[わかみず会資料（2019. 3.27）] 第０版 Edward N. Zalta: Frege's Theorem and Foundations for Arithmetic First published Wed Jun 10, 1998; substantive revision Tue Jun 26, 2018 , paraphrased 柳生 孝昭 要旨： 本稿は、首記解説論文（SEP 掲載）の要約・釈義である。 歴史・背景・周辺的事情の大半を省き、記法を、 標準論理の慣用に沿うべく改めた。 従って分量は大幅（約 1/4）に減じたが、（数学、論理または哲学的な）内容は 変わわらない。 主題は F.L.Gottlob Frege の二主著による業績、即ち [0] Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens (Begr, 1879) の、 2 階述語論理の展開、それに拠る数学的概念の定義と、諸命題の導出、及び [1] Grundgesetze der Arithmetik (Gg, 1893/1903) に於ける、論理則としての Basic Law V の措定、それに基く ‘Hume's Principle (HP)’ の証明と、それからの算術の公理系の導出、である。 Basic Law V が「論理的」とは言えず、しかも矛盾を惹起すことは、広く知られており、Gg の深い理論的成果を 覆って来たが、近年、Hume の原理からの（Dedekind/Peano）算術体系の導出（Frege の定理）の正当性の認識を 核に、再評価が進んでいる。 論文はこの事情を論理・数学的に詳しく検討し、哲学的な考察を加える。 0 目次 1. The Second-Order Predicate Calculus and Theory of Concepts 1.1 The Language 1.2 The Logic 1.3 The Theory of Concepts 2. Frege’s Theory of Extensions: Basic Law V 2.1 Notation for Courses-of-Values of Functions 2.2 Notation for Extensions of Concepts 2.3 Membership in an Extension 2.4 Basic Law V for Concepts 2.5 First Derivation of the Contradiction 2.6 Second Derivation of the Contradiction 2.7 How the Paradox is Engendered 3. Frege’s Analysis of Cardinal Number 3.1 Equinumerosity 3.2 Contextual Definition of ‘The Number of F s’: Hume’s Principle 3.3 Explicit Definition of ‘The Number of F s’ 3.4 Derivation of Hume’s Principle 4. Frege's Analysis of Predecessor, Ancestrals, and the Natural Numbers 4.1 Predecessor 4.2 The Ancestral of a Relation R 4.3 The Weak Ancestral of R 4.4 The Concept Natural Number 5. Frege’s Theorem 5.1 Zero is a Number 5.2 Zero Isn’t the Successor of Any Number 5.3 No Two Numbers Have the Same Successor 5.4 The Principle of Mathematical Induction 5.5 Every Number Has a Successor 5.6 Arithmetic 6. Philosophical Questions Surrounding Frege’s Theorem 6. 1 Frege’s Goals and Strategy in His Own Words 6.2 The Basic Problem for Frege’s Strategy 6.3 The Existence of Concepts 6.4 The Existence of Extensions 6.5 The Existence of Numbers and Truth-Values: The Julius Caesar Problem 6.6 Final Observations Bibliography 1 1. The Second-Order Predicate Calculus and Theory of Concepts 1.1 The Language 1.1.0 Type, Instance and Term ＊ [0] Term ::= Object (Name | Variable)(, Term) [1] Object Name: a , b , •• [2] Object Variable: x,y, •• [3] Predicate Name: P,Q •• , Identity (＝, a binary relation) in particular [4] Predicate Variable: F,G, •• 1.1.1 Formula [0] Aromic Formula ::= Predicate (Name | Variable) Term, the infix expression Term ＝ Term of Identity [1] Formula ::＝ Aromic Formula | (¬ | (∀ | ∃) (Object | Predicate) Variable) Formula | ＋ Formula (∧ | ∨ |→ )Formula)) 注意： Frege は多項述語一般を定義せず、関数《真偽を値とする》に替える。 本論文は逆に、述語のみを導入す る。 Frege の定理の証明に、関数は不要の故である。 一般には、n 変数関数は、下記の条件を満たす n 項述語 として、定義し得る： [2] R is a functional relations ＝df R (x0, •• , xn, y )∧R (x0, •• , xn, z ) → y ＝z [3] 特に重要な論理式は、1 個の自由変数を有する開論理式、即ち真理値を値域とする１変数関数であり、 Frege は、これを‘concept ’（概念）と名付けた。 1.2 The Logic 1.2.0 Axioms [0] Axioms for Propositional Logic [1] Universal Instantiation: [1.0] ∀xP ( ••, x, ••) → P ( ••, a, ••) [1.1] ∀F ( ••, x, ••) → P ( ••, x, ••) [2] Existential Introduction: [2.0] P ( ••, a, ••) → ∃xP ( ••, x, ••) [2.1] F ( ••, x, ••) → ∃FF ( ••, x, ••) [3] Quantifier Distribution: ∀x (ϕ →ψ) → (ϕ → ∀xψ) for any formulae ϕ and ψ where x isn’t free in ϕ [4] Laws of Identity: [4.0] x ＝ x [4.1] x ＝ y → (ϕ ( ••, x, ••) → ϕ ( ••, y, ••)) 1.2.1 Rules of Inference [0] Modus Ponens (MP): ϕ, ϕ →ψ ├ ψ [1] Generalization (GEN): ϕ ├ ∀xϕ 1.3 The Theory of Concepts [0] Comprehension Principle for Concepts: ∃G∀x (Gx ≡ ϕ) for any formula ϕ which has no free G s derivable and constitute very important generalizations within Frege's system. ここに存在を表明した述語 G は、性質（一般に関係） ϕ によって特徴付けられる、対象の内包としての概念を表 す。 