Principia Logico-Metaphysica (Draft/Excerpt)

CONTENTSix12.6.4 Validating Judgments of Logical Consequence . . . . . . . 59912.6.5 Final Issues Concerning Fictional Individuals . . . . . . . 60113 Concepts13.1 The Calculus of Concepts . . . . . . . . . . . . . .13.1.1 Concept Addition . . . . . . . . . . . . . .13.1.2 Concept Inclusion and Containment . . .13.1.3 Concept Inclusion, Addition, and Identity13.1.4 The Algebra of Concepts . . . . . . . . . .13.1.5 The Mereology of Concepts . . . . . . . .13.2 Concepts of Properties and Individuals . . . . . .13.2.1 Concepts of Properties . . . . . . . . . . .13.2.2 Concepts of Ordinary Individuals . . . . .13.2.3 Concepts of Generalizations . . . . . . . .13.3 The Containment Theory of Truth . . . . . . . . .13.4 The Modal Metaphysics of Concepts . . . . . . .13.4.1 Realization, Appearance, Mirroring . . . .13.4.2 Individual Concepts . . . . . . . . . . . .13.4.3 Compossibility . . . . . . . . . . . . . . .13.4.4 Counterparts . . . . . . . . . . . . . . . . .13.4.5 World-Relative Concepts of Individuals .13.4.6 Fundamental Theorems . . . . . . . . . 6366466814 Natural Numbers (with Uri Nodelman)14.1 Philosophical Context . . . . . . . . . . . . . . . . . .14.2 Equinumerosity and Discernible Objects . . . . . . .14.3 Numbering Properties and Natural Cardinals . . . .14.4 Ancestrals and Relations on Discernibles . . . . . . .14.5 Predecessor . . . . . . . . . . . . . . . . . . . . . . . .14.6 Deriving the Number-Theoretic Postulates . . . . . .14.7 Number Theory . . . . . . . . . . . . . . . . . . . . .14.8 Functions and Recursive Definitions . . . . . . . . .14.8.1 Total Functions . . . . . . . . . . . . . . . . .14.8.2 Functions From Domains to Codomains . . .14.8.3 Restricted Functions and Functions Generally14.8.4 Numerical Operations . . . . . . . . . . . . .14.8.5 Recursively-Defined Relations and Functions14.9 Deriving 2nd-Order Peano Arithmetic . . . . . . . .14.10 Infinity . . . . . . . . . . . . . . . . . . . . . . . . .673673686696709718722727738739746760763766779782.

xCONTENTS15 Typed Object Theory and its Applications15.1 The Language and Its Interpretation . . . . . . . . . . . . .15.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .15.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15.4 The Deductive System and Basic Theorems . . . . . . . . .15.4.1 The System, Negations, Conditionals, Quantification15.4.2 Logical Existence, Identity, and Uniqueness . . . . .15.4.3 The Theory of Actuality and Descriptions . . . . . .15.4.4 The Theory of Necessity . . . . . . . . . . . . . . . .15.4.5 The Typed Theory of Relations . . . . . . . . . . . .15.4.6 The Typed Theory of Objects . . . . . . . . . . . . .15.4.7 Discernible Objects Of Every Type . . . . . . . . . .15.4.8 Propositional Properties . . . . . . . . . . . . . . . .15.5 Logical Objects and Worlds . . . . . . . . . . . . . . . . . . .15.6 Mathematics and Mathematical Language . . . . . . . . . .15.6.1 Mathematical Theories and Truth in a Theory . . . .15.6.2 Mathematical Individuals and Relations . . . . . . .15.6.3 Analyzing the Data from Mathematics . . . . . . . .15.7 Higher-Order Fictions and Logical Objects . . . . . . . . . .15.7.1 Higher-Order Fictions . . . . . . . . . . . . . . . . .15.7.2 Higher-Order Logical Objects . . . . . . . . . . . . .15.8 Propositional Attitudes and Reports Thereof . . . . . . . . .15.9 Type-Lowering . . . . . . . . . . . . . . . . . . . . . . . . . .15.9.1 Objectified Properties . . . . . . . . . . . . . . . . . .15.9.2 Objectified States of Affairs . . . . . . . . . . . . . 3584184284384384384684784784784784984984984985016 Philosophical Issues and Observations16.1 Puzzles . . . . . . . . . . . . . . . . . . . . . . .16.2 Carnap and a Metaontological Framework . . .16.3 Naturalism and Dependent Abstracta . . . . . .16.4 Explanatory Power . . . . . . . . . . . . . . . .16.5 Philosophical Interpretations of the Formalism16.6 Meinongian Ideal Objects . . . . . . . . . . . . .85285285285385385385517 Philosophy of Mathematics17.1 The Julius Caesar Problem . . . . . . . . .17.2 Unification of Mathematical Philosophies17.2.1 Naturalism . . . . . . . . . . . . . .17.2.2 Fictionalism and Nominalism . . .857858858858860.

