Inference Belief and Interpretation in Science
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Reliability of Mathematical Inference
Reliability of mathematical inference Jeremy Avigad Department of Philosophy and Department of Mathematical Sciences Carnegie Mellon University January 2020 Formal logic and mathematical proof An important mathematical goal is to get the answers right: • Our calculations are supposed to be correct. • Our proofs are supposed to be correct. Mathematical logic offers an idealized account of correctness, namely, formal derivability. Informal proof is viewed as an approximation to the ideal. • A mathematician can be called on to expand definitions and inferences. • The process has to terminate with fundamental notions, assumptions, and inferences. Formal logic and mathematical proof Two objections: • Few mathematicians can state formal axioms. • There are various formal foundations on offer. Slight elaboration: • Ordinary mathematics relies on an informal foundation: numbers, tuples, sets, functions, relations, structures, . • Formal logic accounts for those (and any of a number of systems suffice). Formal logic and mathematical proof What about intuitionstic logic, or large cardinal axioms? Most mathematics today is classical, and does not require strong assumptions. But even in those cases, the assumptions can be make explicit and formal. Formal logic and mathematical proof So formal derivability provides a standard of correctness. Azzouni writes: The first point to observe is that formalized proofs have become the norms of mathematical practice. And that is to say: should it become clear that the implications (of assumptions to conclusion) of an informal proof cannot be replicated by a formal analogue, the status of that informal proof as a successful proof will be rejected. Formal verification, a branch of computer science, provides corroboration: computational proof assistants make formalization routine (though still tedious). -
Misconceived Relationships Between Logical Positivism and Quantitative Research: an Analysis in the Framework of Ian Hacking
DOCUMENT RESUME ED 452 266 TM 032 553 AUTHOR Yu, Chong Ho TITLE Misconceived Relationships between Logical Positivism and Quantitative Research: An Analysis in the Framework of Ian Hacking. PUB DATE 2001-04-07 NOTE 26p. PUB TYPE Opinion Papers (120) ED 2S PRICE MF01/PCO2 Plus Postage. 'DESCRIPTORS *Educational Research; *Research Methodology IDENTIFIERS *Logical Positivism ABSTRACT Although quantitative research methodology is widely applied by psychological researchers, there is a common misconception that quantitative research is based on logical positivism. This paper examines the relationship between quantitative research and eight major notions of logical positivism:(1) verification;(2) pro-observation;(3) anti-cause; (4) downplaying explanation;(5) anti-theoretical entities;(6) anti-metaphysics; (7) logical analysis; and (8) frequentist probability. It is argued that the underlying philosophy of modern quantitative research in psychology is in sharp contrast to logical positivism. Putting the labor of an out-dated philosophy into quantitative research may discourage psychological researchers from applying this research approach and may also lead to misguided dispute between quantitative and qualitative researchers. What is needed is to encourage researchers and students to keep an open mind to different methodologies and apply skepticism to examine the philosophical assumptions instead of accepting them unquestioningly. (Contains 1 figure and 75 references.)(Author/SLD) Reproductions supplied by EDRS are the best that can be made from the original document. Misconceived relationships between logical positivism and quantitative research: An analysis in the framework of Ian Hacking Chong Ho Yu, Ph.D. Arizona State University April 7, 2001 N N In 4-1 PERMISSION TO REPRODUCE AND DISSEMINATE THIS MATERIALHAS BEEN GRANTED BY Correspondence: TO THE EDUCATIONAL RESOURCES Chong Ho Yu, Ph.D. -
Applying Logic to Philosophical Theology: a Formal Deductive Inference of Affirming God's Existence from Assuming the A-Priori
