## A Reflection on Russell's Ramified Types and Kripke's Hierarchy of Truths (1996)

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Venue: | Journal of the Interest Group in Pure and Applied Logic 4(2 |

Citations: | 1 - 1 self |

### BibTeX

@INPROCEEDINGS{Kamareddine96areflection,

author = {Fairouz Kamareddine and Twan Laan},

title = {A Reflection on Russell's Ramified Types and Kripke's Hierarchy of Truths},

booktitle = {Journal of the Interest Group in Pure and Applied Logic 4(2},

year = {1996}

}

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### Abstract

Both in Kripke's Theory of Truth ktt [8] and Russell's Ramified Type Theory rtt [16, 9] we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can only depend on objects of lower orders and types. Kripke on the other hand, has a ladder of languages where the truth of a proposition in language Ln can only be made in Lm where m ? n. Kripke finds a fixed point for his hierarchy (something Russell does not attempt to do). We investigate in this paper the similarities of both hierarchies: At level n of ktt the truth or falsehood of all order-n-propositions of rtt can be established. Moreover, there are order-n-propositions that get a truth value at an earlier stage in ktt. Furthermore, we show that rtt is more restrictive than ktt, as some type restrictions are not needed in ktt and more formulas can be expressed in the latter. Looking back at the dou...

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Citation Context ...erarchy of Russell, Ramsey [11], and Hilbert and Ackermann [7] considered the orders to cause the restrictiveness, and therefore removed them. This removal resulted in Church's Simple Type Theory stt =-=[1]-=-. We show however that orders in rtt correspond to levels of truth in ktt. Hence, ktt can be regarded as the dual of stt where types have been removed and orders are maintained. As rtt is more restric... |

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Citation Context ...ematics and Computing Science Eindhoven University of Technology P.O.Box 513, 5600 MB Eindhoven, The Netherlands e-mail laan@win.tue.nl November 13, 1995 Abstract Both in Kripke's Theory of Truth ktt =-=[8]-=- and Russell's Ramified Type Theory rtt [16, 9] we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into differ... |

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Citation Context ...ious. z(k 1 ; : : : ; kn ) is a propositional function of higher order: z is a variable for a propositional 2 It is important to note that a variable is not a propositional function. See for instance =-=[12], Chapter -=-viii: "The variable", p.94 of the 7th impression. 3 function with n free variables; the argument list k 1 ; : : : ; kn indicates what should be substituted 3 for these free variables as soon... |

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Citation Context ...formalisation of logic, was inconsistent 1 . Russell considered self-application to be the cause of the contradictions, and hence excluded all possibilities of self-application in his Theory of Types =-=[13, 16]-=-. As paradoxical sentences in Natural Language play a role similar to that of the paradoxes in Logic and Mathematics, Type Theory eliminated the paradoxical sentences (see for instance [10]). Paradoxe... |

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Citation Context ...versity of Technology P.O.Box 513, 5600 MB Eindhoven, The Netherlands e-mail laan@win.tue.nl November 13, 1995 Abstract Both in Kripke's Theory of Truth ktt [8] and Russell's Ramified Type Theory rtt =-=[16, 9]-=- we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can o... |

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Citation Context ...amous Russell's paradox is logical whereas the famous liar's paradox is semantical. The semantical paradoxes usually involve the truth predicate T which gives the truth value of a proposition. Tarski =-=[14]-=- shows that truth is undefinable and that having the truth predicate inside the language leads to contradictions. For this reason, he distinguishes between (object-) language and meta-language and all... |

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Citation Context ...y well-formed sentence can be expressed but its truth value may not be calculated (think of the paradoxical sentences), one may compare this approach to the implicit typing of Curry's Type Theory ctt =-=[2, 3]-=- where self-referential sentences may be expressed but are not typable. Hence, even though we said that ktt is the dual of stt, it may be the twin-brother of ctt where only truth or falsehood of typab... |

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Citation Context ...e show that rtt is more restrictive than ktt, as some type restrictions are not needed in ktt and more formulas can be expressed in the latter. Looking back at the double hierarchy of Russell, Ramsey =-=[11]-=-, and Hilbert and Ackermann [7] considered the orders to cause the restrictiveness, and therefore removed them. This removal resulted in Church's Simple Type Theory stt [1]. We show however that order... |

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Citation Context ...ed abstraction which makes n-ary predication a primitive of the system, with the unary truth predicate being trivially definable upon this basis. For a useful connection between ktt and NaDSet 1, see =-=[4]-=-. Remark 4.6 The extensions suggested above are of a mere technical character. Therefore, we think that we can still speak of an embedding of rtt within ktt. Notation 4.7 To keep notations uniform, we... |

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Citation Context ...garded as a system based on rtt of which the types and not the orders have been removed. 1 An English translation of Russell's letter to Frege in which this inconsistency is described can be found in =-=[6] 2 2a Prop-=-ositional Functions In this section we shall describe the set of propositions and propositional functions which Whitehead and Russell use in "Principia". We give a modernised, formal definit... |

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Citation Context ...ive than ktt, as some type restrictions are not needed in ktt and more formulas can be expressed in the latter. Looking back at the double hierarchy of Russell, Ramsey [11], and Hilbert and Ackermann =-=[7]-=- considered the orders to cause the restrictiveness, and therefore removed them. This removal resulted in Church's Simple Type Theory stt [1]. We show however that orders in rtt correspond to levels o... |

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Citation Context ...sired results, and to no others. But clearly it is not the sort of axiom with which we can rest content." Though serious efforts have been made to develop Mathematics within rtt (for instance by =-=Weyl [15]-=-), this has not become the usual practice. In 1925, Ramsey [11] shows that, by making distinction between language and meta-language, the orders can be removed from the system without re-introducing a... |

4 |
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Citation Context ... Theory in Logic and Mathematics has always been a restrictive one. The need for restrictions was realised at the beginning of this century, when Bertrand Russell showed that Frege's "Begriffschr=-=ift" [5]-=-, a formalisation of logic, was inconsistent 1 . Russell considered self-application to be the cause of the contradictions, and hence excluded all possibilities of self-application in his Theory of Ty... |

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3 | A formalization of the Ramified Type Theory
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(Show Context)
Citation Context ...versity of Technology P.O.Box 513, 5600 MB Eindhoven, The Netherlands e-mail laan@win.tue.nl November 13, 1995 Abstract Both in Kripke's Theory of Truth ktt [8] and Russell's Ramified Type Theory rtt =-=[16, 9]-=- we are confronted with some hierarchy. In rtt, we have a double hierarchy of orders and types. That is, the class of propositions is divided into different orders where a propositional function can o... |