General models and extensionality

1972 ◽  
Vol 37 (2) ◽  
pp. 395-397 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
General Model ◽  
Type Theory ◽  
Axiom Schema ◽  
Nonstandard Model ◽  
Equality Relation ◽  
Definition Of

It is well known that equality is definable in type theory. Thus, in the language of [2], the equality relation between elements of type α is definable as , i.e., xα = yα iff every set which contains xα also contains yα. However, in a nonstandard model of type theory, the sets may be so sparse that the wff above does not denote the true equality relation. We shall use this observation to construct a general model in the sense of [2] in which the Axiom of Extensionality is not valid. Thus Theorem 2 of [2] is technically incorrect. However, it is easy to remedy the situation by slightly modifying the definition of general model.Our construction will show that the Axiom Schema of Extensionality is independent even if one takes as an axiom schema.We shall assume familiarity with, and use the notation of, [2] and §§2–3 of [1].

scholarly journals Minimal models of Heyting arithmetic

1997 ◽  
Vol 62 (4) ◽  
pp. 1448-1460 ◽  
Author(s):  
Ieke Moerdijk ◽  
Erik Palmgren
Keyword(s):  
Type Theory ◽  
Geometric Theory ◽  
Minimal Models ◽  
Nonstandard Model ◽  
Heyting Arithmetic ◽  
Suitable Site ◽  
Definition Of ◽  
A Site ◽  
Effective Proof ◽  
Intuitionistic Arithmetic

In this paper, we give a constructive nonstandard model of intuitionistic arithmetic (Heyting arithmetic). We present two axiomatisations of the model: one finitary and one infinitary variant. Using the model these axiomatisations are proven to be conservative over ordinary intuitionistic arithmetic. The definition of the model along with the proofs of its properties may be carried out within a constructive and predicative metatheory (such as Martin-Löf's type theory). This paper gives an illustration of the use of sheaf semantics to obtain effective proof-theoretic results.The axiomatisations of nonstandard intuitionistic arithmetic (to be called HAI and HAIω respectively) as well as their model are based on the construction in [5] of a sheaf model for arithmetic using a site of filters. In this paper we present a “minimal” version of this model, built instead on a suitable site of provable filter bases. The construction of this site can be viewed as an extension of the well-known construction of the classifying topos for a geometric theory which uses “syntactic sites”. (Such sites can in fact be used to prove semantical completeness of first order logic in a strictly constructive framework, see [6].)We should mention that for classical nonstandard arithmetics there are several nonconstructive methods of proving conservativity over arithmetic, e.g. the compactness theorem, Mac Dowell–Specker's theorem [3].

General models, descriptions, and choice in type theory

1972 ◽  
Vol 37 (2) ◽  
pp. 385-394 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
General Model ◽  
Type Theory ◽  
Combinatory Logic ◽  
Logical Constants ◽  
Simple Idea ◽  
Nonstandard Models ◽  
Selection Operator ◽  
Definition Of

In [4] Alonzo Church introduced an elegant and expressive formulation of type theory with λ-conversion. In [8] Henkin introduced the concept of a general model for this system, such that a sentence A is a theorem if and only if it is true in all general models. The crucial clause in Henkin's definition of a general model ℳ is that for each assignment φ of values in ℳ to variables and for each wff A, there must be an appropriate value of A in ℳ. Hintikka points out in [10, p. 3] that this constitutes a rather strong requirement concerning the structure of a general model. Henkin draws attention to the problem of constructing nonstandard models for the theory of types in [9, p. 324].We shall use a simple idea of combinatory logic to find a characterization of general models which does not directly refer to wffs, and which is easier to work with in certain contexts. This characterization can be applied, with appropriate minor and obvious modifications, to a variety of formulations of type theory with λ-conversion. We shall be concerned with a language ℒ with extensionality in which there is no description or selection operator, and in which (for convenience) the sole primitive logical constants are the equality symbols Qoαα for each type α.

Setoids in type theory

2003 ◽  
Vol 13 (2) ◽  
pp. 261-293 ◽  
Author(s):  
GILLES BARTHE ◽  
VENANZIO CAPRETTA ◽  
OLIVIER PONS
Keyword(s):  
Type Theory ◽  
Type Systems ◽  
The Other ◽  
Dependent Type Theory ◽  
Common Definition ◽  
Equality Relation ◽  
The Common ◽  
Definition Of ◽  
Dependent Type

Formalising mathematics in dependent type theory often requires to represent sets as setoids, i.e. types with an explicit equality relation. This paper surveys some possible definitions of setoids and assesses their suitability as a basis for developing mathematics. According to whether the equality relation is required to be reflexive or not we have total or partial setoid, respectively. There is only one definition of total setoid, but four different definitions of partial setoid, depending on four different notions of setoid function. We prove that one approach to partial setoids in unsuitable, and that the other approaches can be divided in two classes of equivalence. One class contains definitions of partial setoids that are equivalent to total setoids; the other class contains an inherently different definition, that has been useful in the modeling of type systems. We also provide some elements of discussion on the merits of each approach from the viewpoint of formalizing mathematics. In particular, we exhibit a difficulty with the common definition of subsetoids in the partial setoid approach.

