Translations: generalizing relative expressiveness between logics
aa r X i v : . [ m a t h . L O ] J un Translations: generalizing relative expressivenessbetween logics
Diego Pinheiro Fernandes ∗ Friday 3 rd August, 2018
Abstract
There is a strong demand for precise means for the comparison oflogics in terms of expressiveness both from theoretical and from appli-cation areas. The aim of this paper is to propose a sufficiently generaland reasonable formal criterion for expressiveness, so as to apply not onlyto model-theoretic logics, but also to Tarskian and proof-theoretic log-ics. For model-theoretic logics there is a standard framework of relativeexpressiveness, based on the capacity of characterizing structures, and astraightforward formal criterion issuing from it. The problem is that itonly allows the comparison of those logics defined within the same classof models. The urge for a broader framework of expressiveness is not new.Nevertheless, the enterprise is complex and a reasonable model-theoreticformal criterion is still wanting. Recently there appeared two criteria inthis wider framework, one from Garc´ıa-Matos & V¨a¨an¨anen and other fromL. Kuijer. We argue that they are not adequate. Their limitations areanalysed and we propose to move to an even broader framework lack-ing model-theoretic notions, which we call “translational expressiveness”.There is already a criterion in this later framework by Mossakowski et al.,however it turned out to be too lax. We propose some adequacy criteriafor expressiveness and a formal criterion of translational expressivenesscomplying with them is given.
Contents gv . . . . . . . . . . . . . . . . . . . . . 62.5 L. Kuijer on multi-class expressiveness . . . . . . . . . . . . . . . 102.6.1 A trivial translation . . . . . . . . . . . . . . . . . . . . . 102.6.2 Defining expressiveness g . . . . . . . . . . . . . . . . . . 11 ∗ PhD student at University of Salamanca, Spain expressiveness g . . . . . . . . . . . . . . 122.12 Single-class expressiveness vs multi-class expressiveness vs trans-lational expressiveness . . . . . . . . . . . . . . . . . . . . . . . . 15 expressiveness gg : a sufficient condition for expressiveness . . . . 293.19.1 Adequacy criterion 1 . . . . . . . . . . . . . . . . . . . . . 303.20.1 Adequacy criterion 2 . . . . . . . . . . . . . . . . . . . . . 323.23.1 Adequacy criterion 3 . . . . . . . . . . . . . . . . . . . . . 343.27 Corroborating expressiveness gg : the structure preserving trans-lations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 It is very common for those who work with logic to make comparisons such as“the logic L ′ is more expressive than L ”, “ L ′ is stronger than L ”, “ L is includedin L ′ ”, “ L can be reduced to L ′ ”, etc. Such assertions are often made on impre-cise grounds and, though possibly being non-ambiguous and non-problematic,the lack of clarity around the usage of these concepts can generate terminolog-ical confusion across the literature (e.g. [Hum05]) and harden the comparisonof formal results.In the literature, the notion of logic inclusion or sub-logic (these terms will beused interchangeably here) is pretty much linked with language and axiomaticextensions, which on their turn are linked with “strength”, that is, the capacityof proving theorems or having valid formulas. Now the concept of sub-logic issometimes associated with strength and sometimes associated with expressive-ness, and sometimes with both (e.g. in [B´ez99]), which is known to be the case ofparadoxes [MDT09]. Three kinds of systems are relevant here: model-theoreticlogics, Tarskian and proof-theoretic logics, they will now be briefly defined. Alogic L is called model-theoretic if it is defined semantically and presented as asequence ( F , M , (cid:15) ), where F is a set of formulas, M is a class of models and (cid:15) is a satisfaction relation on M × F . A logic L is Tarskian if it is defined as( F , ⊢ ), where ⊢ is a consequence relation on F (possibly multi-consequence).Finally, L is a proof-theoretic logic if it is defined as ( F , R ), where R is a set ofinference rules. Some additional criteria are usually imposed for a system to qualify as one of these threekinds, but they are immaterial here.
In model-theoretic logics there is a straightforward approach to expressive-ness that is also reasonably taken as a definition of logic inclusion: a logic L is at least as expressive/includes L if every class of structures characterizablein L is also characterizable in L (see e.g. [Lin74, p. 129] and [BF85]). Thisnaturally only holds for logics defined within the same class of structures. Ifone wants also to compare logics defined within different classes of structures,then it does not seem adequate to use the concept of sub-logic, as we shall seebelow. It is better to use the concept of expressiveness.There is no straightforward approach to expressiveness for Tarskian andproof-theoretic logics (TPL, for short). As for sub-logic, in TPL it is also linkedwith language and axiomatic extensions. However, we can often see “sub-logic”relations taken in a wider sense, i.e. when, for two given logics L and L ′ ,it happens that L ′ is not a language/axiomatic extension of L , but there isa certain mapping of L -formulas into L ′ -formulas respecting the consequencerelation. These cases are normally interpreted as saying that L is included-/embeddable/reconstructible/interpretable/can be simulated in L ′ . We proposeto call these as expressiveness relations whenever they can be seen as modelingthe following intuition( E ) For every L -sentence φ , there is an L ′ -sentence ψ with the samemeaning.This same intuitive explanation of expressiveness holds for model-theoreticlogics, and is used as a basis for formal criteria therein (e.g. [BF85, p. 42]).Thus we can have a reasonably homogeneous concept for comparing logics:that of expressiveness. We shall reserve the term “sub-logic” just when thereare axiomatic or language extensions, and we shall not use the term “strength”because it is ambiguous between expressive and deductive strength.A precise definition for the notion of relative expressiveness for model-theoreticlogics was given already in the 1970s (e.g. in [Lin74] and [Bar74]). As we said,this definition is based on the capacity of characterizing structures and under-lies each of the so-called Lindstr¨om-type theorems, which form the basis ofabstract model theory. Single-class expressiveness
Considering model-theoretic logics defined withinthe same class of structures, the above intuition can be captured easily sincethere is a common ground where sentences can be compared. This commonground is easily achieved by defining the meaning of a sentence φ in a logic L = ( F , M , (cid:15) L ) as { U ∈ M | U (cid:15) L φ } ( M od L ( φ ), for short). Thus we call thisframework single-class expressiveness . Since every sentence in L is mapped toa sentence in L having the same meaning, this framework of expressivenesscan be seen as consisting of certain formula-mappings between model-theoreticlogics. A formal definition for it is then straightforward. Let τ be a signatureand let L = ( F , M , (cid:15) L ) and L = ( F , M , (cid:15) L ) be model-theoretic logics. That is, theorems of the form “If a logical system L ′ is at least as expressive as L andhave properties P , ..., P n , then L ′ is as expressive as L ”; see e.g. [BF85], [vBTCV09] and[OP10]. Definition 1.1 ( EC ) . L is at least as expressive as L ( L EC L ) if andonly if (iff, for short) for every τ − sentence φ ∈ F there is a τ − sentence ψ ∈ F such that M od L ( φ ) = M od L ( ψ ) . Notice that here the class of models M is the same for both L and L ,and φ, ψ share the same non-logical symbols. The above definition can be para-phrased in terms of elementary classes: L EC L ′ iff every elementary class of L is an elementary class of L ′ .Despite being the basis for many important results, EC is very limited.It is not only restricted to model-theoretic logics, but it requires the classes ofstructures being compared to share the same signature. As a consequence, itonly allows the comparison of logics defined within the same class of structures.The urge for a broader definition is not new. A straightforward means ofextension already appears in [BF85] and is examined in [Sha91]. Using thenotion of projective class, one can loosen the above definition allowing that L ′ is at least as expressive as L iff every elementary class of L is a projective classin L ′ ( L P C L ′ ) ( ibid , p. 232).Even among those expressiveness results using EC , we can notice someflexibility in its application. One such example appears in [AFFM11], where thedefinition of EC above is given, but afterwards (p. 307) it is informally relaxedin order to allow changes of signature, thus the proper definition being usedappears to be the one based on projective classes ( P C ). The problem is thatelsewhere we get different results depending on whether we use EC or P C ,as Shapiro showed [Sha91, p. 232]: L ( Q ) EC L ( A ) and L ( A ) EC L ( Q ),but L ( Q ) P C L ( A ) and L ( A ) P C L ( Q ). Remaining within model-theoretic logics, a wider framework —let us call itmulti-class— would comprise besides formula-mappings also structure-mappings,thus allowing structures of one logic to be mapped to structures of the other.This would enable the comparison of logics defined within different classes ofstructures. Recently there appeared two formal definitions of multi-class ex-pressiveness, to wit [GMV07] and [Kui14]. In the sequence we will present themand argue that they are not adequate.There have been also early claims outside abstract model-theory relatinglogics in the sense of (E) above, but no explicit definitions of the main conceptsinvolved were given. G¨odel used his result on the interpretation of classical intointuitionistic logic to infer that, contrary to the appearances, it is classical logicthat is contained in intuitionistic logic [G¨od01, p. 295]. Since then, there fol-lowed many results of interpretations, embeddings, reconstructions, simulations,etc. among Tarskian and proof-theoretic logics. Such results have often been For some signature τ , a class K of τ -structures is elementary in a logic L iff there is an L -sentence φ such that K = { U | U (cid:15) L φ } . A class K of τ -structures is a projective class of L if for some τ ′ ⊇ τ there is an L -elementary τ ′ -class K ′ such that K = { U ′ ↾ τ | U ′ ∈ K ′ } , where U ′ ↾ τ is the τ -reduct of U ′ . See [Tar86, p. 358], [Mes89, p. 299], [Sha91, p. 232] and [CK90, p. 130]. The logic L ( Q ) is the first-order logic extended with the quantifier “there exists infinitelymany”, and L ( A ) is the first-order logic with the “ancestral” operation A , i.e. Axy ( Rxy ) saysthat x is an ancestor of y in the relation R . used to justify some statement of inclusion or relative expressiveness betweenthe logics at issue. We proposed to call those with the underlying intuition ( E )as expressiveness results. Naturally, this notion of expressiveness is no longer di-rectly linked with the capacity of characterizing structures as in model-theoreticlogics, rather it resides in the capacity of a logic to “encode” another. Let theframework of expressiveness based on such capacity be named “translationalexpressiveness”. As opposed to the case of model-theoretic logics, until recently there wasno attempt to give a precise definition of relative expressiveness in this frame-work. To the best of our knowledge, Mossakowski et al. [MDT09] were the firstto give an explicit formal definition of translational expressiveness for logics,that is, an expressiveness relation based on the existence of certain kinds offormula-mappings. We will expose their definition and show that it is still notadequate. Then, some adequacy criteria for expressiveness are proposed and aformal criterion for translational expressiveness is given.
Structure of the paper
This paper presents the following panorama on relative expressiveness betweenlogics:(*) Relative expressiveness between logics (intuitive concept as given by (E))(a) Adequacy criteria for expressiveness → Approaches to (*) hopefully satisfying (a) (cid:30) single-class (cid:30) formal proposals: EC , P C (cid:30) multi-class (cid:30) formal proposals: gv , expressiveness g (cid:30) translational (cid:30) formal proposals: Mossakowski et al.’s and expressiveness gg .In § gv ) and other from [Kui14]( expressiveness g ). We argue that, using the intuitive explanation of expressive-ness given above, there are counterexamples to both. In the sequence, we inves-tigate what is wrong with them and propose that moving to an even wider frame-work, encompassing a greater range of logics and lacking structure-mappings,might be promising. E.g. [Tho74b, p. 154], [W´oj88, p. 67], [Hum00, p. 441], [Hum05, p. 163], [Con05, p.233], [CCD09, p. 15] and the recent [AA17, p. 207]. The term is borrowed from [Pet12]. Curiously, the same kind of problem appeared incomputer science: there was a multitude of programming languages and process calculi andmany informal claims relating the expressive power of such, through the existence of certainencodings of one into another. This situation fomented a series of works aiming at a stan-dardization of such “expressibility results” (e.g. [Fel90], [Par08] and [Gor10]). Though aimedat different objects, it is still possible to learn from this enterprise and propose the first stepsof a standardization of a definition of relative expressiveness. In § expressiveness gg ) is proposed. We will argue that expressiveness gg satisfiesthe criteria and is materially adequate. Garc´ıa-Matos and V¨a¨an¨anen gave a multi-class definition of sub-logic. Theirdefinition is similar to one given in [Mes89] but is laxer. Seemingly, theytreat the term “sub-logic” as synonymous with “expressiveness” (exchanging theorder of terms, naturally), since they present the Lindstr¨om theorems as beingabout sub-logic, whereas they are presented by one of the authors elsewhere asbeing about expressiveness (e.g. [vBTCV09]). We shall argue that the relationdefined must be seen as an expressiveness relation, and it will be shown thatas an expressiveness relation, it has important downsides. Let us consider theirdefinition of sub-logic [GMV07, p. 21]:
Definition 2.2.