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Tomsk State University Repository Вестник Томского государственного университета Философия. Социология. Политология. 2020. № 55 ОНТОЛОГИЯ, ЭПИСТЕМОЛОГИЯ, ЛОГИКА УДК 16 + 17 + 2 + 51-7 DOI: 10.17223/1998863Х/55/1 V.O. Lobovikov1 APPLYING LOGIC TO PHILOSOPHICAL THEOLOGY: A FORMAL DEDUCTIVE INFERENCE OF AFFIRMING GOD’S EXISTENCE FROM ASSUMING THE A-PRIORI-NESS OF KNOWLEDGE IN THE SIGMA FORMAL AXIOMATIC THEORY For the first time a precise definition is given to the Sigma formal axiomatic theory, which is a result of logical formalization of philosophical epistemology; and an interpretation of this formal theory is offered. Also, for the first time, a formal deductive proof is constructed in Sigma for a formula, which represents (in the offered interpretation) the statement of God’s Existence under the condition that knowledge is a priori. Keywords: formal axiomatic epistemology theory; two-valued algebra of formal axiology; formal-axiological equivalence; a-priori knowledge; existence of God. 1. Introduction Since Socrates, Plato, Aristotle, Stoics, Cicero, and especially since the very beginning of Christianity philosophy, the possibility or impossibility of logical proving God’s existence has been a nontrivial problem of philosophical theology. Today the literature on this topic is immense. However, even in our days, the knot- ty problem remains unsolved as all the suggested options of solving it are contro- versial from some point of view. Some respectable researchers (let us call them “pessimists”) believed that the logical proving of God’s existence in theoretical philosophy was impossible on principle, for instance, Occam, Hume [1], and Kant [2] believed that any rational theoretic-philosophy proof of His existence was a mistake (illusion), consequently, a search for the logical proving of His existence was wasting resources and, hence, harmful; only faith in God was relevant and useful; reason was irrelevant and use- less. -
Issues in the Normativity of Logic and the Logic As Model View By
Issues in the Normativity of Logic and the Logic as Model View by Haggeo Cadenas A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Arts Approved November 2017 by the Graduate Supervisory Committee Angel Pinillos, Chair Bernard Kobes Shyam Nair Richard Creath ARIZONA STATE UNIVERSITY December 2017 ABSTRACT After surveying the literature on the normativity of logic, the paper answers that logic is normative for reasoning and rationality. The paper then goes on to discuss whether this constitutes a new problem in issues in normativity, and the paper affirms that it does. Finally, the paper concludes by explaining that the logic as model view can address this new problem. i TABLE OF CONTENTS Page INTRODUCTION………………………………………………………………………...1 A ROMP THROUGH THE NORMATIVITY OF LOGIC: HARMAN AND THE NORMATIVITY OF LOGIC ……………...……………………………………………..6 Can Harman-esque Worries Lead to Normative Skepticism? ..............................17 Is Logic Normative?..............................................................................................19 WHY THINK THERE IS A UNIQUE PROBLEM?.......................................................22 Field’s account of the Normativity of Logic………………..…………………..35 Field and the Instrumental Conception of Epistemic Rationality………………..38 THE LOGIC AS MODEL VIEW AND NORMATIVITY……………………………...47 Three Ways Logic Might Relate to Reasoning……………………….………….47 An Elaboration of the Logic as Model View…………………………………….50 The Logic as Model View: A Layout, Application, and Analysis………….……55 A Layout………………………………………………………………………....55 The Application and Analysis……………………………………………………58 CONCLUSION…………………………………………………………………………..62 BIBLIOGRAPHY………………………………………………………………………..64 ii Issues in the Normativity of Logic and the Logic as Model View §1: Introduction What was the charge that Plato and Socrates levied against to sophists? Simply put, it’s that they, “made the weaker argument appear the stronger?” Surprisingly, this charge was brought against Socrates in his trial. -
The Routledge Companion to Islamic Philosophy Reasoning in the Qurn
This article was downloaded by: 10.3.98.104 On: 25 Sep 2021 Access details: subscription number Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: 5 Howick Place, London SW1P 1WG, UK The Routledge Companion to Islamic Philosophy Richard C. Taylor, Luis Xavier López-Farjeat Reasoning in the Qurn Publication details https://www.routledgehandbooks.com/doi/10.4324/9781315708928.ch2 Rosalind Ward Gwynne Published online on: 03 Sep 2015 How to cite :- Rosalind Ward Gwynne. 