Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory

1980 ◽  
Vol 1 (1-2) ◽  
pp. 95-137 ◽  
Author(s):  
Gregory H. Moore
Keyword(s):  
Mathematical Logic ◽  
Set Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Axiomatic Set Theory

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

Truth Definitions, Skolem Functions and Axiomatic Set Theory

1998 ◽  
Vol 4 (3) ◽  
pp. 303-337 ◽  
Author(s):  
Jaakko Hintikka
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Negative Results ◽  
Skolem Functions ◽  
Order Language ◽  
Definition Of ◽  
Axiomatic Set Theory

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski (1935). In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore (1994, p. 635) asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable.

scholarly journals Strong Logics of First and Second Order

2010 ◽  
Vol 16 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Peter Koellner
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Close Correspondence ◽  
First Order ◽  
Second Order Logic

AbstractIn this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

scholarly journals Much Ado About the Many

2021 ◽  
Vol 25 (1) ◽  
Author(s):  
Jonathan Mai
Keyword(s):  
Set Theory ◽  
Formal Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Plural Quantification ◽  
First Order ◽  
Order Language ◽  
The Many ◽  
Second Order Logic ◽  
Alternative Theories

English distinguishes between singular quantifiers like "a donkey" and plural quantifiers like "some donkeys". Pluralists hold that plural quantifiers range in an unusual, irreducibly plural, way over common objects, namely individuals from first-order domains and not over set-like objects. The favoured framework of pluralism is plural first-order logic, PFO, an interpreted first-order language that is capable of expressing plural quantification. Pluralists argue for their position by claiming that the standard formal theory based on PFO is both ontologically neutral and really logic. These properties are supposed to yield many important applications concerning second-order logic and set theory that alternative theories supposedly cannot deliver. I will show that there are serious reasons for rejecting at least the claim of ontological innocence. Doubt about innocence arises on account of the fact that, when properly spelled out, the PFO-semantics for plural quantifiers is committed to set-like objects. The correctness of my worries presupposes the principle that for every plurality there is a coextensive set. Pluralists might reply that this principle leads straight to paradox. However, as I will argue, the true culprit of the paradox is the assumption that every definite condition determines a plurality.

Logical Methods for Self-Configuration of Network Devices

2011 ◽  
pp. 189-216 ◽  
Author(s):  
Sylvain Hallé ◽  
Roger Villemaire ◽  
Omar Cherkaoui
Keyword(s):  
Mathematical Logic ◽  
Order Logic ◽  
Point Of View ◽  
First Order Logic ◽  
Declarative Languages ◽  
First Order ◽  
User Intervention

The goal of self-configuration consists of providing appropriate values for parameters that modulate the behaviour of a device. In this chapter, self-configuration is studied from a mathematical logic point of view. In contrast with imperative means of generating configurations, characterized by scripts and templates, the use of declarative languages such as propositional or first-order logic is argued. In that setting, device configurations become models of particular logical formulæ, which can be generated using constraint solvers without any rigid scripting or user intervention.

Gödel, Kurt (1906–78)

2018 ◽  
Author(s):  
John W. Dawson
Keyword(s):  
Twentieth Century ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Mathematical Platonism ◽  
The Axiom Of Choice ◽  
Fundamental Theorems ◽  
The Continuum

The greatest logician of the twentieth century, Gödel is renowned for his advocacy of mathematical Platonism and for three fundamental theorems in logic: the completeness of first-order logic; the incompleteness of formalized arithmetic; and the consistency of the axiom of choice and the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory.

Infinitary logics

2018 ◽  
Author(s):  
Bernd Buldt
Keyword(s):  
Mathematical Logic ◽  
Order Logic ◽  
Logical Analysis ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Essential Tool

An infinitary logic arises from ordinary first-order logic when one or more of its finitary properties is allowed to become infinite, for example, by admitting infinitely long formulas or infinitely long or infinitely branched proof figures. The need to extend first-order logic became pressing in the late 1950s when it was realized that many of the fundamental notions of mathematics cannot be expressed in first-order logic in a way that would allow for their logical analysis. Because infinitary logics often do not suffer the same limitation, they have become an essential tool in mathematical logic.

Minimal predicates, fixed-points, and definability

2005 ◽  
Vol 70 (3) ◽  
pp. 696-712 ◽  
Author(s):  
Johan Van Benthem
Keyword(s):  
Mathematical Logic ◽  
Fixed Points ◽  
Expressive Power ◽  
Correspondence Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Technical Result ◽  
Unique Existence ◽  
First Order ◽  
Preservation Theorem

AbstractMinimal predicates P satisfying a given first-order description ϕ(P) occur widely in mathematical logic and computer science. We give an explicit first-order syntax for special first-order ‘PIA conditions’ ϕ(P) which guarantees unique existence of such minimal predicates. Our main technical result is a preservation theorem showing PIA-conditions to be expressively complete for all those first-order formulas that are preserved under a natural model-theoretic operation of ‘predicate intersection’. Next, we show how iterated predicate minimization on PIA-conditions yields a language MIN(FO) equal in expressive power to LFP(FO), first-order logic closed under smallest fixed-points for monotone operations. As a concrete illustration of these notions, we show how our sort of predicate minimization extends the usual frame correspondence theory of modal logic, leading to a proper hierarchy of modal axioms: first-order-definable, first-order fixed-point definable, and beyond.

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