Newton C. A. da Costa. On the theory of inconsistent formal systems. Notre Dame journal of formal logic, vol. 15 (1974), pp. 497–510. - Newton C. A. da Costa. The philosophical import of paraconsistent logic. The journal of non-classical logic (Campinas), vol. 1 (1982), pp. 1–19. - Newton C. A. da Costa. On paraconsistent set theory. Logique et analyse, n.s. vol. 29 (1986), pp. 361–371. - Newton C. A. da Costa, Jean-Yves Béziau, and Otávio Bueno. Paraconsistent logic in a historical perspective. Logique et analyse, vol. 38 (for 1995, pub. 1997), pp. 111–126.

1998 ◽  
Vol 63 (3) ◽  
pp. 1183-1184
Author(s):  
Henry Kyburg
Keyword(s):  
Set Theory ◽  
Classical Logic ◽  
Historical Perspective ◽  
Paraconsistent Logic ◽  
Formal Logic ◽  
Formal Systems

scholarly journals On Development of Formal Systems Starting from Primitive Logic

1966 ◽  
Vol 28 ◽  
pp. 79-83 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Classical Logic ◽  
Formal Systems

It has been my program to develop fundamental theories of mathematics starting from TABOOS and standing on the primitive logic LO at first instead of starting from AXIOMS and standing on the fairly brought up logic, the lower classical logic LK. This was proposed in my work [1].

Logic and Philosophical Methodology

2016 ◽  
Author(s):  
John P. Burgess
Keyword(s):  
Set Theory ◽  
Model Theory ◽  
Classical Logic ◽  
Proof Theory ◽  
Recursion Theory ◽  
Theory Model ◽  
Philosophical Methodology ◽  
Paraconsistent Logics ◽  
Philosophical Logic

This article explores the role of logic in philosophical methodology, as well as its application in philosophy. The discussion gives a roughly equal coverage to the seven branches of logic: elementary logic, set theory, model theory, recursion theory, proof theory, extraclassical logics, and anticlassical logics. Mathematical logic comprises set theory, model theory, recursion theory, and proof theory. Philosophical logic in the relevant sense is divided into the study of extensions of classical logic, such as modal or temporal or deontic or conditional logics, and the study of alternatives to classical logic, such as intuitionistic or quantum or partial or paraconsistent logics. The nonclassical consists of the extraclassical and the anticlassical, although the distinction is not clearcut.

A weak absolute consistency proof for some systems of illative combinatory logic

1983 ◽  
Vol 48 (3) ◽  
pp. 771-776 ◽  
Author(s):  
M.W. Bunder
Keyword(s):  
Set Theory ◽  
Combinatory Logic ◽  
Weak Consistency ◽  
Consistency Proof ◽  
First Order ◽  
Formal Systems ◽  
Elementary Methods ◽  
Order Arithmetic ◽  
Strongly Consistent ◽  
Order Predicate Calculus

A large number of formal systems based on combinatory logic or λ-calculus have been extended to include first order predicate calculus. Several of these however have been shown to be inconsistent, all, as far as the author knows, in the strong sense that all well formed formulas (which here include all strings of symbols) are provable. We will call the corresponding consistency notion—an arbitrary wff ⊥ is provable—weak consistency. We will say that a system is strongly consistent if no formula and its negation are provable.Now for some systems, such as that of Kuzichev [11], the strong and weak consistency notions are equivalent, but in the systems of [5] and [6], which we will be considering, they are not. Each of these systems is strong enough to have all of ZF set theory, except Grounding and Choice, interpretable in it, and the system of [5] can also encompass first order arithmetic (see [7]). It therefore seems unlikely that a strong consistency result could be proved for these systems using elementary methods. In this paper however, we prove the weak consistency of both these systems by means that could be formulated, at least within the theory of [5]. The method also applies to the typed systems of Curry, Hindley and Seldin [10] and to Seldin's generalised types [12].

