Translating Non-Classical Logics Into the Classical Logic by Using Hidden Variables

Translating non-classical logics into the classical logic by using hidden variables

Juan C. Agudelo-Agudelo Institute of Mathematics, University of Antioquia, Medellin, Colombia

Abstract It is here proposed a method for constructing conservative translations of logics characterized by ‘dyadic semantics’ (a kind of non-truth-functional bivalued semantics) into the classical logic. The method particularly works for several ﬁnite many-valued logics and paraconsistent logics. The translation method uses ‘hidden variables’, which are propositional variables used to represent the indeterminism that arises when non-classical logics are provided of bivalued semantics. Then, it is shown that intuitionistic logic, for instance, is not characterizable by dyadic semantics, then the translation method here proposed do not applied for this logic. Moreover, it is provided an alternative method (not based on dyadic semantics) for constructing conservative translations of any ﬁnite many-valued logic into the classical logic. In this translation method ‘hidden variables’ are also used, but in this case to represent the degree of true or falsehood of propositions.

Introduction

Suszko’s thesis, as it is nowadays known, states that there are only two logical values, true and false, and that any many-valued logic can be provided with a bivalent semantics (cf. [12]). In [2], it is pointed out that Suszko’s original presentation of this reductive result is quite non-constructive and it is provided a method for effectively construct a kind of bivalent semantics, which the authors call gentzenian semantics, for logics that has a truth-functional finite-valued semantics and a ‘sufficiently expressive language’. Then, the dyadic semantics is defined as a specialization of gentzenian semantics where ‘quasi-tabular’ decision procedures can be stablished, “in a deliberate intent to capture the computable class of such semantics”. Moreover, it is shown that some logics that are not characterizable by truth-functional finite-valued semantics are characterizable by dyadic semantics, as it is the case of da Costa’s paraconsistet logic C1 (in [6], dyadic semantics for other paraconsistent logics are also defined). It is important to mention that dyadic semantics are generally non-truth-functional, but they are usefull, for instance, to define tableau decision procedures (cf. [3] and [4]). The non-truth-functional feature of dyadic semantics can be viewed as a kind of indeterminism. In [5], ‘hidden variables’ are introduced for defining translations of paraconsistent logics, characterized by dyadic semantics, into polynomials with coefficients in Z2, providing in this way a proof method for these logics based on polynomial reduction rules. In such translations, hidden variables

1 are (algebraic) variables included in polynomials for representing the indeterminism of dyadic semantics. Here, in Section 1, instead of translating non-classical logics into polynomials, we will take advantage of the idea of hidden variables for constructing translations of non-classical logics characterized by dyadic semantics into the Classical Propositional Logic (CPL), showing in this way that non-classical reasoning can be interpreted in classical terms, but at the cost of introducing new (hidden) propositional variables and possibly increasing the complexity of formulas. Hidden variables are, in this context, used to represent the indeterminism that arises when non-classical logics are provided of bivalued semantics. The following question naturally arises, ¿is any propositional logic characterizable by a dyadic semantics and translatable into CPL by the method here proposed? A negative answer for this question is presented in Section 2, by showing that intuitionistic logic, for instance, is not characterizable by dyadic semantics and consequently the method here proposed for constructing conservative translations into CPL does not applied for this logic. However, the scope of dyadic semantics is really wide, including many finite-valued logics (cf. [2]), logics not characterizable by a two-valued Avron’s non-deterministic matrix (as it is the case of several paraconsistent logics in the hierarchy of logics of formal inconsistency introduced in [6], that are only characterizable by non-deterministic matrices with more than two values, cf. [1]), and even logics not characterizable by any finite non-deterministic matrix (as it is the case of C1, cf. [1]). As it was mentioned above, the method introduced in [2] for constructing a dyadic semantics for a finite-valued logic L demands that the logic has a ‘sufficiently expressive language’. This condition consists in requiring that, for any pair of truth values v1 and v2, there exists a formula ϕ(p) in the language of L such that ϕ(p) takes a designated truth value when p takes value v1 and takes an undesignated truth value when p takes value v2 (or vice versa). A condition that is not too restrictive, because many (if not all) genuinely finite-valued logics satisfy this condition. Consequently, by using the method in [2] and the method described in Section 1 below, conservative translations for many finite-valued logics into CPL can be obtained. However, in Section 3, we present a direct method (not based on dyadic semantics) for constructing conservative translations from any finite-valued logic into CPL, independently of the expressive power of its language. Hidden variables are in this method used for representing the degree of truth or falsehood of atomic propositions. A brief discussion of the translation methods here proposed and some possible future works are included in Section 4.

1 From dyadic semantics to conservative translations into the classical logic

Before introducing the method for constructing conservative translations for logics characterized by dyadic semantics into CPL, some definitions are in order. For a finite set of connectives (or signature) C, an arity function A: C → N and a denumerable set of propositional variables V = {p0, p1, p2,...}, the set of formulas F orC generated by V over C is defined, as usual, as the term algebra of type C generated by V (it is, F orC is the least set containing V and closed under connectives in C). Having defined a set of formulas F orC , a logic is a structure L = hF orC , i, where ⊆ P(F orC ) × F orC is called a consequence relation. A gentzenian semantics for a logic L, as defined in [2], is a set of bivalutions for formulas of

2 L determined by a set of ‘conditional clauses’. The following deﬁnitions consider a logic L = hF orC , i.

Deﬁnition 1. A bivaluation is any function v: F orC → {T,F } (where T represents true and F represents false).

