Three-valued Logics in Modal Logic

Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5.


Introduction
Translations of one logic into another logic might serve as a bridge to carry over technical results and philosophical insights.The translations of classical propositional logic into intuitionistic propositional logic that appeared in the literature around the 1930s all proved the relative consistency of classical propositional logic: if intuitionistic propositional logic is consistent, then so is classical propositional logic.The translation of intuitionistic propositional logic into the modal logic S4, based on Gödel's [9], strongly supports the provability interpretation of intuitionistic logic.Not all of these mappings of one logic into another are translations in the same sense.Some of them are only conservative mappings, others are conservative translations. 1A conservative mapping of a logic L 1 into a logic L 2 is a function that preserves valid formulas in both ways: 1 We take over Feitosa and D'Ottaviano's [8] terminology.They also provide a review of a variety of notions of translations of one logic into another logic and a discussion of the historical translations of the 1930s [8, pp. 211-217].See also D'Ottaviano and Feitosa [7].The notion of a conservative mapping is too blunt an instrument for studying translations of three-valued logics into modal logic.Let us give two examples.First, the identity function (ϕ) = ϕ conservatively maps the Logic of Paradox (LP ) [14] into classical propositional logic, because LP has exactly the same valid formulas as classical propositional logic.Second, the function (ϕ) = p ∧ ¬p (where p is some atomic formula) conservatively maps Strong Three-valued Logic (K 3 ) [10,11] into classical propositional logic, because K 3 does not have any valid formulas.For our present purposes we therefore need a notion of translation that makes sharper distinctions.

Presented by
The notion of a conservative translation gives us precisely what is required for our current investigations.A conservative translation of a logic L 1 into a logic L 2 is a function that preserves valid arguments in both ways: It is easy to see that the conservative mappings of LP and K 3 into classical propositional logic are no conservative translations.The set-up of our paper is as follows.First, we give a general definition of a truth-functional three-valued propositional logic.Second, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula.Third, we show that because for every S5-model there is a translationally equivalent three-valued valuation and vice versa, every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5.Finally, we present a linear translation that conservatively translates both LP and K 3 into S5.

Three-valued Logics
A propositional language is a set of formulas built from a set P = {p, p , . ..} of atomic formulas and a set O = {⊗ 0 , ⊗ 0 , . . ., ⊗ 1 , ⊗ 1 , . ..} of operators (the subscript indicates an operator's arity).The propositional language L O P is the smallest set (in terms of set-theoretical inclusion) satisfying the following two conditions: We refer to elements of this propositional language as propositional formulas.
P is just the familiar language of propositional logic.)In a three-valued logic propositional formulas are interpreted by way of three-valued valuations.A three-valued valuation v on P is a function v : P → {a, b, c}, where {a, b, c} is a set of three distinct truth-values.A valuation v therefore assigns to each atomic formula p in P exactly one of the three truth-values a, b, and c.A valuation v can be extended to a valuation v * that assigns to each propositional formula ϕ in L O P exactly one of the three truth-values, using truth-tables for all operators that occur in ϕ.A truth-table for an operator ⊗ n is nothing but an n-placed function f ⊗ n : {a, b, c} n → {a, b, c} that yields the truth-value of a complex formula ⊗ n (ϕ 1 , . . ., ϕ n ) on the basis of the truth-values of its constituent formulas ϕ 1 , . . ., ϕ n .Definition 2.1.Let v be a three-valued valuation on P. Then v can be extended to a three-valued valuation v * on L O P as follows: Extended valuations do not suffice to define three-valued validity.We need to stipulate which truth-values are the designated truth-values for the premises and which ones are designated for the conclusion. 2 Accordingly, the concept |= XY

