## Many-valued logic (1986)

Venue: | Handbook of Philosophical Logic |

Citations: | 53 - 1 self |

### BibTeX

@INPROCEEDINGS{Pelletier86many-valuedlogic,

author = {Francis Jeffry Pelletier and Alasdair Urquhart},

title = {Many-valued logic},

booktitle = {Handbook of Philosophical Logic},

year = {1986},

pages = {71--116},

publisher = {Publishing Company}

}

### Years of Citing Articles

### OpenURL

### Abstract

ABSTRACT. This paper discusses the general problem of translation functions between logics, given in axiomatic form, and in particular, the problem of determining when two such logics are “synonymous ” or “translationally equivalent. ” We discuss a proposed formal definition of translational equivalence, show why it is reasonable, and also discuss its relation to earlier definitions in the literature. We also give a simple criterion for showing that two modal logics are not translationally equivalent, and apply this to well-known examples. Some philosophical morals are drawn concerning the possibility of having two logical systems that are “empirically distinct ” but are both translationally equivalent to a common logic. KEY WORDS: modal logic, synonymy, translation 1.

### Citations

713 |
A New Introduction to Modal Logic
- Hughes, Cresswell
- 1996
(Show Context)
Citation Context ...lidated by such frames are theorems of this logic, we employ the usual canonical model construction. We omit the details, since they can be found in any standard textbook of modal logic, for example (=-=Hughes and Cresswell, 1996-=-, Chapter 6). To prove completeness for KU ∗ , it suffices to show that if v,w are maximal consistent sets of formulas of KU ∗ ,then(vRw ⇒ C ∈ w) and (C ∈ w ⇒ wRw),whereRis the relation defined on the... |

692 |
Introduction to metamathematics
- Kleene
- 1952
(Show Context)
Citation Context ...logic of L2? PROBLEM 10. Develop methods to prove that there is no exact one-way translation from one logic into another. One of the few examples known to the authors of such a method is sketched in (=-=Kleene, 1952-=-, p. 495). ACKNOWLEDGEMENTS The authors wish to express their thanks to their friends and colleagues in the Society for Exact Philosophy and the Western Canadian Philosophical Association; earlier ver... |

653 |
Word and Object
- Quine
- 1960
(Show Context)
Citation Context ...tructions in the same way as they themselves do. And so they continue to “talk past one another.” The preceding discussion should bring to mind Quinean doctrines about “indeterminacy of translation” (=-=Quine, 1960-=-), especially the issue of whether there can be distinct translation manuals that are “empirically adequate” but nonetheless incompatible with one another. Examples of such cases are given in (Massey,... |

634 |
Modal Logic: An Introduction
- Chellas
- 1980
(Show Context)
Citation Context ... w. For the second condition, if C ∈ w,thenwRw,bytheaxiomC → (✷A → A). ✷sSYNONYMOUS LOGICS 281 The logic KU ∗ is an extension of the logic KU that results from K by adding the axiom scheme ✷(✷A → A) (=-=Chellas, 1980-=-, p. 140); KU is complete with respect to the relational condition of weak reflexivity: xRy ⇒ yRy. We obtain KU ∗ from KU by adding the constant C that represents the reflexive worlds. To distinguish ... |

559 |
Lattice theory
- Birkhoff
- 1967
(Show Context)
Citation Context ... algebras) has been discussed in the literature of abstract algebra as a ‘cryptomorphism’ (the terminology is due to Garrett Birkhoff, who actually used the less euphonious term ‘crypto-isomorphism’ (=-=Birkhoff, 1967-=-, p. 154)). The terminology is widely used in the literature of matroid theory (White, 1986, pp. 33–34, 298–312). The translation invariant that we have just sketched could in principle be used as a c... |

187 |
Boolean algebras with operators
- Jónsson, Tarski
- 1951
(Show Context)
Citation Context ...conditions for modal operators in relational frames. We say that the modal algebra defined in this way on all the subsets of U is the modal algebra A(R) determined by the relation R. It can be shown (=-=Jónsson and Tarski, 1951-=-, 1952) that any modal algebra is a sub-algebra of a modal algebra determined by a relation. The following weaker result is general enough for our purposes. LEMMA 4.2. Any finite modal algebra is isom... |

