Paraconsistent Reasoning in Science and Mathematics

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The detailed program of Paraconsistent Reasoning in Science and Mathematics is also available as printable overview (PDF, 54kb).

11 June

09:00 - 09:15 Opening speech by Holger Andreas and Peter Verdée
09:15 - 10:15 Diderik Batens: “Transitory and Permanent Applications of Paraconsistency”
10:15 - 10:30 Coffee Break
10:30 - 11.30 Heinrich Wansing: “On the methodology of paraconsistent logic”
11:30 - 11:45 Coffee Break
11:45 - 12:45 Andreas Kapsner: “Why designate gluts?”
12:45 - 14:30 Lunch Break
14:30 - 15:00 Hitoshi Omori: “From paraconsistent logic to dialetheic logic ”
15:00 - 15:30 Cian Chartier: “Revision-Theoretic Truth and Degrees of Paradoxicality”
15:30 - 16:00 Coffee Break
16:00 - 17:00 Franz Berto: “Inconsistent Thinking, Fast and Slow”

12 June

09:15 - 10:15 Graham Priest: “Models of Naive Set Theory Validating ZF”
10:15 - 10:30 Coffee Break
10:30 - 11:30 Itala Maria Loffredo D’Ottaviano: “Can a paraconsistent differential calculus extend the classical calculus?”
11:30 - 11:45 Coffee Break
11:45 - 12:15 Zach Weber: “Recursive functions for paraconsistent reasoners"
12:15 - 12:45 João Marcos: “What makes for a good paraconsistent negation?”
12:45 - 14:30 Lunch Break
14:30 - 15:00 Luis Estrada-González: “Prospects for triviality”
15:00 - 15:30 Nick Thomas: “On the interpretation of classical mathematics in naïve set theory.”
15:30 - 16:00 Coffee Break
16:00 - 17.00 Maarten McKubre-Jordens: “Doing Mathematics Paraconsistently. A manifesto.”
17:00 - 17:30 Fenner Tanswell: “Saving Proof from Paradox: Against the Inconsistency of Informal Mathematics”
19:00 Conference dinner in Restaurant “Schlosswirtschaft Schwaige”
Address: Schloss Nymphenburg 30, 80638 München

13 June

09:15 - 10:15 Bryson Brown: “On the Preservation of Reliability”
10:15 - 10:30 Coffee Break
10:30 - 11:30 Holger Andreas: “A Paraconsistent Generalization of Carnap’s Logic of Theoretical Terms”
11:30 - 11:45 Coffee Break
11:45 - 12:15 Corry Shores: “Instantaneous Contradiction in Motion and Perception: Modeling the Phenomenal Present with a Dialetheic Logic of Time”
12:15 - 12:45 David Ripley: “Confusion and contradiction”
12:45 - 14:30 Lunch Break
14:30 - 15:00 Maria Martinez: “Inconsistent and Functional Theories: A Classification”
15:00 - 15:30 Diego Tajer: “Revenge for Berto’s Law of Non-Contradiction”
15:30 - 16:00 Coffee Break
16:00 - 17.00 Otávio Bueno: “Inconsistent Scientific Theories: A Framework”

Abstracts of invited talks

Holger Andreas: A Paraconsistent Generalization of Carnap’s Logic of Theoretical Terms

Some scientific theories are inconsistent, yet non-trivial and meaningful. How is that possible? The present paper aims to show that we can analyse the inferential use of such theories in terms of consistent compositions of the applications of universal axioms. This technique will be represented by a modular semantics, which allows us to accept the instances of universal axioms selectively. For such a semantics to develop, the framework of partial structures by da~Costa and French will be extended by a few elements of the Sneed formalism, also known as the structuralist approach to

Diderik Batens: Transitory and Permanent Applications of Paraconsistency

The advent of paraconsistency offers an excellent opportunity to unveil past prejudices. These do not only concern the truth or sensibility of inconsistencies, but many aspects of the nature of logic(s). My aim will be to raise questions (and possibly arrive at insights) on such topics as the following: methodological versus descriptive application contexts of logics, logical pluralism (in those application contexts), arguments for considering a specific application as suitable, truth-functionality of logical operators, sensibility of (certain uses of) classical negation, fallibilism in