A logic L = ( F , M , (cid:15) ) is a sub-logic of L ′ = ( F ′ , M ′ , (cid:15) ′ ) (insymbols L gv L ′ ) if there are a sentence θ ∈ F ′ and functions f : M ′ −→ M , T : F −→ F ′ such that:(a) For every U ∈ M exists a U ′ ∈ M ′ such that f ( U ′ ) = U and U ′ (cid:15) ′ θ (b) For every φ ∈ F and for every U ′ ∈ M ′ , if U ′ (cid:15) ′ θ , then ( U ′ (cid:15) ′ T ( φ ) iff f ( U ′ ) (cid:15) φ ) Thus, if the class of structures M ′ of a logic L ′ is richer than the class ofstructures M of a logic L , one could still allow a comparison between L and L ′ , by restricting M ′ to the translatable structures, i.e. those U ′ which satisfysome condition θ and then use a function f to translate this reduced class of L ′ -structures into L -structures. gv Let L = ( F , M , (cid:15) ) be a trivial propositional logic in some given signature,and let ( M , v ) be the set of is truth tables together with a valuation. Let L ′ = ( F ′ , M ′ , (cid:15) ′ ) be any logic that has at least one valid sentence δ and let theformula θ of the definition above be such δ . Define the following mappings Garc´ıa-Matos and V¨a¨an¨anen’s approach is a non-signature indexed version of the “mapof logics” in [Mes89, p. 299]. In Meseguer’s paper, it is not allowed for sub-logic mappingsthat sentences in the source logic be mapped to theories in the target logic, and the formula-mappings must be injective. .1 M. Garc´ıa-Matos and J. V¨a¨an¨anen on sub-logic 7 • f : M ′ −→ M . For every U ′ ∈ M ′ , f ( U ′ ) = ( M , v ). • T : F −→ F ′ . For every φ ∈ F , T ( φ ) = δ .Then it is easily seen that both items (a) and (b) above are satisfied.Thus, according to this definition of sub-logic, every logic containing at leastone valid formula has a trivial sub-logic. If we think on the usual meaninggiven to “sub-logic”, this not plausible at all, since the logic ( F ′ , M ′ , (cid:15) ′ ) couldbe non-trivial and might even lack a trivializing particle, so how come it couldhave a trivial sub-logic? It is not enough to require that the mapping T be injective. Using an ideaof [CCD09, p. 14], take for target logic any L ∗ = ( F ∗ , M ∗ , (cid:15) ∗ ) that has adenumerable number of valid formulas δ , δ , ... and define the mapping fromthe formulas of the trivial logic F = { φ , φ , ... } to L ∗ -formulas as T ( φ i ) = δ i .Still we have that L ∗ has a trivial sub-logic, once more, L ∗ may be any logicwith a denumerable number of validities, also lacking a trivializing particle.Naturally, the usual senses of logic inclusion, that is, through language oraxiomatic extensions do not apply here. The only way to make sense of this isto interpret the above cases as saying that a trivial logic can be simulated in anylogic containing at least one validity. This capacity of simulating a logic is anexpressive capacity, therefore the definition above is better seen as a definitionof expressiveness. Yet, as an expressiveness relation, it is noteworthy that norestriction on the translation functions f and T are imposed, so one may wonderwhether the definition over-generates.We are not in position to settle definitively this question. However we willgive a plausibility argument to the effect that we should impose stricter con-ditions on model- and formula-mappings, since there is a natural and reason-able extension of the above definition that indeed over-generates. Though not,strictly speaking, a counter-example, the case to be presented below shall giveevidence that there is an intrinsic problem with the above proposal for multi-class expressiveness.As we said, the sentence θ on the above definition of gv is intended to cut L ′ -structures that are meaningless from the point of view of L . Apparently, itwould do no harm to the idea behind gv to allow θ to be a recursive set ofsentences, as it is normally done in works dealing with translations of logicsand conversion of structures (e.g. [Man96, p. 270]). This would be useful if thelogics at issue have no conjunction, so that θ could be a finite set of sentences; orif the low expressive power of the logics L and L ′ makes that the L ′ -structuresto be reduced into L -structures be only characterizable through an infinite butrecursive set of L ′ -sentences. This happens in the case of many-sorted logic( MSL ) and
FOL . If θ is not allowed to be an infinite set of sentences, then MSL would not be a sub-logic, in the above sense, of
FOL , which is implausible.Though the conversion of
FOL -structures into
MSL -structures is mentioned This counter-example was based on another one given in [CC02, p. 385-6], which wasgiven as an argument for strengthening the notion of translation used. .1 M. Garc´ıa-Matos and J. V¨a¨an¨anen on sub-logic 8[GMV07, p. 23], the case of a given
FOL -signature τ containing infinitely-many unary symbols S , S , ... is not considered. To convert τ -structures into MSL -structures then one needs to make sure that unary predicates S , S , .. to be converted to many-sorted domains are non-empty. This would only beaccomplished by setting θ = {∃ xS ( x ) , ∃ xS ( x ) , ... } [Man96, p. 260].However, if one allows such modification another implausible situation oc-curs. Consider the classical propositional logic ( CPL ) and a propositional logic
WPL , defined by B´eziau [B´ez99].
WPL shares all the definitions of the classicalpropositional connectives, except for negation, where it has only one “half” ofits clause: for a
WPL -model M and formula φ , if M ( φ ) = T , then M ( ¬ φ ) = F ;the converse direction does not hold.B´eziau shows that there is a translation from CPL into
WPL . Below wewill give Mossakowski et al.’s presentation of it, which includes also a modeltranslation [MDT09, p. 107]. Given an n-ary connective T isliteral for T ( φ , ..., φ n )) = T ( φ ) , ..., T ( φ n )); for an atomic formula p , T is literal when T ( p ) = p . Define the mapping ( T , f ) : CPL −→ WPL asfollows: • T : F CPL −→ F
WPL – T ( ¬ φ ) = T ( φ ) → ¬ ( T ( φ )), – literal for ∧ , ∨ , → and atomic formulas; • and f : M WPL −→ M
CPL – f ( M WPL , v ) = ( M CPL , v ),where M comprises the truth-tables for each connective and v a valuation on thepropositional variables. Notice that f takes a WPL -model, keeps the valuation v and replaces the truth-tables for the corresponding CPL ones.Then we have that
Theorem 2.3 (Mossakowski et al.) . f ( M WPL , v ) (cid:15) CPL φ if and only if ( M WPL , v ) (cid:15) WPL T ( φ ) . The model mapping f is surjective, so that it obeys (a) above.Now Mossakowski et al. ( ibid , p. 100) define a mapping also from WPL to CPL using an auxiliary set of formulas ∆ constructed out of
CPL -formulas.Define the mapping ( T ′ , f ′ , ∆) : WPL −→ CPL as follows: • T ′ : F WPL −→ F
CPL – For every φ ∈ F WPL , T ′ ( φ ) = p φ , where p φ is a propositional vari-able.Define ∆ as the following set of formulas, for φ, ψ ∈ F WPL :.1 M. Garc´ıa-Matos and J. V¨a¨an¨anen on sub-logic 9 • T ′ ( φ ∧ ψ ) ↔ T ′ ( φ ) ∧ T ′ ( ψ ) • T ′ ( φ ∨ ψ ) ↔ T ′ ( φ ) ∨ T ′ ( ψ ) • T ′ ( φ → ψ ) ↔ T ′ ( φ ) → T ′ ( ψ ) • T ′ ( φ ) → ¬T ′ ( ¬ φ ).The purpose of ∆ is to encode the semantics of WPL into the proposi-tional variables { p , p , ... } , since every WPL -formula is translated into one ofsuch p i , in a CPL -model satisfying ∆ the valuation of the propositional vari-ables p i is forced to respect the semantics of WPL . For example, in
WPL , if( M WPL , v ) (cid:15) WPL r , then it holds that ( M WPL , v ) (cid:15) WPL ¬ r , but the conversedirection does not hold. This is simulated in the CPL -models satisfying ∆ bythe fourth clause above: if ( M CPL , v ) (cid:15) CPL p r , then ( M CPL , v ) (cid:15) CPL ¬ p ¬ r which implies that ( M CPL , v ) (cid:15) CPL p ¬ r . But, as in WPL , it does not hold thatif ( M CPL , v ) (cid:15) CPL p ¬ r , then ( M CPL , v ) (cid:15) CPL p r .Now define the model-translation f ′ : M CPL −→ M
WPL : • Let ( M , v ) be a CPL -model satisfying ∆. Then f ′ ( M , v ) is defined asfollows: – For every
WPL -formula φ , f ′ ( M , v ) (cid:15) WPL φ iff ( M , v ) (cid:15) CPL T ′ ( φ ). f ′ is also surjective (so it obeys (a) in the criterion for sub-logic above).Then we have that Theorem 2.4 (Mossakowski et al.) . f ′ ( M CPL , v ) (cid:15) WPL φ iff ( M CPL , v ) (cid:15) CPL ∆ and ( M CPL , v ) (cid:15) CPL T ′ ( φ ) . Therefore, by the above results and according to the extended definition ofsub-logic, we would have that
WPL and
CPL are one sub-logic of another,which is not plausible.
CPL is not a sub-logic of
WPL in the sense of lan-guage/axiomatic extension. Neither they are expressively equivalent, using ( E )above, since the “half-negation” present in WPL is not available in
CPL .The problem is that the translation from
WPL to CPL uses a trick to sneakin the semantics of
WPL into ∆. Restricting the
CPL -models that satisfy ∆,one simulates the behaviour of
WPL -formulas in the propositional variables p i and sustain such behaviour through the model-translation.The modified version of gv , allowing θ to be a recursive set of sentenceslooks at least as “natural” as the original one. Even considering the originaldefinition 2.2 we can see that there is something wrong with it, in not requiringany kind of preservation of the structure of formulas e.g. by forcing T to beinductively defined through the formation of formulas. Then one may conjecturethat, among more expressive logics, there be translations ( T , f ) where T mapsentire formulas φ to propositional variables p φ and, with a sentence θ restrictingthe target structures, f is able to mimic the semantic behavior of φ . Then it isvery doubtful that the obtained p φ would have the same meaning as φ .Thus, we think we have good reasons to consider that Garc´ıa-Matos andV¨a¨an¨anen’s definition of sub-logic is not adequate. It would certainly be betterto use a stronger notion of translation, paying attention to the structure of.5 L. Kuijer on multi-class expressiveness 10formulas. Only then the meaning of the target-formulas could be said to matchthe meaning of the source-formulas. Below we will see that a development alongthis line appeared in the literature. Nevertheless, there is still a structure-attentive translation that “cheats” similarly as the one above, mimicking thesemantics of one logic into the other. In his doctorate thesis [Kui14] Kuijer studies the expressiveness of various logicsof knowledge and action, these logics are taken in the model-theoretic sense. Henotices that there are some results relating logics similarly as in single-classexpressiveness. These works were selected as prototypical for a criterion inthe wider framework of multi-class expressiveness.The purpose is to investigate features shared by all the results and constructa criterion, to be called “ expressiveness g ”, based on these features. Similarlywith the work of Garc´ıa-Matos and V¨a¨an¨anen exposed above, these prototypesinvolve translations of sentences and translations of structures. So a translationfrom L to L is a pair ( T , f ), with T : F → F and f : M → M or f : M → M , such that ( T , f ) satisfies some given conditions.A first plausible condition is that ( T , f ) must preserve and respect truth: Definition 2.6 (Truth preserving) . A translation ( T , f ) : L → L with T : F → F and f : M → M is truth preserving if, for every φ ∈ F and U ∈ M U (cid:15) L φ if and only if f ( U ) (cid:15) L T ( φ ) . Then a tentative definition of expressiveness g could be L is at least expressive g as L iff there is a ( T , f ) : L → L thatis truth preserving.The problem is that the requirement of truth preservation is very weak, indeedthere are several trivial truth-preserving translations among almost every logic.Kuijer gives the following example [Kui14, p. 88]. Let L = ( F , M , (cid:15) L ) be any logic on possible world semantics such that F is countable and let L = ( F , M , (cid:15) L ) be a logic where F is a countable setof propositional variables but with no connectives and where M is a class ofmodels with possible worlds. Thus, every U ′ ∈ M is a set of possible worldswith a valuation.Define a truth-preserving translation ( T t , f t ) from L to L in the follow-ing way: map every φ ∈ F to a propositional variable p φ ∈ F , f t mapsa model U ∈ M to a model U ′ ∈ M taking the set of possible worldsof U and removing every other structure, and with the following valuation The referred results are: [Tho74a], [GH96], [GJ05], [BHT06b] and [BHT06a]. .5 L. Kuijer on multi-class expressiveness 11 v ( p φ ) = { w ∈ U | ( U , w ) (cid:15) L φ } . Then clearly, by definition, ( T t , f t ) : L −→ L is a truth preserving translation. expressiveness g Since L in the above example is an arbitrary logic on possible world models,if truth preservation were the only condition for multi-class expressiveness, L would be at least as expressive as L , which is absurd, given that L has scarceexpressive means. Nevertheless, truth-preservation is clearly a necessary condi-tion. Thus, one must find other features P , ..., P n a translation must satisfy inorder to serve as a formal elucidation of the notion of multi-class expressiveness.Another immediate criterion that comes to mind in order to avoid the trivialtranslations is to require the preservation of validities and entailment relations.However, some of the chosen prototypical translations do not preserve validityand some do not preserve entailment. Since the idea was to capture the essentialfeatures shared by all prototypical translations in expressiveness g , none of thesecan be imposed as a necessary condition.Kuijer then goes through a number of tentative criteria, e.g. preservation ofatomic formulas, of sub-formulas, etc., and shows that they are either too laxor too restrictive. Among the lax criteria, that is, the ones that are satisfied bysome trivial translation, is one that Kuijer considers nonetheless important, thecriterion of being model based: Definition 2.7 (Model based) . A translation ( T , f ) is model based if there aretwo functions f , f such that, for all ( M , w ) ∈ M , we have that f ( M , w ) =( f ( M ) , f ( M , w )) . A model based translation would force f to preserve some structure of M andprevent that the pointed models ( M , w ) and ( M , w ′ ) be translated to completelyunrelated models.Finally, the condition that apparently divides the good from bad translationsand gives a reasonable notion of multi-class expressiveness is the criterion ofbeing finitely generated. For the sake of simplicity, some aspects of the definitionbelow are not completely formalized. Let F be a set of formulas generated by aset P of propositional variables and a set C of connectives. Let X = { x , x , ... } be a set of variables with P ∩ X = ∅ , and let F X be the set of formulas generatedby P ∪ { x , x , ... } with the connectives C . Then we have ( ibid , p. 115): Definition 2.8 (Finitely Generated) . Let L and L be such that F i is generatedby a set P i of propositional variables and a finite set C i of connectives, for i ∈ { , } . Let φ X ∈ F X , then a translation ( T , f ) : L → L is finitelygenerated if T can be inductively defined by a finite number of clauses of theform T ( φ X ) = ψ X for ( x , ..., x n ) ∈ Ψ For the complete formal definition, the reader may consult [Kui14, p. 115]. .5 L. Kuijer on multi-class expressiveness 12 where ψ X is an F X -sentence constructed out of x , ..., x n and possibly containing T ( x i ) , for x i ∈ F X ; and where Ψ is the range of the x i , e.g. if a given x i is tobe replaced by a formula or only by an atomic formula. The set X contains the special propositional variables to be used in thetranslation clauses, for which one can substitute formulas. An example of sucha translation clause is: T ( x → x ) = ¬ ( T ( x ) ∧¬T ( x )) for ( x , x ) ∈ F ×F ;and T ( x ) = x for x ∈ P .The idea is that ( ibid , p. 110) it is the fact of being inductively definedand thus respecting (some) of the structure of the formulas that sets the finitelygenerated translations apart from the trivial translations. Thus Kuijer concludesthat the truth-preserving translations giving rise to an expressiveness relationcould be characterized as the ones being finitely generated and model-based.Therefore, the final criterion given for multi-class expressiveness is ( ibid , p.111) Definition 2.9 (Expressiveness g ) . Let L and L be such that F i is generated bya set P i of propositional variables and a finite set C i of connectives for i ∈ { , } .Then L is at least as expressive g as L iff there is a translation ( T , f ) from L to L that is model based, finitely generated and truth preserving. expressiveness g Kuijer had no pretensions that his multi-class definition were to be the gener-alization of expressiveness as given by the single-class framework. The aim wasto find only a “reasonable generalization” ( ibid , p. 83). While keeping this inmind, we would like to argue that his proposal is still not good enough as acriterion for multi-class expressiveness. This is because one can find a pair oflogics L , L ′ such that L ′ is intuitively more expressive than L , although L is atleast as expressive g as L ′ .The logics at issue are Epstein’s relatedness logic ( R ) [Eps13, p. 80] andclassical propositional logic ( CPL ). The logic R besides the truth-functionalconnectives, has a relevant implication “ → ”, which is the reason it is intuitivelymore expressive than CPL , which lack such a connective. The referred transla-tion would imply that
