A Completeness Theorem for Higher-Order Intuitionistic Logic: An Intuitionistic Proof

1987 ◽  
pp. 107-124 ◽  
Author(s):  
A. G. Dragalin
Keyword(s):  
Intuitionistic Logic ◽  
Completeness Theorem ◽  
Higher Order

On the validity of hilbert's nullstellensatz, artin's theorem, and related results in grothendieck toposes

1988 ◽  
Vol 53 (4) ◽  
pp. 1177-1187
Author(s):  
W. A. MacCaull
Keyword(s):  
Intuitionistic Logic ◽  
Main Idea ◽  
Rational Functions ◽  
Completeness Theorem ◽  
Point Of View ◽  
Polynomial Rings ◽  
Von Neumann ◽  
Ordered Fields ◽  
Von Neumann Regular ◽  
Theoretic Point

Using formally intuitionistic logic coupled with infinitary logic and the completeness theorem for coherent logic, we establish the validity, in Grothendieck toposes, of a number of well-known, classically valid theorems about fields and ordered fields. Classically, these theorems have proofs by contradiction and most involve higher order notions. Here, the theorems are each given a first-order formulation, and this form of the theorem is then deduced using coherent or formally intuitionistic logic. This immediately implies their validity in arbitrary Grothendieck toposes. The main idea throughout is to use coherent theories and, whenever possible, find coherent formulations of formulas which then allow us to call upon the completeness theorem of coherent logic. In one place, the positive model-completeness of the relevant theory is used to find the necessary coherent formulas.The theorems here deal with polynomials or rational functions (in s indeterminates) over fields. A polynomial over a field can, of course, be represented by a finite string of field elements, and a rational function can be represented by a pair of strings of field elements. We chose the approach whereby results on polynomial rings are reduced to results about the base field, because the theory of polynomial rings in s indeterminates over fields, although coherent, is less desirable from a model-theoretic point of view. Ultimately we are interested in the models.This research was originally motivated by the works of Saracino and Weispfenning [SW], van den Dries [Dr], and Bunge [Bu], each of whom generalized some theorems from algebraic geometry or ordered fields to (commutative, von Neumann) regular rings (with unity).

A completeness theorem for higher order logics

2000 ◽  
Vol 65 (2) ◽  
pp. 857-884 ◽  
Author(s):  
Gábor Sági
Keyword(s):  
Set Theory ◽  
Representation Theorem ◽  
Completeness Theorem ◽  
Higher Order ◽  
Algebraic Solution ◽  
Algebraic Proof ◽  
Relation Algebras ◽  
Cylindric Algebras ◽  
Strong Representation ◽  
Algebraic Counterpart

AbstractHere we investigate the classes of representable directed cylindric algebras of dimension α introduced by Németi [12]. can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, “purely cylindric algebraic” proof for the following theorems of Németi: (i) is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain a strong representation theorem for if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories.

An intuitiomstic completeness theorem for intuitionistic predicate logic

1976 ◽  
Vol 41 (1) ◽  
pp. 159-166 ◽  
Author(s):  
Wim Veldman
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Logic ◽  
Completeness Theorem ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Strong Completeness ◽  
Recursive Predicate ◽  
Image Position ◽  
Primitive Recursive ◽  
Intuitionistic Mathematics

The problem of treating the semantics of intuitionistic logic within the framework of intuitionistic mathematics was first attacked by E. W. Beth [1]. However, the completeness theorem he thought to have obtained, was not true, as was shown in detail in a report by V. H. Dyson and G. Kreisel [2]. Some vague remarks of Beth's, for instance in his book, The foundations of mathematics, show that he sustained the hope of restoring his proof. But arguments by K. Gödel and G. Kreisel gave people the feeling that an intuitionistic completeness theorem would be impossible [3]. (A (strong) completeness theorem would implyfor any primitive recursive predicate A of natural numbers, and one has no reason to believe this for the usual intuitionistic interpretation.) Nevertheless, the following contains a correct intuitionistic completeness theorem for intuitionistic predicate logic. So the old arguments by Godel and Kreisel should not work for the proposed semantical construction of intuitionistic logic. They do not, indeed. The reason is, loosely speaking, that negation is treated positively.Although Beth's semantical construction for intuitionistic logic was not satisfying from an intuitionistic point of view, it proved to be useful for the development of classical semantics for intuitionistic logic. A related and essentially equivalent classical semantics for intuitionistic logic was found by S. Kripke [4].

Intuitionism

2018 ◽  
Author(s):  
David Charles McCarty
Keyword(s):  
Twentieth Century ◽  
Intuitionistic Logic ◽  
A Priori ◽  
Higher Order ◽  
Natural Languages ◽  
Higher Order Logic ◽  
Bounded Interval ◽  
Higher Order Functions ◽  
Uniformly Continuous ◽  
Intuitionistic Mathematics

Ultimately, mathematical intuitionism gets its name and its epistemological parentage from a conviction of Kant: that intuition reveals basic mathematical principles as true a priori. Intuitionism’s mathematical lineage is that of radical constructivism: constructive in requiring proofs of existential claims to yield provable instances of those claims; radical in seeking a wholesale reconstruction of mathematics. Although partly inspired by Kronecker and Poincaré, twentieth-century intuitionism is dominated by the ‘neo-intuitionism’ of the Dutch mathematician L.E.J. Brouwer. Brouwer’s reworking of analysis, paradigmatic for intuitionism, broke the bounds on traditional constructivism by embracing real numbers given by free choice sequences. Brouwer’s theorem – that every real-valued function on a closed, bounded interval is uniformly continuous – brings intuitionism into seeming conflict with results of conventional mathematics. Despite Brouwer’s distaste for logic, formal systems for intuitionism were devised and developments in intuitionistic mathematics began to parallel those in metamathematics. A. Heyting was the first to formalize both intuitionistic logic and arithmetic and to interpret the logic over types of abstract proofs. Tarski, Beth and Kripke each constructed a distinctive class of models for intuitionistic logic. Gödel, in his Dialectica interpretation, showed how to view formal intuitionistic arithmetic as a calculus of higher-order functions. S.C. Kleene gave a ‘realizability’ interpretation to the same theory using codes of recursive functions. In the last decades of the twentieth century, applications of intuitionistic higher-order logic and type theory to category theory and computer science have made these systems objects of intense study. At the same time, philosophers and logicians, under the influence of M. Dummett, have sought to enlist intuitionism under the banner of general antirealist semantics for natural languages.

scholarly journals A logic for reasoning about qualitative probability

2010 ◽  
Vol 87 (101) ◽  
pp. 97-108 ◽  
Author(s):  
Angelina Ilic-Stepic
Keyword(s):  
Completeness Theorem ◽  
Higher Order ◽  
Probabilistic Logic ◽  
Qualitative Probability ◽  
Probability Operators ◽  
Probability Operator

We offer extended completeness theorem for probabilistic logic that combines higher-order probabilities (nesting of probability operators) and the qualitative probability operator.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Quantitative EPMA of nitrogen : A tricky element in the electron-probe microanalyzer

1992 ◽  
Vol 50 (2) ◽  
pp. 1622-1623
Author(s):  
G.F. Bastin ◽  
H.J.M. Heijligers
Keyword(s):  
Background Subtraction ◽  
Electron Probe ◽  
Detection System ◽  
Higher Order ◽  
Light Elements ◽  
Strong Suppression ◽  
Electron Probe Microanalyzer ◽  
Quantitative Epma ◽  
Peak Count ◽  
Lead Stearate

Among the ultra-light elements B, C, N, and O nitrogen is the most difficult element to deal with in the electron probe microanalyzer. This is mainly caused by the severe absorption that N-Kα radiation suffers in carbon which is abundantly present in the detection system (lead-stearate crystal, carbonaceous counter window). As a result the peak-to-background ratios for N-Kα measured with a conventional lead-stearate crystal can attain values well below unity in many binary nitrides . An additional complication can be caused by the presence of interfering higher-order reflections from the metal partner in the nitride specimen; notorious examples are elements such as Zr and Nb. In nitrides containing these elements is is virtually impossible to carry out an accurate background subtraction which becomes increasingly important with lower and lower peak-to-background ratios. The use of a synthetic multilayer crystal such as W/Si (2d-spacing 59.8 Å) can bring significant improvements in terms of both higher peak count rates as well as a strong suppression of higher-order reflections.

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