Franz Berto: Inconsistent Thinking, Fast and Slow

This plays on Kahneman’s Thinking Fast and Slow. We implement two reasoning systems: our Slow system is logical-rule-based. Our Fast system is associative, context-sensitive, and integrates what we conceive via background information Slow inconsistent thinking may rely on paraconsistent logical rules, but I focus on Fast inconsistent thinking. I approach our Fast-conceiving inconsistencies in terms of ceteris paribus intentional operators: variably restricted quantifiers on possible and impossible worlds. The explicit content of an inconsistent conception is similar to a ceteris paribus relevant conditional antecedent. I discuss how such operators invalidate logical closure for conceivability, and how similarity works when impossible worlds are

Bryson Brown:On the Preservation of Reliability

…all models are wrong, but some are useful. (G E.P. Box and N. R. Draper, 1987)

C.S. Peirce examined several broad methods for arriving at beliefs in “On the Fixation of Belief”; the central theme of his essay is the importance of having a method that leads to stable agreement amongst the members of a society. Peirce argues that the ‘scientific method meets this standard, generalizing Hobbes’ observation that Harvey’s hypothesis of the circulation of the blood is an important example of a once-controversial view that came to be accepted even by those who initially rejected it, and a demonstration of the special epistemic success of science, in contrast with other forms of inquiry. But stable agreements are sometimes overturned. The brilliant, wide-ranging successes of classical physics made the basic principles of Newton’s theory laws of nature in the eyes of physicists, philosophers and the educated public. Yet they have been superseded by new principles. This pattern of success followed by failure and replacement is the key premise of Laudan’s pessimistic induction argument against scientific realism. Yet we have confidence, often justified confidence, in many scientific inferences, including inferences that begin with models and theoretical principles known to be false. Though the equations used and descriptions of actual systems our models are applied to are false, the results of the calculations (even when only approximate) are considered reliable for many purposes, and with good reason. Thus models provided by orbital mechanics are used to place probes in desired orbits and even land on other planets, while atmospheric GCMs are used to estimate large-scale climate changes likely to occur given various scenarios for human GHG emissions. Neither the equations used nor the descriptions of the systems such models include are true; the results of the calculations (even assuming the calculations are exact) cannot reasonably be taken to be true, either. Yet we do rely on them, and with good reason. Many measurable physical quantities that models allow us to calculate values for turn out to be very close to the results of actual observations under a wide range of specifiable conditions. This paper develops a pragmatic view of theories, models and the inferences we use our models and theories to make. The preservation of reliability allows for the use of incompatible theories and principles in our inferences, along with contextually-determined levels of acceptable approximation, and conceptual shifts in how measurement results are interpreted in the light of the different principles relied on at different points. But it is also compatible with a modest scientific realism: the criteria by which we decide when a theory can be relied on generally include measurable parameters whose values, even though our understanding of them remains imperfect, reliably indicate when and to what extent a given theory is reliable and constitutes reliably settled science in a sense that Peirce might have found satisfying even without an accepted background theory in which we can explain both that reliability and its