CPL is at least as expressive g as R .Despite the circumscribed character of Kuijer’s criterion, we think that areasonable generalization of single-class expressiveness should be able to dealwith a reasonable amount of logics, not only with a handful of them. Particularlywhen the logics at issue are in the literature, and have not been constructed in anad-hoc fashion just to give a counter-example. Finally, there is nothing specificabout the logics appearing in the counter-example, so it is quite possible thatthere are also modal counter-examples.Epstein presents R with the connectives ¬ , ∧ , → . The first two are defined asusual and the underlying idea for interpreting the relevant implication symbol“ → ” is as follows. It holds that p → q whenever p materially implies q andboth are subject-matter related to each other through a relation R defined onall propositional variables. Specifically, for propositional variables p i , p j and.5 L. Kuijer on multi-class expressiveness 13 R -sentences φ and ψ , R ( φ, ψ ) holds if and only if for some p i occurring in φ ,and p j occurring in ψ , it holds that R ( p i , p j ). Thus, the truth table for “ → ”is the one for material implication with an additional column for R , so that if R ( φ, ψ ) holds and ¬ ( φ ∧ ¬ ψ ) is true, then φ → ψ is true; else, if R ( φ, ψ ) doesnot hold, then φ → ψ is false.Let τ = { p , p , ..., ¬ , → , ∧} be a signature for R . An R -model ( M , R , v )is formed by the truth-tables for ∧ , ¬ , → , a symmetric and reflexive relation R on τ -formulas and a valuation v . For propositional variables d i,j , let τ + = { p , p , ... } ∪ { d i,j | i, j ∈ N } ∪ {¬ , ∧ , ⊃} . Let CPL be defined on τ + (note weuse ⊃ here to emphasize that it is a material implication). We will see below that there is a truth-preserving, model-based and finitelygenerated translation ( T E , f E ) : R −→ CPL . The mapping T E is defined asfollows: • T E ( φ → ψ ) = ( T E ( φ ) ⊃ T E ( ψ )) ∧ d φ,ψ • literal for ¬ , ∧ and atomic formulas. Here the basic idea for the translation of φ → ψ comes from the definition of“ → ”: φ materially implies ψ and both formulas are related through R . As thetranslation is defined inductively through the formation of formulas by a finitenumber of clauses, it is finitely generated .Now, from an R -model ( M , R , v ), one easily defines a transformation f E from R -models to CPL -models. Let f E ( M , R , v ) = ( M ∗ , v ∗ ), where, for M ∗ take all the truth-tables in M , excluding the one for → . Define v ∗ as follows(adapted from [Eps13, p. 300]): • v ∗ ( p i ) = v ( p i ); • v ∗ ( d φ,ψ ) = T iff R ( φ, ψ ) holds.Clearly f E is model-based .Both CPL and R satisfy a semantic deduction theorem ( ibid , p. 299). Toprove that ( T E , f E ) is truth-preserving, one has to prove only that, for anarbitrary R -model ( M , R , v ), it holds that The use of new propositional variables is for the sake of simplicity, as we could arrangethe p , p , ... in CPL so as to assign some of the p i s the role of such d i,j . The mapping presented was adapted from ( ibid , p. 299). It was given a simpler formwhich makes the proof of the theorem below straightforward. We refer to Epstein’s mappingas T E ∗ , which is identical with T E except for → , where T E ∗ ( φ → ψ ) =( T E ∗ ( φ ) ⊃ T E ∗ ( ψ )) ∧ [( W p i in φ, p j in ψ d i,j ) ∨ ( W p n in φ, p n in ψ ( d n,n ∨ ¬ d n,n ))].Notice that our mapping T E below is only truth-preserving while Epstein’s T E ∗ is alsovalidity-preserving, as e.g. T E ( p → p ) = ( p ⊃ p ) ∧ d p,p and T E ∗ ( p → p ) = ( p ⊃ p ) ∧ [ d p,p ∨ ( d p,p ∨ ¬ d p,p )]. Kuijer requires also that no propositional variable occurs outside the scope of a translationfunction, so for atomic formulas one should use additional functions s : P −→ P . Thus wecan take the identity function as such s . .5 L. Kuijer on multi-class expressiveness 14 Theorem 2.10 (adapted from Epstein) . ( M , R , v ) (cid:15) R φ if and only if f E ( M , R , v ) (cid:15) CPL T E ( φ ) . Corollary 2.11.
CPL is at least as expressive g as R . The main question now is: does ( T E , f E ) : R −→ CPL show that
CPL is atleast as expressive as R ? We do not think it is reasonable to say so, since theextra expressiveness brought about by the implication connective in R is onlyby a trick mimicked in CPL . Independently of the model-translation f E to givethe intended truth values for the “relevance-mimicking” variables d φ,ψ , it is notpossible to have a relevant conditional in CPL , by say, adjoining to a conditional φ ⊃ ψ such variables d φ,ψ . To do so, would require too much for the intendedmeaning of such variables. Surely this would not augment the expressive powerof the propositional logic, as it concerns only an interpretation of propositionalvariables, and intuitively, specific interpretations of propositional variables donot influence the expressiveness of a logic.Anyway, the model-mappings are not essential for these translations usingindexed variables, they only facilitate their definition. An early example wasgiven by Richard Statman in [Sta79] where a translation of IPL into its implica-tional fragment
IPL ↾ {→} is presented. There, the conjunctions p ∧ q are mappedto implications containing x p ∧ q , among formulas of the sort x p → ( x q → x p ∧ q ), x p ∧ q → x p , etc. Here the situation is entirely different since the proof-theoreticbehaviour of individual conjunctions are encoded in specific variables using im-plicational axioms.Coming back to Kuijer’s criterion, we argued above that it is not enough togive an intuitively adequate account of expressiveness. If the model mappingwere not from the source logic to the target logic but vice-versa, then therewould not be such truth preserving mappings from R to CPL , as there wouldbe no way to construct the relatedness predicate R out of a CPL -model. Kui-jer discarded such a definition of the model mappings f since it implies thatany truth-preserving translation is also validity preserving, and some of hisparadigmatic examples of multi-class expressiveness are not validity preserving.Let us analyse a possible strengthening on the formula translation. We willnot give a detailed analysis of features of translations since it suffices to noticethat Epstein’s translation preserves completely the structure of the formulas,except for → . For this case, additional propositional variables d φ,ψ must beintroduced to bear the intended meaning of R (variables whose interpretationin CPL is sustained by the model translation.) If one required that T be com-positional , that is, every n -ary connective C ( φ , ..., φ n ) of the source logic istranslated by a schema C T ( T ( φ ) /ξ , ..., T ( φ n ) /ξ n ) of the target logic, then theabove translation would not pass the test. This is because p → p is trans-lated through the schema ¬ ( ξ ∧ ¬ ξ ) ∧ d p ,p , and p → p by the schema Suppose that for logics L = ( F , M , (cid:15) L ) and L ′ = ( F ′ , M ′ (cid:15) L ′ ) that ( T , f ) : L −→ L ′ istruth-preserving, with T : F −→ F ′ and f : M ′ −→ M . Suppose φ is L -valid, then for anymodel U ′ ∈ M ′ , f ( U ′ ) (cid:15) L φ , thus, by truth-preservation, U ′ (cid:15) L ′ T ( φ ), but U ′ is any L ′ -model,thus, T ( φ ) is L ′ -valid. .12 Single-class expressiveness vs multi-class expressiveness vs translationalexpressiveness 15 ¬ ( ξ ∧ ¬ ξ ) ∧ d p ,p . If the translation were compositional, dealing with thesame connective, the same translation schema would be used.The problem of adopting this criterion is that it implies that the connectivesbe translated one at a time, and again some of the paradigmatic translationsselected by Kuijer takes into consideration sequences of connectives, so theywould not satisfy it.Therefore, to prevent translations such as those above from passing thetest for multi-class, one would have to use a criterion for T that is strongerthan being finitely generated, but weaker than being compositional. Neverthe-less, the enterprise of placing restrictions on the formula translations T aloneseems not to be promising, as the model-translations play a major role in thecounter-examples presented above. On the other hand, placing also restrictionson model-translations and making them fit with the restrictions on formula-translations is a very complex enterprise, and there may be better alternatives.Given this situation, we would like to suggest a change of perspective asregards relative expressiveness between logics. Below, some comments will bemade regarding the nature of the notion of expressiveness and its relation withthe concept of logical system it applies to. Now we would like to make some remarks on the study of the relation of ex-pressiveness between logics. As we commented before, in the single-class frame-work it is very simple to define relative expressiveness, since there is a commonground, the structures, where one can compare whether the sentences have thesame meaning. Now consider the multi-class framework, if L = ( F , M , (cid:15) ) and L ′ = ( F ′ , M ′ , (cid:15) ′ ) are defined on different classes of structures, how would weknow whether an L -sentence φ and an L ′ -sentence ψ have the same meaning?After all, in this case it trivially holds that M od L ( φ ) = M od L ′ ( ψ ).As we saw, for this task new tools are needed: a model-mapping f : M −→M ′ or f ′ : M ′ −→ M ; and a formula-mapping T : F −→ F ′ or T ′ : F ′ −→ F .Now, for an L -formula φ and L ′ -formula ψ , we would have some possibilities forguessing when φ and ψ have the same meaning: • M od L ( φ ) = M od L ( T ′ ( ψ )), • M od L ′ ( ψ ) = M od L ′ ( T ( φ )), • f [ M od L ( φ )] = M od L ′ ( ψ ), • f ′ [ M od L ′ ( ψ )] = M od L ( φ ).Thus, now the weight goes on the notion of translation ( T , f ). As we sawin the examples presented above, for ( U , φ ) in L , and ( U ′ , ψ ) in L ′ , the task ofestablishing the congruence between the pairs ( U , φ ) and ( U ′ , ψ ) by means oftranslations is very difficult. Basing it on satisfaction is far away from being The translation f presupposes a mapping σ of signatures: for each L [ τ ]-structure, therewould correspond a L ′ [ σ ( τ )]-structure, respectively for f ′ . Let f [ Mod L ( φ )] = { f ( U ) | U ∈ Mod L ( φ ) } . .12 Single-class expressiveness vs multi-class expressiveness vs translationalexpressiveness 16sufficient, since we can easily devise translation functions such that U satisfies φ iff U ′ satisfies ψ .On the other hand, imposing conditions on ( T , f ) is a complex enterprise,because either it under-generates or, by a little breach, it over-generates. More-over, the need to have model-mappings besides formula-mappings may open upa back door to undesirable translations, to see it, consider again the examples of-fered against Garc´ıa-Matos & V¨a¨an¨anen’s and Kuijer’s approaches. All of themuse some “trick” in the formula-translation function and sustain it through themodel-translation. Then it is of little help to place structural restrictions onformula-translations, as did Kuijer. He also tried placing restrictions on model-translations, but it did not help either.Therefore, it might be more promising to move to a wider framework ofrelative expressiveness, dispensing with the semantic notions altogether. Inthis framework, to be called “translational expressiveness”, we would then con-centrate the investigations on the conditions on formula translations. Theaim is to find the set of conditions that better preserve/respect the theorem-hood/consequence relation and the structure of formulas of each logic. This waya reasonable formal criterion of expressiveness for Tarskian and proof-theoreticlogics (TPL, for short) would be obtained, and a bigger range of logics wouldbe comparable. Finally, these advantages would arguably come at no cost, sincethis wider enterprise would not be more difficult than multi-class expressiveness.The big difference between the approaches of expressiveness is not in thedivision between expressiveness for model-theoretic logics and for TPL, but inthe division, in model-theoretic logics, of expressiveness within the same andwithin different classes of structures. Naturally the most direct concepts ofexpressiveness are linked with the capacity of characterizing structures, but thisonly applies when comparing the same class of structures.If one allows translations between structures, such capacity is no longer atissue. Once we depart from the safe harbour of a single class of structuresfor comparing logics, then all bets are off. Multi-class expressiveness does notguarantee a firmer grasp of the intuitive concept of expressiveness anymore thantranslational expressiveness. Since the move to a wider framework might notonly free us from problems inherent to multi-class expressiveness, but also allowa bigger range of comparison of logics, then the prospects for the enterprise arebetter.As we said in the introduction, people have been using informally some con-cepts of translational expressiveness between logics. However, as opposed towhat happens with model-theoretic logics, to the best of our knowledge, in theliterature there is only one explicit and formal criterion in this framework, thatof [MDT09]. In the next section, their proposal will be analysed and we willshow that it is not adequate. We shall then propose some adequacy criteria forexpressiveness and a formal criterion in the framework of translational expres-siveness will be given. We then argue that the criterion satisfies the adequacycriteria.7 In this section we will deal with logics in the Tarskian and proof-theoretic sense.We also mention logics taken as a closed set of theorems/validities, to be calledsimply “formula logics”. Let L and L be logics, Γ ∪{ φ } be a set of L -formulasand T a translation mapping L -formulas into L -formulas in such a way thatfor each L -formula φ : ⊢ L φ if and only if ⊢ L T ( φ ).In this case L is translatable into L with respect to theoremhood.If it is the case thatΓ ⊢ L φ if and only if T (Γ) ⊢ L T ( φ )then L is translatable into L with respect to derivability [PM68, p. 216]. Thelater translations are known as conservative translations [FD01]. Definition 3.1 (Conservative translation) . A conservative translation is a trans-lation with respect to derivability.
