## New dimensions on translations between logics (2009)

### BibTeX

@MISC{Carnielli09newdimensions,

author = {Walter A. Carnielli and Marcelo E. Coniglio and Itala M. L. D’ottaviano},

title = {New dimensions on translations between logics},

year = {2009}

}

### OpenURL

### Abstract

After a brief promenade on the several notions of translations that appear in the literature, we concentrate on three paradigms of translations between logics: conservative translations, transfers and contextual translations. Though independent, such approaches are here compared and assessed against questions about the meaning of a translation and about comparative strength and extensibility of a logic with respect to another.

### Citations

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(Show Context)
Citation Context ... translations in the sense of Prawitz, Wójcicki and Epstein. By its turn, Gödel’s ones are translations only in Prawitz’ sense (cf. Feitosa and D’Ottaviano [17]). On the other hand, in [23] (see also =-=[24]-=- and [32]), Goguen and Burstall define a general notion of logic system and his morphisms called institutions, within the framework of Category Theory. Institutions generalize Tarski’s notion of truth... |

36 |
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Citation Context ...formulas are called “translations”, but we may say that Szczerba is only concerned with translations between models, and so do not coincide with the approach described in Section 2 below. Wójcicki in =-=[36]-=- and Epstein in [15] can be considered as the first proposals towards a general systematic study on translations between logics. For Wójcicki, logics are seen as algebras with consequence operators: a... |

29 |
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Citation Context ...uage, acting as a kind of sequent calculus whose rules govern the consequence relation of the logic. This accords with the idea of inferential basis of a logic introduced by J. ̷Lo´s and R. Suszko in =-=[30]-=-. In this section we present a simplified version of the definitions and results given in [6]. Consider a set X = {Xi : i ∈ N} of set variables, a set Ξ = {ξi : i ∈ N} of schema variables and a set V ... |

26 |
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Citation Context ...ility if, for every set Γ ∪ {α} of formulas in S1, Γ ⊢S1 α if and only if t(Γ) ⊢S2 t(α), where t(Γ) = {t(β) : β ∈ Γ}. The concept of schematic translation is also defined in [33]. Brown and Suszko in =-=[3]-=- construct “a general framework of the theory of abstract logics”, concerned with algebraic properties of abstract logics of a same similarity type. Though motivating, they are not interested in the s... |

17 |
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(Show Context)
Citation Context ... consistency of classical arithmetic with respect to intuitionistic arithmetic.2 W.A. Carnielli, M.E. Coniglio and I.M. Loffredo D’Ottaviano Still related to intuitionism, Glivenko in 1929 proves in =-=[20]-=- that, if α is a theorem of classical propositional logic (CPL), the double negation of α is a theorem of intuitionistic propositional logic (IPL). In 1932, during the Mathematical Colloquium held in ... |

17 |
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(Show Context)
Citation Context .... For him, this result attests that intuitionistic number theory and arithmetic are only apparently weaker than the classical versions, and that the former “contain” the latter. He also introduces in =-=[22]-=- an interpretation i that preserves theoremhood from IPL into a system G, which is “equivalent” to Lewis’ system of strict implication S4 plus an additional axiom. The aim of Gentzen in [19] is to sho... |

15 |
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(Show Context)
Citation Context ... with our intuition (shared with [3], [9] and [25]) that topological spaces can be seen as particular cases of logics. Other developments of the wider notion of translation sprung forth: Carnielli in =-=[4]-=- proposed a new approach to formal semantics for non-classical logics using translations, the so-called possible-translations semantics, further investigated by 3The logic co-induced by A, Q is 〈A�∼ ,... |

15 |
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Citation Context ...d in Vienna, Gödel (apparently not aware of [29]) proved that if α is a theorem of CPL then, under a specific translation G, the interpretation G(α) of α is a theorem of Heyting’s IPL. Gödel shows in =-=[21]-=- that this result is also valid relatively to intuitionistic arithmetic and classical number theory. For him, this result attests that intuitionistic number theory and arithmetic are only apparently w... |