A Translation Manual for Three-valued Logics
We now present a Translation Manual that translates any three-valued logic thus defined into S5.For clarity we first briefly discuss S5.The modal logic S5 interprets formulas from the modal language ML that is built from the set P = {p, p , . ..} of atomic formulas and the falsum (⊥) using negation (¬), disjunction (∨), conjunction (∧), possibility (♦), and necessity ( ).We refer to elements of this modal language as modal formulas.The semantics of S5 is as follows: Definition 3.1.An S5-model M (= W, V ) consists of a nonempty set W of possible worlds and a valuation function V that assigns to each atomic formula p in P a subset V (p) of W .Let w ∈ W , let p ∈ P, and let ϕ, ψ ∈ ML.
We write In an S5-model there are three mutually exclusive and jointly exhaustive possibilities for each atomic formula p : either p is true in all possible worlds, or p is true in some possible worlds and false in others, or p is false in all possible worlds.Our Translation Manual first maps the three possible truthvalues of any atomic formula p to these three possibilities, and then maps the three possible truth-values of any complex formula ⊗ n (ϕ 1 , . . ., ϕ n ) to truth-functional combinations of the mapped truth-values of its constituent formulas according to the strictures of ⊗ n 's truth-table: Given a formula ϕ and a set X ⊆ {a, b, c} of truth-values, we write X(ϕ) for x∈X x(ϕ).(Again, note that X(ϕ) = ⊥, if X = ∅.)Similarly, given a set Π of formulas and a set X ⊆ {a, b, c} of truth-values, we write X(Π) for {X(ψ) : ψ ∈ Π}.Using our Translation Manual, we can now state the conditions under which a three-valued valuation and an S5-model are translationally equivalent.
Definition 3.3.Let v be a three-valued valuation on P and let M be an S5-model.Then v and M are 3-equivalent, if for all ϕ in Note that if a three-valued valuation v and an S5-model M are 3-equivalent, it holds for all ϕ in L O P and for all subsets X of {a, b, c} that v * (ϕ) ∈ X if and only if M |= S5 X(ϕ).
For each S5-model there is a 3-equivalent three-valued valuation and for each three-valued valuation there is a 3-equivalent S5-model.To show this, we need a lemma about the specific type of modal formulas that are generated by our Translation Manual.In fact, all translations of propositional formulas are fully modalized, that is, for all ϕ in L O P it holds that a(ϕ), b(ϕ), and c(ϕ) are in the modal sublanguage ML , where ML is built from {♦p : p ∈ P} ∪ { p : p ∈ P} ∪ {⊥} using negation (¬), disjunction (∨), and conjunction (∧).Fully modalized formulas have a special property: they are true somewhere in an S5-model if and only if they are true everywhere in that model (a straightforward structural induction proves this): For each S5-model there is a 3-equivalent three-valued valuation.
Proof.Let M (= W, V ) be an S5-model.We construct a three-valued valuation v M by stipulating that for all atomic formulas p in It is easy to see that v M is a three-valued valuation: v M assigns to each atomic formula p exactly one of the truth-values a, b, and c.
We now show by structural induction on ϕ that v M and M are 3-equivalent.
Basis.That the claim holds for atomic formulas follows directly from the definition of v M , Definition 2.1, Lemma 3.1, and the semantics of S5.
Induction Hypothesis.Suppose that our theorem holds for all formulas ϕ with less operators than the formula ⊗ n (ϕ 1 , . . ., ϕ n ).
Take an arbitrary ϕ i and consider v * M (ϕ i ) = x i .By the Induction Hypothesis, it must be that The cases for b and c are analogous.Lemma 3.3.For each three-valued valuation there is a 3-equivalent S5model.
Proof.Let v be a three-valued valuation.We construct an S5-model M v (= W v , V v ) by stipulating that (1) W v = {w, w } and (2) for all atomic formulas p in P Obviously, M v is an S5-model.An adaption of the inductive proof of Lemma 3.2 shows that v and M v are 3-equivalent.
We now prove our theorem that every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5: In general, the length of the translations produced by our Translation Manual is exponential, because a single step in the production of, say, the a-translation of a complex formula built from an n-ary operator ⊗ n might comprise 3 n clauses: if f ⊗ n ( x 1 , . . ., x n ) = a for all x 1 , . . ., x n ∈ {a, b, c} n , then This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5.The answer is affirmative: there is an elegant linear conservative translation of the three-valued logics LP and K 3 into the modal logic S5.

A Linear Translation of LP and K 3 into S5
Translations of the Logic of Paradox (LP ) and Strong Three-Valued Logic (K 3 ) into modal logic have been offered in the literature.Batens [1, p. 284] gives essentially the same translation of LP into S5 as we do, but fails to include a proof.Busch's [3, p. 72] translation of K 3 into modal logic only applies to the fragment of the language of propositional logic that is built from (negations of) atomic formulas using disjunction and conjunction, and introduces unnecessary s in each step where a disjunction or a conjunction is treated.We present a single linear translation that conservatively translates both LP and K 3 into S5.
LP and K 3 evaluate formulas and arguments from a propositional language L O P , where P = {p, p , . ..} and O = {¬ 1 , ∨ 2 , ∧ 2 }.LP adds a third truth-value 'both' to the classical pair 'false' and 'true'.An LP -valuation is a function v LP from the set P of atomic formulas to the set {{0}, {1}, {0, 1}} of truth-values 'false', 'true', and 'both'.K 3 adds a third truth-value 'none' to the pair 'false' and 'true'.A K 3 -valuation is a function v K 3 from P to the set {∅, {0}, {1}} of truth-values 'none', 'false', and 'true'.An LP -valuation v LP (a K 3 -valuation v K 3 ) is extended to a valuation v * LP (a valuation v * K 3 ) on L O P as follows (where X stands for both LP and K 3 ): Note that in LP the set of designated truth-values for both the premises and the conclusion is {{1}, {0, 1}}, and that in K 3 the set of designated truth-values for both the premises and the conclusion is {{1}}.

