Aczel, P. (1980). Frege Structures and the Notion of Proposition, Truth and Set. In Barwise, J. et. al. (eds.), The Kleene Symposium. North Holland: Dordrecht.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... : It is necessary to have the strong elimination rule, since we want to inductively de ne indexed families of sets U : I set and functions T : i : I) U(i) D[i] where D[i] depends non trivially on i, as in the de nition of Palmgren s higher order universe (where for instance D[0] set, D[1] = Fam(set) Fam(set) see 3.2 for more explanation) We will also add a level between set and type, which we call stype for small types: stype : type. The reason for the need for stype is discussed in [8] If a : set then a : stype. Moreover, stype is also closed under dependent function ....

....an early version of his intuitionistic type theory and proves a normalization theorem using such Tait style computability predicates. He works in an informal intuitionistic metalanguage but gives no explicit justi cation for the meaningfulness of these computability predicates. Later Aczel [1] has shown how to model a similar construction in classical set theory. Since the metalanguage is informal the inductive recursive nature of this de nition is implicit. One of the objectives of the current work is indeed to formalize an extension of Martin L of type theory where the ....

P. Aczel. Frege Structures and the Notions of Proposition, Truth, and Set, pages 31-59. North-Holland, 1980.

....a truth predicate gives rise to logical paradoxes in formal theories. Property Theory can avoid these paradoxes by effectively limiting the application of the truth predicate to only non problematic terms. In Turner s Property Theory (PT) which axiomatises Aczel s Frege Structures (Turner, 1992; Aczel, 1980), the axioms of truth governing the truth predicate can only be applied to those terms which we can prove to be propositions . The axioms governing propositionhood do not allow a proof that paradoxical expressions are propositions, despite their appearance. The paradoxes are thus taken to be ....

Aczel, P. (1980). Frege structures and the notions of proposition, truth and set. In Barwise, Keisler, and Keenan, editors, The Kleene Symposium, North Holland Studies in Logic, pages 31--39. North Holland.

....to more powerful systems. More work is needed to see whether this approach can be extended to more recalcitrant discourse phenomena. A PT (Property Theory) The version of Property Theory presented here is Ray Turner s axiomatisation of Aczel s Frege Structures (Turner, 1990; Turner, 1992; Aczel, 1980), PT. The Language of terms, basic vocabulary: Individual variables: x; y; z; Individual constants: c; d; e; Logical constants: Xi; Theta; Inductive Definition of Terms: 1. Every variable or constant is a term. 2. If t is a term and x is a variable then x:t is a ....

Aczel, P. (1980). Frege structures and the notions of proposition, truth and set. In Barwise, Keisler, and Keenan, editors, The Kleene Symposium, North Holland Studies in Logic, pages 31--39. North Holland.

....term. Martin Lof presumably considered this definition intuitionistically valid, but did not provide an explicit discussion of why this is so. It is in fact a non trivial problem to give classical mathematical meaning to Martin Lof s computability predicates. One approach is due to Aczel [1] for the closely related construction of a Frege structure. Other approaches have been proposed by Allen [2] and by Lofvall and Sjodin [15] Although Martin Lof s computability predicates nowadays can be regarded as an informal example of an inductive recursive definition and therefore as a ....

P. Aczel. Frege structures and the notions of proposition, truth, and set. In J. Barwise, H. J. Keisler, and K. Kunen, editors, The Kleene Symposium, pages 31--59. North-Holland, 1980.

....formulas in general and those expressing genuine propositions, and which characterize this predicate (i.e. that which is true of a well formed formula if it expresses a proposition) by proof104 theoretic means. Notable among these approaches are in particular Aczel s theory of Frege structures [ Aczel, 1980 ] the constructive type theory, e.g. along the lines developed by Martin Lof [ Martin Lof, 1984 ] and the Property Theory of Turner [ Turner, 1988 ] Turner, 1992 ] and others. Reversibility Can the semantic representation be used for generation This is connected to the issue of ....

Aczel, P. 1980. Frege structures and the notions of proposition, truth and set. In Barwise, J.; Keisler, H.J.; and Kunen, K., editors 1980, The Kleene Symposium. North-Holland, Amsterdam. 31--59.

....doesn t for internal definability. For this we need to introduce the theory we are working with and the model that we are working inside. The theory can be found in [Kamareddine 89] and it is basically a type free intensional theory. The models of such a theory are known as Frege structures (see [Aczel 90] The theory or the model will not be described here as they can be referred to in the original papers. Yet to make things clear for the reader, the structure of the model is described in what follows: We assume an explicitly closed family (i.e. function space is closed under the constant ....

....for understanding this paper and hence one can easily now move to the set theory. Again if the reader would like to better understand the theory then she is referred to [Kamareddine 89] 2 If she would like to better understand the model (which is shortly described above) she is referred to [Aczel 90] 3 A SET THEORY AND PLURALS We introduce the operator Delta, understanding DeltaP to mean that P is a set. Delta is defined as follows: DeltaP = df 8x Omega Gamma x 2 P ) where Omega a stands for a is a proposition That is, something is a set iff whenever it applies to an object, the ....