Cubical Agda: A dependently typed programming language with univalence and higher inductive types

2021 ◽  
Vol 31 ◽  
Author(s):  
ANDREA VEZZOSI ◽  
ANDERS MÖRTBERG ◽  
ANDREAS ABEL
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Type Checking ◽  
Dependent Type Theory ◽  
Wide Range ◽  
General Schema ◽  
Direct Definition ◽  
Inductive Types ◽  
Definition Of ◽  
Cubical Type Theory

Abstract Proof assistants based on dependent type theory provide expressive languages for both programming and proving within the same system. However, all of the major implementations lack powerful extensionality principles for reasoning about equality, such as function and propositional extensionality. These principles are typically added axiomatically which disrupts the constructive properties of these systems. Cubical type theory provides a solution by giving computational meaning to Homotopy Type Theory and Univalent Foundations, in particular to the univalence axiom and higher inductive types (HITs). This paper describes an extension of the dependently typed functional programming language Agda with cubical primitives, making it into a full-blown proof assistant with native support for univalence and a general schema of HITs. These new primitives allow the direct definition of function and propositional extensionality as well as quotient types, all with computational content. Additionally, thanks also to copatterns, bisimilarity is equivalent to equality for coinductive types. The adoption of cubical type theory extends Agda with support for a wide range of extensionality principles, without sacrificing type checking and constructivity.

A TYPE-FREE THEORY OF HALF-MONOTONE INDUCTIVE DEFINITIONS

1995 ◽  
Vol 06 (03) ◽  
pp. 203-234 ◽  
Author(s):  
YUKIYOSHI KAMEYAMA
Keyword(s):  
Type Theory ◽  
Characteristic Point ◽  
Logical Theory ◽  
Inductive Definition ◽  
Program Extraction ◽  
First Order ◽  
Inductive Definitions ◽  
Free Theory ◽  
Program Proof ◽  
Definition Of

This paper studies an extension of inductive definitions in the context of a type-free theory. It is a kind of simultaneous inductive definition of two predicates where the defining formulas are monotone with respect to the first predicate, but not monotone with respect to the second predicate. We call this inductive definition half-monotone in analogy of Allen’s term half-positive. We can regard this definition as a variant of monotone inductive definitions by introducing a refined order between tuples of predicates. We give a general theory for half-monotone inductive definitions in a type-free first-order logic. We then give a realizability interpretation to our theory, and prove its soundness by extending Tatsuta’s technique. The mechanism of half-monotone inductive definitions is shown to be useful in interpreting many theories, including the Logical Theory of Constructions, and Martin-Löf’s Type Theory. We can also formalize the provability relation “a term p is a proof of a proposition P” naturally. As an application of this formalization, several techniques of program/proof-improvement can be formalized in our theory, and we can make use of this fact to develop programs in the paradigm of Constructive Programming. A characteristic point of our approach is that we can extract an optimization program since our theory enjoys the program extraction theorem.

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 Zoocentrism in the weeds? Cultivating plant models for cognitive yield

2020 ◽  
Vol 35 (5) ◽  
Author(s):  
Adam Linson ◽  
Paco Calvo
Keyword(s):  
Natural Science ◽  
General Model ◽  
Animal Cognition ◽  
Plant Science ◽  
Turn Point ◽  
Cognitive Agents ◽  
Action Memory ◽  
Definition Of ◽  
And Behavior ◽  
Perception Action

Abstract It remains at best controversial to claim, non-figuratively, that plants are cognitive agents. At the same time, it is taken as trivially true that many (if not all) animals are cognitive agents, arguably through an implicit or explicit appeal to natural science. Yet, any given definition of cognition implicates at least some further processes, such as perception, action, memory, and learning, which must be observed either behaviorally, psychologically, neuronally, or otherwise physiologically. Crucially, however, for such observations to be intelligible, they must be counted as evidence for some model. These models in turn point to homologies of physiology and behavior that facilitate the attribution of cognition to some non-human animals. But, if one is dealing with a model of animal cognition, it is tautological that only animals can provide evidence, and absurd to claim that plants can. The more substantive claim that, given a general model of cognition, only animals but not plants can provide evidence, must be evaluated on its merits. As evidence mounts that plants meet established criteria of cognition, from physiology to behavior, they continue to be denied entry into the cognitive club. We trace this exclusionary tendency back to Aristotle, and attempt to counter it by drawing on the philosophy of modelling and a range of findings from plant science. Our argument illustrates how a difference in degree between plant and animals is typically mistaken for a difference in kind.

