Russelian Simple Type Theory

2013 ◽  
pp. 231-244
Author(s):  
Alonzo Church
Keyword(s):  
Type Theory ◽  
Simple Type ◽  
Simple Type Theory

A proof of the cut-elimination theorem in simple type theory

1973 ◽  
Vol 38 (2) ◽  
pp. 215-226
Author(s):  
Satoko Titani
Keyword(s):  
Boolean Algebra ◽  
Type Theory ◽  
Arbitrary Number ◽  
Formal System ◽  
Cut Elimination ◽  
Simple Type ◽  
Inductive Definition ◽  
Simple Type Theory ◽  
Maximal Ideals ◽  
Definition Of

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schütte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations.The logical system we are concerned with is a modification of Schütte's formal system of simple type theory in [1] into Gentzen style.Inductive definition of types.0 and 1 are types.If τ1, …, τn are types, then (τ1, …, τn) is a type.Basic symbols.a1τ, a2τ, … for free variables of type τ.x1τ, x2τ, … for bound variables of type τ.An arbitrary number of constants of certain types.An arbitrary number of function symbols with certain argument places.

scholarly journals Basic simple type theory

1998 ◽  
Vol 32 (1-3) ◽  
pp. 211-213
Author(s):  
Nissim Francez
Keyword(s):  
Type Theory ◽  
Simple Type ◽  
Simple Type Theory

On nonstandard models in higher order logic

1984 ◽  
Vol 49 (1) ◽  
pp. 204-219
Author(s):  
Christian Hort ◽  
Horst Osswald
Keyword(s):  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
Elementary Embedding ◽  
Simple Type ◽  
Order Model ◽  
Higher Order Logic ◽  
First Order ◽  
Simple Type Theory ◽  
Nonstandard Models

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.

Theory of types

2018 ◽  
Author(s):  
Nino B. Cocchiarella
Keyword(s):  
Type Theory ◽  
Vicious Circle ◽  
Bertrand Russell ◽  
Simple Type ◽  
Two Dimensional ◽  
Logical Theory ◽  
Simple Type Theory ◽  
Concrete Objects ◽  
Different Types

The theory of types was first described by Bertrand Russell in 1908. He was seeking a logical theory that could serve as a framework for mathematics, and, in particular, a theory that would avoid the so-called ‘vicious-circle’ antinomies, such as his own paradox of the property of those properties that are not properties of themselves – or, similarly, of the class of those classes that are not members of themselves. Such paradoxes can be thought of as resulting when logical distinctions are not made between different types of entities, and, in particular, between different types of properties and relations that might be predicated of entities, such as the distinction between concrete objects and their properties, and the properties of those properties, and so on. In ‘ramified’ type theory, the hierarchy of properties and relations is, as it were, two-dimensional, where properties and relations are distinguished first by their order, and then by their level within each order. In ‘simple’ type theory properties and relations are distinguished only by their orders.

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