Deﬁnition 2. A conditional clause is an expression φ =⇒ ψ, where both φ and ψ are metaformulas of the form:

1 1 n1 n1 1 1 nm nm v(ϕ1) = w1, . . . , v(ϕ1 ) = w1 | ... |v(ϕm) = wm, . . . , v(ϕm ) = wm ,

j j where every ϕi is a formula scheme of F orC and every wi is a truth value in {T,F }. In a conditional clause, ‘=⇒’ represents implication, ‘,’ represents conjunction and ‘|’ represents disjunction. The metalogic that governs conditional clauses is the first order classical logic. Definition 3. Given a set of conditional clauses B, the set of bivaluations that satisfy all conditions in B is called a gentzenian semantics and is denoted by Sem(B) (i.e. Sem(B) = {v: F orC → {T,F } | v satisfies all conditions in B}). A gentzenian semantics allows to define the notion of model and a consequence relation in the usual way: Definition 4. Let B be a set of conditional clauses:

• A bivaluation v ∈ Sem(B) is a model of (or satisﬁes) a formula ϕ ∈ F orC if v(ϕ) = T . The set of models of ϕ, with respect to B, will be denoted by ModB(ϕ).

• A bivaluation v is a model of (or satiﬁes) a set of formulas Γ ⊆ F orC if it is a model of every formula in Γ. The set of models of Γ, with respect to B, will be denoted by ModB(Γ).

• The consequence relation B is deﬁned as follows: a formula ϕ is a consequence of a set of formulas Γ in B, which is denoted by Γ B ϕ, if and only if ModB(Γ) ⊆ ModB(ϕ).

• The logic L is characterized by the gentzenian semantics Sem(B) if = B. In this case, it is also said that Sem(B) is an adequate semantics for L. In shake of simplicity, the concept of dyadic semantics is here defined in a slightly different way than in [2]. Before of that, other two definitions are presented.

Deﬁnition 5. A complexity measure is a function l: F orC → N, such that l(ϕ) < l(δ) whenever ϕ is a subformula of δ.

A function l: F orC → N, deﬁned by

l(pi) = 0, for each propositional variable pi; and

l(∗(ϕ1, . . . , ϕn)) = max{l(ϕ1), . . . , l(ϕn)} + 1, if ∗ is an n-ary connective in C; is a particular case of a complexity measure and will be called the canonical complexity measure for F orC .

3 Definition 6. A conditional clause c will be called admissible, with respect to a complexity measure l, if all formulas in the antecedent of c have strictly lower complexity than the formulas in the consequent. Definition 7. A gentzenian semantics, defined by a set of conditional clauses B, will be called a dyadic semantics if:

1. B is ﬁnite. 2. The consequent of every conditional clause in B has only one term (i.e. consequent of conditional clauses have the form v(ϕ) = w). 3. Formula schemes on the consequent of conditional clauses in B cover all compound formulas of F orC (i.e. any compound formula of F orC can be obtained by uniform substitution on some formula scheme on the consequent of a conditional clause in B). 4. There exists a recursively deﬁnable complexity measure l with respect to which all conditional clauses in B are admissible.

The following examples show some logics characterized by dyadic semantics. It will be used φ ⇐⇒ v(ϕ) = T as an abbreviation of two conditional clauses c1 : φ =⇒ v(ϕ) = T and c2 : ψ =⇒ v(ϕ) = F such that ψ is equivalent (in classical logic) to the negation of φ (i.e. when c2 is equivalent to the reciprocal of c1).

Example 1. Lukasiewicz three-valued logic L3 is deﬁne over the signature CL3 = {¬, →}, where A(¬) = 1 and A(→) = 2. The following conditional clauses deﬁne an adequate dyadic semantics for L3 (cf. [11, p. 233]):

v(α) = F =⇒ v(¬α) = T ;(L3–1)

v(α) = T ⇐⇒ v(¬¬α) = T ;(L3–2)

v(α) = T, v(¬β) = T ⇐⇒ v(¬(α → β)) = T ;(L3–3)

v(¬α) = T | v(β) = T | (v(α) = F, v(¬β) = F ) ⇐⇒ v(α → β) = T. (L3–4)

Conditional clauses (L3–1)–(L3–4) are admissible with respect to the complexity measure l: F or → , deﬁned by CL3 N

l(pi) = 0, for each propositional variable pi; l(¬α) = l(α) + 1; and l(α → β) = max{l(α), l(β)} + 2.

Note that in the previous example, the truth value of ¬α is only partially determined by the truth value of α. By (L3–1), if α is false, then ¬α have to be true; however, if α is true, the truth value of ¬α can be false or true. In other words, the truth value of ¬α is not functionally determined by the truth value of α (and of any other formula in F or ). This kind of situation CL3 is that shows the indeterministic nature of dyadic semantics.

4 1 Example 2. Sette’s paraconsistent logic P is deﬁne over the signature CP 1 = {¬, →}, where A(¬) = 1 and A(→) = 2. The following conditional clauses deﬁne an adequate dyadic semantics for P 1 (cf. [2, Example 6.1]):

v(α) = F =⇒ v(¬α) = T ;(P 1–1) v(¬α) = T =⇒ v(¬¬α) = F ;(P 1–2) v(α) = F | v(β) = T =⇒ v(¬(α → β)) = F ;(P 1–3) v(α) = F | v(β) = T ⇐⇒ v(α → β) = T. (P 1–4)

Conditional clauses (P 1–1)–(P 1–4) are admissible with respect to the canonical complexity measure for F or . CP 1 Example 3. The paraconsistent logic mbC (a fundamental logic in the hierarchy of logics of formal inconsistency introduced in [6]) is deﬁne over the signature CmbC = {¬, ◦, ∨, ∧, →}, where A(¬) = A(◦) = 1 and A(∨) = A(∧) = A(→) = 2. The following conditional clauses deﬁne an adequate dyadic semantics for mbC (cf. [6, Section 3.3]):

v(α) = F =⇒ v(¬α) = T ;(mbC–1) v(α) = T, v(¬α) = T =⇒ v(◦α) = F ;(mbC–2) v(α) = T | v(β) = T ⇐⇒ v(α ∨ β) = T ;(mbC–3) v(α) = T, v(β) = T ⇐⇒ v(α ∧ β) = T ;(mbC–4) v(α) = F | v(β) = T ⇐⇒ v(α → β) = T. (mbC–5)

Conditional clauses (mbC–1)–(mbC–5) are admissible with respect to the complexity measure l: F orCmbC → N, deﬁned by

l(pi) = 0, for each propositional variable pi; l(¬α) = l(α) + 1; l(◦α) = l(α) + 2; and l(α#β) = max{l(α), l(β)} + 1, for # ∈ {∨, ∧, →}.