Whenever we want to refer indistinctly to translations with respect to theo-remhood or conservative translations, the term back-and-forth will be employed.
Definition 3.2 (Back-and-forth translation) . A translation is back-and-forth ifit is either a theoremhood preserving or a conservative translation.
As far as we know, Mossakowski et al. [MDT09] proposed the first explicitcriterion for the concept of sub-logic and expressiveness in the framework oftranslational expressiveness:
Definition 3.4 (Sub-logic) . L is a sub-logic of L if and only if there is aninjective conservative translation from L to L ; Definition 3.5 (Expressiveness) . L is at most as expressive as L iff there isa conservative translation α : L −→ L . The authors do not explain why sub-logic requires injective conservativemappings while expressiveness does not. Anyway, we will see that these criteriafor sub-logic and expressiveness via conservative mappings do not work.The conception that conservative translations could give rise to a notion ofexpressiveness and also a notion of logic inclusion has been supported more thanonce. For example, in [Con05, p. 233] it is said that the existence of a conser-vative translation (maybe injective or bijective) would give rise to some kind of.3 Mossakowski et. al.’s approach 18logic inclusion between Tarskian logics. Also for Kuijer, conservative transla-tions give an adequate concept of expressiveness for Tarskian logics [Kui14, p.86]. Unfortunately, conservative translations will not make a reasonable conceptneither of sub-logic nor of expressiveness. Due to a result of Jeˇr´abek [Jeˇr12],explaining expressiveness and sub-logic through conservative translations wouldmake
CPL include and be at least as expressive as many familiar logical systems,e.g. first-order logic. He proved the following result ( ibid , p. 668), where for alogic L , a translation is most general whenever it is equivalent to a substitutioninstance of every other translation of L to CPL . Theorem 3.6 (Jeˇr´abek) . For every finitary deductive system L = ( F , ⊢ ) over acountable set of formulas F , there exists a conservative most general translation T : L → CPL . If ⊢ is decidable, then f is computable. The defined mapping is injective. Let a logic be called “reasonable” if it isa countable finitary Tarskian logic. Jeˇr´abek managed to generalize even morehis results so that almost any reasonable logic can be conservatively translatedinto the usual logics dealt with in the literature. Now one would hardly acceptthat every countable finitary logic has the same expressiveness or is one sub-logicof the other.The author criticizes the notion of conservative translation for not requiringthe preservation of neither the structure of the formulas nor the properties ofthe source logic [Jeˇr12, p. 666]. Thus, it must be strengthened in order toserve for an expressiveness measure. This could be done in a simpler way byrequiring injective, surjective or bijective mappings. As Jeˇr´abek’s mapping isinjective, only requiring injectiveness will not do. As a matter of fact, it seemsthat already requiring injectiveness one is overshooting the mark. Since inthis way
CPL ↾ {∧ , ¬} would not be as expressive as CPL ↾ {∧ , ¬ , ∨} . Any mapping g : CPL ↾ {∧ , ¬ , ∨} −→ CPL ↾ {∧ , ¬} would have to map both CPL ↾ {∧ , ¬ , ∨} -sentences φ ∨ ψ and ¬ ( ¬ φ ∧ ¬ ψ ) to the same CPL ↾ {∧ , ¬} -sentence ¬ ( ¬ g ( φ ) ∧ ¬ g ( ψ )), so itwould not be injective.Other kinds of strengthening hinted by Jeˇr´abek’s ( ibid ) are: The author says ( ibid ):If we assume (...) a Tarskian perspective, then a logic system is nothingmore than a set of formulas together with a [consequence] relation (...) Thus,the preservation of that relation by a conservative translation [from L to L ]would reveal that, as structures, L “contains” L (Probably we should add therequirement that f is an injective or even a bijective mapping.) The author says ( ibid ):There is a conservative translation from L to L if and only if everythingthat can be said in L can also be said in L . For the sake of brevity, we omit the definition of the translation and simply point outthat it is a non-general-recursive translation (to be defined below). Among others, classical, intuitionistic, minimal and intermediate logics, modal logics (clas-sical or intuitionistic), substructural logics, first-order (or higher-order) extensions of the for-mer logics. .7 Adequacy criteria for expressiveness 191. force the mappings to preserve more structure of the source logic sentencesin the target logic;2. force the mappings to preserve more properties of the source logic.The adequacy criteria for expressiveness to be given below will require tosome extent (1) and (2).
As we saw above, Mossakowski et al. [MDT09] gave a proposal for a wide notionof expressiveness: by means of the existence of conservative translations. Dueto Jeˇr´abek’s results on the ubiquity on this kind of translation, their definitionis not adequate. Maybe we should step back and think about some adequacycriteria every approach to expressiveness ought to accomplish.The intuitive explanation for expressiveness ( E ) given in the beginning elu-cidates relative expressiveness in terms of a certain congruence of meanings. Itappears already in a more direct form in W´ojcicki’s Theory of Logical Calculi[W´oj88, p. 67], and we place it as the first adequacy criterion[ Adequacy Criterion 1 ] L is at least as expressive as L only ifeverything that can be said in terms of the connectives of L canalso be said in terms of the connectives of L .Here, for “being said in terms of the connectives” there can be stricter inter-pretations (as proposed by W´ojcicki, Humberstone, Epstein) and wider inter-pretations (as proposed by Mossakowski et al. and us), to be developed below.There are some meta-properties of logics that are intuitively known to limitor increase expressiveness. Thus, the presence/absence of such properties can beused to test whether there can be or not an expressiveness relation between thegiven logics. A first one coming to mind is that nothing can be expressed in atrivial logic, so it cannot be more expressive than any logic. Another one has todo with the relation between expressiveness and computational complexity. Thisrelation has even been stated as the “Golden Rule of Logic” by van Benthemin [vB06, p. 119], where he says “gains in expressive power are lost in highercomplexity”. Nevertheless, the “Golden Rule” is not quite useful here, since weknow that in general neither a low expressiveness means low complexity, nora high complexity means high expressiveness. Nevertheless the complexity levels of decidability/undecidability can be use-ful for expressiveness comparisons: if a logic is decidable, then it cannot describeTuring machines, Post’s normal systems, or semi-Thue systems. Therefore, adecidable logic L cannot be more expressive than an undecidable logic L ′ , oth-erwise, L would not be decidable! For example, there are propositional logics whose complexity is in each arbitrary degreeof unsolvability (e.g. see [Gla69]). There can be equally expressive logics that, though both decidable, have very differentcomputational complexities (e.g. see [LB87]). .7 Adequacy criteria for expressiveness 20The third meta-property that could be useful when evaluating expressivenessrelations (except, naturally, when dealing with formula-logics) is the deductiontheorem. Though involved in many formulation issues, as we shall see, a logichas a deduction theorem when it has the capacity to express in the object lan-guage its deductibility relation. Thus, other things being equal, a logic havingthis capability is intuitively more expressive than another one lacking it. There-fore, it is desirable that an expressiveness relation carries with it the deductiontheorem, so that (a) below apparently should hold(a) if L is more expressive than L , and L has a deductiontheorem, then so does L .We have some issues here. Being formulation sensitive, it is complicated todefine in which circumstances the existence of a deduction theorem for a logicimplies its existence in another logic, whenever there is an expressiveness relationbetween them. For example, a less expressive logic might have the standarddeduction theorem, while the more expressive logic has only a general versionof it, or perhaps lacks it completely. This happens with Mendelson’s FOL , the propositional fragment of it still satisfies the standard deduction theorem,though it fails for quantified formulas. So, it does not seem reasonable to saythat this formulation of FOL is not more expressive than
CPL , because itdoes not satisfy the standard deduction theorem, since the fragment of
FOL asexpressive as
CPL satisfies it. Cases like these constrain us to limit the role of the deduction theorem inexpressiveness relations, admitting wider formulations of it. Thus we are forcedto adapt (a) accordingly so as to be able to take into account such phenomena.Finally, we have the meta-property related adequacy criterion.[
Adequacy Criterion 2 ] It cannot hold that L be more expressivethan L when • L is non trivial and L is trivial; • L is undecidable and L is decidable; • L satisfies the standard deduction theorem and the languagefragment of L purportedly as expressive as L does not satisfy(not even) the general deduction theorem;The last criterion reflects the intuition that expressiveness is a transitiverelation and there are logics that are more expressive than others.[ Adequacy Criterion 3 ] (Taken from [Kui14]) The expressivenessrelation should be a non-trivial pre-order, that is, it should be atransitive and reflexive relation, and there must be some pair oflogics L and L such that L is not at least as expressive as L .We now analyse with greater detail the criteria 1 and 2. To be defined below. A Hilbert-style first-order calculus with the generalization rule “from φ infer ∀ xφ ”. Formore, see [Men97, p. 76]. The same considerations apply to L TK described in [FNG10] and [Mor16]. .7 Adequacy criteria for expressiveness 21 We can understand this criterion as saying “every connective of L is definable in L ”. But the usual notion of definability is either treated within the same logic,or between different logics within the same class of structures. As we intendto deal with translations between logics, the usual notion of definability is toorigid. We must give a broader reading of the criterion 1 in order to understandit as imposing an intuitive restriction on translations between logics. Thus theidea is to impose restrictions P , P , ... on translations so that T : L −→ L satisfies P , P , ... only if, intuitively, everything thatcan be said in terms of the connectives of L can also be said interms of the connectives of L ; let us say in shorter terms that thishappens only if the connectives of L are generally preserved in L .In the sequence some candidates for such P , P , ... are listed, the back-and-forth condition was given before. Definition 3.8 (Compositional) . A translation T : L −→ L is compositionalwhenever for every n -ary connective of L there is an L -formula ψ suchthat T ( φ , ..., φ n )) = ψ ( T ( φ ) , ..., T ( φ n )) . Definition 3.9 (Grammatical) . A grammatical translation T is a back-and-forth compositional translation such that, for a sentence φ , T ( φ ) may containno other formulas other than the ones appearing in T ( p ) , where p appears in φ (thus, no parameters are allowed). Definition 3.10 (Definitional) . A definitional translation T is a grammaticaltranslation for which T ( p ) = p for every atomic p . We have four proposals for filling the above list of restrictions. All of themrequire basically two conditions, taking as P the back-and-forth condition. Indecreasing order of strictness, there is divergence in taking P as a1. definitional translation (W´ojcicki and Humberstone),2. grammatical translation (Epstein and apparently Koslow),3. general-recursive translation (to be defined below),4. surjective conservative translation (Mossakowski et al.).Humberstone [Hum05], recalling W´ojciki’s definitional translations and in-tuitions about expressiveness, guessed that if there is a definitional translationbetween L and L , then all connectives in L are preserved in L . For us, theexistence of a definitional translation from L to L is the strongest guaranteethat the connectives of L are generally preserved in L . Nevertheless, it is toostrict a requirement, and there are weaker forms of translations that can alsodo the job. However, it seems that in [Hum05, p. 147] he allows that connectives are preserved in aweaker way, through compositional translations. .7 Adequacy criteria for expressiveness 22For Epstein [Eps13, p. 302], a grammatical translation is a homomorphismbetween languages and thus it yields a translation of the connectives. Thejustification is that such translations are only possible when for each connectivein the source logic, there corresponds a specific structure in the target logic thatbehaves similarly. Thus, through a grammatical translation, the connectives ofthe source logic are generally preserved in the target logic. Koslow [Kos15, p.48] also allows that a connective from one logic L “persists” in L if there is ahomomorphism from L to L .According to Mossakowski et al. [MDT09], grammatical translations aretoo demanding for the task, as many useful and important translations are non-grammatical (e.g. the standard modal translation). For them, instead of seekingto preserve the structure of the formulas, it would be better to preserve theproof-theoretic behaviour of the connectives and to treat the connectives onlyas regards this behaviour ( ibid , p. 100). In this paper, some proof-theoreticconditions on the connectives are listed, e.g. for conjuntction the condition isΓ ⊢ φ ∧ ψ iff Γ ⊢ φ and Γ ⊢ ψ . This formulation may lead one to think that ∧ here shall be a logical constant, and not possibly a formula γ ( φ, ψ ) (think of CPL ↾ {¬ , ∨} , where γ ( φ, ψ ) = ¬ ( ¬ φ ∨ ¬ ψ )); naturally in the first case, the wholeproposal would make no sense. In table 1 we reformulate the conditions toreflect their proposal more clearly, where δ is an arbitrary formula that standsfor the connective δ ⊥ ( ξ ) ⊢ φ , for every φ conjunction Γ ⊢ δ ∧ ( φ, ψ ) iff Γ ⊢ φ and Γ ⊢ ψ disjunction δ ∨ ( φ, ψ ) , Γ ⊢ χ iff φ, Γ ⊢ χ and ψ, Γ ⊢ χ implication Γ ⊢ δ → ( φ, ψ ) iff Γ , φ ⊢ ψ negation Γ , φ ⊢ δ ⊥ ( ξ ) iff Γ ⊢ δ ¬ ( φ ).Table 1: Reformulation of proof theoretic connectives as given by Mossakowskiet al. Definition 3.11 (presence of a proof-theoretic connective) . A proof-theoreticconnective is present in a logic if it is possible to define the corresponding oper-ations on sentences satisfying the conditions given in table 1.