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Aczel P. (1980), Frege structures and the notions of proposition, truth and set; The Kleene Synposium, Edited by Barwise et.al, Studies in Logic 101, NorthHolland, New York, pp 31-60.

....theory could be constructed where the axioms of truth need never apply to sentences with false presuppositions. 4.1. 1 The Basic Theory 4 The particular version of property theory to be introduced here is PT, Ray Turner s axiomatisation of Aczel s Frege Structures [ Turner, 1990; Turner, 1992; Aczel, 1980 ] Other formalisations of property theory will not be discussed, except to mention that equating predication with application may prove problematic for some examples [ Turner, 1989; Bealer, 1989 ] A short example will be used to illustrate how the paradoxes are avoided. 4.1.1.1 General ....

....Similarly, in the language of wff, properties may hold of the same terms, yet they may be distinct. The equality of terms is thus weaker than the notion of logical equivalence obtained when considering truth conditions in the meta language. It can be seen that PT characterises a Frege Structure [ Aczel, 1980 ] two classes of terms are defined by P and T as below: P Propositions T True Propositions ill formed and paradoxical terms Diagram: A Frege Structure 4.1.1.2 The Formal Theory The following presents a formalisation of the languages of terms and wff, together with the ....

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Aczel, P. 1980. Frege structures and the notions of proposition, truth and set. In Barwise, J.; Keisler, H.J.; and Kunen, K., editors 1980, The Kleene Symposium. North-Holland, Amsterdam. 31--59.

....here. Several more examples of simultaneous inductive recursive definitions of large universes and universe operators can be found in the recent papers by Palmgren [36] and Rathjen, Griffor, and Palmgren [39] 5 Frege structures The notion of a Frege structure was introduced by Peter Aczel [4]. One purpose was to provide an appropriate setting for calculus (or abstract realisability) interpretations of Martin Lof type theory. Another was to provide a model for a foundational framework where the notions of proposition and truth are primitive. We shall here show how to construct a ....

....such that 8xf(x) is true iff f(a) is true for all objects a. Note that a oe b is a non standard notion of implication, since b is required to be a proposition only when a is true. A structure with an encoding of the logical constants can be obtained by a standard construction in the calculus [4]. To get a Frege structure we need to construct the collections of propositions and truths. The basic idea is to view the logical schemata as inductively defining these collections. But a direct interpretation as a positive inductive definition is not possible, since the notion of truth appears ....

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P. Aczel. Frege Structures and the Notions of Proposition, Truth, and Set, pages 31--59. NorthHolland, 1980.

....is crucial for the formalization of metamathematical notions. Simultaneous induction recursion gives a type theoretic explanation of half positive inductive definitions, which are used for defining universes a la Tarski [14] computability predicates for dependent types [10] and Frege structures [1]. In this paper we consider the pure theory of dependent types, which contains the rules of assumption and substitution. We define (i) a calculus of normal terms simultaneously with substitution functions and (ii) a substitution calculus. We also prove the equivalence of these two calculi. A ....

P. Aczel. Frege Structures and the Notions of Proposition, Truth, and Set, pages 31--59. NorthHolland, 1980.

....t j t and s Omega (t Omega u) j (s Omega t) Omega u for terms s, t and u (strict associativity) 2 The term K( x 1 )t 1 ; x r )t r ; t) binds the variables from sequence x i in t i . Plotkin has pointed out that this can be viewed as a variant of Aczel s general binding operators [Acz80]. This issue is discussed in [BGHP98] and a concrete example is given in Example 3.12. The term let x be s in t binds the variables from sequence x in t. We write tfu=xg for the standard capture free substitution. Definition 3.2 A term is well typed if it can be shown to annotate a sequent ....

Aczel, P. (1980), Frege structures and the notions of proposition, truth and set, in "The Kleene Symposium", pp.31--59, North-Holland.

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Aczel, P. (1980). Frege Structures and the Notion of Proposition, Truth and Set. In Barwise, J. et. al. (eds.), The Kleene Symposium. North Holland: Dordrecht.

No context found.

P. Aczel. Frege structures and the notions of proposition, truth, and set. In J. Barwise, H. J. Keisler, and K. Kunen, editors, The Kleene Symposium, pages 31--59. North-Holland, 1980.

No context found.

P. Aczel. Frege Structures and the Notions of Proposition, Truth, and Set, pages 31--59. North-Holland, 1980.

No context found.

P. Aczel. Frege Structures and the Notions of Proposition, Truth, and Set, pages 31--59. North-Holland, 1980.

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P. Aczel. Frege Structures and the Notions of Proposition, Truth, and Set, pages 31-59. North-Holland, 1980.

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Aczel, P. 1980. Frege structures and the notions of proposition, truth and set. In Barwise, J.; Keisler, H.J.; and Kunen, K., editors 1980, The Kleene Symposium. North-Holland, Amsterdam. 31--59.

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