THE SECOND-ORDER PREDICATE LOGIC. THEORY OF DEFINITION

1971 ◽  
pp. 83-106
Author(s):  
ROBERT ROGERS
Keyword(s):  
Predicate Logic ◽  
Second Order ◽  
Theory Of Definition

scholarly journals Higher-Order Illative Combinatory Logic

2013 ◽  
Vol 78 (3) ◽  
pp. 837-872 ◽  
Author(s):  
Łukasz Czajka
Keyword(s):  
Strong Consistency ◽  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Model Construction ◽  
Partial Answer ◽  
Combinatory Logic ◽  
First Order ◽  
Propositional Quantifiers

AbstractWe show a model construction for a system of higher-order illative combinatory logic thus establishing its strong consistency. We also use a variant of this construction to provide a complete embedding of first-order intuitionistic predicate logic with second-order propositional quantifiers into the system of Barendregt, Bunder and Dekkers, which gives a partial answer to a question posed by these authors.

scholarly journals Definability theorem for the intuitionistic predicate logic with equality

1982 ◽  
Vol 87 ◽  
pp. 1-15
Author(s):  
Chiharu Mizutani
Keyword(s):  
Predicate Logic ◽  
Second Order ◽  
Order Logic ◽  
Infinitary Logic ◽  
Second Order Logic

Svenonius’ definability theorem and its generalizations to the infinitary logic Lω1ω or to a second order logic with countable conjunctions and disjunctions have been studied by Kochen [1], Motohashi [2], [3] or Harnik and Makkai [4], independently. In this paper, we consider a (Svenonius-type) definability theorem for the intuitionistic predicate logic IL with equality.

Second- and higher-order logics

2018 ◽  
Author(s):  
Shaughan Lavine
Keyword(s):  
Predicate Logic ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Third Order ◽  
First Order ◽  
Proof Systems ◽  
The Universe ◽  
Second Order Logic

In first-order predicate logic there are symbols for fixed individuals, relations and functions on a given universe of individuals and there are variables ranging over the individuals, with associated quantifiers. Second-order logic adds variables ranging over relations and functions on the universe of individuals, and associated quantifiers, which are called second-order variables and quantifiers. Sometimes one also adds symbols for fixed higher-order relations and functions among and on the relations, functions and individuals of the original universe. One can add third-order variables ranging over relations and functions among and on the relations, functions and individuals on the universe, with associated quantifiers, and so on, to yield logics of even higher order. It is usual to use proof systems for higher-order logics (that is, logics beyond first-order) that include analogues of the first-order quantifier rules for all quantifiers. An extensional n-ary relation variable in effect ranges over arbitrary sets of n-tuples of members of the universe. (Functions are omitted here for simplicity: remarks about them parallel those for relations.) If the set of sets of n-tuples of members of a universe is fully determined once the universe itself is given, then the truth-values of sentences involving second-order quantifiers are determined in a structure like the ones used for first-order logic. However, if the notion of the set of all sets of n-tuples of members of a universe is specified in terms of some theory about sets or relations, then the universe of a structure must be supplemented by specifications of the domains of the various higher-order variables. No matter what theory one adopts, there are infinitely many choices for such domains compatible with the theory over any infinite universe. This casts doubt on the apparent clarity of the notion of ‘all n-ary relations on a domain’: since the notion cannot be defined categorically in terms of the domain using any theory whatsoever, how could it be well-determined?

A new deconstructive logic: linear logic

1997 ◽  
Vol 62 (3) ◽  
pp. 755-807 ◽  
Author(s):  
Vincent Danos ◽  
Jean-Baptiste Joinet ◽  
Harold Schellinx
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Predicate Logic ◽  
Linear Logic ◽  
Second Order ◽  
Main Concern ◽  
Simple Pattern ◽  
Strong Normalization ◽  
Viable Systems ◽  
Proof Nets

AbstractThe main concern of this paper is the design of a noetherian and confluent normalization for LK2 (that is, classical second order predicate logic presented as a sequent calculus).The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD ([10, 12, 32, 36]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ ([26, 27]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making.A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these ‘deconstructive purposes’.

scholarly journals Proofs for free

2012 ◽  
Vol 22 (2) ◽  
pp. 107-152 ◽  
Author(s):  
JEAN-PHILIPPE BERNARDY ◽  
PATRIK JANSSON ◽  
ROSS PATERSON
Keyword(s):  
Programming Language ◽  
Predicate Logic ◽  
Type Systems ◽  
Second Order ◽  
Pure Type ◽  
Parametric Polymorphism ◽  
Pure Type Systems ◽  
System F

AbstractReynolds' abstraction theorem (Reynolds, J. C. (1983) Types, abstraction and parametric polymorphism, Inf. Process.83(1), 513–523) shows how a typing judgement in System F can be translated into a relational statement (in second-order predicate logic) about inhabitants of the type. We obtain a similar result for pure type systems (PTSs): for any PTS used as a programming language, there is a PTS that can be used as a logic for parametricity. Types in the source PTS are translated to relations (expressed as types) in the target. Similarly, values of a given type are translated to proofs that the values satisfy the relational interpretation. We extend the result to inductive families. We also show that the assumption that every term satisfies the parametricity condition generated by its type is consistent with the generated logic.

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