53 |
Some theorems about the sentential calculi of Lewis and Heyting
- McKinsey, Tarski
- 1948
(Show Context)
Citation Context ...NYMOUS LOGICS 265 Gödel’s translation scheme from intuitionistic propositional logic was the first such scheme proposed, but there are alternative schemes, of which perhaps the best known is that of (=-=McKinsey and Tarski, 1948-=-). EXAMPLE 2.3. Let L1 be the language of intuitionistic propositional logic, and L2 the language of the modal logic S4. Then the McKinsey– Tarski translation scheme from L1 into L2 is as follows: (1)... |

40 |
Protoalgebraic logics
- Czelakowski
- 2001
(Show Context)
Citation Context ... conditions on the equivalence connective define all of the logics below as equivalential logics in the sense of (Prucnal and Wroński, 1974). The model theory of equivalential logics is discussed in (=-=Czelakowski, 1980-=-). Although in most of what follows, we shall be considering systems of modal logic based on classical logic, so that ↔ can be taken to be the classical biconditional, it should be noted that the noti... |

35 |
Theory of Logical Calculi
- Wójcicki
- 1988
(Show Context)
Citation Context ...ion of translation that we define in the next paragraphs is essentially the same as the concept of schematic interpretation of Prawitz and Malmnäs (Prawitz and Malmnäs, 1968). See also (Kuhn, 1977), (=-=Wójcicki, 1988-=-, p. 69). Let us add to the languages a new font P of variables α,β,γ,δ,..., and call these new variables parameters (the parameters are introduced simply as place-holders in formulas, so that we can ... |

15 |
The Semantic Foundations of Logic
- Epstein
- 1995
(Show Context)
Citation Context ...ms of consequence relations rather than equivalence connectives, as explained in the introduction. Similar notions appear in (Kanger, 1968; Prawitz and Malmnäs, 1968; Segerberg, 1982; Wójcicki, 1988; =-=Epstein, 1990-=-; Pollard, 1998; Da Silva et al., 1999). Let us suppose, as before, that we are dealing with two systems of propositional logic, S1 and S2, expressed in the languages L1 and L2. For reasons to be ment... |

12 |
Reduction of tense logic to modal logic I
- Thomason
- 1974
(Show Context)
Citation Context ...ational fragment. We shall give further examples of exact translations in what follows. Another very interesting translation in the literature is the translation from tense logic into modal logic in (=-=Thomason, 1974-=-, 1975). Thomason’s translation method is very close to, but not identical with our notion of translation scheme, so we do not discuss it in detail here, but it is one of the most technically useful t... |

11 | Translations between logics - Silva, D’Ottaviano, et al. - 1999 |

9 |
An algebraic characterization of the notion of structural completeness
- Prucnal, Wroński
- 1974
(Show Context)
Citation Context ...ed to insure that they are preserved under extensions of the language in question. The conditions on the equivalence connective define all of the logics below as equivalential logics in the sense of (=-=Prucnal and Wroński, 1974-=-). The model theory of equivalential logics is discussed in (Czelakowski, 1980). Although in most of what follows, we shall be considering systems of modal logic based on classical logic, so that ↔ ca... |

5 |
Epistemologische Betrachtungen zu [S4,S5]. Erkenntnis
- Lenzen
- 1979
(Show Context)
Citation Context ....4 and KD45 are equivalent under the translation schemes: ✷1α ↦−→ (✷2α ∧ α) ✷2α ↦−→ ✸1✷1α, where ✷1 is the necessity operator of S4.4 and ✷2 the necessity operator of KD45. The reader is referred to (=-=Lenzen, 1979-=-, Satz 3) for the proof that these schemes establish translational equivalence. Thus these two systems are in effect notational variants of each other. Lenzen gives a philosophical interpretation to t... |

5 |
The not-so-strange modal logic of indeterminacy’. Logique et Analyse 27
- Pelletier
- 1984
(Show Context)
Citation Context ...ame logic” while at the same time believing that some other languages (e.g., the modal logics S4 and S5)arenot “really the same logic.” This paper is a sequel to an earlier paper by the first author (=-=Pelletier, 1984-=-b), in which the topic was called the problem of determining “translational equivalence” between logics, and six open problems are stated. The present paper answers these earlier problems by giving so... |

5 |
Six Problems in “Translational Equivalence”’, Logique et Analyse 27
- Pelletier
- 1984
(Show Context)
Citation Context ...nce, the fact that the two systems are not translationally equivalent follows from (a slight extension of) Theorem 4.3. ✷ 5. CONCLUDING REMARKS Most of this paper concerns the problems enunciated in (=-=Pelletier, 1984-=-b) about “translational equivalence.” The issue raised by (Pelletier, 1984b)’s Problem 1 was whether or not this notion was a reasonable account ofs282 FRANCIS JEFFRY PELLETIER AND ALASDAIR URQUHART “... |

