Logic and Foundations of Mathematics Commons^™

The Principle Of Tolerance And Logical Pluralism, Teresa Kouri Kissel

Philosophy Faculty Publications

Logical pluralism is the view that there is more than one correct logic. This is in contrast with logical monism, which holds that only one is, and logical nihilism, which holds that none are. Monists are typically folks who defend a single logic, like Michael Dummett (2000) or Alan Ross Anderson and Nuel D. Belnap (1975), as correct for guiding deductive reasoning.

The Grim Reaper Argument Is An Argument For No One, Daniel Linford, Alex Malpass

Philosophy Faculty Publications

According to a prominent philosophical argument for the beginning of time – the Grim Reaper Argument (GRA) – had it been possible for the past to be infinite, then a provably impossible scenario could have been constructed. Such a scenario is not possible, so, the GRA concludes, the past cannot be infinite. Here, we show that the GRA includes one premise acceptable only for Humeans and another premise acceptable only for anti-Humeans. Since, plausibly, everyone is either a Humean or an anti-Humean, the GRA is an argument for no one. Additionally, we argue that there may not be a way …

“What Line Can’T Be Measured With A Ruler?” Riddles And Concept-Formation In Mathematics And Aesthetics, William H. Brenner, Samuel J. Wheeler

Philosophy Faculty Publications

We analyze two problems in mathematics – the first (stated in our title) is extracted from Wittgenstein’s “Philosophy for Mathematicians”; the second (“What set of numbers is non-denumerable?”) is taken from Cantor. We then consider, by way of comparison, a problem in musical aesthetics concerning a Brahms variation on a theme by Haydn. Our aim is twofold: first, to bring out and elucidate the essentially riddle-like character of these problems; second, to show that the comparison with riddles does not reduce their solution to an exercise in bare subjectivity

Pluralistic Perspectives On Logic: An Introduction, Colin R. Caret, Teresa Kouri Kissel

Philosophy Faculty Publications

(First paragraph) Logical pluralism is the view that there are distinct, but equally good logics. Recent years have witnessed a sharp upswing of interest in this view, resulting in an impressive literature. We only expect this trend to continue in the future. More than one commentator has, however, expressed exasperation at the view: what can it mean to be a pluralist about logic of all things? [see, e.g., Eklund (2017); Goddu (2002); Keefe (2014)]. In this introduction, we aim to set out the basic pluralist position, identify some issues over which pluralists disagree amongst themselves, and highlight the topics at …

Logical Instrumentalism And Concatenation, Teresa Kouri Kissel

Philosophy Faculty Publications

Logical pluralism is the theory that there is more than one right logic. Logical instrumentalism is the view that a logic is a correct logic if it can be used to fruitfully pursue some deductive inquiry. Logical instrumentalism is a version of logical pluralism, since more than one logic can be used fruitfully. In this paper, I will show that a logical instrumentalist must accept linear logic as a correct logic, since linear logic is useful for studying natural language syntax. I further show that this means that the logical instrumentalist must accept a wide range of connectives, in particular …

Susan Stebbing, Teresa Kouri Kissel

Philosophy Faculty Publications

Susan Stebbing (1885-1943) was a founder of Analysis and had a large influence on philosophy during the early 20th century. Recently, the work of Michael Beaney (2000), Siobhan Chapman (2013) and Frederique Janssen- Lauret (2017), amongst others, has begun a resurgence of interest in Stebbing. This paper serves as a brief introduction to some of the major features of her philosophical work.

Classical Logic, Stewart Shapiro, Teresa Kouri Kissel

Philosophy Faculty Publications

[From introductory section]

Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language has components that correspond to a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record arguments that are valid for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions for at least part of the language.

The following sections provide the basics of a typical logic, sometimes called “classical elementary logic” or “classical first-order logic”....

Against The Intentional Definition Of Argument, G. C. Goddu

Philosophy Faculty Publications

Intentional definitions of argument, i.e. the conclusion being intended to follow from the premises, abound. Yet, there are numerous problem cases in which we appear to have arguments, but no intention. One way to try to avoid these problem cases is to appeal to acts, in which case one has to give up on the repeatability of arguments. One can keep repeatability and intentions if one resorts to act types, but then it appears that the problem cases re-emerge.

Logical Pluralism From A Pragmatic Perspective, Teresa Kouri Kissel

Philosophy Faculty Publications

This paper presents a new view of logical pluralism. This pluralism takes into account how the logical connectives shift, depending on the context in which they occur. Using the Question-Under-Discussion Framework as formulated by Craige Roberts, I identify the contextual factor that is responsible for this shift. I then provide an account of the meanings of the logical connectives which can accommodate this factor. Finally, I suggest that this new pluralism has a certain Carnapian flavour. Questions about the meanings of the connectives or the best logic outside of a specified context are not legitimate questions.

