ∨ t χ 2 } is unsatisfiable, while from b. and d. it follows with the help of (∨ 2 t ) that Θ 2 ∪ {z : χ 1 ∨ t χ 2 } is. Therefore the sets j. and k. are unsatisfiable and hence ∼ t (∼ t χ 1 ∧ t ∼ t χ 2 ) is the z, p-interpolant that was sought after.
Let us turn to the entailment relations we are interested in and to the auxiliary relations in terms of which they are characterised. We first define what it means for these relations to have the interpolation property on a sublanguage of L tfi .
R is said to have the perfect interpolation property on L if the condition that Voc(ϕ) ∩ Voc(ψ) = ∅ can be dropped, i.e. if, for any ϕ, ψ ∈ L such that ϕRψ, there is a χ ∈ L with Voc(χ) ⊆ Voc(ϕ) ∩ Voc(ψ) such that ϕRχ and χRψ.
The auxiliary relations |= T , |= F , |= N , and |= B indeed have the interpolation property on all sublanguages L of L tfi (note that for the functionally complete language itself this also follows from Takano [10]). If at least one of the negations is missing from L, they have the perfect interpolation property. Can this result be extended to the entailment relations |= t , |= f , |= i , and |= that we are after? The answer is that in many cases we can find interpolants for these entailment relations that are certain truth-functional combinations of interpolants for the auxiliary relations in terms of which they can be analysed. Before we show the general procedure, let us first make a few simple observations. The first has to do with perfect interpolation. Proof. Let R be as described. Suppose ϕ and ψ are L sentences such that ϕRψ. Then ϕ |= T ψ and Lemma 1 gives an interpolant χ such that Voc(χ) ⊆ Voc(ϕ)∩Voc(ψ). Since no sentence can have an empty vocabulary, it follows that Voc(ϕ) ∩ Voc(ψ) = ∅. So, since R has the interpolation property on L, it has the perfect interpolation property on L.
The second observation concerns the relation |=.
From this proposition the following useful lemma follows directly.
Lemma 3. If |= t or |= f has the (perfect) interpolation property on a language L, then |= likewise has the (perfect) interpolation property on L.

Interpolation Results for Sublanguages of L tfi
The language L tfi is functionally complete and hence maximally expressive given the underlying semantics. This makes it relatively easy to construct interpolants. Do less expressive languages still have the interpolation property? The question is not without interest, as it concerns languages such as L tf : Shramko and Wansing [8], and [4] we have shown to be expressively equivalent to the languages L → t tf and L → f tf considered in Odintsov [5].
We will give affirmative answers for these and a range of other languages here, but will restrict attention to those sublanguages of the functionally complete one that are closed under duals in the following sense.
So, in all languages under consideration conjunctions and disjunctions come in pairs. Let us first discuss languages that do not contain all of these pairs. For these certain dualities arise. First a definition.
Definition 16. For each sign x and each k ∈ {t, f, i}, x * k will denote the unique sign such that The reader may want to compare this definition with the side conditions of the (∼ k ) tableau expansion rules. On languages that do not have all conjunction/disjunction pairs some entailment relations are coextensive.
Proposition 7. Let L ⊆ L tfi be a language such that, for some k ∈ {t, f, i}, An immediate consequence of this duality (and Proposition 1) is that certain entailment relations collapse to equivalence and as a consequence have the interpolation property. {t, f, i}). Then ϕ |= k ψ implies V (ϕ) = V (ψ), for all valuations V and L sentences ϕ and ψ. It follows that |= k enjoys interpolation on L.
From this the following proposition about the limiting case of languages only containing negations follows immediately.
Another consequence of Propositions 1 and 7 is that in the absence of ∧ k and ∨ k (k ∈ {t, f, i}) the characterisations of entailment relations |= , where = k, can be simplified.
as before, and let ϕ and ψ be L sentences. Then the following equivalences hold.