Example 4. Da Costa’s paraconsistent logic C1 is define over the signature CC1 = {¬, ∨, ∧, →}, where A(¬) = 1 and A(∨) = A(∧) = A(→) = 2, and an unary consistency connective ‘◦’ is defined def by ◦α = ¬(α ∧ ¬α). The following conditional clauses define an adequate dyadic semantics for C1 (it is easy to check that these conditional clauses are equivalent to those of [6, Example 65]):

v(α) = F =⇒ v(¬α) = T ;(C1–1)

v(α) = F =⇒ v(¬¬α) = F ;(C1–2)

v(α) = F | v(¬α) = F ⇐⇒ v(◦α) = T ;(C1–3)

v(◦α) = T, v(◦β) = T =⇒ v(◦(α ∨ β)) = T ;(C1–4)

v(◦α) = T, v(◦β) = T =⇒ v(◦(α ∧ β)) = T ;(C1–5)

v(◦α) = T, v(◦β) = T =⇒ v(◦(α → β)) = T ;(C1–6)

v(α) = T | v(β) = T ⇐⇒ v(α ∨ β) = T ;(C1–7)

5 v(α) = T, v(β) = T ⇐⇒ v(α ∧ β) = T ;(C1–8)

v(α) = F | v(β) = T ⇐⇒ v(α → β) = T. (C1–9)

Conditional clauses (C1–1)–(C1–9) are admissible with respect to the canonical complexity measure for F or (taking into consideration that ‘◦’ is not a primitive connective, but an abbreviation CC1 of ¬(α ∧ ¬α)). The concept of ‘conservative translation’ is introduced in [8] as follows:1

Deﬁnition 8. A conservative translation from a logic L1 = hF orC1 , 1i into a logic L2 = hF orC2 , 2i is a function T : F orC1 → F orC2 such that Γ 1 ϕ if and only if T [Γ] 2 T (ϕ), for every Γ ∪ {ϕ} ⊆ F orC1 .

Now, let’s suppose that L = hF orC , Li (where F orC is generated by the set of propositional variables V = {p0, p1, p2,...}) is a logic characterized by a dyadic semantics Sem(B), whose conditional clauses are admissible with respect to a complexity measure l. For translating L into CPL, we will consider that formulas of CPL are generated over the signature CCPL = {¬c, ∨c, ∧c, →c} (where connectives represent, respectively, classical negation, disjunction, conjunction and impli- 0 cation) and by the set of propositional variables V = V ∪ Vh, where Vh = {pα | α ∈ F orC } is a set of hidden variables which will be used to represent the indeterminism of Sem(B). The following procedure is a (partial) recipe for constructing a recursive deﬁnition of a conservative translation

TL: F orC → F orCCPL :

1. Each propositional variable of F orC must be translated into itself (i.e. TL(pi) = pi, for every propositional variable pi ∈ V ). 2. For each formula scheme δ on the right side of a conditional clause in B do the following:

(a) If φ ⇐⇒ v(δ) = T can be derived in Sem(B), for some metaformula φ with formula schemes ϕ1, . . . , ϕn of lower complexity measure than δ (i.e. if the truth value of δ is totally determined by the truth values of ϕ1, . . . , ϕn), then use the fact that the metalogic of conditional clauses is the classical logic to obtain a CPL-formula CPL(δ) that represents the truth value of δ in terms of ϕ1, . . . , ϕn. (b) If φ ⇐⇒ v(δ) = T can not be derived in Sem(B), for any metaformula φ with formula schemes of lower complexity measure than δ: i. Construct the set D of formula schemes and hidden variables that will be used to determine the truth value of δ in the following way: A. Add to D the formula schemes on the antecedent of conditional clauses where δ is on the consequent. B. If there is some formula scheme γ on the right side of a conditional clause c such that δ = γ[αi/βi], for some formula schemes αi and βi, then for each ϕ in the antecedent of c add ϕ[αi/βi] to D.

C. Add the hidden variable Pδ to D. 1For more information about the notion of translation between logics see [7].

6 ii. Construct a truth table including a column for each ϕ ∈ D. Use the rows of the table to write all the possible assignations of truth values (in {T,F }) to the elements of D (taking into account that the truth value of some elements of D may be restricted by the truth values of other elements of D) and set the respective truth values of δ in accordance with the conditional clauses in B. In the rows where the truth value of δ is not uniquely determine by the truth values of D \{Pδ}, set to δ the truth value of pδ. In this way, it is obtained a bivalued truth function for δ, in terms of the truth values of the elements of D, validating the conditions in B and representing the indeterminism of the truth value of δ by including the new propositional variable pδ. iii. Construct a CPL-formula CPL(δ) that represents the bivalued truth function obtained for δ. This is always possible because CPL is functionally complete.

(c) Define the translation of δ by TL(δ) = CPL(δ)[ϕi/TL(ϕi)], where CPL(δ)[ϕi/TL(ϕi)] denotes the expression obtained by uniformly substituting TL(ϕi) for ϕi in CPL(δ). When a formula satisfy more than one scheme, it must be translated using the rule defined for the more restrictive scheme.2 Note that the procedure above basically consists in internalizing the (classical) metalogic of conditional clauses into CPL. Moreover, the set of hidden variables Vh, as it was defined, is just a denumerable set of propositional variables. Then, V 0 is no more than a denumerable set of propositional variables conveniently partitioned into two sets in order to allow the construction of

TL: F orC → F orCCPL . The following examples put in practice the procedure for constructing conservative translations 1 from L3,P , mbC and C1 into CPL. Example 8 shows that, in some cases, a particular presentation of a dyadic semantics does not allow a direct application of the recipe and some (meta)logical manipulations have to be made in order to obtain an equivalent dyadic semantics whose conditional clauses make possible the application of the procedure.