Linear X-translations
Our translation transforms propositional formulas from L O P into modal formulas from ML.Each propositional formula ϕ has both an LP -translation LP (ϕ) and a K 3 -translation K 3 (ϕ).Both X and Y may stand for LP and K 3 , but if X is the one logic, then Y is the other.In the course of finding a propositional formula's X-translation, we may have to use Y -translations of its subformulas. 3Our translation is given by the following rules: Let us give two examples: LP (p ∧ ¬p) = ♦p ∧ ¬ p and K 3 (p ∧ ¬p) = p ∧ ¬♦p.An X-translation X (ϕ) hence leaves a propositional formula ϕ's negations, disjunctions, and conjunctions untouched, and only affects its atomic formulas: if an atomic formula p in ϕ is under the scope of an even number of negations, then its translation is X (p).If p in ϕ is under the scope of an odd number of negations, then its translation is Y (p).Hence, the length of an X-translation X (ϕ) equals the length of ϕ plus the number of atomic formula occurrences in ϕ.Therefore, X-translations are linear translations.
Using our X-translations, we can now state the conditions under which an X-valuation and an S5-model are translationally equivalent (again, X and Y may stand for both LP and K 3 , but if X is the one logic, then Y is the other): Definition 5.1.Let v X be an X-valuation on P and let M be an S5-model.
To show that the three-valued logics LP and K 3 can be embedded in the modal logic S5, it suffices to show that for every S5-model there is a Xequivalent X-valuation and that for every X-valuation there is a X -equivalent S5-model.The embedding then follows easily.
Lemma 5.1.For each S5-model there is a X -equivalent X-valuation.
Proof.Let M (= W, V ) be an S5-model.We construct a three-valued X-valuation v X M by stipulating that for all atomic formulas p in P It is easy to see that v X M is an X-valuation: v X M assigns to each atomic formula p exactly one of X's three truth-values.By simultaneous structural induction and Lemma 3.1 it is easy to show that v X M and M are X -equivalent.
Lemma 5.2.For each X-valuation there is a X -equivalent S5-model.
Obviously, M v X is an S5-model.An adaption of the inductive proof of Lemma 5.1 shows that v X and M v X are X -equivalent.logic and LP .Priest's [15] arguments to the contrary notwithstanding, we showed that negation in LP can be understood as a negation that "satisfies all the proof-theoretic rules of classical negation" [15, p. 203], provided we determine the meaning of LP 's atomic formulas by way of our X-translation of LP into S5.As a consequence, Priest tells only half the story, as far as LP 's "theoretical account of negation" is concerned, when he says: "Dialetheic logic, unlike modal logic, does [...] provide a genuine rival theory to that provided by classical logic" [15, p. 210].He might just as well have said that dialetheic logic provides a 'rival theory' of the interpretation of atomic formulas.

Conclusion
Our Translation Manual can easily be adapted to handle n-valued logics as long as we have a modal logic in which there are n mutually exclusive and jointly exhaustive possibilities for each atomic formula p.For example, every truth-functional four-valued propositional logic can be conservatively translated into a modal logic interpreted on minimal models M = W, N, V where N is a universal neighbourhood such that for each atomic formula p it holds that The adapted Translation Manual's handling of complex formulas and the subsequent proof that the adapted Translation Manual conservatively translates any truth-functional four-valued propositional logic into minimal modal logic closely follow the lines of our treatment of three-valued logics.

3
of three-valued validity is relative to a set X ⊆ {a, b, c} of designated truth-values for the premises and a set Y ⊆ {a, b, c} of designated truth-values for the conclusion.An argument from a set Π of premises to a conclusion ϕ is XY-valid (notation: Π |= XY 3 ϕ) if and only if for each valuation v it holds that if v * (ψ) ∈ X for all ψ ∈ Π, then v * (ϕ) ∈ Y.