14 |
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(Show Context)
Citation Context ...traditional methods. Víctor Fernández, in his Ph.D. Thesis [18] used translations in order to investigate combinations of logics, more particularly fibring of logics (cf. Carnielli, Coniglio et al in =-=[5]-=-). 3. Transfers: a model-theoretic dimension to translations The notion of isomorphism is the indiscernibility principle among algebraic structures, as much as elementary equivalence is the indiscerni... |

14 |
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(Show Context)
Citation Context ...ations, transformations, translations... The method of studying inter-relations between logic systems by the analysis of translations between them was originally introduced by Kolmogorov in 1925 (see =-=[29]-=-). The first known ‘translations’ involving classical logic, intuitionistic logic and modal logic were presented by Kolmogorov, Glivenko, Lewis and Langford, Gödel and Gentzen, most of them developed ... |

13 |
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Citation Context ...New dimensions 3 Nevertheless, this paper can be considered as an important landmark in the development of the theory of logical translations, anticipating some concepts and results later studied in =-=[8]-=- in a more general setting (see Section 2 below). Szczerba in [35] defines the concept of interpretation function, that “map structures onto structures”. The corresponding functions mapping formulas t... |

11 | Transfers between logics and their applications
- Coniglio, Carnielli
(Show Context)
Citation Context ... aim of the notion of transfer. This section briefly explains the idea of transfers and the underlying modeltheoretic approach to translations between logics as developed by Coniglio and Carnielli in =-=[7]-=- (from where all definitions and results in this section are taken). The basic idea is to consider a formal first-order meta-language to express theoretical (meta)properties of (propositional) logics.... |

11 |
The Semantic Foundations of Logic, Volume 1: Propositional Logics
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(Show Context)
Citation Context ...“translations”, but we may say that Szczerba is only concerned with translations between models, and so do not coincide with the approach described in Section 2 below. Wójcicki in [36] and Epstein in =-=[15]-=- can be considered as the first proposals towards a general systematic study on translations between logics. For Wójcicki, logics are seen as algebras with consequence operators: a logic (A, C) is suc... |

11 | What is a logic
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(Show Context)
Citation Context ...ions in the sense of Prawitz, Wójcicki and Epstein. By its turn, Gödel’s ones are translations only in Prawitz’ sense (cf. Feitosa and D’Ottaviano [17]). On the other hand, in [23] (see also [24] and =-=[32]-=-), Goguen and Burstall define a general notion of logic system and his morphisms called institutions, within the framework of Category Theory. Institutions generalize Tarski’s notion of truth, by cons... |

7 |
On the relation between intuitionist and classical arithmetic
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(Show Context)
Citation Context ...oduces in [22] an interpretation i that preserves theoremhood from IPL into a system G, which is “equivalent” to Lewis’ system of strict implication S4 plus an additional axiom. The aim of Gentzen in =-=[19]-=- is to show that “the applications of the law of double negation in proofs of classical arithmetic can in many instances be eliminated”. He introduces a “transformation” t from the language of CPL int... |

6 |
Recherches sur la Logique Universelle (excessivite, negation
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(Show Context)
Citation Context ... ; {a} ∈ P for all a ∈ A; and if Γ ∈ P then {a ∈ A : Γ ⊢L a} ∈ P . It is not hard to see that standard abstract logics are defined along the same lines as Béziau’s Universal Logic (see, for instance, =-=[1]-=-), that is: a logic is basically a pair formed by a set of entities called formulas and a consequence relation, without assuming any properties. As expected, function symbols of L ′ of type form n −→f... |

6 |
Semânticas de Traduções Posśıveis (Possible Translations Semantics
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(Show Context)
Citation Context ... Q is 〈A�∼ , C�∼ 〉 such that a set X ⊆ A�∼ is closed iff the set {x ∈ A : x�∼ ∈ X} is closed in A.8 W.A. Carnielli, M.E. Coniglio and I.M. Loffredo D’Ottaviano João Marcos in his Master Dissertation =-=[31]-=-. D’Ottaviano and Feitosa in [11], [12] and [14] introduced several conservative translations involving classical logic, intuitionistic logic, modal logics, paraconsistent logics and many-valued logic... |

6 |
Interpretability of elementary theories
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(Show Context)
Citation Context ...n important landmark in the development of the theory of logical translations, anticipating some concepts and results later studied in [8] in a more general setting (see Section 2 below). Szczerba in =-=[35]-=- defines the concept of interpretation function, that “map structures onto structures”. The corresponding functions mapping formulas to formulas are called “translations”, but we may say that Szczerba... |