Example 5. For L3, considering the dyadic semantics presented in Example 1, for the formula scheme ¬α, the following truth table is obtained:

α p¬α ¬α F F T F T T T F F T T T and a CPL-formula that represents the bivalued function for ¬α is:

CPL(¬α) = ¬cα ∨c p¬α.

The CPL-formulas for the other formula schemes on the right side of conditions (L3–2)–(L3–4) are: CPL(¬¬α) = α,

2Given two formula schemes γ and δ, it will be said that γ is more general than δ (or that δ is more restrictive than γ) if all formulas satisfying scheme δ also satisfy scheme γ; i.e. if δ = γ[αi/βi], for some formula schemes αi and βi, where γ[αi/βi] denotes the expression obtained by uniformly substituting βi for αi in γ. For instance, ¬α is a more general scheme than ¬¬α (or ¬¬α is a more restrictive scheme than ¬α).

7 CPL(¬(α → β)) = α ∧c ¬β,

CPL(α → β) = ¬α ∨c β ∨c (¬cα ∧c ¬c¬β).

Then, the translation function T : F or → F or is deﬁned by: L3 CL3 CCPL

TL3 (pi) = pi;  ¬ p ∨ p , if ϕ = p ;  c i c ¬pi i TL3 (¬ϕ) = TL3 (α), if ϕ = ¬α;  TL3 (α) ∧c TL3 (¬β), if ϕ = α → β;

TL3 (α → β) = TL3 (¬α) ∨c TL3 (β) ∨c (¬cTL3 (α) ∧c ¬cTL3 (¬β)).

Example 6. For P 1, considering the dyadic semantics presented in Example 2, the CPL-formula for the formula scheme ¬α is the same than in the previous example. For the formula scheme ¬¬α, although its truth value seems to be only partially determined by the truth value of ¬α, from (P 1–1) and (P 1–2) it is obtained that:

v(¬α) = F ⇐⇒ v(¬¬α) = T.

Then: CPL(¬¬α) = ¬c¬α. A similar situation occurs with the formula scheme ¬(α → β). From (P 1–3), (P 1–1) and (P 1–4) it is obtained that: v(α) = T, v(β) = F ⇐⇒ v(¬(α → β)) = T. Then: CPL(¬(α → β)) = α ∧c ¬cβ; or, equivalently: CPL(¬(α → β)) = ¬c(α →c β). For the formula scheme α → β, by (P 1–4) we have that:

CPL(α → β) = α →c β.

Then, the translation function T 1 : F or → F or is deﬁned by: P CP 1 CCPL

TP 1 (pi) = pi;  ¬ p ∨ p , if ϕ = p ;  c i c ¬pi i TP 1 (¬ϕ) = ¬cTP 1 (¬α), if ϕ = ¬α;  ¬c(TP 1 (α) →c TP 1 (β)), if ϕ = α → β;

TP 1 (α → β) = TP 1 (α) →c TP 1 (β).

Example 7. For mbC, considering the dyadic semantics presented in Example 3, the CPL-formula for the formula scheme ¬α is the same than in Example 5. For the formula scheme ◦α, the following truth table is obtained:

8 α ¬α p◦α ◦α F T F F F T T T T F F F T F T T T T F F T T T F and a CPL-formula that represents the bivalued function for ◦α is:

CPL(◦α) = ¬c(α ∧c ¬α) ∧c p◦α.

The CPL-formulas for the other formula schemes on the right side of conditions (mbC–3)–(mbC–5) are:

CPL(α ∨ β) = α ∨c β,

CPL(α ∧ β) = α ∧c β,

CPL(α → β) = α →c β.

Then, the translation function TmbC : F orCmbC → F orCCPL is deﬁned by:

TmbC (pi) = pi;

TmbC (¬α) = ¬cTmbC (α) ∨c p¬α;

TmbC (◦α) = ¬c(TmbC (α) ∧c TmbC (¬α)) ∧c p◦α;

TmbC (α ∨ β) = TmbC (α) ∨c TmbC (β);

TmbC (α ∧ β) = TmbC (α) ∧c TmbC (β);

TmbC (α → β) = TmbC (α) →c TmbC (β).

Example 8. For C1, considering the dyadic semantics presented in Example 4, the CPL-formula for the formula scheme ¬α is the same than in Example 5. For the formula scheme ¬¬α, the following truth table is obtained:

α ¬α p¬¬α ¬¬α F T F F F T T F T F F T T F T T T T F F T T T T and a CPL-formula that represents the bivalued function for ¬¬α is:

CPL(¬¬α) = α ∧c (¬c¬α ∨c p¬¬α).

For the formula scheme ◦α the CPL-formula is (by condition (C1–3)):

CPL(◦α) = CPL(¬(α ∧ ¬α)) = ¬c(α ∧c ¬α).

