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An Argument for a Neutral Free Logic Daniel Yeakel Wayne State University

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An Argument for a Neutral Free Logic Daniel Yeakel Wayne State University Wayne State University Wayne State University Dissertations 1-1-2015 An Argument For A Neutral Free Logic Daniel Yeakel Wayne State University, Follow this and additional works at: http://digitalcommons.wayne.edu/oa_dissertations Recommended Citation Yeakel, Daniel, "An Argument For A Neutral Free Logic" (2015). Wayne State University Dissertations. Paper 1177. This Open Access Dissertation is brought to you for free and open access by DigitalCommons@WayneState. It has been accepted for inclusion in Wayne State University Dissertations by an authorized administrator of DigitalCommons@WayneState. AN ARGUMENT FOR A NEUTRAL FREE LOGIC by DANIEL YEAKEL Submitted to the Graduate School of Wayne State University, Detroit, Michigan in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY 2015 MAJOR: PHILOSOPHY Approved by: Advisor Date DEDICATION For James and Kathryn ii ACKNOWLEDGMENTS Whatever humble benefits might follow from the completion of this thesis are not solely my responsibility to claim. I owe a debt to the members of my committee, Robert Bruner, Greg Novack, Larry Powers, Susan Vineberg, and especially Mike McKinsey. I also owe a great deal to the many others who provided me with help, advice and encouragement: Dan Blaser, John Burn, Marcus Cooper, Mike Gavin, Mark Huston, Herb Granger, Gwen Gordon, Lin Harris, Eric Hiddleston, Renee Hurcomb, Katherine Kim, Sloan Lee, Drew Matzke, Kevin Mowrer, Mark Reynolds, Phyllis Seals, Patrick Smith, Sean Stidd, Bill Stine, Sean Tilson, Bob Yanal, my sister Sarah Yeakel, Jim and Linda Zielke, Matt Zuckero, and Bill from the MHRFC. I am also especially thankful for the support and patience of my parents and my wife Jessica. iii TABLE OF CONTENTS DEDICATION ii ACKNOWLEDGMENTS iii Introduction: Master Argument, Chapter Summaries, and Closure 1 Motivation and Explanation of NFL 1 Chapter Summaries 5 Closure of the A Priori 29 Chapter One: The Existential Quantifier and Ontological Commitment 39 Paraphrase and Ontology 39 Avoiding Ontological Commitment 41 Sentences Meeting I-III are Best Represented by an Objectual Quantifier 45 Sentences Meeting I-III are Univocal 53 Sentences Meeting I-III are Heavyweight 57 Objection One: The Triviality Argument 59 Objection Two: From the Relative Strengths of Realist Claims 61 Objection Three: Ambiguous Domain 66 Two kinds of Sentence that Satisfy I-III 68 Chapter Two: Positive Free Logic 72 Bivalent PFL and the Prototype Schema 72 An Appeal to Authority 74 Ambiguous Quantifiers Again? 78 Non-bivalent PFL: Supervaluations 87 Introduction to SVS 87 The Utility Defense of SVS 92 iv Criticisms of SVS 98 Accounting for CLTs with Corresponding Arguments 112 Saving Contingent Truths 122 Chapter Three: Negative Free Logic 128 Introduction to Negative Free Logic 129 Three Criticisms of Negative Free Logic 131 Three Arguments for Negative Free Logic 134 Bivalence 138 Williamson‘s Argument for Bivalence 139 Another Defense of Bivalence: Argument from the T—Scheme 146 The Law of Excluded Middle 149 Defenses of LEM 150 Criticisms of LEM 152 Chapter Four: Neutral Free Logic 158 Choices for a Neutral Free Logic 158 Domain 158 Identity 162 Relations and Symbols 164 Quantifiers 166 Logical Properties 170 The Semantic Rules 173 Criticism: Hedging 177 General problems 177 v Pryor‘s account 180 Use and Mention 185 BIBLIOGRAPHY 188 ABSTRACT 197 AUTOBIGRAPHICAL STATEMENT 199 vi 1 INTRODUCTION: MASTER ARGUMENT, CHAPTER SUMMARIES AND CLOSURE I will argue that the appropriate semantics for natural language is a neutral free semantics with weak Kleene tables and trivalent quantifiers. The general pattern of my project will be to present an argument with an absurd conclusion and consider the options for avoiding the absurdity. I will try to show that all but one of these options are unacceptable. In the introduction I will present the argument, lay out the possible ways of criticizing it, and explain the approach I endorse. After that, I will summarize the arguments of the following chapters and provide an argument for a premise of the master argument that does not fit neatly elsewhere. Motivation and Explanation of Neutral Free Logic—The Master Argument Call a sentence of the form ‗c exists‘ a simple existence claim. If an appropriate symbolization of a simple existence claim is ‗∃푥 푥 = 푐 ‘ then it is a problem for classical logic that it regards all simple existence claims as logically true. Logical truths are knowable a priori and simple existence claims are not. Michael McKinsey (in conversation) and James Pryor1 have suggested an argument similar to what I will call the ‗master argument:‘ 1. It is logically true that Powers is Powers 2. Logical truths are a priori knowable. 3. So, it is a priori knowable that Powers is Powers. 4. That Powers is Powers logically entails that Powers exists. 5. What is logically entailed by a priori knowable truths is a priori knowable. 2 1 James Pryor, "XIII--Hyper-Reliability and Apriority," Meeting of the Aristotelian Society, (2006): 335-6. Pryor does not formalize his argument, but a rough paraphrase of it would be: ‗Jack is self-identical‘ implies that ‗Jack exists.‘ Since the latter seems not to be knowable a priori, the former is not either. Since logical truths are knowable a priori, ‗Jack is self-identical‘ must not be one. 2 This is just a weakened version of Michael McKinsey‘s principle: Closure of Apriority under Logical Implication (CA) from his ―Forms of externalism and privileged access,‖ Philosophical Perspectives 16, (2002): 207. His is prefixed with ‗necessarily‘ and uses a ‗knows a priori‘ relation between people and propositions instead of my ‗a 2 6. So, it is a priori knowable that Powers exists. The conclusion is absurd even supposing that ‗Powers‘ refers to Larry Powers, the erstwhile professor of philosophy at Wayne State University and author of Non-contradiction. The alternatives are: to accept the absurd conclusion, to reject the validity of the argument from 1 and 2 to 3 or of the argument from 3, 4, and 5 to 6, or to reject a premise (either 1, 2, 4, or 5). The first two of these options are disastrous. Since ‗Powers‘ in the argument above could be replaced with any name, accepting the conclusion is accepting that all simple existence claims are knowable a priori. With a few faintly possible exceptions, it is plausible that no simple existence claims are knowable a priori. The second option is equally undesirable; the inferences are intuitively valid. So much so that any view that rejects them is dubious, for that reason alone. Both inferences are of the form All A are B; x is an A; so, x is a B. For a defense of the validity of this argument form see Chapter Two, on positive free logic. So the remaining options are to reject 1, 2, 4, or 5. First consider premise 2. That logical truths are knowable a priori has its own motivation: logic is the study of correct inferences, and determinations of correct inferences are purely mental activities; so, logic is a priori. However, 2 follows from 5; so it needs no defense above that of 5. Since all logical truths follow from all truths, including a priori-knowable truths, the closure principle 5 implies that all logical truths are knowable a priori. At the conclusion of this introduction, I will briefly offer a pair of arguments for 5 and defend them from criticism. If those arguments are successful then the only remaining options are to reject 1 or reject 4. My defense of premise 4 comes from two premises and the transitivity of logical entailment: priori knowable‘ operator. The reductio argument above is very similar to the reductio argument for which McKinsey uses (CA) in that paper and elsewhere. 3 (EG) That Powers is Powers logically entails that something is Powers. (OC) That something is Powers logically entails that Powers exists. A rejection of premise 4 is a rejection of Existential Generalization or of the ontological commitment of simple existentially quantified claims. In chapter one I will defend (OC) by arguing that certain simple existentially quantified claims are ontologically committed. In chapter two I will consider two ways of denying the validity of (EG): positive free logic and supervaluational logic. Both allow that some statements containing non-referring individual constants can be true. I will argue that both of those views are inadequate and I will suggest that any acceptable semantics will disallow the truth of statements with empty constants. Since not all instances of the schema ‗푥 = 푥‘ are true, ‗푎 = 푎‘ is not a logical truth. So 1 must be rejected. Premise 1 follows from ‗∀푥(푥 = 푥)‘ by an application of Universal Instantiation. So a rejection of 1 will also be a rejection of the universally quantified claim or of UI. There are two ways to reject ‗푎 = 푎‘. We could say that simple statements with empty names are false or that they are neither true nor false. A logic that regards those statements as false is a negative free logic (NgFL); one that makes ‗푎 = 푎‘ neither true nor false (when ‗a‘ is empty) is a neutral free logic (NFL). Notably, on a negative free logic ‗푃푎 ∨ ~푃푎‘ remains logically true but a neutral free logic loses that and all other classical tautologies containing individual constants. My particular NFL, because of its treatment of the empty domain, loses all logical truths, but that result is not required of NFL in general. If there are persuasive arguments for bivalence then a negative free logic would be preferable to one allowing a lack of truth value. In chapter three I will consider arguments for 4 bivalence; most notably, I will consider Timothy Williamson‘s.3 I will also try to justify the loss of the Law of the Excluded Middle (LEM) by NFL.
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