Otávio Bueno: Inconsistent Scientific Theories: A Framework

Four important issues need to be considered when inconsistent scientific theories are under discussion: (1) To begin with, are there–and can there be–such things as inconsistent scientific theories? On standard conceptions of the structure of scientific theories, such as the semantic and the syntactic approaches (Suppe [1989], and van Fraassen [1980]), there is simply no room for such theories, given the classical underpinnings of these views. In fact, both the syntactic and the semantic approaches assume that the underlying logic is classical, and as is well known, in classical logic everything follows from an inconsistent theory. Despite this fact, it seems undeniable that inconsistent scientific theories have been entertained–or, at least, stumbled upon–throughout the history of science. So, it looks as though we need to make room for them. (2) But once some room is made for inconsistent scientific theories, how exactly should they be accommodated? In particular, it seems crucial that we are able to understand the styles of reasoning that involve inconsistencies; that is, the various ways in which scientists and mathematicians reason from inconsistent assumptions without deriving everything from them. It is tempting, of course, to adopt a paraconsistent logic to model some of the reasoning styles in question (see da Costa and French [2003], da Costa, Krause, and Bueno [2007], and da Costa, Bueno, and French [1998]). This is certainly a possibility. However, actual scientific practice is not typically done using paraconsistent logic. And if our goal is to understand that practice in its own terms, rather than to produce a parallel discourse about that practice that somehow justifies the adequacy of the latter by invoking tools that are foreign to it, an entirely different strategy is called for. (3) What are the sources of the inconsistencies in scientific theories? Do such inconsistencies emerge from empirical reasons, from conceptual reasons, from both, or by sheer mistake? By identifying the various sources in question, we can handle and assess the significance of the inconsistencies in a better way. Perhaps some inconsistencies are more important, troublesome, or heuristically fruitful than others—and this should be part of their assessment. (4) Several scientific theories become inconsistent due to the mathematical framework they assume. For example, the theories may refer to infinitesimals, as the latter were originally formulated in the early versions of the calculus (see Robinson [1974] and Bell [2005]), the theories may invoke Dirac’s delta function (Dirac [1958]), or some other arguably inconsistent mathematical framework. The issue then arises as to how we should deal with inconsistent applied mathematical theories. What is the status of these theories? Which commitments do they bring? Are we committed to the existence of inconsistent objects if we use such theories in explaining the phenomena? Can an inconsistent scientific theory ever be indispensable?
Questions of this sort need to be answered so that we can make sense of the role of inconsistent theories in applications. (For an insightful discussion, see Colyvan [2009].)
In this paper, I examine these four issues, and develop a framework–in terms of partial mappings (Bueno, French and Ladyman [2002], and Bueno [2006]), and the inferential conception of the application of mathematics (Bueno and Colyvan [forthcoming])–to represent and interpret inconsistent theories in science. Along the way, I illustrate how the framework can be used to make sense of various allegedly inconsistent theories, from the early formulations of the calculus through Dirac’s delta function and Bohr’s atomic model (Bohr [1913]).top

Itala Maria Loffredo D’Ottaviano: “Can a paraconsistent differential calculus extend the classical calculus?”

In 2000, da Costa proposes the construction of a paraconsistent differential calculus, whose language is the language L of his known paraconsistent logic C1, extended to the language of his paraconsistent set theory CHU1, introduced in 1986. We have studied and improved the calculus proposed by da Costa, having obtained extensions of several fundamental theorems of the classical differential calculus. From the introduction of the concept of paraconsistent super-structure X over a set X of atoms of CHU1 and of the concept of monomorphism between paraconsistent super-structures, we will present a Transference Theorem that “translates” the classical differential calculus into da Costa’s paraconsistent calculus.
Da COSTA, N.C.A. Paraconsistent Mathematics. In: I WORLD CONGRESS ON PARACONSISTENCY, 1998, Ghent, Belgium. Frontiers in paraconsistent logic: proceedings. Edited by D. Batens, C. Mortensen, G. Priest, J.P. van Bendegen. London: King’s College Publications, 2000, p.

Andreas Kapsner: Why designate gluts?

In this talk, I want to explore the following idea: Truth value gluts should be allowed in the semantics of logical systems, as they are in many non-classical systems. However, unlike what is standard in such systems, these gluts should be treated as undesignated values. I shall give my reasons for taking this to be a view worth exploring and discuss its effects on such topics as dialetheism, paraconsistency and relevance. On the whole, it will turn out to be a surprisingly attractive view that deals well with epistemic inconsistencies and semantic paradoxes. Some of the greatest difficulties arise in the attempt to account for interesting inconsistent scientific and mathematical theories; this, then, will be the touchstone for the proposed

Maarten McKubre-Jordens: Doing Mathematics Paraconsistently. A manifesto.