9 Formula schemes ◦(α#β), where # ∈ {∨, ∧, →}, are particular cases of the previous formula scheme, thus their truth values are also totally determine by the truth values of α#β and ¬(α#β), and the CPL-formulas for these schemes have to be also calculated by the previous equation. The question is then: how to include conditions (C1–4)–(C1–6) in the translation? By using the metalogic of conditional clauses, it can be proven that conditions (C1–4)–(C1–6) can be changed by the following equivalent conditions:

(v(α) = F | v(¬α) = F ), (v(β) = F | v(¬β) = F ), v(α ∨ β) = T =⇒ v(¬(α ∨ β)) = F ;(C1–4’)

(v(α) = F | v(¬α) = F ), (v(β) = F | v(¬β) = F ), v(α ∧ β) = T =⇒ v(¬(α ∧ β)) = F ;(C1–5’)

(v(α) = F | v(¬α) = F ), (v(β) = F | v(¬β) = F ), v(α → β) = T =⇒ v(¬(α → β)) = F. (C1–6’)

These alternative conditions allow to associate with each formula scheme ¬(α#β), where # ∈ {∨, ∧, →}, the following CPL-formula (the truth table for helping the construction of this CPL- formula is too long and is not here presented):

CPL(¬(α#β)) = ¬c(α#β) ∨c (((α ∧c ¬α) ∨c (β ∧c ¬β)) ∧c p¬(α#β)).

The CPL-formulas for the other formula schemes on the right side of conditions (C1–7)–(C1–9) are:

CPL(α ∨ β) = α ∨c β,

CPL(α ∧ β) = α ∧c β,

CPL(α → β) = α →c β.

Then, the translation function T : F or → F or is deﬁned by: C1 CC1 CCPL

TC1 (pi) = pi;  ¬cpi ∨c p¬pi , if ϕ = pi;  T (α) ∧ (¬ T (¬α) ∨ p ), if ϕ = ¬α;  C1 c c C1 c ¬¬α TC1 (¬ϕ) = ¬c(TC1 (α) ∧c TC1 (¬α)), if ϕ = (α ∧ ¬α);  ¬c(TC1 (α#β)) ∨c (((TC1 (α) ∧c TC1 (¬α)) ∨c (TC1 (β) ∧c TC1 (¬β))) ∧c p¬(α#β)),  if ϕ = (α#β), with # ∈ {∨, ∧, →} and # 6= ∧ or β 6= ¬α;

TC1 (α ∨ β) = TC1 (α) ∨c TC1 (β);

TC1 (α ∧ β) = TC1 (α) ∧c TC1 (β);

TC1 (α → β) = TC1 (α) →c TC1 (β).

Since the recipe above is only a partial and not rigorous procedure, after constructing a translation TL by using this recipe it is necessary to prove that TL is really a conservative translation, which can be done in the following way:

1. First prove that, for any CPL-valuation vCPL: F orCCPL → {T,F }, the composition vCPL◦TL is an L-valuation (i.e. that vCPL ◦ TL ∈ Sem(B)).

10 2. Then prove that, given an L-valuation vL: F orC → {T,F } (i.e. given vL ∈ Sem(B)), 0 the CPL-valuation vCPL: F orCCPL → {T,F } that set to propositional variables in V the following values:

vCPL(pi) = vL(pi), for every pi ∈ V ;

vCPL(pα) = vL(α), for every pα ∈ Vh;

is such that vL(ϕ) = vCPL(TL(ϕ)), for every ϕ ∈ F orC .

3. From the previous facts, for any L-valuation vL there is a CPL-valuation vCPL such that vL(α) = vCPL(TL(α)), for every α ∈ F orC ; and for any CPL-valuation vCPL there is an L-valuation vL such that vL(α) = vCPL(TL(α)), for every α ∈ F orC ; consequently Γ B ϕ if and only if TL[Γ] CPL TL(ϕ), for every Γ ∪ {ϕ} ⊆ F orC (i.e. TL is a conservative translation).

In the following of this section we prove that the function TL3 (deﬁned in Example 5) is a 1 conservative translation. In an analogous way, it can be proven that functions TP ,TmbC and TC1 (deﬁned in Examples 6, 7 and 8, respectively) are conservative translations (the proof of these facts are easy but tedious, therefore we omit them).

Lemma 1. For any CPL-valuation vCPL, the composition vCPL ◦ TL3 is an L3-valuation.

Proof. We will only prove here that vCPL ◦ TL3 satisﬁes condition (L3–1) (the proof of the other conditions are easier and are left to the reader). We proceed by induction on the complexity measure l(ϕ) (as it is deﬁned in Example 1):

• Base case: if l(ϕ) = 0, then ϕ = pi for some pi ∈ V . Now, let’s suppose that vCPL(TL3 (pi)) =

vCPL(pi) = F , then vCPL(TL3 (¬pi)) = vCPL(¬cpi ∨c p¬pi ) = T . • Induction step: if l(ϕ) > 0 we have the following cases:

– ϕ = ¬α: let’s suppose that vCPL(TL3 (¬α)) = F . By inductive hypothe-

sis vCPL(TL3 (α)) = F implies vCPL(TL3 (¬α)) = T , then vCPL(TL3 (¬¬α)) =

vCPL(TL3 (α)) = T .

– ϕ = α → β: let’s suppose that vCPL(TL3 (α → β)) = vCPL(TL3 (¬α) ∨c TL3 (β) ∨c

(¬cTL3 (α) ∧c ¬cTL3 (¬β))) = F , then vCPL(TL3 (¬α)) = F and vCPL(TL3 (β)) = F . By

inductive hypothesis vCPL(TL3 (α)) = T and vCPL(TL3 (¬β)) = T , and it follows that

vCPL(TL3 (α → β)) = vCPL(TL3 (α) ∧c TL3 (¬β)) = T .

Lemma 2. Let vL3 be an L3-valuation and vCPL be the CPL-valuation that set to propositional variables in V 0 the following values:

vCPL(pi) = vL3 (pi), for every pi ∈ V ;

vCPL(pα) = vL3 (α), for every pα ∈ Vh.