scholarly journals Transporting functions across ornaments

2014 ◽  
Vol 24 (2-3) ◽  
pp. 316-383 ◽  
Author(s):  
PIERRE-ÉVARISTE DAGAND ◽  
CONOR McBRIDE
Keyword(s):  
Type Theory ◽  
Structured Data ◽  
Dependent Types ◽  
Natural Numbers ◽  
Code Reuse ◽  
Domain Specific ◽  
Definition Of ◽  
The Relationship ◽  
Do So ◽  
Structural Invariants

AbstractProgramming with dependent types is a blessing and a curse. It is a blessing to be able to bake invariants into the definition of datatypes: We can finally write correct-by-construction software. However, this extreme accuracy is also a curse: A datatype is the combination of a structuring medium together with a special purpose logic. These domain-specific logics hamper any attempt to reuse code across similarly structured data. In this paper, we capitalise on the structural invariants of datatypes. To do so, we first adapt the notion of ornament to our universe of inductive families. We then show how code reuse can be achieved by ornamenting functions. Using these functional ornaments, we capture the relationship between functions such as the addition of natural numbers and the concatenation of lists. With this knowledge, we demonstrate how the implementation of the former informs the implementation of the latter: The users can ask the definition of addition to be lifted to lists and they will only be asked the details necessary to carry on adding lists rather than numbers. Our presentation is formalised in the type theory with a universe of datatypes and all our constructions have been implemented as generic programs, requiring no extension to the type theory.

Extending Gödel's negative interpretation to ZF

1975 ◽  
Vol 40 (2) ◽  
pp. 221-229 ◽  
Author(s):  
William C. Powell
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Stable Sets ◽  
Class Theory ◽  
Heyting Arithmetic ◽  
Inner Model ◽  
Order Arithmetic ◽  
Definition Of ◽  
Excluded Middle ◽  
Negative Sense

In [5] Gödel interpreted Peano arithmetic in Heyting arithmetic. In [8, p. 153], and [7, p. 344, (iii)], Kreisel observed that Gödel's interpretation extended to second order arithmetic. In [11] (see [4, p. 92] for a correction) and [10] Myhill extended the interpretation to type theory. We will show that Gödel's negative interpretation can be extended to Zermelo-Fraenkel set theory. We consider a set theory T formulated in the minimal predicate calculus, which in the presence of the full law of excluded middle is the same as the classical theory of Zermelo and Fraenkel. Then, following Myhill, we define an inner model S in which the axioms of Zermelo-Fraenkel set theory are true.More generally we show that any class X that is (i) transitive in the negative sense, ∀x ∈ X∀y ∈ x ¬ ¬ x ∈ X, (ii) contained in the class St = {x: ∀u(¬ ¬ u ∈ x→ u ∈ x)} of stable sets, and (iii) closed in the sense that ∀x(x ⊆ X ∼ ∼ x ∈ X), is a standard model of Zermelo-Fraenkel set theory. The class S is simply the ⊆-least such class, and, hence, could be defined by S = ⋂{X: ∀x(x ⊆ ∼ ∼ X→ ∼ ∼ x ∈ X)}. However, since we can only conservatively extend T to a class theory with Δ01-comprehension, but not with Δ11-comprehension, we will give a Δ01-definition of S within T.

Tychonoff's theorem in the framework of formal topologies

1997 ◽  
Vol 62 (4) ◽  
pp. 1315-1332 ◽  
Author(s):  
Sara Negri ◽  
Silvio Valentini
Keyword(s):  
Type Theory ◽  
Constructive Proof ◽  
Topological Product ◽  
Essential Difference ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
Definition Of ◽  
The Axiom Of Choice ◽  
Inductive Property ◽  
Formal Topology

In this paper we give a constructive proof of the pointfree version of Tychonoff's theorem within formal topology, using ideas from Coquand's proof in [7]. To deal with pointfree topology Coquand uses Johnstone's coverages. Because of the representation theorem in [3], from a mathematical viewpoint these structures are equivalent to formal topologies but there is an essential difference also. Namely, formal topologies have been developed within Martin Löf's constructive type theory (cf. [16]), which thus gives a direct way of formalizing them (cf. [4]).The most important aspect of our proof is that it is based on an inductive definition of the topological product of formal topologies. This fact allows us to transform Coquand's proof into a proof by structural induction on the last rule applied in a derivation of a cover. The inductive generation of a cover, together with a modification of the inductive property proposed by Coquand, makes it possible to formulate our proof of Tychonoff s theorem in constructive type theory. There is thus a clear difference to earlier localic proofs of Tychonoff's theorem known in the literature (cf. [9, 10, 12, 14, 27]). Indeed we not only avoid to use the axiom of choice, but reach constructiveness in a very strong sense. Namely, our proof of Tychonoff's theorem supplies an algorithm which, given a cover of the product space, computes a finite subcover, provided that there exists a similar algorithm for each component space.

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