03 Sep 2015, Reasoning in the Qurn from: The Routledge Companion to Islamic Philosophy Routledge Accessed on: 25 Sep 2021 https://www.routledgehandbooks.com/doi/10.4324/9781315708928.ch2 PLEASE SCROLL DOWN FOR DOCUMENT Full terms and conditions of use: https://www.routledgehandbooks.com/legal-notices/terms This Document PDF may be used for research, teaching and private study purposes. Any substantial or systematic reproductions, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 2 REASONING- IN THE QURʾAN Rosalind Ward Gwynne Introduction Muslims consider the Qurʾa-n to be the revealed speech of God—sublime, inimitable and containing information that only God knows. It has been analyzed in every pos- sible way—theologically, linguistically, legally, metaphorically—and some of these analyses have presented their results as the conclusions of reasoning in the Qurʾa-n itself, whether explicit or implicit. -
Logic in the Light of Cognitive Science
STUDIES IN LOGIC, GRAMMAR AND RHETORIC 48 (61) 2016 DOI: 10.1515/slgr-2016-0057 Jan Woleński University of Information, Technology and Management Rzeszow LOGIC IN THE LIGHT OF COGNITIVE SCIENCE Abstract. Logical theory codifies rules of correct inferences. On the other hand, logical reasoning is typically considered as one of the most fundamental cognitive activities. Thus, cognitive science is a natural meeting-point for investigations about the place of logic in human cognition. Investigations in this perspec- tive strongly depend on a possible understanding of logic. This paper focuses on logic in the strict sense; that is, the theory of deductive inferences. Two problems are taken into account, namely: (a) do humans apply logical rules in ordinary reasoning?; (b) the genesis of logic. The second issue is analyzed from the naturalistic point of view. Keywords: logica docens, logica utens, rules of inference, genetic code, natural- ism, information. Logic is traditionally considered as very closely associated with cogni- tive activities. Philosophers and psychologists frequently say that human rationality consists in using logical tools in thinking and other cognitive ac- tivities, e.g. making decisions. In particular, reasoning and its correctness seem to have their roots in a faculty, which, by way of analogy with the lan- guage faculty, could be viewed as logical competence. On the other hand, one might also argue, logical competence itself is a manifestation of logic. Independently, whether logic is prior to logical competence or whether the latter has its grounds in the former, the concept of logic is indispensable for any analysis of the practice of inferences performed by humans as well as perhaps other species.1 The above, a very sketchy introduction, justifies taking the concept of logic as fundamental for my further analysis of logical aspects related to the cognitive space. -
Logical Inference and Its Dynamics
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by PhilPapers Logical Inference and Its Dynamics Carlotta Pavese 1 Duke University Philosophy Department Abstract This essay advances and develops a dynamic conception of inference rules and uses it to reexamine a long-standing problem about logical inference raised by Lewis Carroll's regress. Keywords: Inference, inference rules, dynamic semantics. 1 Introduction Inferences are linguistic acts with a certain dynamics. In the process of making an inference, we add premises incrementally, and revise contextual assump- tions, often even just provisionally, to make them compatible with the premises. Making an inference is, in this sense, moving from one set of assumptions to another. The goal of an inference is to reach a set of assumptions that supports the conclusion of the inference. This essay argues from such a dynamic conception of inference to a dynamic conception of inference rules (section x2). According to such a dynamic con- ception, inference rules are special sorts of dynamic semantic values. Section x3 develops this general idea into a detailed proposal and section x4 defends it against an outstanding objection. Some of the virtues of the dynamic con- ception of inference rules developed here are then illustrated by showing how it helps us re-think a long-standing puzzle about logical inference, raised by Lewis Carroll [3]'s regress (section x5). 2 From The Dynamics of Inference to A Dynamic Conception of Inference Rules Following a long tradition in philosophy, I will take inferences to be linguistic acts. 2 Inferences are acts in that they are conscious, at person-level, and 1 I'd like to thank Guillermo Del Pinal, Simon Goldstein, Diego Marconi, Ram Neta, Jim Pryor, Alex Rosenberg, Daniel Rothschild, David Sanford, Philippe Schlenker, Walter Sinnott-Armstrong, Seth Yalcin, Jack Woods, and three anonymous referees for helpful suggestions on earlier drafts. -