Then v (ϕ) = v (T (ϕ)), for every ϕ ∈ F or . L3 CPL L3 CL3

11 Proof. By induction on the complexity measure l(ϕ) (as it is deﬁned in Example 1):

• Base case: if l(ϕ) = 0, then ϕ = pi for some pi ∈ V , and vCPL(TL3 (pi)) = vCPL(pi) = vL3 (pi). • Induction step: if l(ϕ) > 0 we have the following cases:

– ϕ = ¬δ: here we have the following subcases:

∗ δ = pi for some pi ∈ V : vCPL(TL3 (¬pi)) = vCPL(¬cpi ∨c p¬pi ) = F if only if

vCPL(pi) = T and vCPL(p¬pi ) = F , if and only if vL3 (pi) = T and vL3 (¬pi) = F ,

if and only if vL3 (¬pi) = F (by (L3–1)).

∗ δ = ¬α: vCPL(TL3 (¬¬α)) = vCPL(TL3 (α)) = T if only if vL3 (α) = T (by inductive

hypothesis), if and only if vL3 (¬¬α) = T (by (L3–2)).

∗ δ = α → β: vCPL(TL3 (¬(α → β))) = vCPL(TL3 (α) ∧c TL3 (¬β)) = T if only

if vCPL(TL3 (α)) = T and vCPL(TL3 (¬β)) = T , if and only if vL3 (α) = T and

vL3 (¬β) = T (by inductive hypothesis), if and only if vL3 (¬(α → β)) = T (by (L3–3)).

– ϕ = α → β: vCPL(TL3 (α → β)) = vCPL(TL3 (¬α) ∨c TL3 (β) ∨c (¬cTL3 (α) ∧c

¬cTL3 (¬β))) = T if only if vCPL(TL3 (¬α)) = T or vCPL(TL3 (β)) = T , or both

vCPL(TL3 (α)) = F and vCPL(TL3 (¬β)) = F ; if and only if vL3 (¬α) = T or vL3 (β) = T ,

or both vL3 (α) = F and vL3 (¬β) = F (by inductive hypothesis); if and only if

vL3 (α → β) = T (by (L3–4)).

Theorem 1. The function TL3 is a conservative translation from L3 into CPL. Proof. A direct consequence of Lemmas 1 and 2.

2 Intuitionistic logic is not characterizable by dyadic semantics

The following theorem shows the impossibility of characterizing Intuitionistic Logic (IL) by means of a dyadic semantics. It is important to clarify that this theorem does not contradict results in [10], where IL is characterized by a two-valued valuation semantics. In [10], for deﬁning the class of bivaluations for characterizing IL, it is included a condition (labeled by V →) where the value of a scheme of formulas under a valuation v depends on the existence of another valuation s satisfying some conditions. This kind of ‘global’ conditions on valuations are not allowed on dyadic semantics. It is also important to mention that, although the following theorem avoids the possibility of using the method described in the previous section for constructing a conservative translation from IL into CPL, it does not eliminate at all the possibility of a classical interpretation of IL in terms of hidden variables. Theorem 2. IL is not characterizable by a dyadic semantics.

12 Proof. Let IL = hF orIL, ILi and let’s suppose that there is a set of conditional clauses B such that IL= B. The next table shows some deductions rules of IL and the respective conditional clauses that must be deducible from B in order to Sem(B) validate them. Deduction rules Conditional clauses α, β IL α ∧ β, α ∧ β IL α, α ∧ β IL β v(α) = T, v(β) = T ⇐⇒ v(α ∧ β) = T (IL–1) α IL α ∨ β, β IL α ∨ β v(α) = T | v(β) = T =⇒ v(α ∨ β) = T (IL–2) α ∧ ¬α IL ⊥ v(α) = T =⇒ v(¬α) = F (IL-3) α IL ¬¬α v(α) = T =⇒ v(¬¬α) = T (IL-4) ¬¬¬α IL ¬α v(¬¬¬α) = T =⇒ v(¬α) = T (IL-5) α → β, α IL β v(α → β) = T =⇒ v(α) = F | v(β) = T (IL-6) ¬α IL α → β v(¬α) = T =⇒ v(α → β) = T (IL-7) β IL α → β v(β) = T =⇒ v(α → β) = T (IL-8) The conditional clauses in the previous table are necessary but not suﬃcient for characterizing IL. For instance, we have that IL α → α; however, if we suppose that v(α → α) = F , from (IL-7) and (IL-8) it follows that v(¬α) = F and v(α) = F , but this does not lead to a contradiction, consequently 2B α → α. In order to Sem(B) validate α → α, and other intuitionistic valid implications, we have the following options:

• Strengthen the conditions for the negation connective avoiding the possibility that both v(¬α) = F and v(α) = F (i.e. add to B the conditional clause v(α) = F =⇒ v(¬α) = T ). If this is made, we will have that B α ∨ ¬α (by (IL–2)), but 1IL α ∨ ¬α. • Strengthen the conditions for the implication connective (following some conditions of CPL- valuations). Here we have the following possibilities:

– Add to B the conditional clause v(α) = F =⇒ v(α → β) = T. If this is made, we will have that v(α) = F | v(β) = T ⇐⇒ v(α → β) = T. Then, the intuitionistic implication will behave like the implication of CPL, and any CPL-valid formula containing only the implication connective will be also valid in Sem(B). For instance, we will have that B ((α → β) → α) → α, but 1IL ((α → β) → α) → α. – Add to B the conditional clause v(¬¬α) = F =⇒ v(α → β) = T. This condition is not enough for validating α → α (another condition for avoiding the possibility that both v(¬¬α) = T and v(α) = F would be necessary). However, if it is added to B, then we will have that B ¬¬α ∨ (α → β), but 1IL ¬¬α ∨ (α → β). – Add to B the conditional clause v(¬β) = F =⇒ v(α → β) = T. If this is made, we will have that B ¬¬α → α, but 1IL ¬¬α → α. – Add to B the conditional clause v(¬¬β) = T =⇒ v(α → β) = T. This condition is not enough for validating α → α (another condition for avoiding the possibility that both v(¬α) = F and v(¬¬α) = F would be necessary). However, if it is added to B, then we will have that ¬¬β B α → β, but ¬¬β 1IL α → β. As v(¬α) = T ⇐⇒ v(¬¬¬α) = T is deducible from (IL-4) and (IL-5), conditional clauses having more nested negations are not necessary. Then, all condition clauses concerning negation and implication valid for the analogous connectives in CPL were considered, and

13 all of them lead to invalid deduction in IL. Furthermore, as any valid deduction in IL is also a valid deduction in CPL, we cannot include into B conditional clauses that are not valid in CPL. Consequently, IL is not characterizable by a dyadic semantics.