5 |
Classical Propositional Operators
- Segerberg
- 1982
(Show Context)
Citation Context ...inition, since it is given in terms of consequence relations rather than equivalence connectives, as explained in the introduction. Similar notions appear in (Kanger, 1968; Prawitz and Malmnäs, 1968; =-=Segerberg, 1982-=-; Wójcicki, 1988; Epstein, 1990; Pollard, 1998; Da Silva et al., 1999). Let us suppose, as before, that we are dealing with two systems of propositional logic, S1 and S2, expressed in the languages L1... |

4 |
Equivalent Theories
- Kanger
- 1972
(Show Context)
Citation Context ...omewhat more general than Pelletier’s definition, since it is given in terms of consequence relations rather than equivalence connectives, as explained in the introduction. Similar notions appear in (=-=Kanger, 1968-=-; Prawitz and Malmnäs, 1968; Segerberg, 1982; Wójcicki, 1988; Epstein, 1990; Pollard, 1998; Da Silva et al., 1999). Let us suppose, as before, that we are dealing with two systems of propositional log... |

4 |
1957] Contributions to the Axiomatic Foundations of Set Theory
- Montague
- 1957
(Show Context)
Citation Context ...nslational equivalence. We shall show by an example that this condition is in fact not sufficient. Similar examples are known in the context of classical logic. An example of this kind in the thesis (=-=Montague, 1957-=-) is reported in (De Bouvère, 1965). The two logics constituting our example are formulated with a single modal operator ✷A, together with a propositional constant C. Thefirst logic, T ∗ , is the resu... |

4 |
Homeomorphism and the equivalence of logical systems
- Pollard
- 1998
(Show Context)
Citation Context ...ce relations rather than equivalence connectives, as explained in the introduction. Similar notions appear in (Kanger, 1968; Prawitz and Malmnäs, 1968; Segerberg, 1982; Wójcicki, 1988; Epstein, 1990; =-=Pollard, 1998-=-; Da Silva et al., 1999). Let us suppose, as before, that we are dealing with two systems of propositional logic, S1 and S2, expressed in the languages L1 and L2. For reasons to be mentioned in the ne... |

2 | Synonymous theories - Bouvere - 1965 |

2 |
1995]: Bounded Arithmetic
- Krajíček
- 1995
(Show Context)
Citation Context ...nificance in the area of the complexity of propositional proofs. It is possible to define an indirect translation function from L1 to L2 that avoids the exponential blowup. The reader is referred to (=-=Krajíček, 1996-=-) for the details. EXAMPLE 2.2. Let L1 be the language of intuitionistic propositional logic, and L2 the language of the modal logic S4. Then Gödel’s translation scheme from L1 into L2 is as follows: ... |

1 |
On three alleged rivals to homophonic translation, Philos
- Kirk
- 1982
(Show Context)
Citation Context ...sertion and denial are opposite in the two manuals. (If uttering a sentence counts as asserting it according to one manual, then it counts as denying it according to the other manual.) Critics (e.g., =-=Kirk, 1982-=-) have objected to this last feature of Massey’s method on the grounds that such speech acts should not be allowed as part of a translation manual. Although we ourselves take no stand on that issue, i... |

1 |
Many-sorted Modal Logics
- Kuhn
- 1977
(Show Context)
Citation Context ...recise definitions of the technical terms above, and in subsequent sections, give at least partial answers to these questions. We are indebted to a referee for calling our attention to the monograph (=-=Kuhn, 1977-=-), which already contains some of the results reported here, in particular, a number of the positive results given below. Kuhn uses a somewhat different framework, his results being formulated in term... |

1 |
Indeterminacy, inscrutability, and ontological relativity
- Massey
- 1978
(Show Context)
Citation Context ...e, 1960), especially the issue of whether there can be distinct translation manuals that are “empirically adequate” but nonetheless incompatible with one another. Examples of such cases are given in (=-=Massey, 1978-=-), where he discusses the possibility (in a first-order language with no singular terms) that the sets of individuals assigned to the predicates by one translation manual are the complements of the se... |

1 | Malmnäs (1968): A survey of some connections between classical, intuitionistic and minimal logic - Prawitz |