Ante Rem Structuralism And The No-Naming Constraint, Teresa Kouri

Philosophy Faculty Publications

Tim Räz has presented what he takes to be a new objection to Stewart Shapiro's ante rem structuralism (ARS). Räz claims that ARS conflicts with mathematical practice. I will explain why this is similar to an old problem, posed originally by John Burgess in 1999 and Jukka Keränen in 2001, and show that Shapiro can use the solution to the original problem in Räz's case. Additionally, I will suggest that Räz's proposed treatment of the situation does not provide an argument for the in re over the ante rem approach.

A New Interpretation Of Carnap's Logical Pluralism, Teresa Kouri

Philosophy Faculty Publications

Rudolf Carnap’s logical pluralism is often held to be one in which corresponding connectives in different logics have different meanings. This paper presents an alternative view of Carnap’s position, in which connectives can and do share their meaning in some (though not all) contexts. This re-interpretation depends crucially on extending Carnap’s linguistic framework system to include meta-linguistic frameworks, those frameworks which we use to talk about linguistic frameworks. I provide an example that shows how this is possible, and give some textual evidence that Carnap would agree with this interpretation. Additionally, I show how this interpretation puts the Carnapian position …

Restall's Proof-Theoretic Pluralism And Relevance Logic, Teresa Kouri

Philosophy Faculty Publications

Restall (Erkenntnis 79(2):279–291, 2014) proposes a new, proof-theoretic, logical pluralism. This is in contrast to the model-theoretic pluralism he and Beall proposed in Beall and Restall (Aust J Philos 78(4):475–493, 2000) and in Beall and Restall (Logical pluralism, Oxford University Press, Oxford, 2006). What I will show is that Restall has not described the conditions on being admissible to the proof-theoretic logical pluralism in such a way that relevance logic is one of the admissible logics. Though relevance logic is not hard to add formally, one critical component of Restall’s pluralism is that the relevance logic that gets added must …

Denying Antecedents And Affirming Consequents: The State Of The Art, David Godden, Frank Zenker

Philosophy Faculty Publications

Recent work on conditional reasoning argues that denying the antecedent [DA] and affirming the consequent [AC] are defeasible but cogent patterns of argument, either because they are effective, rational, albeit heuristic applications of Bayesian probability, or because they are licensed by the principle of total evidence. Against this, we show that on any prevailing interpretation of indicative conditionals the premises of DA and AC arguments do not license their conclusions without additional assumptions. The cogency of DA and AC inferences rather depends on contingent factors extrinsic to, and independent of, what is asserted by DA and AC arguments. © David …

A Reply To Heathcote's "On The Exhaustion Of Mathematical Entities By Structures", Teresa Kouri

Philosophy Faculty Publications

In this article I respond to Heathcote’s "On the Exhaustion of Mathematical Entities by Structures". I show that his ontic exhaustion issue is not a problem for ante rem structuralists. First, I show that it is unlikely that mathematical objects can occur across structures. Second, I show that the properties that Heathcote suggests are underdetermined by structuralism are not so underdetermined. Finally, I suggest that even if Heathcote’s ontic exhaustion issue if thought of as a problem of reference, the structuralist has a readily available solution.

Philosophy Of Mathematics In The Twentieth Century: Selected Essays, Stewart Shapiro, Teresa Kouri Kissel

Philosophy Faculty Publications

No abstract provided.

Why We Still Do Not Know What A “Real” Argument Is, G. C. Goddu

Philosophy Faculty Publications

In his recent paper, “What a Real Argument is,” Ben Hamby attempts to provide an adequate theoretical account of “real” arguments. In this paper I present and evaluate both Hamby’s motivation for distinguishing “real” from non-“real” arguments and his articulation of the distinction. I argue that neither is adequate to ground a theoretically significant class of “real” arguments, for the articulation fails to pick out a stable proper subclass of all arguments that is simultaneously both theoretically relevant and a proper subclass of all arguments.

Logic, Truth And Inquiry (Book Review), G. C. Goddu

Philosophy Faculty Publications

Mark Weinstein’s, Logic, Truth and Inquiry is an ambitious and provocative case for a theory of truth and warrant strength that will undergird an “account of argument in the broad sense of current argumentation theory” (p. 12). I begin with a very schematic synopsis of Weinstein’s rich discussion through his six chapters. Weinstein himself notes that his arguments are “frequently presented in broad outline” (p. 1), so my quick sketch will be even broader. I conclude with some brief observations about both what the book leaves unresolved and the merits of Weinstein’s intriguing book.