We shall now investigate this idea in detail and argue that, as it is, thepreservation of connectives would require mappings stricter than conservativetranslations otherwise the notion of the “presence” of a connective must berelaxed.
Drawbacks on the preservation of proof-theoretic connectives
A trans-lation T : L −→ L transports a given L -connective L implies its presence in L , the converse implication is called reflection [MDT09,p. 100]. It is claimed ( ibid ) that if a mapping T : L −→ L is conservativeand surjective, then all proof theoretic connectives of L are transported to L .7 Adequacy criteria for expressiveness 23and all proof-theoretic connectives present in L are reflected in L . However,this claim must be taken with a grain of salt, let us see why.Let L be a logic having a proof-theoretic conjunction according with thetable 1 above and suppose there is a surjective conservative mapping T : L −→L . For L -formulas δ , δ , let Γ ∪ { φ, ψ } be a set of L -formulas with T ( φ ) = δ and T ( ψ ) = δ . Then it holds that( T (Γ) ⊢ L T ( φ ) and T (Γ) ⊢ L T ( ψ )) iff Γ ⊢ L δ ∧ ( φ, ψ ) iff T (Γ) ⊢ L T ( δ ∧ ( φ, ψ )).Thus, L would have proof-theoretic conjunction. The grain of salt is that,once no structural restriction is imposed upon T , it is not necessary that T ( δ ∧ ( φ, ψ )) be constructed out of T ( φ ) and T ( ψ ). In this case, it seems atleast unnatural to say that T ( δ ∧ ( φ, ψ )) is an operation on the sentences T ( φ )and T ( ψ ).Therefore, we must relax what it means for a connective to be present in alogic. One has to say that e.g. the proof-theoretic conjunction is present in alogic L if, for all formulas δ , δ and set of formulas ∆, there is a formula γ such that (∆ ⊢ L δ and ∆ ⊢ L δ ) iff ∆ ⊢ L γ . A similar reformulation shouldbe given for the other connectives. In this case, though, whenever it holds that∆ ⊢ L δ and ∆ ⊢ L δ , then any L -theorem in the place of γ serves to satisfythis condition for conjunction.For example, take a Tarskian logic L defined on the signature { p, q, r, ⊤} ,where p, q, r are propositional variables and ⊤ the constant for logical truth.Then L has proof-theoretic conjunction since p, q ⊢ p and p, q ⊢ q holds iff p, q ⊢ ⊤ . This is probably unproblematic and a consequence of the meaning of ⊤ . Nevertheless, for some cases this approach to the presence of connectiveshas some downsides. For example, restrict L to the signature { p, ⊤} . Then L has the proof-theoretic conditional, since it holds that p ⊢ p iff ⊢ ⊤ , ⊤ ⊢ p iff ⊢ p , p ⊢ ⊤ iff ⊢ ⊤ and ⊤ ⊢ ⊤ iff ⊢ ⊤ .But if the signature were incremented by another variable q , then the resultingsystem would no longer have a proof-theoretic conditional, since for no δ itwould hold that p ⊢ q iff ⊢ δ . This volatility of the presence of proof-theoreticconnectives is unreasonable.Recapitulating, the idea of this approach is that one shall define the map-pings so as to preserve the proof-theoretic connectives, instead of requiring themappings themselves to preserve the structure of the formulas. But if the map-pings do not respect the structure of the formulas, what shall be called thepresence of a connective, must also be relaxed.Besides the inconvenients mentioned above, this proposal would be too re-strictive in some cases. For example, Statman’s translation [Sta79] of IPL intoits implicational fragment shows how can one “express” (in some sense of theterm) conjunctions using only implicational formulas; recent works have general-ized this result so that any logic having a certain natural deduction formulation.7 Adequacy criteria for expressiveness 24and having the sub-formula principle is translatable into the implicational frag-ment of minimal logic [Hae15]. Nevertheless, not even in the weaker sensegiven above the conjunctions are “present” in
IPL ↾ {→} .Anyway, it must be borne in mind that to give a good and general def-inition of when a connective or operator is generally preserved is a difficultand spinous topic. Below we give another proposal, which is at the same timeweaker (the translation mentioned above would enter) and stronger (requiresstructure-attentive mappings).Let us now consider the structure-attentive translations and think on theminimum conditions on the preservation of the structure of formulas that wouldallow for a reasonable and general notion of preservation of connectives. General-recursive translations: allowing context-sensitivity in a gen-eral preservation of connectives
The criterion of compositionality givenabove a priori seems a reasonable condition for the preservation of connectivesthrough translations. Notice that in the criterion the function T that translates φ , ..., φ n ) is the same that translates the sub-formulas φ i . From this comesthe compositionality: a translation T of a formula is obtained through the sametranslation T of its sub-formulas.Thinking about the issue of translating a connective, it is also reasonable thatthe translation be sensitive to the context where the connective is inserted. Thisis the case in the translation ( T + ) : Grz −→ S T + ( ¬ (cid:3) p ) = ¬ (cid:3) p , but T + ( (cid:3) p ) = (cid:3) ( (cid:3) ( p → (cid:3) p ) → p ). Therefore, T + distinguishes betweentranslating (cid:3) -formula and ¬ (cid:3) -formula, and this is done through the help of anauxiliary translation (see complete definition in section 3.27). Thus, there aretranslations between some logics where the mappings must be context-sensitive,so as to convey the proper meaning of some source connectives in the target logic.There are also those cases where the connectives can be dealt context--independently but auxiliary translations are needed anyway. The standardtranslation of modal logic to FOL , besides some parameters, needs n auxil-iary translations for each formula of modal degree n e.g. as T x ( p ) = P x but T x ( (cid:3) φ ) = ∀ y ( Rxy → T y ( φ )).For the sake of simplicity, we will restrict our notion of context-sensitivityto whether or not the connective to be translated is in the scope of an unaryoperator. When the translation of a n -ary connective ◦ , a simple solution is to treat ◦ n -ary connective to be translated. With the aim of capturing these cases, let usconsider a sufficiently general kind of translation.French in [Fre10] presents a concept of recursively interdependent translationthat includes non-compositional translations that are still defined recursivelythrough the formation of formulas. A generalization of his concept will beemployed here, since the original has an unmotivated restriction allowing only The idea of these translations is the following: for a given
IPL -formula φ , take all sub-formulas δ , δ and associate to it implicational axioms of the sort x δ ∧ δ → x δ , x δ ∧ δ → x δ and x δ → ( x δ → x δ ∧ δ ), where x δ , x δ and x δ ∧ δ are fresh variables. .7 Adequacy criteria for expressiveness 25unary auxiliary mappings. The generalization allows auxiliary mappings of anyarity and also has a simpler notation. Let L = ( F , ⊢ L ) and L = ( F , ⊢ L )be logics, Definition 3.12 (General-Recursive) . Let T ′ , ..., T ′ w be auxiliary mappings ofany arity defined inductively on F -formulas. A translation T : F −→ F from L to L is general-recursive if, for every n -ary connective and formulas φ , ..., φ n ∈ F , there is an L -formula T ( p , ..., p m ) containing only theshown propositional variables p , ..., p m , such that T ( φ , ..., φ n )) = T ( T ′ ( φ i , ..., φ j ) /p , ..., T ′ w ( φ h , ..., φ l ) /p m ) where { φ i , ..., φ j } ∪ { φ h , ..., φ l } ⊆ { φ , ..., φ n } . Notice that the clauses must be given for each single connective in the sourcelogic. If there is a need to translate a composite connective, an additional clausefor it should be given.Therefore, the general-recursive translations are still structure-preservingand must be defined inductively through the formation of formulas. Later in sec-tion 3.19.1 we argue that, together with some other conditions, general-recursivetranslations preserve, in a general but reasonable sense of the term, the connec-tives of the source in the target logic.
Another issue with translated connectives
One might insist whether thebehaviour of the defined connective in the target logic would indeed be equiva-lent with the behavior of the original connective. Corcoran argues that this isoften not the case. In [Cor69, p. 172] he defines a notion of “deductive strength”which is based on the capacity of a logic to introduce and eliminate a connectiveoccurring as a principal sign in a formula.Considering this notion, it can be that a given connective L be definable in a logic L through a translation, nevertheless, the “deductivestrength” of L as regards L . For ex-ample, consider the two classical propositional logics CPL ↾ {¬ , →} and CPL ↾ {¬ , ∧} ,formulated as natural deduction systems. Translating the conditional from CPL ↾ {¬ , →} to CPL ↾ {¬ , ∧} one obtains the following rule of inference: from thepattern of reasoning from φ to ψ , infer ¬ ( φ ∧ ¬ ψ ). Contrary to the rule for → in L (the usual natural deduction rule), according to Corcoran this rule of L is not rigorous, since it depends on the rules of the other connectives.Corcoran’s considerations are very interesting, but we think that despite thefact that the defined connectives can lose “deductive strength” (in his terms) it isreasonable to say that they maintain expressive strength. The loss of “deductivestrength” can influence other issues such as modularity, normalization, etc. butit does not affect directly expressiveness.Having revised the literature linked with the adequacy criterion 1 and statedour proposal, now the same will be done with respect to the adequacy criterion2..7 Adequacy criteria for expressiveness 26 There are two important issues here:(i) what is being understood as a meta-property(ii) what does it mean for a translation to preserve a meta-property of onelogic into anotherIt would seem desirable to have a general formal framework so that one couldgive precise answers to (i) and (ii). Nevertheless, the adequacy criterion 2 asksfor preservation of specific meta-properties, and not of every meta-property ofa certain kind. Thus, there is no need to place them in a fixed framework.Moreover the first two (non-triviality and decidability) have simple and exactformulations, so it is straightforward to stablish whether they are preserved bya translation.The only meta-property whose statement and definition of preservation needelucidation is the deduction theorem. As the framework(s) of (hyper) contextualtranslations offers exact answers to (i) and (ii) above, we will investigatewhether they are adequate for our purposes. In both, a logic is taken as anassertion calculus containing a set of formulas and a set of rules of inferencebetween sequents. The difference between the frameworks is the kind of sequentallowed. The language of the assertion calculus includes schematic variablesfor sentences ξ , ξ , ... and set of sentences X , X , ... , so the calculus has alsosubstitution and instantiation rules for dealing with those.In these frameworks, P is a meta-property of a logic L defined in the aboveterms whenever P can be formulated as an inference between sequents (or hyper-sequents), that is, if P can be formulated as a derived rule of L . For example, thededuction theorem can be formulated this way: from Γ , φ ⊢ ψ , infer Γ ⊢ φ → ψ .Now for the disjunctive property, one needs the richer framework of the hyper-sequents: from Γ ⊢ φ ∨ ψ , infer Γ ⊢ φ or infer Γ ⊢ ψ .A (hyper) contextual translation T : L −→ L is a mapping that is trans-parent to the schematic variables such that if P is a meta-property of L then, T ( P ) is a meta-property of L . The transparency to schematic variables impliesthat (hyper) contextual translations by definition preserve structural propertiessuch as left weakening: from X ⊢ ξ , infer X, X ′ ⊢ ξ .Consider now the finiteness property, i.e. if Γ ⊢ φ , then for a finite ∆ ⊆ Γ,∆ ⊢ φ . It cannot be formulated in neither of the cited frameworks, indeedthe same holds for the majority of other relevant meta-properties of logics:decidability, interpolation, cut-elimination, etc.Despite the limitation on expressible meta-properties, the framework of (hy-per) contextual translations has also a clear answer to item (ii) above: a trans-lation T preserves a meta-property P of the source logic if T ( P ) is a derivedrule of the target logic. See [CCD09], [CF15] and [Mor16] for a detailed presentation of both. .7 Adequacy criteria for expressiveness 27In the sequence we use the example of the deduction theorem to argue thatthere can be some problems even with this strict notion of meta-property preser-vation.