CONTENTS17.2.3 Inferentialism . . . . . . . . . . .17.2.4 Structuralism . . . . . . . . . . .17.2.5 If-thenism or Deductivism . . . .17.2.6 Logicism . . . . . . . . . . . . . .17.2.7 Finitism . . . . . . . . . . . . . .17.2.8 Formalism . . . . . . . . . . . . .17.2.9 Psychologism and Conceptualism17.2.10 Conclusion . . . . . . . . . . . .17.3 Analyzing the Mathematics in Physics .xi.86086086086186186186286286218 Epistemological Considerations86618.1 Thin Conception of Objects . . . . . . . . . . . . . . . . . . . . . 86718.2 Carnap Again . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86718.3 Resnik’s Postulational Epistemology . . . . . . . . . . . . . . . . 86719 Future Research86819.1 Classical and Nonclassical Conjunction and Negation . . . . . . 86819.2 The Context Principle . . . . . . . . . . . . . . . . . . . . . . . . . 868IVTechnical Appendices, Bibliography, Index869Appendix to Chapter 5871Appendix to Chapter 6897Appendix to Chapter 7935Appendix to Chapter 9939Appendix to Chapter 10951Appendix to Chapter 12954Appendix: Proofs of Theorems and Metarules957Bibliography1332

AcknowledgmentsObject theory would not be where it is today without the help of numerouscolleagues. I’ve coauthored papers with many of these colleagues, had enlightening discussions with others, and received email messages with suggestedcorrections from still others.Daniel Kirchner and Uri Nodelman deserve special mention; they are listedon the title page as having made critical theoretical contributions. This is nounderstatement. Though I authored the text of this monograph (with the exception of Chapter 14, which was coauthored with Nodelman), both Kirchnerand Nodelman are coauthors of the theory described herein. I’d also like toacknowledge Daniel West, who read early drafts of nearly all of Part II, andcontributed a number of interesting observations, proofs, and even a helpfulmeta-metarule, all of which led to improvements. Wes Anderson also readsignificant parts of Part II and notified me of a number of errors.Until this monograph gets into a state closer to publication, I am simply going to list the names of my other colleagues who’ve contributed in some way tothis monograph, in alphabetical order: Peter Aczel, Jesse Alama, Colin Allen,C. Anthony Anderson, David James Anderson, Guillermo Badía Hernández,Christoph Benzmüller, Johannes Brandl, Otávio Bueno, Mark Colyvan, KitFine, Branden Fitelson, Richard Grandy, Allen Hazen, Alexander Hieke, Michael Jubien, Jeffrey Kegler, Johannes Korbmacher, Fred Kroon, Hannes Leitgeb, David Lewis, Godehard Link, Bernard Linsky, Alan McMichael, Christopher Menzel, Edgar Morscher, Michael Nelson, Karl-Georg Niebergall, PaulE. Oppenheimer, Terence Parsons, Barbara Partee, Francis Jeffrey Pelletier, Gideon Rosen, Tony Roy, Dana Scott, Merel Semeijn, Peter Simons, Chris Swoyer,Johan van Benthem, and Kai Wehmeier.Finally, and most importantly, I’d like to thank my wife, Susanne Z. Riehemann, who has joined me in so many adventures, both intellectual and in theoutdoors, and brought me such joy and happiness. I extremely grateful to herfor sharing her life and making it possible for me to write this book. I’ve beenlucky to have a life partner who is keenly interested in my work, and whoseown fields of study and investigation so nicely dovetail with my own.Stanford, CA and Mountain View, CAMay 5, 2022

Part IProphilosophy1

Part IIPhilosophy171

Chapter 7The LanguageThroughout most of Part II, we prove theorems that are formulated in a secondorder modal language. This will be the language used in Chapters 7 – 14.However this language is just a fragment of a more general, type-theoreticmodal language. We postpone the definition of the type-theoretic languageuntil Chapter 15, where we investigate typed object theory and its applications.In the present and subsequent chapters, our metalanguage makes use ofsome basic notions and principles of number theory and set theory, so as tomore precisely articulate certain definitions. But none of these notions andprinciples are used in the object language defined in this chapter. Ultimately,the philosophical system sketched over the next few chapters will offer us ananalysis of the basic notions and principles of number theory and set theoryused in the metalanguage, but we won’t be in a position to see this until Chapter 10 (where we define and prove facts about natural classes) and Chapter 14(where we define and prove facts about natural cardinals and natural numbers).7.1Metatheoretical DefinitionsThe definitions that generate a particular second-order language are given overthe next several items.(1) Metadefinitions: Simple Terms. A simple term of our second-order language is any expression that is a simple individual term or a simple n-place relation term (n 0), where these are listed as follows:(.1) Simple Individual Terms:Individual Constants (Names):a1 , a 2 , . . .173(Less Formal)(a, b, c, b1 , b2 , . . . , c1 , c2 , . . .)

