Bryson Brown

Chunk and Permeate, a Paraconsistent Inference Strategy. Part I: The Infinitesimal Calculus

In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind – specifically concerning the preservation of the consistency of each chunk – and concludes with some other possible applications and technical questions.

Yes, Virginia, there Really are Paraconsistent Logics

B. H. Slater has argued that there cannot be any truly paraconsistent logics, because its always more plausible to suppose whatever “negation” symbol is used in the language is not a real negation, than to accept the paraconsistent reading. In this paper I neither endorse nor dispute Slaters argument concerning negation; instead, my aim is to show that as an argument against paraconsistency, it misses (some of) the target. A important class of paraconsistent logics — the preservationist logics — are not subject to this objection. In addition I show that if we identify logics by means of consequence relations, at least one dialetheic logic can be reinterpreted in preservationist (non-dialetheic) terms. Thus the interest of paraconsistent consequence relations — even those that emerge from dialetheic approaches — does not depend on the tenability of dialetheism. Of course, if dialetheism is defensible, then paraconsistent logic will be required to cope with it. But the existence (and interest) of paraconsistent logics does not depend on a defense of dialetheism.

A Solution to the Completeness Problem for Weakly Aggregative Modal Logic

We are accustomed to regarding K as the weakest modal logic admitting of a relational semantics in the style made popular by Kripke. However, in a series of papers which demonstrates a startling connection between modal logic and the theory of paraconsistent inference, Ray Jennings and Peter Schotch have developed a generalized relational frame theory which articulates an infinite hierarchy of sublogics of K , each expressing a species of “weakly aggregative necessity”. Recall that K is axiomatized, in the presence of N and RM , by the schema of “binary aggregation” For each n ≥ 1, the weakly aggregative modal logic K n is axiomatized by replacing K with the schema of “ n -ary aggregation” which is an n -ary relaxation, or weakening, of K . Note that K 1 = K . In [3], the authors claim without proof that K n is determined by the class of frames F = ( W, R ), where W is a nonempty set and R is an ( n + 1)-ary relation on W , under the generalization of Kriples truth condition according to which □ α is true at a point w in W if and only if α is true at one of x 1 ,…, x n for all x 1 ,…, x n in W such that Rw, x 1 ,…, x n .

On preserving : essays on preservationism and paraconsistent logic

Paraconsistent logic is a theory of reasoning in philosophy that studies inconsistent data. The discipline has several different schools of thought, including preservationism, which responds to the problems that arise when human beings continue to reason when faced with inconsistent data. On Preserving is the first complete account of the Preservationist School, which developed in Canada out of the early work of Raymond Jennings, Peter Schotch, and their students. Assembling the previously scattered works of the Preservationist School, this collection contains all of the most significant works on the basic theory of the preservationist approach to paraconsistent logic. With essays both written and rewritten specifically for this volume, the contributors cover topics that include the motivation for the preservationist approach, as well as more technical results of their research. Concise and unified, On Preserving is the ideal introduction to a distinct philosophical field.

Logic and aggregation

Paraconsistent logic is an area of philosophical logic that has yet to find acceptance from a wider audience. The area remains, in a word, disreputable. In this essay, we try to reassure potential consumers that it is not necessary to become a radical in order to use paraconsistent logic. According to the radicals, the problem is the absurd classical account of contradiction: Classically inconsistent sets explode only because bourgeois classical semantics holds, in the face of overwhelming evidence to the contrary, that both A and ∼ A cannot simultaneously be true! We suggest (more modestly) that there is, at least sometimes, something else worth preserving, even in an inconsistent, unsatisfiable premise set. In this paper we present, in a new guise, a very general version of this “preservationist” approach to paraconsistency.

Smoke and Mirrors: A Few Nice Tricks

Two aims are at work in James Browns Smoke and Mirrors: to defend realism against some of its recent detractors, and to expound his own programmatic commitment to a Platonic form of realism. I am sympathetic to his first goal, and dubious about the second, so, as Brown himself predicts, I am enthusiastic about the critical part of the book but critical of his Platonic project. But I will begin this review with a hearty recommendation. Smoke and Mirrors is clear, articulate, perceptive, occasionally provocative, and a healthy antidote to the sceptical pessimism about science that one encounters so often today.

On the Preservation of Reliability

“Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but, nevertheless, what you get out depends upon what you put in; and as the grandest mill in the world will not extract wheat-flour from peascod, so pages of formulae will not get a definite result out of loose date” (Thomas Huxley (1869) Geological Reform, Presidential Address to the Geological Society). Reasoning in science is a rich and complex phenomenon. On one hand, we find detailed, sophisticated and rigorous calculations. But on the other, we encounter a multiplicity of models and approximations whose status has been the subject of extensive debate (See [6] How the Laws of Physics Lie (Oxford, New York, Oxford University Press) and [7] The Dappled World (Cambridge, Cambridge University Press)). Detailed and demanding calculations give the appearance of mathematical rigour, and from a practical perspective, inferences and calculations based on successful models have proven to be reliable guides to our world, predicting the results of many measurements and suggesting interventions in the world that produce startling and impressive novel phenomena ranging from laser light to transistors to monoclonal antibodies and new types of sub-atomic particles. But the logical incompatibility of different models, each making different assumptions and approximations, together with the application of distinct, conflicting models in the course of deriving important results, raise serious questions about the nature and status of the both the premises and the conclusions of scientific reasoning.

Stipulation and Symmetrical Consequence

In this paper I lay some of the groundwork for a naturalistic, empirically oriented view of logic, attributing the special status of our knowledge of logic to the power of stipulation and expressing the stipulations that constitute the vocabulary of formal logic by rules of inference. The stipulation hypothesis does nothing by itself to explain the usefulness of logic. However, though I do not argue for it here, I believe the selective adoption and application of stipulations can. My concern here is with an issue that has already received a good bit of attention: it seems that we are free to make whatever stipulations we care to make, but we also know that logical stipulations must be carefully constrained, to avoid trivialization, as well as subtler impositions on the already established inferential practices to which we apply our logical vocabulary. I propose three increasingly stringent criteria that fully conservative extensions of a language should meet, and apply them to evaluate three symmetrical, multiple-conclusion logics. A new result, proven first for classical multiple-conclusion logics and then modified and extended to all reflexive, monotonic, and transitive consequence relations, undergirds the focus on proof-theoretic approaches to the consequence relation I adopt here.

Consequence as Preservation: Some Refinements

Preservationism generalises the idea of consequence beyond the standard focus on ‘preservation of truth’ or, syntactically, preservation of consistency. Instead, we preservationists suggest that other properties of premise sets can also worthy of preservation by a consequence relation. This paper presents a broader view of the properties that consequence relations can preserve, focusing on symmetrical treatments of consequence relations in multiple-conclusion logics that preserve variations on proof-theoretic consistent deniability from right to left as well as consistent assertability from left to right. The paper closes with remarks on another approach to producing preservationist logics, viz. the preservation of a ‘base’ consequence relation across a range of images of premise and conclusion sets, rather than preservation of properties of premise and conclusion sets. Further formal definitions and results appear in two appendices.