scholarly journals On Arbitrary sets andZFC

2011 ◽  
Vol 17 (3) ◽  
pp. 361-393 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Set Theory ◽  
Axiom Of Choice ◽  
Real Numbers ◽  
The Real ◽  
Formal Systems ◽  
Infinite Sets ◽  
The Axiom Of Choice ◽  
The Continuum

AbstractSet theory deals with the most fundamental existence questions in mathematics-questions which affect other areas of mathematics, from the real numbers to structures of all kinds, but which are posed as dealing with the existence of sets. Especially noteworthy are principles establishing the existence of some infinite sets, the so-called “arbitrary sets.” This paper is devoted to an analysis of the motivating goal of studying arbitrary sets, usually referred to under the labels ofquasi-combinatorialismorcombinatorial maximality. After explaining what is meant by definability and by “arbitrariness,” a first historical part discusses the strong motives why set theory was conceived as a theory of arbitrary sets, emphasizing connections with analysis and particularly with the continuum of real numbers. Judged from this perspective, the axiom of choice stands out as a most central and natural set-theoretic principle (in the sense of quasi-combinatorialism). A second part starts by considering the potential mismatch between the formal systems of mathematics and their motivating conceptions, and proceeds to offer an elementary discussion of how far the Zermelo–Fraenkel system goes in laying out principles that capture the idea of “arbitrary sets”. We argue that the theory is rather poor in this respect.

scholarly journals Recovery operators, paraconsistency and duality

2019 ◽  
Vol 28 (5) ◽  
pp. 624-656 ◽  
Author(s):  
Walter Carnielli ◽  
Marcelo E Coniglio ◽  
Abilio Rodrigues
Keyword(s):  
Common Core ◽  
Classical Logic ◽  
Propositional Logic ◽  
Inference Rules ◽  
Object Language ◽  
Logical Operators ◽  
Formal Systems ◽  
Excluded Middle

Abstract There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express metalogical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the logics of formal inconsistency (LFIs) and by the logics of formal undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core—in the case studied here, this common core is classical positive propositional logic. The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of LFI and undeterminedness, namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion one at a time. The last sections offer an algebraic account for such logics by adapting the swap structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite nondeterministic matrices.

Category theory, applications to the foundations of mathematics

2018 ◽  
Author(s):  
Colin McLarty
Keyword(s):  
Set Theory ◽  
Category Theory ◽  
Classical Logic ◽  
Elementary Theory ◽  
Mathematical Structure ◽  
Point Of View ◽  
Foundations Of Mathematics ◽  
Logical Foundations ◽  
Classical Mathematics ◽  
The 1960S

Since the 1960s Lawvere has distinguished two senses of the foundations of mathematics. Logical foundations use formal axioms to organize the subject. The other sense aims to survey ‘what is universal in mathematics’. The ontology of mathematics is a third, related issue. Moderately categorical foundations use sets as axiomatized by the elementary theory of the category of sets (ETCS) rather than Zermelo–Fraenkel set theory (ZF). This claims to make set theory conceptually more like the rest of mathematics than ZF is. And it suggests that sets are not ‘made of’ anything determinate; they only have determinate functional relations to one another. The ZF and ETCS axioms both support classical mathematics. Other categories have also been offered as logical foundations. The ‘category of categories’ takes categories and functors as fundamental. The ‘free topos’ (see Lambek and Couture 1991) stresses provability. These and others are certainly formally adequate. The question is how far they illuminate the most universal aspects of current mathematics. Radically categorical foundations say mathematics has no one starting point; each mathematical structure exists in its own right and can be described intrinsically. The most flexible way to do this to date is categorically. From this point of view various structures have their own logic. Sets have classical logic, or rather the topos Set has classical logic. But differential manifolds, for instance, fit neatly into a topos Spaces with nonclassical logic. This view urges a broader practice of mathematics than classical. This article assumes knowledge of category theory on the level of Category theory, introduction to §1.

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