174CHAPTER 7. THE LANGUAGEIndividual Variables:x1 , x 2 , . . .(x, y, z, u, v, w, y1 , y2 , . . . , z1 , z2 , . . .)(.2) Simple n-ary Relation Terms (n 0):n-ary Relation Constants (Names)P1n , P2n , . . .n-ary Relation Variables:F1n , F2n , . . .(P n , Qn , Rn , S n , T n )(F n , Gn , H n , I n , J n )(.3) Distinguished Unary Relation Constant:E! (read: ‘being concrete’ or ‘concreteness’)where E! is just a rewritten version of the unary relation constant P11In what follows, we shall use the technical term primitive constant as follows:(.4) A primitive constant is any simple individual constant or simple n-aryrelation constant (for some n 0) that occurs in the lists in (.1), (.2), and(.3) above.This helps us to distinguish primitive constants from new constants introducedby definition. No similar distinction is needed for variables, since we won’t introduce new variables into the language by definition. To facilitate readability,we often use the expressions listed in the column labeled ‘Less Formal’ as replacements for the official expressions of the language.(2) Metadefinitions: Syncategorematic Expressions. A syncategorematic expression represents a primitive notion of the language but is neither a term(i.e., the kind of expression that may have a denotation) nor a formula (i.e.,the kind of expression that has truth conditions). To list the syncategorematicexpressions of our language, we use α as a metavariable that ranges over allvariables and use ν (Greek nu), sometimes decorated with a numerical subscript, as a metavariable that ranges just over individual variables:(.1) Unary Formula-Forming Operators: (‘it is not the case that’ or ‘it is false that’) (‘necessarily’ or ‘it is necessary that’)𝒜 (‘actually’ or ‘it is actually the case that’)(.2) Binary Formula-Forming Operator: (‘if . . . , then . . . ’)(.3) Variable-Binding Formula-Forming Operator: α (‘every α is such that’)for every variable α

7.1. METATHEORETICAL DEFINITIONS175(.4) Variable-Binding Individual-Term-Forming Operator:ıν (‘the ν such that’)for every individual variable ν(.5) Variable-Binding n-ary Relation-Term-Forming Operators (n 0):λν1 . . . νn (‘being ν1 , . . . , νn such that’)for any distinct individual variables ν1 , . . . , νn , andλ (‘that . . . ’)where no variables follow the λ.These primitive, syncategorematic expressions are referenced in the definitionof the syntax of our language and are used to define complex formulas andcomplex terms. In what follows, we sometimes call: 𝒜 ıλthe negation operatorthe necessity operatorthe actuality operatorthe conditional operatorthe universal quantifierthe definite description operatorthe relation abstraction, or λ, operatorBy convention, in any conditional formula of the form ϕ ψ, we say ϕ is theantecedent and ψ the consequent of the conditional.(3) Metadefinitions: Syntax of the ‘Second-order’ Language. We present thesyntax of our second-order language by a simultaneous recursive definition ofthe following four kinds of expressions: individual term, n-place relation term,formula, and term.(.1) Every simple individual term (i.e., every individual constant and individual variable) is an individual term and every simple n-ary relation term(i.e., every n-ary relation constant and n-ary relation variable, is an n-aryrelation term (n 0)(.2) Every 0-ary relation constant and 0-ary relation variable is a formula(.3) If Πn is any n-ary relation term (n 1) and κ1 , . . . , κn are any individualterms, then(.a) Πn κ1 . . . κn is a formula(‘κ1 , . . . , κn exemplify Πn ’)(.b) κ1 . . . κn Πn is a formula(‘κ1 , . . . , κn encode Πn ’)(.4) If ϕ and ψ are formulas and α any variable, then[λ ϕ], ( ϕ), (ϕ ψ), αϕ, ( ϕ), and (𝒜ϕ) are formulas.