5 |
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- 2001
(Show Context)
Citation Context ...lmogorov’s and Gentzen’s interpretations are translations in the sense of Prawitz, Wójcicki and Epstein. By its turn, Gödel’s ones are translations only in Prawitz’ sense (cf. Feitosa and D’Ottaviano =-=[17]-=-). On the other hand, in [23] (see also [24] and [32]), Goguen and Burstall define a general notion of logic system and his morphisms called institutions, within the framework of Category Theory. Inst... |

5 |
Fibrilação de lógicas na hierarquia de Leibniz (Fibring of logis in Leibniz hierarchy), in Portuguese
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(Show Context)
Citation Context ...lations algebraic semantics, in which translations play an essential role for providing an algebraic approach to logics non-algebraizable by traditional methods. Víctor Fernández, in his Ph.D. Thesis =-=[18]-=- used translations in order to investigate combinations of logics, more particularly fibring of logics (cf. Carnielli, Coniglio et al in [5]). 3. Transfers: a model-theoretic dimension to translations... |

5 |
A survey of some connections between classical, intuitionistic and minimal logic
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Citation Context ...tinct terms having been used by their authors such as interpretation and transformation. Since then, interpretations between logics have been used to different purposes. Prawitz and Malmnäs survey in =-=[33]-=- these historical papers, and theirs is the first paper in which a general definition for the concept of translation between logic systems is introduced. For them a translation from a logic system S1 ... |

4 | Recovering a logic from its fragments by meta-fibring
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- 2007
(Show Context)
Citation Context ... (a function symbol of L ′) must be translated into another connective. This is a bit restrictive, because often translations between languages are defined in a more liberal way. Coniglio proposes in =-=[6]-=- a different approach to translations by means of contextual translations, 4 which are mappings between languages preserving certain meta-properties of the source logic. As in the case of transfers, t... |

4 |
Traduções conservativas (Conservative tranlations). Campinas: UNICAMP. (Tese de Doutorado em Lógica
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(Show Context)
Citation Context ...ly continuous functions between the spaces of their theories are also investigated. An important subclass of translations, the conservative translations, is investigated by D’Ottaviano and Feitosa in =-=[16]-=-, [17] and [10]. Definition 2.5. Let A and B be logics. A conservative translation from A into B is a function t : A−→B such that, for every set X ∪ {x} ⊆ A, x ∈ CA(X) if and only if t(x) ∈ CB(t(X)).... |

4 |
Béziau’s translation paradox
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(Show Context)
Citation Context ...meta-rules of the former are globally valid in the latter. A translation tradeoff can be identified here: translations, intended in the sense of Definition 2.3, help to clarify the maxim expressed in =-=[26]-=- and [27] according to which the weaker a logic is (in its deductive strength) the stronger it is in its ability to draw distinctions (in its discriminatory strength). So the weaker a logic is, the mo... |

4 |
Logical Discrimination
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- 2005
(Show Context)
Citation Context ...ntences of a given language). The notion of translation between logics seeks to identify the deductive capability and the ability to draw distinctions (the “discriminatory strength”, in Humberstone’s =-=[27]-=- terminology) of a logic inside another (a non-surjective translation) or onto another (a surjective translation). Now, propositional logics can be advantageously seen as special first-order structure... |

2 |
Conservative translations and model-theoretic translations
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- 1999
(Show Context)
Citation Context ...unctions between the spaces of their theories are also investigated. An important subclass of translations, the conservative translations, is investigated by D’Ottaviano and Feitosa in [16], [17] and =-=[10]-=-. Definition 2.5. Let A and B be logics. A conservative translation from A into B is a function t : A−→B such that, for every set X ∪ {x} ⊆ A, x ∈ CA(X) if and only if t(x) ∈ CB(t(X)).6 W.A. Carniell... |

2 |
Paraconsistent logics and translations
- D’OTTAVIANO, FEITOSA
- 2000
(Show Context)
Citation Context ...t X ⊆ A�∼ is closed iff the set {x ∈ A : x�∼ ∈ X} is closed in A.8 W.A. Carnielli, M.E. Coniglio and I.M. Loffredo D’Ottaviano João Marcos in his Master Dissertation [31]. D’Ottaviano and Feitosa in =-=[11]-=-, [12] and [14] introduced several conservative translations involving classical logic, intuitionistic logic, modal logics, paraconsistent logics and many-valued logics. Mauro César Scheer, in his Mas... |

2 |
Deductive systems and translations
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- 2007
(Show Context)
Citation Context ...osed iff the set {x ∈ A : x�∼ ∈ X} is closed in A.8 W.A. Carnielli, M.E. Coniglio and I.M. Loffredo D’Ottaviano João Marcos in his Master Dissertation [31]. D’Ottaviano and Feitosa in [11], [12] and =-=[14]-=- introduced several conservative translations involving classical logic, intuitionistic logic, modal logics, paraconsistent logics and many-valued logics. Mauro César Scheer, in his Master Dissertatio... |

1 |
Semântica algébrica de traduções possíveis (Possible-translations semantic algebraizability
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- 2004
(Show Context)
Citation Context ...many-valued logics. Mauro César Scheer, in his Master Dissertation [34] initiated the study of translations involving cummulative non-monotonic logics. Juliana Bueno-Soler, in her Master Dissertation =-=[2]-=- introduced the possible-translations algebraic semantics, in which translations play an essential role for providing an algebraic approach to logics non-algebraizable by traditional methods. Víctor F... |

1 |
Fechos caracterizados por interpretações (Closures characterized by interpretations
- D’Ottaviano
- 1973
(Show Context)
Citation Context ...lowing condition holds: Mod(f)(M ′ ) |=Σ ϕ iff M ′ |=Σ ′ Sen(f)(ϕ) for every model M ′ ∈ |Mod(Σ ′ )| and every sentence ϕ ∈ Sen(Σ). 2. Running definitions of (conservative) translation D’Ottaviano in =-=[9]-=- studies variants of Tarskian closure operators characterized by interpretations. Hoppmann in [25] 1 uses this characterization and claims that, from a logical point of view, continuous functions betw... |

1 | Translations from Lukasiewicz logics into classical logic: Is it possible - D’Ottaviano, Feitosa - 2006 |

1 |
Introducing institutions. In: Logics of programs (Carnegie-Mellon
- Goguen, Burstall
- 1983
(Show Context)
Citation Context ...rpretations are translations in the sense of Prawitz, Wójcicki and Epstein. By its turn, Gödel’s ones are translations only in Prawitz’ sense (cf. Feitosa and D’Ottaviano [17]). On the other hand, in =-=[23]-=- (see also [24] and [32]), Goguen and Burstall define a general notion of logic system and his morphisms called institutions, within the framework of Category Theory. Institutions generalize Tarski’s ... |

1 |
Fecho e imersão (Closure and embedding, in Portuguese
- Hoppmann
- 1973
(Show Context)
Citation Context ... )| and every sentence ϕ ∈ Sen(Σ). 2. Running definitions of (conservative) translation D’Ottaviano in [9] studies variants of Tarskian closure operators characterized by interpretations. Hoppmann in =-=[25]-=- 1 uses this characterization and claims that, from a logical point of view, continuous functions between closure structures correspond to functions that preserve deductions, in a remarkable coinciden... |

1 | Compiler correctness and the translation of logics
- Janssen
(Show Context)
Citation Context ...r it is in its ability to draw distinctions (in its discriminatory strength). So the weaker a logic is, the more logics could be translated into it. This point of view is also supported by Janssen in =-=[28]-=- But of course, in the opposite direction, conservative contextual translations are much stronger: examples are hard to find, but widely expressive. In the direction of the second and third questions ... |

1 |
uma teoria de traduções entre lógicas cumulativas (Towards a theory of translations between cumulative logics
- Para
- 2002
(Show Context)
Citation Context ...constructive proofs of the existence of a conservative translation from IPL into CPL, and from the finite ̷Lukasiewicz’ logics into CPL. Conservative translations do not exist in all cases: Scheer in =-=[34]-=-, for instance, showed that there is no conservative translation from a cumulative non-monotonic logic into a Tarskian logic, and that there is no surjective conservative translation from a Tarskian l... |