Introduction
i i OUP CORRECTED PROOF – FINAL, //, SPi i i PART I Introduction i i i i i i OUP CORRECTED PROOF – FINAL, //, SPi i i i i i i i i OUP CORRECTED PROOF – FINAL, //, SPi i i Logical Consequence Its Nature, Structure, and Application Colin R. Caret and Ole T. Hjortland . Introduction Recent work in philosophical logic has taken interesting and unexpected turns. It has seen not only a proliferation of logical systems, but new applications of a wide range of different formal theories to philosophical questions. As a result, philosophers have been forced to revisit the nature and foundation of core logical concepts, chief amongst which is the concept of logical consequence. This volume collects together some of the most important recent scholarship in the area by drawing on a wealth of contributions that were made over the lifetime of the AHRC-funded Foundations of Logical Consequence project. In the following introductory essay we set these contributions in context and identify how they advance important debates within the philosophy of logic. Logical consequence is the relation that obtains between premises and conclu- sion(s) in a valid argument. Validity, most will agree, is a virtue of an argument, butwhatsortofvirtue?Orthodoxyhasitthatanargumentisvalidifitmustbethe case that when the premises are true, the conclusion is true. Alternatively, that it is impossible for the premises to be true and the conclusion false simultaneously. In short, the argument is necessarily truth preserving. These platitudes, however, leave us with a number -
Passmore, J. (1967). Logical Positivism. in P. Edwards (Ed.). the Encyclopedia of Philosophy (Vol. 5, 52- 57). New York: Macmillan
Passmore, J. (1967). Logical Positivism. In P. Edwards (Ed.). The Encyclopedia of Philosophy (Vol. 5, 52- 57). New York: Macmillan. LOGICAL POSITIVISM is the name given in 1931 by A. E. Blumberg and Herbert Feigl to a set of philosophical ideas put forward by the Vienna circle. Synonymous expressions include "consistent empiricism," "logical empiricism," "scientific empiricism," and "logical neo-positivism." The name logical positivism is often, but misleadingly, used more broadly to include the "analytical" or "ordinary language philosophies developed at Cambridge and Oxford. HISTORICAL BACKGROUND The logical positivists thought of themselves as continuing a nineteenth-century Viennese empirical tradition, closely linked with British empiricism and culminating in the antimetaphysical, scientifically oriented teaching of Ernst Mach. In 1907 the mathematician Hans Hahn, the economist Otto Neurath, and the physicist Philipp Frank, all of whom were later to be prominent members of the Vienna circle, came together as an informal group to discuss the philosophy of science. They hoped to give an account of science which would do justice -as, they thought, Mach did not- to the central importance of mathematics, logic, and theoretical physics, without abandoning Mach's general doctrine that science is, fundamentally, the description of experience. As a solution to their problems, they looked to the "new positivism" of Poincare; in attempting to reconcile Mach and Poincare; they anticipated the main themes of logical positivism. In 1922, at the instigation of members of the "Vienna group," Moritz Schlick was invited to Vienna as professor, like Mach before him (1895-1901), in the philosophy of the inductive sciences. Schlick had been trained as a scientist under Max Planck and had won a name for himself as an interpreter of Einstein's theory of relativity. -
Field on the Normative Role of Logic∗
Field on the Normative Role of Logic∗ Gilbert Harman Harman (1986) asserts that logic is neither a normative nor a psychological theory and, although immediate implication and inconsistency may play a role in reasoning, there is nothing special about logic in this connection. Field (2009) purports to reject these claims but actually accepts them and concludes that, because of the semantic paradoxes, logic is ‘somehow connected to norms of thought’! A real nonsequitur! Field (2009) purports to reject my arguments (Harman 1986) that there is not (as he puts it) ‘a close connection between logic and rationality’ (p. 252). Here I briefly reply. I begin by summarizing the first two chapters of (Harman 1986). The first chapter stresses the importance