176CHAPTER 7. THE LANGUAGE(.5) If ϕ is any formula and ν any individual variable, thenıνϕ is an individual term(.6) If ϕ is any formula and ν1 , . . . , νn are any distinct individual variables(n 0), then(a) [λν1 . . . νn ϕ] is an n-ary relation term, and(b) ϕ itself is a 0-ary relation term.(.7) A term is any individual term or n-ary relation term (n 0).Though it should be clear how to read the formulas and terms of this language, there are two interesting facts to note about reading certain expressions,namely, 0-ary λ-expressions of the form [λϕ] are both formulas and terms, andevery formula ϕ is a 0-ary relation term. We discuss these in turn.By (.4), expressions of the form [λ ϕ] are formulas and, by (.6.a), they arealso 0-ary relation terms. So expressions of the form [λ ϕ] should be read inone of two ways, depending on the context. On the one hand, when [λ ϕ]stands by itself or occurs in formula position, it asserts that ϕ is true, sincetruth is the 0-ary case of predication. So, we read the formula [λ ϕ] ϕ as:that-ϕ is true if and only if ϕ. On the other hand, there will be contexts inwhich [λ ϕ] functions as a term, and in those contexts, we read [λ ϕ] simply as‘that-ϕ’. So, for example, when we define the notion τ exists (where τ is anyterm) and represent it as τ , then [λ ϕ] would be read as: that-ϕ exists. Andwhen we define the notion τ is identical to σ and represent it a τ σ , then theformula p [λϕ] would be read: p is identical to that-ϕ. And when we definethe notion Contingent(p), where p can be instantiated by any 0-ary relationterm, the formula Contingent([λ ϕ]) would be read: that-ϕ is contingent. If ithelps, one can always preface ‘the proposition’ to the reading of [λϕ] when thelatter is being used as a term. For instance, one could read our last example,Contingent([λ ϕ]), as: the proposition that-ϕ is contingent.The second interesting fact about reading the language is that (.6.b) stipulates that every formula is a 0-ary relation term. This feature allows us toregard formulas of the form ϕ ϕ as substitution instances of the universalclaim p(p p). But it also means that formulas can occur in contexts wherethey are simply naming rather than asserting propositions. For example, inthe previous paragraph, we noted that τ is defined for every term. So whena 0-ary relation term such as Fx is substituted for τ, the resulting formula,Fx , is well-formed. This claim asserts existence of the proposition Fx. So inthe claim Fx , the formula Fx is not asserting that x exemplifies F, but rathernaming the proposition x-exemplifies-F. Again, if it helps, one can add ‘theproposition’ to the reading of a formula being used as term, so that Fx maybe read: the proposition Fx exists. Similarly, identity is defined for all terms,

7.1. METATHEORETICAL DEFINITIONS177including 0-ary relation terms. So a claim of the form ϕ ψ is well-formed.In claims of this form, the instances of ϕ and ψ are not making assertions. Ifit helps, read such claims as: the proposition ϕ is identical to the propositionψ. For example, Fx [λz Fz]x would be read: the proposition x exemplifies Fis identical to the proposition x exemplifies being an object z that exemplifiesF. It should always be clear, in what follows, when a formula is being used,and is to be read, as a term denoting a proposition instead of as an expressionmaking an assertion.When n 1, we call formulas of the form Πn κ1 . . . κn (atomic) exemplificationformulas and formulas of the form κ1 . . . κn Πn (atomic) encoding formulas. Inwhat follows, we say:(.8) ϕ is the matrix of αϕ, ıνϕ, and [λν1 . . . νn ϕ] (n 0).Once we define αϕ, then one should extend the above definition so that ϕcounts as its matrix.According to clause (.6.a), any formula ϕ can serve as the matrix of a relation term of the form [λν1 . . . νn ϕ], where n 0. If we allow ourselves tospeak informally about the denotation of a term, then it is important to alertthe reader to the following facts about the system we shall be developing: Not every λ-expression is guaranteed to have a denotation (indeed, someλ-expressions will provably fail to have denotations), and so these expressions, like definite descriptions, will be governed by a free logic. Every 0-ary λ-expression [λ ϕ] is guaranteed to have a denotation, byaxiom (40.2), and every formula ϕ is guaranteed to have a denotation, bytheorem (105.2). It will be provable that [λϕ] and ϕ always denote the same 0-ary relation,by theorem (112.1).In general, λ-expressions are not to be interpreted as terms that potentiallydenote functions, but rather as terms that potentially denote relations. Thus,when we introduce axioms and rules of inference governing λ-expressions inthe next two chapters, the resulting λ-calculus is to be understood as a calculusof relations.Finally, in what follows, we use τ to range over terms. The simple termslisted in (1) are terms in virtue of clause (.1). We say:(.9) A term τ is complex if and only if τ is not a simple term.So the constants and variables listed in (1) are not complex terms. Given (.9),clauses (.5), and (.6) introduce kinds of complex terms: definite descriptions are complex individual terms, by (.5) and (.9)

178CHAPTER 7

iv Principia Logico-Metaphysica Draft / Excerpt NOTE: This is an excerpt from an incomplete draft of the monograph Prin- cipia Logica-Metaphysica.The monograph draft currently has four parts: