Some logical and syntactical observations concerning the first-order dependent type system λP

1999 ◽  
Vol 9 (4) ◽  
pp. 335-359 ◽  
Author(s):  
HERMAN GEUVERS ◽  
ERIK BARENDSEN
Keyword(s):  
Predicate Logic ◽  
Type System ◽  
Logical Framework ◽  
First Order ◽  
Logical Derivation ◽  
Dependent Type ◽  
First Order Predicate Logic

We look at two different ways of interpreting logic in the dependent type system λP. The first is by a direct formulas-as-types interpretation à la Howard where the logical derivation rules are mapped to derivation rules in the type system. The second is by viewing λP as a Logical Framework, following Harper et al. (1987) and Harper et al. (1993). The type system is then used as the meta-language in which various logics can be coded.We give a (brief) overview of known (syntactical) results about λP. Then we discuss two issues in some more detail. The first is the completeness of the formulas-as-types embedding of minimal first-order predicate logic into λP. This is a remarkably complicated issue, a first proof of which appeared in Geuvers (1993), following ideas in Barendsen and Geuvers (1989) and Swaen (1989). The second issue is the minimality of λP as a logical framework. We will show that some of the rules are actually superfluous (even though they contribute nicely to the generality of the presentation of λP).At the same time we will attempt to provide a gentle introduction to λP and its various aspects and we will try to use little inside knowledge.

Syllogism and quantification

1962 ◽  
Vol 27 (1) ◽  
pp. 58-72 ◽  
Author(s):  
Timothy Smiley
Keyword(s):  
Predicate Logic ◽  
Modern Logic ◽  
First Order ◽  
First Order Predicate Logic

Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.In the familiar first-order predicate logic generality is expressed by means of variables and quantifiers, and each interpretation of the system is based upon the choice of some class over which the variables may range, the only restriction placed on this ‘domain of individuals’ being that it should not be empty.

First order predicate logic

1986 ◽  
pp. 155-183
Author(s):  
Igor Aleksander ◽  
Henri Farreny ◽  
Malik Ghallab
Keyword(s):  
Predicate Logic ◽  
First Order ◽  
First Order Predicate Logic

A Test of a Two-Stage Model of the Evaluation of Hypotheses from Quantified First-Order Predicate Logic in Information Use Tasks

1992 ◽  
Vol 71 (3_suppl) ◽  
pp. 1091-1104 ◽  
Author(s):  
Peter E. Langford ◽  
Robert Hunting
Keyword(s):  
Predicate Logic ◽  
Information Use ◽  
Stage Model ◽  
Two Stage ◽  
First Order ◽  
Alternative Hypotheses ◽  
Age Range ◽  
Hypothesis Evaluation ◽  
Universe Of Discourse ◽  
First Order Predicate Logic

480 adolescents and young adults between the ages of 12 and 29 years participated in an experiment in which they were asked to evaluate hypotheses from quantified first-order predicate logic specifying that certain classes of event were necessarily, possibly, or certainly not included within a universe of discourse. Results were used to test a two-stage model of performance on hypothesis evaluation tasks that originated in work on the evaluation of conditionals. The two-stage model, unlike others available, successfully predicted the range of patterns of reply observed. In dealing with very simple hypotheses subjects in this age range tended not to make use of alternative hypotheses unless these were explicitly or implicitly suggested to them by the task. This tells against complexity of hypothesis as an explanation of the reluctance to use alternative hypotheses in evaluating standard conditionals.

scholarly journals Deontic Logics as Axiomatic Extensions of First-Order Predicate Logic: An Approach Inspired by Wolniewicz’s Formal Ontology of Situations

Axioms ◽  
2019 ◽  
Vol 8 (4) ◽  
pp. 109 ◽  
Author(s):  
Malec
Keyword(s):  
Deontic Logic ◽  
Predicate Logic ◽  
The Other ◽  
Choice Situation ◽  
Formal Ontology ◽  
Legal Rules ◽  
First Order ◽  
The Third ◽  
Other Hand ◽  
First Order Predicate Logic

The aim of this article is to present a method of creating deontic logics as axiomatic theories built on first-order predicate logic with identity. In the article, these theories are constructed as theories of legal events or as theories of acts. Legal events are understood as sequences (strings) of elementary situations in Wolniewicz′s sense. On the other hand, acts are understood as two-element legal events: the first element of a sequence is a choice situation (a situation that will be changed by an act), and the second element of this sequence is a chosen situation (a situation that arises as a result of that act). In this approach, legal rules (i.e., orders, bans, permits) are treated as sets of legal events. The article presents four deontic systems for legal events: AEP, AEPF, AEPOF, AEPOFI. In the first system, all legal events are permitted; in the second, they are permitted or forbidden; in the third, they are permitted, ordered or forbidden; and in the fourth, they are permitted, ordered, forbidden or irrelevant. Then, we present a deontic logic for acts (AAPOF), in which every act is permitted, ordered or forbidden. The theorems of this logic reflect deontic relations between acts as well as between acts and their parts. The direct inspiration to develop the approach presented in the article was the book Ontology of Situations by Boguslaw Wolniewicz, and indirectly, Wittgenstein’s Tractatus Logico-Philosophicus.

Ultraproducts which are not saturated

1967 ◽  
Vol 32 (1) ◽  
pp. 23-46 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Predicate Logic ◽  
Continuum Hypothesis ◽  
First Order ◽  
Generalized Continuum ◽  
First Order Predicate Logic

In this paper we continue our study, begun in [5], of the connection between ultraproducts and saturated structures. IfDis an ultrafilter over a setI, andis a structure (i.e., a model for a first order predicate logicℒ), the ultrapower ofmoduloDis denoted byD-prod. The ultrapower is important because it is a method of constructing structures which are elementarily equivalent to a given structure(see Frayne-Morel-Scott [3]). Our ultimate aim is to find out what kinds of structure are ultrapowers of. We made a beginning in [5] by proving that, assuming the generalized continuum hypothesis (GCH), for each cardinalαthere is an ultrafilterDover a set of powerαsuch that for all structures,D-prodisα+-saturated.

scholarly journals On A New Semantics for First-Order Predicate Logic

2017 ◽  
Vol 46 (3) ◽  
pp. 259-267 ◽  
Author(s):  
Hajnal Andréka ◽  
Johan van Benthem ◽  
István Németi
Keyword(s):  
Predicate Logic ◽  
First Order ◽  
First Order Predicate Logic
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