A limitation of the formulation of meta-property in the framework of(hyper) contextual translations
Even in the strict framework of (hyper)contextual translations, meta-properties are formulation-sensitive, so that oneformulation of a meta-property P may hold for a logic L while other formu-lation P ′ fails for L . A paradigmatic example is the deduction theorem. Inmost formulations, e.g. for classical propositional logic ( CPL ) and intuitionisticpropositional logic (
IPL ), it is read as: If Γ , φ ⊢ ψ , then Γ ⊢ φ → ψ . Never-theless, only a generalized version holds for Lukasiewicz L : If Γ , φ ⊢ ψ , thenΓ ⊢ φ → ( φ → ψ ) [Pog64].A similar issue occurs in systems containing proof-rules besides inference-rules, for example, Mendelson’s FOL and modal logic with the necessitationrule. In both cases, only a modified version of the deduction theorem holds. Formodal logic K , among other possibilities, the following deduction theorem holds[Zem67, p. 58]: if Γ , φ ⊢ K ψ , and each of the propositional variables appearingin hypothesis Γ ∪ { φ } is in the scope of a modal operator, then Γ ⊢ K φ → ψ . For Mendelson’s system, the formulation of the deduction theorem is alsoclumsy [Men97, p. 80]: “Assume that, in some deduction showing that Γ , φ ⊢ ψ ,no application of [the generalization rule] to a wff that depends upon φ has asits quantified variable a free variable of φ . Then, Γ ⊢ φ → ψ .”So what is a deduction theorem? According to Zeman, the general statementof it might be [Zem67, p. 56] Definition 3.13 (DT) . If there is a proof from the hypotheses φ , ..., φ n for theformula ψ , then there is a proof from the hypotheses φ , ..., φ n − for the formula φ n ⊃ ψ . For Zeman, the problem of the formulation of the deduction theorem foreach system lies in the proper understanding in the system of what it is meantby a “proof from hypotheses”. Thus, the different results cited above for L ,modal logic and Mendelson’s FOL are different ways —seemingly equivalentmodulo the specificities of each system— of capturing the idea of DT above.The situation is explained by Hakli and Negri [HN12] as follows. For some logics, either one modifies their rules in order for them to deal adequately withassumptions, and get the “standard formulation” of the deduction theorem, or A proof rule is of the form: from ⊢ φ , infer ⊢ ψ . An inference rule is of the form: from φ ,infer ψ . The necessitation rule and generalization rules are sometimes defined as proof-rules:from ⊢ φ , infer ⊢ (cid:3) φ ; from ⊢ ψ ( x ) infer ⊢ ∀ xψ ( x ). Notice that both φ and ψ ( x ) must betheorems in their respective systems, otherwise one gets implausible inferences: from p itfollows (cid:3) p , and from P ( x ) it follows that ∀ xP ( x ). Other formulation is given in [HN12]: if Γ , φ ⊢ K ψ , and the rule of necessitation is applied m ≥ φ , then Γ ⊢ K ( (cid:3) φ ∧ ... ∧ (cid:3) m φ ) → ψ , where (cid:3) φ = φ , (cid:3) φ = (cid:3) φ , etc. .7 Adequacy criteria for expressiveness 28leave the rules from the logic intact and obtain a “non-standard” form of thededuction theorem. Now let us come back to the issue of preservation of meta-properties bytranslations. Consider
IPL and modal logic S
4, presented in the frameworkof (hyper) contextual translations, e.g. both equipped with a common set ofpropositional variables p , p , ... and schematic variables ξ , ξ , ..., X , X , ... , forformulas and sets of formulas, respectively, etc. Consider G¨odel’s translation T g : IPL −→ S T g ( p i ) = (cid:3) p i T g ( X i ) = X i T g ( ξ i ) = ξ i T g ( ¬ φ ) = (cid:3) ¬T g ( φ ) T g ( φ → ψ ) = (cid:3) ( T g ( φ ) → T g ( ψ ))literal for ⊥ , ∧ , ∨ .Then the deduction theorem for IPL is defined as the following meta-property( P ) if X, ξ ⊢ ξ , then X ⊢ ξ → ξ .Carnielli et al. ([CCD09, p. 13]) notice that T g above is not a contextualtranslation since S T g ( P ) if X, ξ ⊢ ξ , then X ⊢ (cid:3) ( ξ → ξ ).To see why, instantiate X to p → p and ξ i to p i , for i ∈ { , } . In S p → p , p ⊢ p , but it does not hold that p → p ⊢ (cid:3) ( p → p ).One can see clearly that this is caused by the transparency given in T g tothe schematic variables of P . If P were formulated in terms of non-schematicformulas, e.g.( P ′ ) if Γ , p ⊢ p , then Γ ⊢ p → p ,then its translation T g ( P ′ ) into S T g ( P ′ ) if T g [Γ] , (cid:3) p ⊢ (cid:3) p , then T g [Γ] ⊢ (cid:3) ( (cid:3) p → (cid:3) p ),which is satisfied in S P is a correct formulation of DT (see above) for IPL , T g ( P ) isnot the correct formulation of DT for S
4. Therefore, the claim that contex-tual and hyper-contextual translations preserve the meta-properties of logics(expressible in the framework) is not entirely justified. The opacity given forthe schematic variables may give a“false negative” as regards the presence ofsome meta-property in the target logic, this would prevent the definition of suchtranslations. Therefore this framework is not adequate for our purposes.
General statement and preservation of the deduction theorem
Recallour discussion on the deduction theorem. We saw that there are many formu-lations of it, and it depends on how the notion of proof from assumptions istreated in each logic. Now to talk about the preservation of deduction theorem Although the moral of the story holds for first-order logic, as shown above, they only men-tioned modal logics. Nevertheless, we do not know of any such rectification for the formulationof the deduction theorem in Lukasiewicz L . .17 expressiveness gg : a sufficient condition for expressiveness 29through the translations, we have to give a sufficiently general formulation ofit, but such that it still carries the spirit of Zeman’s definition. Let us give it amore direct formulation: Definition 3.14 ( standard deduction theorem) . A logic L has the stan-dard deduction theorem whenever it holds that φ , ..., φ n ⊢ L ψ if and only if φ , ..., φ n − ⊢ L φ n → ψ . The general formulation has to be lax enough so as to enable one to say that,for example, the translation T l : CPL −→ L preserves the deduction theorem,since it holds that “if Γ , φ ⊢ L ψ , then Γ ⊢ L φ → ( φ → ψ )”; analogously forthe translation of IPL into S
4. The general version of the deduction theoremwe propose is the following:
Definition 3.15 ( general deduction theorem) . A logic L has the general deduction theorem whenever φ , ..., φ n ⊢ L ψ iff φ , ..., φ n − ⊢ L α → ( φ n , ψ ) ,where α → is an L -formula, with one or more occurrences of φ n and ψ . In abstract algebraic logic this formulation is known as the uniterm globaldeduction-detachment theorem [FJP03, p. 36].
Definition 3.16 (preservation of the general deduction theorem) . A translation T : L −→ L is said to preserve the general deduction theorem whenever L hasthe standard deduction theorem and T ( L ) has the general deduction theorem. The case where L satisfies only the general deduction theorem is morecomplex, as it will be seen below. expressiveness gg : a sufficient condition for expressive-ness In the adequacy criteria we proposed for expressiveness, there appears two in-formal necessary conditions: preserving the connectives and behaving in theappropriate way as regards the selected meta-properties. The other conditionof being a non-trivial pre-order is already given precisely. The first two con-ditions are open to interpretation, so we proposed a precise formulation of theminimal requirements such interpretations would have to satisfy. This amountedon requiring the translation to preserve the general deduction theorem, and tobe back-and-forth general-recursive.The one-way mappings between logics are a very weak in the sense theyare almost omnipresent, so that requiring back-and-forth mappings as a formalnecessary condition for expressiveness is rather uncontroversial. Nevertheless,requiring structure-attentive translations in order to preserve the connectiveshas been questioned by Mossakowski et al. as we have seen above. They pro-posed other way to preserve the connectives without requiring such translations.We think this approach has some downsides and proposed a different one, basedon general-recursive translations..17 expressiveness gg : a sufficient condition for expressiveness 30Being a general-recursive mapping is a relatively weak condition on transla-tions. If some translation does not comply with it is because at least some con-nective of the source logic is only translated “globally”, i.e. a formula containingit is translated as a whole, and the translation ignores its eventual sub-formulas,e.g. Glivenko’s double negation translation of CPL into
IPL [Gli29].Thus, it is reasonable to require general-recursiveness as a formal necessarycondition for expressiveness, along with the back-and-forth condition. One veryimportant issue to be dealt with in a future work is already pointed out byMossakowski et al. ([MDT09]): this minimal notion of structure preservationis up to now only defined for propositional logics. It is to be investigated howit should deal with quantifiers. Notice, however, that this limitation does notweaken the necessary character of general-recursive translations, as an eventualwider approach should include it.Before we present a sufficient formal criterion for our concept, whose contentsurely is no surprise by now, some adjustments must be made concerning thepreservation of the general deduction theorem. The general statement of thededuction theorem still involves compositionality: the formula α → ( φ n , ψ ) atissue should have φ n and ψ as sub-formulas. In order to assure this, we haveto define a slightly stricter notion of general-recursive translation, which we callgeneral-recursive C : Definition 3.18 (general-recursive C ) . A translation T is general-recursive C iff T is general-recursive and it is compositional for the conditional symbol, that is,for a formula φ → ψ in the source logic and a template-formula C T ( p , ..., p n ) inthe target logic, T ( φ → ψ ) = C T ( T ( φ ) , ..., T ( ψ )) . Thus the translated formulamay contain as sub-formulas one or more occurrences of T ( φ ) and T ( ψ ) . The restriction on general-recursive translations is intended to assure thatfor the logics satisfying the standard deduction theorem, at least the translationclause for the conditional is compositional. This will rule out clauses suchas T ( φ → ψ ) = C T ( T i ( φ ) , ..., T i n ( φ ) , ..., T j ( ψ ) , ..., T j n ( ψ )), for T ij differentfrom T . Otherwise, it could happen that the resulting translation C T of theconditional φ → ψ does not contain T ( φ ) and T ( ψ ) as sub-formulas. Then itwould not be reasonable to say that such formula C T expresses the deductibilityrelation between T ( φ ) and T ( ψ ).Now we present a sufficient criterion for expressiveness Definition 3.19 ( expressiveness gg ) . A logic L is at least as expressive gg as L if and only if there is a back-and-forth general-recursive (for short, B&F-GR)translation T from L to L , such that T does not require model-mappings. If L satisfy the standard deduction theorem, then T must be general-recursive C . Below it will be shown that expressiveness gg satisfies the adequacy criteriagiven above. As we mentioned before (section 3.7.1) there is some consensus in the literaturethat preservation of connectives requires at least compositional back-and-forth.17 expressiveness gg : a sufficient condition for expressiveness 31translations. The various non-compositional translations show that we couldhave a wider notion of preservation of connectives. We now argue that throughgeneral-recursive translations and some other conditions, it is still guaranteedthat whatever can be said in terms of the source connectives can be said interms of the target connectives. Proposition 3.20 (Connetive preservation (general sense)) . If a translation T : L −→ L is back-and-forth and general-recursive (B&F-GR), and T doesnot require model translations to convey the meaning of some connective in L ,then the connectives of L are preserved (in a general sense) in L . The back-and-forth condition shall mean either a theoremhood- or deriva-bility-preserving translation, depending on whether one is considering formulalogics or Tarskian logics, respectively. A back-and-forth translation assures acertain similarity between the global deductive behaviour of the source andtarget formulas. Though, as Jeˇr´abek’s result shows, it is not enough for anyreasonable notion of connective preservation and there must be some extent ofstructure preservation. In this sense, the advantage of compositional transla-tions is that they are particularly regular, so that each connective in the sourceis associated to a fixed schema in the target logic, and the translation clauses areclearer. But this can be also a limitation on the means of translation, compara-ble to restricting translations in ordinary language to word-to-word mappings.There are many cases of logics where the translation of certain operatorsmust consider their context, so that they have to be translated in block. We gavebefore some examples, for another one consider Balbani and Herzig’s [BH94]translation T bh of modal provability logic G into K4. For T bh the result oftranslating (cid:3) φ also depends on whether it occurs in the scope of a negation sign: T bh ( ¬ (cid:3) p ) = ¬ (cid:3) p , but T ( (cid:3) p ) = (cid:3) ( (cid:3) p → p ). These cases cannot be capturedin compositional translations and can only be dealt with in more complex non-compositional ones.The following clause is proposed as a refinement of adequacy criterion 1,capturing more precisely when a connective or group of connectives is generallypreserved by a translation from L to L .( α ) for each n -ary (composite) connective ⊗ in L and L -formulas φ , ..., φ n , there must be L -formulas δ ⊗ ( p , ..., p m ) (possibly m = n )and ψ , ..., ψ m such that ⊗ ( φ , ..., φ n ) has a similar deductive behaviourwith δ ⊗ ( ψ /p , ..., ψ m /p m ).It is easy to see that back-and-forth general-recursive (B&F-GR) transla-tions satisfy clause ( α ). In general-recursive translations, every connective ofthe source logic must be given inductive translation clauses, and translations forcomposite connectives may be given either as extra clauses, or by means of auxil-iary translations. Thus, for a n -ary (composite) connective ⊗ in L , the formula ⊗ ( φ , ..., φ n ) must be mapped by a GR-translation to a formula δ ⊗ ( ψ , ..., ψ m ),where each ψ i is obtained from the translation of some φ k . Now if the transla-tion is back-and forth, then ⊗ ( φ , ..., φ n ) will have a similar deductive behaviourwith δ ⊗ ( ψ , ..., ψ m ), since Γ ⊢ L ⊗ ( φ , ..., φ n ) iff T (Γ) ⊢ L δ ⊗ ( ψ , ..., ψ m )..17 expressiveness gg : a sufficient condition for expressiveness 32Nevertheless, the satisfaction of ( α ) is still not enough guarantee for generalpreservation of connectives, as Epstein’s translation T E : R −→ CPL we sawabove (section 2.9.1) is B&F-GR. We asserted then that the relatedness impli-cation “ → ” is not expressible in CPL . The issue is that T E uses the backdoorof the model-mapping to make the translated formula ( p ⊃ q ) ∧ d p,q true when-ever the source formula p → q is true. However, if uninterpreted, the translatedformula does not have the same meaning as the original formula. Thus, themeaning of the relatedness implication is not really expressed in terms of theconnectives of CPL .Thus, the last part of the proposition above is intended to force the sourceconnectives to be defined entirely in terms of the target connectives and notsmuggled by the model-translations. Therefore, if T : L −→ L is B&F-GR, then the clause ( α ) is satisfied. If be-sides the translations of the connectives are not aided by model-mappings, thenit is reasonable to say that everything expressible in terms of the connectives of L are expressible in terms of the connectives of L .In the beginning of the section we cited proposals for preservation of con-nectives via translations by W´ojcicki, Epstein and Mossakowski et al. The pro-posal above is much weaker than the first two. As regards the Mossakowski etal.’s, this approach is both weaker and stronger: stronger since it requires somepreservation of structure; and weaker since it does not require the preservationof the proof-theoretic connectives. In the adequacy criteria we asked that if a logic L hasthe standard deduction theorem, and L is at least as expressive as L , then thelanguage fragment of L as expressive as L has the general deduction theorem.We formulated above the standard deduction theorem and a general version ofit. As it will be seen below, in order to guarantee the preservation of the generaldeduction theorem, the source logic must have the standard deduction theorem.We fist remark that conservative general-recursive C translations preserve thegeneral deduction theorem: Proposition 3.21.