Propositional Quantification, Ryan Christensen

Philosophy Faculty Publications

Ramsey deWned truth in the following way: xz is true if and only if 'pzz(xz = [zpz] & pz). This deWnition is ill-formed in standard Wrst-order logic, so it is normally interpreted using substitutional or some kind of higher-order quanti-Wer. I argue that these quantiWers fail to provide an adequate reading of the deWnition, but that, given certain adjustments, standard objectual quantiWcation does provide an adequate reading.

What Is A “Real” Argument?, G. C. Goddu

Philosophy Faculty Publications

Numerous informal logicians and argumentation theorists restrict their theorizing to what they call “real” arguments. But is there a clear distinction to be made between “real” and “non-real” arguments? Here I explore four possible accounts of the alleged distinction and argue that none can serve the theoretical uses to which the distinction is most often put.

A Theory Of Argument (Book Review), G. C. Goddu

Philosophy Faculty Publications

A Theory of Argument is an advanced textbook “written for upper-level undergraduate students who have completed at least one prior course in argumentation theory, critical thinking, informal logic, formal logic, or some other related discipline” (ix). This puts Vorobej’s book in a unique position since, to my knowledge, there are no other second course undergraduate textbooks with a nonsymbolic focus. (Second course symbolic logic textbooks written for undergraduates, rather than primarily for graduate students, were relatively rare until recently; the past decade has seen a proliferation in such texts.)

Walton On Argument Structure, G. C. Goddu

Philosophy Faculty Publications

In previous work I argued against (i) the likelihood of finding a theoretically sound foundation for the linked/convergent distinction and (ii) the utility of the distinction even if a sound theoretical basis could be found. Here I subject Douglas Walton’s comprehensive discussion of the linked/convergent distinction found in Argument Structure: A Pragmatic Theory to careful scrutiny and argue that at best Walton’s theory remains incomplete and that attempts to fill out the details will run afoul of at least one of the problems adduced above—i.e., result in either a theoretically unsound distinction or a theoretically sound, but unnecessary distinction.

The Reach Of Abduction: Insight And Trial (Book Review), G. C. Goddu

Philosophy Faculty Publications

The Reach of Abduction is the second volume in a planned three-volume set - A Practical Logic of Cognitive Systems - spanning relevance, abduction, and fallacious reasoning. Despite this fact, Gabbay and Woods write in the preface that "we have written the individual volumes with a view to their being read either as stand-alone works or as linked and somewhat overlapping items in the series" (p. xvii). The aim of The Reach of Abduction is to embed abduction "within a practical logic of cognitive systems" and in so doing, provide "an adequate stand-alone characterization of abduction itself' (p. 9). At …

Sylvester: Ushering In The Modern Era Of Research On Odd Perfect Numbers, Steven Gimbel, John Jaroma

Philosophy Faculty Publications

In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester's bound stood …

Against The "Ordinary Summing" Test For Convergence, G. C. Goddu

Philosophy Faculty Publications

One popular test for distinguishing linked and convergent argument structures is Robert Yanal's Ordinary Summing Test. Douglas Walton, in his comprehensive survey of possible candidates for the linked/convergent distinction, advocates a particular version of Yanal's test. In a recent article, Alexander Tyaglo proposes to generalize and verify Yanal's algorithm for convergent arguments, the basis for Yanal's Ordinary Summing Test. In this paper I will argue that Yanal's ordinary summing equation does not demarcate convergence and so his Ordinary Summing Test fails. Hence, despite Walton's recommendation or Tyaglo's generalization, the Ordinary Summing Test should not be used for distinguishing linked argument …

The 'Most Important And Fundamental' Distinction In Logic, G. C. Goddu

Philosophy Faculty Publications

In this paper I argue that the debate over the purported distinction between deductive and inductive arguments can be bypassed because making the distinction is unnecessary for successfully evaluating arguments. I provide a foundation for doing logic that makes no appeal to the distinction and still performs all the relevant tasks required of an analysis of arguments. I also reply to objections to the view that we can dispense with the distinction. Finally, I conclude that the distinction between inductive and deductive arguments is not one of the most important and fundamental ideas in logic, but rather is unnecessary.

Critical Thinking By Alec Fisher (Book Review), G. C. Goddu

Philosophy Faculty Publications

The aim of Critical Thinking is to explicitly and directly teach critical thinking skills and to facilitate the use of these skills to subjects and contexts beyond critical thinking (v, 1). Though the book is primarily intended as an introductory textbook for the teaching of critical thinking, Fisher maintains that the "material is presented in such a way that it can be worked through on a self-study basis"(vi).