3 An alternative method for translating many-valued logics into the classical logic

Before defining the alternative method for translating finite many-valued logics into CP l, some definitions are in order. By a finite many-valued logic we mean a logic L = hF orC , Li characterizable by a finite logical matrix. A logical matrix is a structure M = hW, D, (verc)c∈C i, where: W is a non-empty set of truth degrees; D ⊆ W , D 6= ∅ is a set of designated truth degrees; and m verc is a truth degree function verc: W → W , where m is the arity of c (when m = 0, verc is really a constant element of W , but it is common and useful to consider it as a function of arity 0, which is called a nullary function). It is said that M is finite if W is finite. A valuation in M is a function v: F orC → W such that v(c(ϕ1, . . . , ϕm)) = verc(v(ϕ1), . . . , v(ϕm)) if c is a connective with arity m. A valuation v is a model of (or satisfies) a formula ϕ ∈ F orC if v(ϕ) ∈ D, and is a model of a set of formulas Γ ⊆ F orC if it is a model of every formula in Γ. Denoting by Mod(X) the set of models of X, where X is a formula or a subset of formulas of F orC , the consequence relation generated by M is defined by Γ M ϕ if and only if Mod(Γ) ⊆ Mod(ϕ). Finally, a logic L = hF orC , Li is characterized by a logical matrix M if L = M. Now, let L = hF orC , Li be a finite-valued logic characterized by a logical matrix M = hW, D, (verc)c∈C i, n = max{|D|, |W \ D|} (where |X| denotes the cardinal of X) and l = dlog2 ne. Considering that formulas of CPL are generated over the signature CCPL = {¬c, ∨c, ∧c, →c} (like 0 j in the previous section) and by the set of propositional variables V = V ∪Vh, where Vh = {pi | pi ∈ V and 1 ≤ j ≤ l} is a set of hidden variables which will be used to represent the degree of truth or falsehood of propositional variables in V , a translation function T : F orC → CPL can be defined in the following way: 1. Codify the elements of W by sequences of zeros and ones of length l+1. A code for an element a ∈ W will be denoted by a and (a)i will denote the element in position i of a (positions will begin in 0). A code for an element a ∈ W must satisfy the following condition: (a)0 = 1 if and only if a ∈ D. In order that any sequence of zeros and ones of length l + 1 represents an element of W , some a ∈ W may have several codes. The set of all codes will be denoted by W (i.e. W = {a0a1 . . . al | aj = 0 or aj = 1, for 0 ≤ j ≤ l}). 2. For each c ∈ C do the following: m (a) Let m be the arity of c, then define a function c: W → W such that c(a1,..., am) = verc(a1, . . . , am) (where verc(a1, . . . , am) is any of the possible codes for verc(a1, . . . , am)).

(b) For each 0 ≤ i ≤ l, construct a CPL-formula CPL((c(a1,..., am))i) that represents the function (c)i in terms of a1,..., am (taking 0 as representing false and 1 as representing true). This is always possible because CPL is functionally complete.

14 (c) Deﬁne the functions Tj: F orC → CPL, for 0 ≤ j ≤ l, by simultaneous recursion as follows:

j 0 Tj(pi) = pi (where pi = pi), Tj(c(ϕ1, . . . , ϕm)) = CPL((c)j)[(ai)0/T0(ϕi),..., (ai)l/Tl(ϕi)].

Then, deﬁne T : F orC → CPL by T (ϕ) = T0(ϕ).

Example 9. Lukasiewicz three-valued logic L3 is characterized by the logical matrix ML3 = 1 h{0, /2, 1}, {1}, ver¬, ver→i, where the truth degree functions ver¬ and ver→ are presented in the following tables:

1 a1 ver¬(a1) ver→(a1, a2) 0 /2 1 0 1 0 1 1 1 1/2 1/2 1/2 1/2 1 1 1 0 1 0 1/2 1

1 In this case n = 2 and l = 1, then Vh = {pi | pi ∈ V }. Codifying the truth degree values by 0 = 00, 1/2 = 01 and 1 ∈ {10, 11}, the functions ¬ and → are presented in the following tables:

a1 ¬(a1) →(a1, a2) 00 01 10 11 00 10 00 10 10 10 10 01 01 01 01 10 10 10 10 00 10 00 01 10 10 11 00 11 00 01 10 10 The CPL-formulas for representing the projections on ¬ and → are:

CPL((¬(a1))0) =¬c((a1)0) ∧c ¬c((a1)1),

CPL((¬(a1))1) =¬c((a1)0) ∧c (a1)1,

CPL((→(a1, a2))0) =((a1)0 →c (a2)0) ∧c ((a1)0 ∨c ¬c(a1)1 ∨c (a2)0 ∨c (a2)1),

CPL((→(a1, a2))1) =(¬c(a1)0 ∧c (a1)1 ∧c ¬c(a2)0 ∧c ¬c(a2)1) ∨c ((a1)0 ∧c ¬c(a2)0 ∧c (a2)1). The functions T : F or → CPL and T : F or → CPL are then deﬁned by: 0 CL3 1 CL3

T0(pi) = pi,

T0(¬α) = ¬cT0(α) ∧c ¬cT1(α),

T0(α → β) = (T0(α) →c T0(β)) ∧c (T0(α) ∨c ¬cT1(α) ∨c T0(β) ∨c T1(β)), 1 T1(pi) = pi ,

T1(¬α) = ¬cT0(α) ∧c T1(α),

T1(α → β) = (¬cT0(α) ∧c T1(α) ∧c ¬cT0(β) ∧c ¬cT1(β)) ∨c (T0(α) ∧c ¬cT0(β) ∧c T1(β)).