Let L with conditional symbol “ → ” satisfy the standard deduction theorem. If T : L −→ L is a conservative general-recursive C trans-lation, then T ( L ) has the general deduction theorem.Proof. Let the hypotheses of the proposition be satisfied.Then T ( φ ),..., T ( φ n ) ⊢ L T ( ψ ) iff φ , ..., φ n ⊢ L ψ iff φ , ..., φ n − ⊢ L φ n → ψ ,(by the standard deduction theorem) iff T ( φ ) , ..., T ( φ n − ) ⊢ L T ( φ n → ψ ). This might be seen as forcing the connectives in a certain logic to be given first anadequate set of axioms/rules of inference in order to be translatable. This would agree withZucker maxim that the meanings of the connectives must not be imposed from the outside[Zuc78, p. 518]. Nevertheless, this restriction does not prohibit non-axiomatizable model-theoretic logics to be translated into each other. For example, the identity mapping from L ( Q ) to L ( Q , Q ) is a perfectly reasonable translation and would comply with the criterionabove. .17 expressiveness gg : a sufficient condition for expressiveness 33By the definition of general-recursive C translations, T ( φ n → ψ ) is a formulacontaining one or more occurrences of T ( φ n ) and T ( ψ ). Thus, the image of L under T have a general deduction theorem.To preserve the general deduction theorem the source logic must have thestronger one. If L only satisfies the general deduction theorem and T : L −→L is B&F-GR, we cannot guarantee that T ( L ) satisfies the general deductiontheorem.If L satisfies the general deduction theorem, then φ , ..., φ n ⊢ L ψ iff itholds that φ , ..., φ n − ⊢ L δ → ( φ n , ψ ), for some L -formula δ → . Then we havethat T ( φ ) , ..., T ( φ n ) ⊢ L T ( ψ ) iff T ( φ ) , ..., T ( φ n − ) ⊢ L T ( δ → ). As δ → con-tains φ n and ψ as sub-formulas and T is general-recursive, T ( δ → ) will contain T j ( φ n ) , ..., T jn ( ψ ) as sub-formulas. If T j , ..., T jn are equal to T , then the com-positionality of the deduction theorem is saved and T ( L ) has also a generaldeduction theorem. Else, if T j , ..., T jn are distinct from T , then the general-deduction theorem is not preserved in T ( L ).Therefore the present approach is limited in that the source logics to beanalysed in terms of expressiveness have to be put in “proper form” so thatthey satisfy the standard deduction theorem or an even wider version of it mustbe defined, dropping the compositionality requirement. Non-triviality
A logic L is non-trivial if for some L -formulas φ and ψ it holdsthat φ L ψ . By the adequacy criteria, a trivial logic cannot be more expressivethan any logic. Thus, we have to make sure that a translation intended to induceexpressiveness must reflect triviality or, alternatively, preserve non-triviality.That is, if T : L −→ L , and L is trivial, then L is trivial. Proposition 3.22 (triviality reflection) . All back-and-forth translations reflecttriviality.
Undecidability
We commented before that the presence of decidability couldbe used as an indicator of the adequacy of our definition. This is becausedecidability is a limitation of expressiveness of a logic. Thus the conditionrequires thatif L is undecidable and L is decidable, then L is not more expres-sive than L .The result below shows this condition is normally satisfied. Proposition 3.23 ([FD01]) . If L is undecidable, then there is no computableback-and-forth translation T : L −→ L , where L is decidable. If a logic L is undecidable, L is decidable and T : L −→ L is B&F, thenit follows that T would not be computable. In this case, T apparently wouldnot be general-recursive..17 expressiveness gg : a sufficient condition for expressiveness 34 The non-triviality part of the pre-order is alreadyfulfilled by a proposition above: there are two logics L , L , such that L is non-trivial, L is trivial and there is no back-and-forth translation from L to L .We have to prove that back-and-forth general-recursive (recall the abbreviationB&F-GR) translations are transitive, that they are reflexive is clear.If T : L −→ L is surjective back-and-forth and T ′ : L → L is back-and-forth, then T ′ ◦ T : L −→ L is a back-and-forth translation [FD01].Now if T : L −→ L and T ′ : L −→ L are both B&F-GR, being T additionally surjective, then T ′ ◦ T is also B&F-GR. To see it, let C , ..., C n and C ′ , ..., C ′ n be the translation clauses for T and T ′ , respectively. By surjectivity,each L -formula is reached by some L -formula through the applications of C , ..., C n . Thus to obtain a general-recursive mapping, is just to combine theapplication of the two set of clauses. Take a L -formula φ and obtain through C , ...C n an L -formula T ( φ ). Now apply C ′ , ..., C ′ n to T ( φ ) to obtain an L -formula T ′ ( T ( φ )). This translation is B&F, and is general-recursive C , since itis obtained through the clauses C , ..., C n , C ′ , ..., C ′ n .Nevertheless, the surjectiveness requirement may be difficult to comply with,if one is comparing increasingly expressive logics (through language extension),e.g. propositional logic, modal logic, first-order logic. We should find a way toguarantee that whenever T : L −→ L and T ′ : L −→ L are B&F-GR, thenthere is a B&F-GR translation T ∗ (not necessarily T ′ ◦ T ) from L to L .Let us suppose T : L −→ L is a non-surjective B&F-GR and that T ′ : L −→ L is B&F-GR. There is naturally a weakened version T w of T ′ ◦ T , theweaker part being in the way back of L -formulas to L -formulas. That is, for L -formulas φ , ψ , it holds that if φ ⊢ L ψ , then T ′ ( T ( φ )) ⊢ L T ′ ( T ( ψ )).But the converse direction only holds partially.In order that the converse direction hold, instead of normally taking L -formulas φ , ψ in the range of T ′ ( L ), one has to take L -formulas in theintersected range of T ′ ◦ T . For such L -formulas φ , ψ , with T ′ ◦ T ( φ ) = φ and T ′ ◦ T ( ψ ) = ψ for some L -formulas φ , ψ , it holds that if φ ⊢ L ψ ,then φ ⊢ L ψ .But this is exactly what we wanted. The backward direction should hold onlyfor those L formulas that are linked with L -formulas through L -formulas. Totranslate L -formulas into L formulas T w takes only the T ′ -clauses C ′ , ...C ′ n involved in translating T ( L ) formulas.Thus, this weakened version of T ′ ◦ T will suffice for us to conclude thatwhenever there are B&F-GR translations T : L −→ L and T ′ : L −→ L ,then there is a B&F-GR translation T w : L −→ L .There remains the question whether this T w will preserve the general de-duction theorem. This will happen whenever T is compositional for “ → ” andthe translation clause of T ′ for the formula T ( φ → ψ ) is compositional. Then T ′ ( T ( φ → ψ )) is an L -formula containing T ′ ( T ( φ )) and T ′ ( T ( ψ )) as sub-formulas, which implies that T w preserves the general deduction theorem.Therefore we have that.27 Corroborating expressiveness gg : the structure preserving translations35 Proposition 3.24.
Back-and-forth general-recursive translations form a non-trivial pre-order on logics.
From the above propositions we can conclude that
Corollary 3.25.
Every back-and-forth general-recursive translation preservesthe connectives and the selected meta-properties (undecidability, non-trivialityand general deduction theorem) and form a non-trivial pre-order on logics.
This implies that
Corollary 3.26.
Every back-and-forth general-recursive translation not aidedby model-mappings agrees with adequacy criteria 1,2 and 3.
Many well known translations intuitively giving rise to an expressivenessrelation satisfy expressiveness gg . In the sequence, we briefly present them. expressiveness gg : the structure preserv-ing translations For the sake of supporting our notion of translational expressiveness, there fol-lows some translations obeying the criterion that are reasonably taken as induc-ing an expressiveness relation. • (W´ojcicki) from CPL into L : T l ( p i ) = p i T l ( ¬ φ ) = T l ( φ ) → ¬T l ( φ ) T l ( φ → ψ ) = T l ( φ ) → ( T l ( φ ) → T l ( ψ )) • (Gentzen) from classical first-order logic CL into intuitionistic first-orderlogic ( IL ), and also from CL to Minimal first-order logic ( M ) [PM68, p.218]: T c ( P t ...t n ) = ¬¬ P t ...t n T c ( φ ∨ ψ ) = ¬ ( ¬T c ( φ ) ∧ ¬T c ( ψ )) T c ( ∃ xφ ) = ¬∀ x ¬T c ( φ ) literal for ⊥ , ∧ , → and ∀ ; • (G¨odel) from IL to CL extended with the modal system S T s ( R i t ...t n ) = (cid:3) R i t ...t n T s ( ∀ xφ ) = (cid:3) ∀ x ( T s ( φ )) T s ( φ → ψ ) = (cid:3) ( T s ( φ ) → T s ( ψ )) literal for ⊥ , ∧ , ∨ and ∃ ; • (Prawitz and Malmn¨as) from IL to M (for ∈ {∧ , ∨ , →} ): T m ( R i t , ..., t n ) = R i t , ..., t n ∨ ⊥ T m ( ∀ xφ ) = ∀ x ( T m ( φ ) ∨ ⊥ ) T m ( φ ψ ) = ( T m ( φ ) T m ( ψ )) ∨ ⊥ T m ( ⊥ ) = ⊥ ; M is the intuitionistic logic without the rule of ex falso quodlibet. An interesting resultdue to Luiz Carlos Pereira and Herman Haeusler [PH16] is that this translation maps CL to any intermediate logic between M and CL . • (Demri and Gor´e) from Grz to S T + ( (cid:3) φ ) = (cid:3) ( (cid:3) [ T + ( φ ) → (cid:3) T − ( φ )] → T + ( φ )) T − ( (cid:3) φ ) = (cid:3) T − ( φ ) T + ( ¬ φ ) = ¬T − ( φ ) T − ( ¬ φ ) = ¬T + ( φ ) T + ( φ → ψ ) = T − ( φ ) → T + ( ψ ) T − ( φ → ψ ) = T + ( φ ) → T − ( ψ ) T + and T − are literal for ∧ and atomic formulas; • (Van Benthem) Standard translation from modal logic to FOL : T x ( p i ) = P i x T x ( ♦ φ ) = ∃ y ( Rxy ∧ T y ( φ )) T x ( (cid:3) φ ) = ∀ y ( Rxy → T y ( φ )) literal for ¬ , ∧ , ∨ , → , ⊥ . The commonly used precise notions of expressiveness are defined within a frame-work which is based on the capacity of characterizing structures, thus they applyonly to model-theoretic logics. As this framework of expressiveness is definedonly with respect to logics sharing the same class of structures, it was called inthis text “single-class expressiveness”. This framework can be seen as consistingof certain formula-mappings between model-theoretic logics.We saw two formal criteria for expressiveness, due to Garc´ıa-Matos andV¨a¨an¨anen and Kuijer, constructed in a wider framework which we called “multi-class expressiveness”. This wider framework encompasses besides formula-map-pings, also model-mappings. We argued that both criteria are inadequate formulti-class expressiveness. Then it was defended that moving to an even broaderframework might be more promising, this is because the possibility of usingmodel-mappings, as it happens with the counter-examples presented, opens abackdoor for “undesirable” translations. In the broader framework, which wecalled “translational expressiveness”, a criterion for expressiveness would lacksemantic notions and be based exclusively in terms of the existence of certainformula-mappings preserving the consequence relations of the logics at issue.A proposal in this direction due to Mossakowski et al. was analysed and criti-cized, since it also over-generates. Studying the reasons for the over-generation,we proposed some adequacy criteria for relative expressiveness and a formalcriterion of translational expressiveness satisfying them. The criterion is stilllimited in some aspects, as the notion of a structure-preserving translation is upto now only precisely defined with respect to propositional logics. The definitionof structure-preserving translation is only intuitively extrapolated to quantifiers,that is, one would normally recognize a structure-preserving translation clausefor a quantifier, though there is still no formal definition of it. Therefore, a trulybroad formal criterion for expressiveness encompassing preservation of quanti-fiers is sill wanting, and we leave it for a future work. As regards this limitation, it is curious to see that in Lindstr¨om’s characterization of first-order logic as the most expressive logic satisfying countable compactness and the L¨owenheim- eferences 37
References [AA17] Juan C. Agudelo-Agudelo. Translating non-classical logics into clas-sical logic by using hidden variables.