Moreover, if two conjunction/disjunction pairs are missing, the only remaining entailment relation that does not collapse to equivalence will in fact be coextensive with |= T , as the following proposition shows.
Proof. Let ϕ and ψ be L sentences. Again use Propositions 1 and 7 in order to show that ϕ |= k ψ ⇐⇒ ϕ |= T ψ. That |= k has the interpolation property follows from Lemma 1. That |= enjoys interpolation, for ∈ {t, f, i} and k = , follows from Proposition 8.
Propositions 9 and 11 imply that |= t , |= f , and |= i enjoy interpolation on all relevant L that have at most one conjunction/disjunction pair. So, from this point on we can focus on languages closed under duality that contain at least two conjunction/disjunction pairs.
But what if negations are missing? We have already seen that interpolation results for languages lacking one or more negations can immediately be strengthened to results about perfect interpolation, but now must take into account that it is no longer a given that formulas constantly denoting elements of 16 are definable. Suppose, for example, that L is a language not containing ∼ i and ϕ is an L-sentence. Then a straightforward induction on sentence complexity gives that if V (p) = 0, 0, 0, 0 for every p ∈ Voc(ϕ), we also have that V (ϕ) = 0, 0, 0, 0 . Similarly, V (ϕ) = 1, 1, 1, 1 , if V (p) = 1, 1, 1, 1 for every p ∈ Voc(ϕ). It follows that no L-formula can have a constant denotation. Since formulas with constant denotation were used to 'glue' interpolants together in Proposition 6, we need to adapt the method.
In languages that contain only a single negation we see a property similar to the one just described. Consider, for example, a language L that only contains the ∼ i negation and let ϕ be any sentence of L. Then we see that, if V B (p) = 0 and V N (p) = 1 for every p occurring in ϕ, we also have V B (ϕ) = 0 and V N (ϕ) = 1.
Let us analyse the situation a bit further. Here are some useful definitions.
Definition 17. A form is a partial function F : {B, F, T, N} {0, 1} with a non-empty domain. If V is a valuation and ϕ is a formula then V is called an F -valuation on ϕ if, for all x ∈ dom(F ), V x (ϕ) = F (x). If P is a set of propositional letters then V is an F -valuation on P if V is an F -valuation on all p ∈ P . A form F is fixed for a formula ϕ if V is an F -valuation on ϕ whenever V is an F -valuation on Voc(ϕ), for all V . F is fixed for a language L if F is fixed for all L-sentences. Table 3 gives, for each L ⊆ L tfi , a collection of forms fixed for L, depending on the value of For example, { B, 0 , N, 1 } is a form fixed for languages containing only the ∼ i negation, while for languages that contain only ∼ t and ∼ f { B, 0 , F, 0 , T, 0 , N, 0 } is fixed. This corresponds to two of the situations just described. The proof of the following proposition is a straightforward induction on the complexity of L formulas in each case. as in the left column  of Table 3, then the corresponding forms on the right are fixed for L.  B, 0 , F, 0 , T, 0 , N, 0 },  { B, 1 , F, 1 , T, 1 , N, 1 }  {∼t, ∼i}  { B, 1 , F, 1 , T, 0 , N, 0 },  { B, 0 , F, 0 , T, 1 , N We will use certain conjunctions and disjunctions of literals for 'glueing' interpolants together. Here is a definition.
Definition 18. A literal over the propositional letter p is any formula σp, where σ is a (possibly empty) string of negations. A literal σp is in canonical form if σ ∈ C, where C is as in Definition 11. Let L ⊆ L tfi . A literal over p in canonical form that is also an L-formula is called a canonical L-literal over p. If P is a set of propositional letters, we let Lit L (P ) := {ϕ | ϕ is a canonical L-literal over some p ∈ P } .
The following proposition makes a connection between values that are not fixed by some form and literals witnessing that fact.
Proposition 13. Let L ⊆ L tfi while p is a propositional letter and x ∈ {B, F, T, N}. For each valuation V , one of the two following statements holds.