Finally, the function T : F orC → CPL is deﬁned by T (ϕ) = T0(ϕ).

Theorem 3. Given a ﬁnite-valued logic L = hF orC , Li, characterized by a logical matrix M = hW, D, (verc)c∈C i, the function T : F orC → CPL obtained by the procedure above is a conservative translation.

15 Proof. Given an L-valuation vL: F orC → W , if we consider the CPL-valuation vcpl: F orCCPL → {0, 1} (here we used 0 and 1 instead of F and T , respectively) that set to propositional variables in V 0 the following values:

j 0 vcpl(pi ) = (vL(pi))j (where pi = pi), by construction of functions Tj: F orC → CPL, for 0 ≤ j ≤ l, we have that:

vL(ϕ) = a ⇐⇒ vCPL(T0(ϕ))vCPL(T1(ϕ)) . . . vCPL(Tl(ϕ)) = a, for every ϕ ∈ F orC . Then, by the condition imposed in the codiﬁcation of truth values, it follows that: vL(ϕ) ∈ D ⇐⇒ vCPL(T0(ϕ)) = 1, for every ϕ ∈ F orC .

In the other direction, given a CPL-valuation vcpl: F orCCPL → {0, 1}, if we consider the L-valuation vL: F orC → W that set to propositional variables the following values:

1 l vL(pi) = a if vCPL(pi)vCPL(pi ) . . . vCPL(pi) = a, by construction of functions Tj: F orC → CPL, for 0 ≤ j ≤ l, we also have that:

vL(ϕ) = a ⇐⇒ vCPL(T0(ϕ)) . . . vCPL(Tj(ϕ)) = a, for every ϕ ∈ F orC ; and consequently that:

vL(ϕ) ∈ D ⇐⇒ vCPL(T0(ϕ)) = 1, for every ϕ ∈ F orC .

From the previous facts, it follows that Γ L α if and only if TL[Γ] CPL TL(α), for every Γ ∪ {α} ⊆ F orC (i.e. TL is a conservative translation).

4 Final remarks

The translation methods described in Sections 1 and 3 show how some non-classical logics can be interpreted in terms of classical logic by means of introducing hidden variables. Historically, (local) hidden variable theories emerge in physics within the debate about indetermism in quantum mechanics, suggesting that quantum mechanics gives only an incomplete statistical description of reality, but that there must be a complete “deterministic (and local) theory describing nature, where the precise value of all observables of a physical system are ﬁxed by some unknown variables (the hidden variables).” (cf. [9]). In an analogous way, we can thought that non-classical logics are logical calculus for reasoning in contexts where truth depends on some unknown (or hidden) propositions. Then, the translation methods here proposed can be viewed as methods to make explicit the hidden propositions necessary to have a complete description of truth in contexts where truth is only partially determine by known propositions.

16 Concerning truth-functionality, we can say that a semantics for a logic L is truth-functional if the truth value of any complex formula of L is uniquely determined by the truth value of its subformulas and some truth rules associated with connectives (for a rigorous characterization of truth- functionality see [11]). Many logics characterized by dyadic semantics are not truth-functional in this sense, because the truth value of some complex formulas is only partially determined by the truth values of its subformulas. The truth-value of formulas of many paraconsistent logics, for instance, can not be described in a finite-valued truth-functional way (cf. [6]); however they can be characterized by dyadic semantics in a bivalued but not non-truth functional way. Moreover, when many-valued logics are characterized by means of dyadic semantics, the truth functionality is usually lost (cf. [2]). What we show here is that even in logics that are not characterizable by means of bivalued truth-functional semantics, the truth value of formulas can be understood as a bivalued truth function, that does not depend only on the truth value of subformulas and truth rules associated with connectives, but on the truth values of less complex formulas (under some appropriate complexity measure), the truth values of some hidden propositional variables, and truth rules associated with some formula schemes (in some cases depending on more than one connective). Now, we can ask the following question: what is the advantage of having a hidden-variable classical interpretation of a non-classical logic? From a philosophical point of view, it can be seen as an argument in favor of the universality of classical logic: while more non-classical logics can be interpreted in terms of the classical logic, it seems to be more reasonable to think that any king of reasoning can be substituted by classical reasoning. However, if some non-classical logic can not be definitely interpreted in terms of classical logic by using hidden variables, this might show some limitations of classical reasoning. From this perspective, the translation methods here introduced are technical tools to be taking into consideration in philosophical debates about the universality of classical logic. From a pragmatic point of view, the translation methods here proposed put all the technical tools developed for classical logic at the service of non-classical logics. The practical utility of these tools depends, however, on the increase in complexity introduced by the translations obtained. As it can be seen in the examples above, translations using hidden variables can increase the complexity of formulas in terms of number of propositional variables, number of connectives and depth of formulas. A quantification of such increases is necessary to determine if a particular translation and the methods used for solving problems in classical logic provide efficient ways of solving problems for the original logic. In order to advance in the potentialities of hidden-variables interpretations of non-classical logics, some questions must be answered in future works, among others: Are the methods here proposed applicable to other non-classical logics? Are there other forms of using hidden variables for providing classical interpretations of non-classical logics? Are these methods extensible to first order (and higher order) logics?

Acknowledgments

The author would like to thank Marcelo E. Coniglio for having pointed out the existence of Lopari´c’s article ([10]). This research was supported by the University of Antioquia, Colombia (grant number IN627CE).

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