In this talk, we outline several motivations for conducting mathematics–in the style of the working mathematician–without dependence on assumptions of non-contradiction. The story involves a short analysis of theorem and counterexample, what it is to reason paraconsistently within mathematics, and takes note of some non-traditional obstacles and attempts to resolve them. In part, this will provide motivation to the mathematician to think outside the box when approaching surprising conclusions within the usual framework. Then, as we delve into the mathematics, we survey some recent results in elementary analysis when performed paraconsistently, and outline some conjectures for future research.
This talk is of interest both to provide reasons and techniques for paraconsistent mathematics, and to show how rich a picture can be painted without recourse to assumptions of

Graham Priest: Models of Naive Set Theory Validating ZF

Any adequate paraconsistent set theory must be able to validate at least a major part of the standard results of orthodox set theory. One way to achieve this is to take the universe or universes of sets to be such as to validate not only the naïve principles, but also all the theorems of Zermelo Fraenkel set theory. In this talk I will discuss various constructions of models of set theory which do just

Heinrich Wansing: On the methodology of paraconsistent logic

The present note contains a critical discussion of the methodology of paraconsistent logic in general and “the central optimisation problem of paraconsistent logics” in particular. It is argued that there exist several reasons not to consider classical logic as the reference logic for developing systems of paraconsistent logic, it is suggested to weaken a certain maximality condition that may be seen as essential for “optimisation”, and it is briefly pointed out that there are other notions of maximality. Moreover, it is argued that the guiding motivation for the development of paraconsistent logics should be neither epistemological nor ontological, but

Abstracts of contributed talks

Cian Chartier: Revision-Theoretic Truth and Degrees of Paradoxicality

I show a variant of the Revision Theory of Truth can partially define truth for sentences in a paraconsistent logic with predicates for truth and (a proper class of) degrees of truth-paradoxicality, in a manner which can be defined in a classical metalanguage. I propose a new Paraconsistent Limit Rule Principle to establish fixed points for locally-defined extensions of the truth and paradoxicality predicates, from which I find incidental support for the view of Roy T. Cook that the concept of truth value is indefinitely

Luis Estrada-González: Prospects for triviality

In this talk I argue, pace Mortensen, that there is a case in which no mathematical catastrophe is implied by mathematical triviality, namely that of a degenerate topos, an extremely simple mathematical universe in which everything is true. I will show that either one of the premises of Dunn’s trivialization result for real number theory on which Mortensen mounts his case cannot obtain (from a point of view external to the universe) or that it obtains in calculations internal to such trivial universe and the theory associated, yet the calculations are possible and meaningful albeit extremely

João Marcos: What makes for a good paraconsistent negation?

Consequence relations for paraconsistent logics are quite simple to specify, once a negation ¬ with the appropriate behavior is available. However, deciding whether such symbol ¬ deserves to be called a negation, to start with, and guaranteeing that ⁄ has good logical properties, are often much more involved tasks. This talk will deal with a large class of paraconsistent logics which are negation-decreasing, have natural modal semantics, and moreover have a rich language that is able to express the very notion of consistency at the object-language level. All along, a very inclusive yet forceful definition of negation will be assumed, designed to help us settling what sort of opposition is a paraconsistent negation, after

Maria del Rosario Martinez Ordaz: Inconsistent and Functional Theories: A Classification

What do our scientific theories can tell us about consistency in science? Recent research has suggested that if a scientific theory was inconsistent, it could only be in an inner sense. Here I will argue that, by appealing to certain philosophical analysis regarding the operation of the functional empirical theories, it is possible to recognize at least three types of inconsistencies in the empirical sciences, and also to provide a classification for what we can call functional and inconsistent empirical

Hitoshi Omori: From paraconsistent logic to dialetheic logic

Dialetheism is the metaphysical view that there are true contradictions. And in order to develop dialetheic theories that handle contradictions, dialetheists call for paraconsistent logics. One of the examples is Logic of Paradox developed by Graham Priest. Based on these, the aim of the talk is twofold. First, we review the modern origin of paraconsistency and try to clarify the real challenge of paraconsistency. Second, we argue that we need more than paraconsistent logic in developing dialetheic theories, with a special attention on the dialetheic approach to the problem of naive set