of not confusing inference with implication and of not confusing reasoning with the sort of argument studied in deductive logic. Inference and reasoning are psychological events or processes that can be done more or less well. The sort of implication and argument studied in deductive logic have to do with relations among propositions and with structures of propositions distinguished into premises, intermediate steps, and conclusion. Deductive logic is not a particular psychological subject and is not a particularly normative subject, although one might attempt to develop a logic of belief or a deontic logic, for example. The second chapter begins by considering the suggestion that logic might be specially relevant to reasoning in two ways, via implication and inconsistency. It seems that any relevant principle would have to be defeasible, holding only other things being equal. Furthermore, it would apply only to someone who recognized the implication or inconsistency. -
What Is Inference? Penalty
WHAT IS INFERENCE? OR THE FORCE OF REASONING by Ulf Hlobil Dipl.-Psych., University of Trier, 2008 Magister Artium in Philosophy, University of Trier, 2009 M.A. in Philosophy, University of Pittsburgh, 2012 Submitted to the Graduate Faculty of the Kenneth P. Dietrich School of Arts and Sciences in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Pittsburgh 2016 UNIVERSITY OF PITTSBURGH KENNETH P. DIETRICH SCHOOL OF ARTS AND SCIENCES This dissertation was presented by Ulf Hlobil It was defended on May 31, 2016 and approved by Robert Brandom, Distinguished Professor of Philosophy John McDowell, Distinguished University Professor of Philosophy Kieran Setiya, Professor of Philosophy James Shaw, Associate Professor of Philosophy Dissertation Director: Robert Brandom, Distinguished Professor of Philosophy ii Copyright © by Ulf Hlobil 2016 iii WHAT IS INFERENCE? OR THE FORCE OF REASONING Ulf Hlobil, PhD University of Pittsburgh, 2016 What are we doing when we make inferences? I argue that to make an inference is to attach inferential force to an argument. Inferential force must be understood in analogy to assertoric force, and an argument is a structure of contents. I call this the “Force Account of inference.” I develop this account by first establishing two criteria of adequacy for accounts of inference. First, such accounts must explain why it is absurd to make an inference one believes to be bad. The upshot is that if someone makes an inference, she must take her inference to be good. Second, accounts of inference must explain why we cannot take our inferences to be good—in the sense that matters for inference—by merely accepting testimony to the effect that they are good. -
The “Positive Argument” for Constructive Empiricism and Inference to the Best Explanation Moti Mizrahi Florida Institute Of
The “Positive Argument” for Constructive Empiricism and Inference to the Best Explanation Moti Mizrahi Florida Institute of Technology [email protected] Abstract: In this paper, I argue that the “positive argument” for Constructive Empiricism (CE), according to which CE “makes better sense of science, and of scientific activity, than realism does” (van Fraassen 1980, 73), is an Inference to the Best Explanation (IBE). But constructive empiricists are critical of IBE, and thus they have to be critical of their own “positive argument” for CE. If my argument is sound, then constructive empiricists are in the awkward position of having to reject their own “positive argument” for CE by their own lights. Keywords: Constructive Empiricism; Inference to the Best Explanation; Positive Argument; Scientific Realism According to Bas van Fraassen (1980, 73), the “positive argument” for Constructive Empiricism (CE), is that “it makes better sense of science, and of scientific activity, than realism does and does so without inflationary metaphysics.” Although van Fraassen would not characterize it as such, this “positive argument” for CE looks like an Inference to the Best Explanation (IBE), for in IBE, “a hypothesis is accepted on the basis of a judgment that it best explains the available evidence” (Psillos 2007, 442). And a “good explanation makes sense out of that which it is intended to explain” (Sinnott-Armstrong and Fogelin 2015, 198; emphasis added). In van Fraassen’s “positive argument,” CE is said to be supported by the premise that it “makes better 1 sense of,” or provides a better explanation for, scientific activity than Scientific Realism (SR) does.