(a) For some F that is fixed for L, V is an F -valuation on p and x ∈ dom(F ).
Proof. Note that (a) holds in case L∩{∼ t , ∼ f , In all other cases, we suppose that (a) does not hold, pick the unique form F that is fixed for L such that x, V x (p) ∈ F , conclude that V is not an F -valuation on p, and construe the desired literal that witnesses (b). We give two examples.
• Consider the case that Other cases are left to the reader, but are each very similar to one of these two.
While we will not use the fact, it is worthwile to note that whenever L ∩ {∼ t , ∼ f , ∼ i } is as in the left column of Table 3 and some form F is fixed for L, F is the union of corresponding forms on the right. This can be proved in a way akin to the proof of the preceding proposition. Here is a sketch. If {∼ t , ∼ f , ∼ i } ⊆ L then no F is fixed for L (for the reason we have just seen) and the statement is trivially true. Suppose that {∼ t , ∼ f , ∼ i } ⊆ L and F is not a union of forms in the entry for L on the right of Table 3. Then there is a x, y ∈ F , such that the unique form F on the right with x, y ∈ F is not a subset of F . This means that there is a x , y ∈ F such that x , y / ∈ F . In each concrete case it is now easy to find an F -valuation V on some p and a canonical L literal λ over p such that V x (λ) = y, which shows that F is not fixed for L. Details are left to the reader. It is now easy to see that the forms fixed for a given L are exactly those unions of forms mentioned in the entry for L in Table 3 that are functions.
Proposition 13 can be used to show that, while it is impossible to define the top and bottom elements of the three lattices if not all negations are present, we can have approximations.
For each nonempty but finite set P of propositional letters, there are L-formulas τ k P and β k P , containing only propositional letters from P , such that, for each x ∈ {B, F, T, N} and each valuation V , one of the following two statements holds.

(a) There is an F that is fixed for L, x ∈ dom(F ) and V is an F -valuation on P . [In this case
Proof. Define τ k P as k Lit L (P ) and β k P as k Lit L (P ). Let V be a valuation, let x ∈ {B, F, T, N}, and suppose that (a) does not hold, so that V is not an F -valuation on P for any F fixed for L with x ∈ dom(F ). By Proposition 13 there are a p ∈ P and a canonical L-literal λ over p such that V x (p) = V x (λ). Inspection of Definition 2 reveals that (b) holds.
Let us stress that in the (b) case of the preceding proof it is not necessarily the case that V (τ k P ) = k or V (β k P ) = ⊥ k . Counterexamples are easily arrived at. The 'pointwise' formulation is really essential here, as it is in the applications of the proposition below.
So we have formulas that approximate the constantly denoting formulas that we want, modulo certain exceptions. Will the exceptions spoil our game? They will not and the following proposition gives the essential reason.
Proposition 15. Let L ⊆ L tfi and let ϕ, ψ, and χ be L formulas such that ϕ |= x ψ for some x ∈ {T, F, N, B}, while Voc(χ) ⊆ Voc(ϕ) ∩ Voc(ψ). Let F be fixed for L and let V be an F -valuation on Voc(ϕ) ∩ Voc(ψ). Then, if is shown similarly. If F (x) = 1 then V x (χ) = 1 and we are done. Assume that F (x) = 0. Define the valuation V by letting, for each y ∈ {T, F, N, B} and each p, V y (p) = F (y) if p ∈ Voc(ψ) and y ∈ dom(F ), while V y (p) = V y (p) otherwise. Then V is an F -valuation on Voc(ψ) and V x (ψ) = 0. Since ϕ |= x ψ, it follows that V x (ϕ) = 0. But V and V agree on Voc(ϕ), so V x (ϕ) = 0 and the statement holds.
We now have enough material to prove the remaining interpolation statements. Let us first consider the case that all conjunctions and disjunctions are present. We then get a generalisation of Proposition 6 whose proof is close to the latter's, but with the twist that it uses the considerations above in order to get the necessary 'glue'.
Then the entailment relations |= t , |= f , and |= i each have the interpolation property on L.
Let P be short for Voc(ϕ) ∩ Voc(ψ) and let τ t P , τ f P , β t P , and β f P be as in Proposition 14. Let b ≈ := τ t P ∧ i β f P , f ≈ := β t P ∧ i β f P , t ≈ := τ t P ∧ i τ f P , n ≈ := β t P ∧ i τ f P , and let χ be the following sentence.
With the help of Proposition 14 is easily seen that, for all valuations V , and all x ∈ {B, F, T, N}, at least one of the two following statements is true.
(a) V is an F -valuation on P , for some F fixed for L with x ∈ dom(F ).
In the (b) case, note that, in view of Definition 2, only the V x values of t ≈ , b ≈ , f ≈ , and n ≈ are relevant for the value of V x (χ), so that we have the following.
It can be concluded that, for all It follows that |= t enjoys interpolation on L. That |= f and |= i also do follows by almost identical argumentation.
The remaining case is the one in which exactly one conjunction and its dual are absent from the language. Its proof makes essential use of Proposition 10. Otherwise it is very much like the previous proof.
Proposition 17. Let L ⊆ L tfi be a language closed under duals such that Proof. Let L be as described. Consider the case that L ∩ {∧ i , ∨ i } = ∅, so that {∧ t , ∨ t , ∧ f , ∨ f } ⊆ L.
The proof for |= f is virtually identical, and also gives an interpolant of the form (χ 1 ∧ f τ t P ) ∨ f (χ 2 ∧ f β t P ). That |= i enjoys interpolation on L follows from Proposition 8.
This concludes the case that L ∩ {∧ i , ∨ i } = ∅. The two remaining cases are very similar. In case L ∩ {∧ f , ∨ f } = ∅ one arrives at interpolants of the form (χ 1 ∧ t τ i P ) ∨ t (χ 2 ∧ t β i P ) , while the case that L ∩ {∧ t , ∨ t } = ∅ leads to interpolants of the form (χ 1 ∧ i τ f P ) ∨ i (χ 2 ∧ i β f P ) . In all cases χ 1 and χ 2 are interpolants for appropriate auxiliary entailment relations. Details are left to the reader.
We sum up our results in the following theorem, which is just a combination of Propositions 9, 11, 16, 17, and Lemmas 2 and 3. The theorem affirmatively answers the question that was asked in Takano [11]-does |= enjoy perfect interpolation on L tf ? Concrete interpolants are easily extracted from our proofs. In particular, if ϕ and ψ are L tf sentences such that ϕ |= ψ, we can conclude that also ϕ |= t ψ. From the proof of Proposition 17 it follows that ϕ |= t χ |= t ψ, where χ is . Here χ 1 and χ 2 are perfect interpolants for ϕ |= T ψ and ϕ |= B ψ respectively and can be extracted from the proof of Theorem 1. τ t P is the ∨ t disjunction of all canonical L tf literals over the (nonempty) shared vocabulary P of ϕ and ψ, while β t P is a similar ∧ t conjunction. From Lemma 3 it follows that in fact ϕ |= χ |= ψ, so that we have extracted the interpolant that was sought after.

Conclusion
The analytic tableau calculus PL 16 provides several propositional logics based on the trilattice SIXTEEN 3 with a syntactic characterisation. Entailment relations of interest are typically characterisable as intersections of certain auxiliary entailment relations and/or their converses and verifying or disproving an entailment may require the development of several tableaux.
In this paper we have shown that several entailment relations of obvious interest enjoy interpolation. Our methods have been constructive-in concrete cases interpolants can be found by first finding interpolants for some of the relevant auxiliary entailment relations and by then glueing these together in certain ways. The method works for a language that can express all truth functions over PL 16 , but also for all sublanguages closed under duals. This includes the language originally considered by Shramko and Wansing [8].