David Ripley: “Confusion and contradiction”

Confusion is what happens when multiple objects, properties, substances, or propositions are treated as though they were one. This can happen for a number of reasons: sometimes we don’t distinguish things because we don’t realize there is a distinction to be drawn, sometimes because we have decided that distinguishing them wouldn’t be worth our time, etc. In previous work, I have argued that logical systems appropriate for confused languages will often be nontransitive. This is the reflection within the system of what, from the outside, looks like equivocation: each step of reasoning is legitimate, but they cannot be legitimately combined. In the present paper, I turn to the case of propositional languages built from conjunction, disjunction, and negation in the usual way. It emerges that contradictions are often provable in the resulting ‘confused’ systems. As in the case of nontransitivity, this might at first seem unwelcome, but on reflection it is exactly what should be wanted: if a is P and b is not P, a system that confuses a with b, calling them both c, ought to ‘think’ that c is both P and not P. Indeed, this prediction is forthcoming. Further, the resulting systems are often nontrivial, while accommodating contradictions as theorems, without being paraconsistent (in the usual sense). Their nontransitivity allows them to handle contradictions in a novel way; there is no need to invalidate ECQ to avoid total collapse, even in the presence of contradictions. I compare these systems favorably to the (paraconsistent, transitive) treatment of confusion favored by [Camp, 2002].top

Corry Shores: Instantaneous Contradiction in Motion and Perception: Modeling the Phenomenal Present with a Dialetheic Logic of Time

One difficulty phenomenologists encounter when describing motion-perception is explaining how the multiple positions of a moving object appear co-present without being physically simultaneous. After examining Graham Priest’s ‘spread hypothesis’ and Barry Dainton’s ‘overlap model’, I will propose a dialetheic account of the specious present consistent with the concept of the instant as it is understood in

Diego Tajer: Revenge for Berto’s Law of Non-Contradiction

Francesco Berto (2008, 2013) proposed a reformulation of the Law of Non-Contradiction which involves the notion of material exclusion. In this talk, I argue against the possibility of adopting this new principle in a dialetheic context. First, I claim that the notion of material exclusion should not be taken as a primitive, because it can be better understood as necessary incompatibility. Then I show that, once this analysis is adopted, Berto’s Law of Non-Contradiction is affected by a variation of Berry’s

Fenner Tanswell: Saving Proof from Paradox: Gödel’s Paradox and the Inconsistency of Informal Mathematics

In In Contradiction, Priest presents what he calls “Gödel’s Paradox”, claiming that this demonstrates that the informal proof, and hence informal mathematics, is inconsistent and inherently so. Furthermore, Priest applies the first incompleteness theorem to informal mathematics to get the same result. In this talk I will give a number of objections to these two arguments, based on the notion of formalisation that is used, the nature of informal proof and the reliance on a single system for all

Nick Thomas: On the interpretation of classical mathematics in naive set theory

I develop an informal hypothesis stating that there is no natural system of logic in which naïve set theory can interpret a significant fragment of classical mathematics. I formalize this hypothesis as a mathematical conjecture, by first developing a theory of what it means for one formal theory to interpret another formal theory, and operationalizing the notion of a “natural system of logic” by drawing on the previously developed theory of “generalized Routley-Meyer logics.” I then state the conjecture that there is no generalized Routley-Meyer logic in which naïve set theory can interpret Σ

Zach Weber: Recursive functions for paraconsistent reasoners

Are there are any properly paraconsistent algorithms—computations that are recognizable as such, but which are not recognized by non-paraconsistent logic? It has been suggested [by Sylvan, Priest, and others] that there are, on the basis of the ‘naive’ arguments of mathematical provers. In a slogan, ‘naive proof is recursive’. This urges us to reconsider the notion of computations in a paraconsistent setting. We look at some simple formulations of paraconsistent machines and their basic properties, and discuss these in relation to the halting problem.