Logica Universalis , 11(2):205–224, 2017.[AFFM11] Carkis Areces, Diego Figueira, Santiago Figueira, and Sergio Mera.The expressive power of memory logics.
The Review of SymbolicLogic , 4(2):290–318, 2011.[Bar74] Jon Barwise. Axioms for abstract model theory.
Annals of Mathe-matical Logic , 7(2-3):221 – 265, 1974.[B´ez99] Jean-Yves B´eziau. Classical negation can be expressed by one ofits halves.
Logic Journal of the Igpl , 7(2):145–151, 1999.[BF85] Jon Barwise and Solomon Feferman, editors.
Model-theoretic logics .Perspectives in mathematical logic. Springer-Verlag, 1985.[BH94] Philippe Balbiani and Andreas Herzig. A translation from themodal logic of provability into k4.
Journal of Applied Non-ClassicalLogics , 4(1):73–77, 1994.[BHT06a] Jan Broersen, Andreas Herzig, and Nicolas Troquard. Embeddingalternating-time temporal logic in strategic stit logic of agency.
Journal of Logic and Computation , 16:559–578, 2006.[BHT06b] Jan Broersen, Andreas Herzig, and Nicolas Troquard. From coali-tion logic to stit.
Electronic Notes in Theoretical Computer Science ,157:23–35, 2006.[CC02] M.E. Coniglio and W.A. Carnielli. Transfers between logics andtheir applications.
Studia Logica , 72(3):367–400, 2002.[CCD09] Walter A. Carnielli, Marcelo E. Coniglio, and Itala M.L.D’Ottaviano. New dimensions on translations between logics.
Log-ica Universalis , 3(1):1–18, 2009.[CF15] M. E. Coniglio and M. Figallo. A formal framework for hyper-sequent calculi and their fibring. In Arnold Koslow and ArthurBuchsbaum, editors,
The Road to Universal Logic: Festschrift for50th Birthday of Jean-Yves B´eziau Volume I , pages 73–93. SpringerInternational Publishing, 2015.[CK90] C.C. Chang and H.J. Keisler.
Model Theory . Studies in Logic andthe Foundations of Mathematics. Elsevier Science, 1990.
Skolem theorem [Lin69], the notion of logic used does not have anything like a quantifier.Perhaps this is due to the wide interpretation of “quantifier” as a class of structures. Some-thing like quantification is only required to prove a characterization with respect to the upwardTarski-L¨owenheim-Skolem theorem [Lin74]. Only after sometime a proper “quantifier prop-erty” was required of an abstract logic extending first-order logic [Bar74]. eferences 38[Con05] M.E. Coniglio. Towards a stronger notion of translation betweenlogics.
Manuscrito , 28(2):231–262, 2005.[Cor69] J. Corcoran. Three logical theories.
Philosophy of Science ,36(2):153–177, 1969.[DG00] St´ephane Demri and Rajeev Gor´e. An O (( n · log n ) )-time trans-formation from grz into decidable fragments of classical first-orderlogic. In Ricardo Caferra and Gernot Salzer, editors, AutomatedDeduction in Classical and Non-Classical Logics: Selected Papers ,pages 152–166. Springer Berlin Heidelberg, Berlin, Heidelberg,2000.[Eps13] R.L. Epstein.
The Semantic Foundations of Logic Volume 1: Propo-sitional Logics . Nijhoff International Philosophy Series. SpringerNetherlands, 2013. Contributors: Carnielli, W. and d’Ottaviano,I.M. and Krajewski, S. and Maddux, R.D.[FD01] H´ercules A. Feitosa and Itala M. Loffredo D’Ottaviano. Conserva-tive translations.
Annals of Pure and Applied Logic , 108(1-13):205 –227, 2001. XI Latin American Symposium on Mathematical Logic.[Fel90] Matthias Felleisen. On the expressive power of programming lan-guages. In Neil Jones, editor,
ESOP ’90: 3rd European Symposiumon Programming Copenhagen, Denmark, May 15–18, 1990 Proceed-ings , pages 134–151. Springer Berlin Heidelberg, Berlin, Heidelberg,1990.[FJP03] J. M. Font, R. Jansana, and D. Pigozzi. A survey of abstractalgebraic logic.
Studia Logica , 74(1):13–97, 2003.[FNG10] H´ercules Feitosa, Mauri Nascimento, and Maria Gr´acio. Logic tk:Algebraic notions from tarski’s consequence operator.
Principia:an international journal of epistemology , 14(1):47–70, 2010.[Fre10] Rohan French.
Translational Embeddings in Modal Logic . PhDthesis, Monash University, Melbourne, Australia, 2010. Availableat http://rohan-french.github.io/writing/phd-thesis.pdf .[GH96] Olivier Gasquet and Andreas Herzig. From classical to normalmodal logics. In Heinrich Wansing, editor,
Proof Theory of ModalLogic , pages 293–311. Springer Netherlands, Dordrecht, 1996.[GJ05] Valentin Goranko and Wojciech Jamroga. Comparing semanticsof logics for multi-agent systems. In
Information, Interaction andAgency , pages 77–116. Springer Netherlands, Dordrecht, 2005.[Gla69] M. D. Gladstone. Some ways of constructing a propositional cal-culus of any required degree of unsolvability.
Journal of SymbolicLogic , 34(3):505–506, 1969.eferences 39[Gli29] V. I. Glivenko. Sur quelque points de la logique de m. brouwer.
Bulletin de la classe des sciences , 15(4), 1929.[GMV07] Marta Garc´ıa-Matos and Jouko V¨a¨an¨anen. Abstract model theoryas a framework for universal logic. In Jean-Yves Beziau, editor,
Logica Universalis , pages 19–33. Birkh¨auser Basel, 2007.[G¨od01] Kurt G¨odel. On intuitionistic arithmetic and number theory -1933e. In K. G¨odel, S. Feferman, J.W. Dawson, S.C. Kleene,G. Moore, R. Solovay, and J. van Heijenoort, editors,
Kurt G¨odel:Collected Works: Volume I: Publications 1929-1936 , CollectedWorks. OUP USA, 2001.[Gor10] Daniele Gorla. Towards a unified approach to encodability and sep-aration results for process calculi.
Information and Computation ,208(9):1031 – 1053, 2010.[Hae15] Edward Hermann Haeusler. Propositional logics complexity andthe sub-formula property. In Ugo Dal Lago and Russ Harmer, ed-itors,
Proceedings Tenth International Workshop on Developmentsin Computational Models , 2015.[HN12] Raul Hakli and Sara Negri. Does the deduction theorem fail formodal logic?
Synthese , 187(3):849–867, 2012.[Hum00] Lloyd Humberstone. Contra-classical logics.
Australasian Journalof Philosophy , 78(4):438–474, 2000.[Hum05] Lloyd Humberstone. B´eziau’s translation paradox.
Theoria ,71(2):138–181, 2005.[Jeˇr12] E. Jeˇr´abek. The ubiquity of conservative translations.
The Reviewof Symbolic Logic , 5:666–678, 12 2012.[Kos15] Arnold Koslow. Implicit definitions, second order quantifiers andthe robustness of logical operators. In A. Torza, editor,
Quanti-fiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics,and Language , Synthese Library. Springer International Publishing,2015.[Kui14] Louwe Bart Kuijer.
Expressivity of Logics of Knowl-edge and Action . PhD thesis, University of Gronin-gen, Groningen, Netherlands, 2014. Available at .[LB87] Hector J. Levesque and Ronald J. Brachman. Expressiveness andtractability in knowledge representation and reasoning.
Computa-tional Intelligence , 3(1):78–93, 1987.eferences 40[Lin69] P. Lindstr¨om. On extensions of elementary logic.
Theoria , 35(1):1–11, 1969.[Lin74] P. Lindstr¨om. On characterizing elementary logic. In S. Sten-lund, A.-M. Henschen-Dahlquist, L. Lindahl, L. Nordenfelt, andJan Odelstad, editors,
Logical Theory and Semantic Analysis: Es-says Dedicated to STIG KANGER on His Fiftieth Birthday , pages129–146. Springer Netherlands, Dordrecht, 1974.[Man96] M. Manzano.
Extensions of First-Order Logic . Cambridge Tracts inTheoretical Computer Science. Cambridge University Press, 1996.[MDT09] Till Mossakowski, Razvan Diaconescu, and Andrzej Tarlecki. Whatis a logic translation?
Logica Universalis , 3(1):95–124, 2009.[Men97] E. Mendelson.
Introduction to Mathematical Logic, Fourth Edition .Chapman and Hall, London, 1997.[Mes89] Jos´e Meseguer. General logics. In H.D. Ebbinghaus, J. Fernandes-Prida, M. Garrido, D. Lascar, and M. Rodrigues Artalejo, edi-tors,
Logic Colloquium ’87: Proceedings of the Colloquium Held inGranada, Spain, July 20-25, 1987 . North-Holland, 1989.[Mor16] Angela P. R. Moreira.
Sobre tradu¸c˜oes entre l´ogicas: rela¸c˜oes entretradu¸c˜oes conservativas e tradu¸c˜oes contextuais abstratas . PhD the-sis, Instituto de Filosofia e Ciencias Humanas, UNICAMP, Camp-inas, 2016.[OP10] M. Otto and R. Piro. A lindstr¨om characterisation of the guardedfragment and of modal logic with a global modality. In L. Beklem-ishev, V. Goranko, and V. Shehtman, editors,
Advances in ModalLogic, Volume 8 . College Publications, 2010.[Par08] Joachim Parrow. Expressiveness of process algebras.
ElectronicNotes in Theoretical Computer Science , 209:173 – 186, 2008.[Pet12] Kristin Peters.
Translational Expressiveness- Comparing Pro-cess Calculi using Encodings . PhD thesis, Tenchichen Universit¨atBerlin, Berlin, Germany, 2012.[PH16] Luiz Carlos Pereira and Edward Hermann Haeusler. Two basic re-sults on translations between logics. Unpublished typescript, 2016.[PM68] D. Prawitz and P. E. Malmn¨as. A survey of some connectionsbetween classical, intuitionistic and minimal logic. In H. ArnoldSchmidt, K. Sch¨utte, and H. J. Thiele, editors,
Contributions toMathematical Logic, Proceedings of the Logic Colloquium, Han-nover 1966 , pages 215–229. North-Holland Publishing Company,1968.eferences 41[Pog64] Witold A. Pogorzelski. The deduction theorem for lukasiewiczmany-valued propositional calculi.
Studia Logica: An InternationalJournal for Symbolic Logic , 15:7–23, 1964.[Sha91] S. Shapiro.
Foundations without Foundationalism : A Case forSecond-Order Logic: A Case for Second-Order Logic . Oxford LogicGuides. Clarendon Press, 1991.[Sta79] Richard Statman. Intuitionistic propositional logic is polynomial-space complete.
Theoretical Computer Science , 9(1):67 – 72, 1979.[Tar86] A Tarlecki. Bits and pieces of the theory of institutions. In
Proceed-ings of a Tutorial and Workshop on Category Theory and ComputerProgramming , pages 334–363, New York, NY, USA, 1986. Springer-Verlag New York, Inc.[Tho74a] S. K. Thomason. Reduction of tense logic to modal logic. i.
J.Symbolic Logic , 39(3):549–551, 09 1974.[Tho74b] S. K. Thomason. Reduction of tense logic to modal logic ii.
Theoria ,40(3):154–169, 1974.[vB06] Johan van Benthem. Where is logic going, and should it?
Topoi ,25(1):117–122, 2006.[vBTCV09] J. van Benthem, B. Ten Cate, and J. V¨a¨an¨anen. Lindstr¨om theo-rems for fragments of first-order logic.
Logical Methods in ComputerScience , 5(3), 2009.[W´oj88] R. W´ojcicki.
Theory of Logical Calculi: Basic Theory of Conse-quence Operations . Synthese Library. Springer Netherlands, 1988.[Zem67] J. Jay Zeman. The deduction theorem in s4, s4.2, and s5.
NotreDame Journal of Formal Logic , 8(1,2):56, 1967.[Zuc78] J. I. Zucker. The adequacy problem for classical logic.