xϕ が、その名である。 [1] λ-Conversion: ∀y ([xϕ ] y ≡ ϕxy ) Note: Frege equivalently postulated Rule of Substitution that allowed him to substitute any formula ϕ for free concept variables F . The system described, i.e., second-order logic with Identity and Comprehension Principles, extended with λ- expressions and λ-Conversion, is consistent. The domain of objects contains a single object, say b , and the do- main of 1-place relations contains two concepts (i.e., one which b falls under and one which nothing falls under), then all of the above axioms are true. 上記の、拡張された 2 階述語論理は、無矛盾である。 2 2. Frege’s Theory of Extensions: Basic Law V Second-order logic with comprehension gives rise to Russell’s paradox when one adds Frege’s theory of courses -of-values and extensions. The course-of-values of a function f is the set of ordered pairs that records the value fx for every argument x , and for a concept, is called extension, the set of all objects that fall under that concept. {(x, fx ):x ∈定義域} {x : fx ＝True } 2.1 Notation for the Course-of-Values ϵf of a Function f Basic Law V: ϵf ＝ ϵg ≡∀x [f (x) ＝ g (x)] 左辺は、特性関数 f の再帰的適用を許す ϵf を措定、矛盾の原因。 2.2 Notation for Extensions of Concepts ϵ[xϕ ] designates the extension of the concept [xϕ ] ϵ[xϕ ] ＝df {x : ϕx } Prefixed to a concept variable F, ϵF is a variable that ranges over extensions: for each value f of F , ϵF denotes the extension ϵf . 述語変数 F の外延 ϵF は、F の各々の値 f の外延 ϵF を値とする変数である。 2.3 Membership in an Extension x ∈y ＝df ∃G (y ＝ϵ G ∧Gx ) 即ち x を要素とする集合 y は、x が例示する或る概念 G の外延である。 この定義から、任意の H について Hx ├ x ∈ϵH の成立は、自明。 The following principle, which asserts that every concept has an extension, is derived as a theorem: Existence of Extensions ∀G∃x (x ＝ ϵG ) x ＝x ├ ∀x (x ＝x ) ├ ϵF ＝ ϵF ├ ∃x (x ＝ϵF ) この「証明」は、表記 ϵF が如何なる対象を指示するのか、そもそも存在するのかが不明という点に於いて、非公式 的、しかも等置の表明が、存在を証明すべき項を予め措定しているのは、全くの循環論と言う他無い。 2.4 Basic Law V for Concepts [0] Basic Law V (Special Case): ϵF ＝ ϵG ≡ ∀x (Fx ≡Gx ) asserts that the set of F s is identical to the set of G s if and only if F and G are materially equivalent: {x ∣ Fx } = {y ∣ Gy } ≡} ∀z (Fz ≡Gz ) in modern guise [1] Two Corollaries [1.0] Law of Extensions: ∀F ∀x (x ∈ ϵF ≡Fx ) Extension (x) ＝df ∃F (x ＝ ϵF ) [1.1] Principle of Extensionality: Extension (x ) ∧Extension (y ) → [∀z (z∈x ≡ z∈y ) → x ＝y ] (Derivation of the Principle of Extensionality; https://plato.stanford.edu/entries/frege-theorem/proof3.html) The above show why Basic Law V may not be consistently added to second-order logic with comprehension. The following is a volatile mix: (a) the Comprehension Principle for concepts, which ensures that there is a con- cept corresponding to every open formula, (b) the Existence of Extensions principle, which ensures every con- cept is correlated with an extension, and (c) Basic Law V, which ensures that the correlation of concepts with extensions behaves in a certain way. 2.5 First Derivation of the Contradiction The extension ϵ[x ∃F (x ＝ ϵF ∧¬Fx )] falls under the concept [λx ∃F (x ＝ ϵF ∧¬Fx )] iff it does not. 「何らかの概念 F の外延で、自身は F を満たさない」対象 x は、概念 […] を満たし、且つ満たさない。 2.6 Second Derivation of the Contradiction Naive Comprehension Axiom for Extensions: ∀F ∃y ∀x (x ∈y ≡Fx ) ∀F ∀x (x ∈ ϵF ≡Fx ) ├ ∀x (x ∈ ϵF ≡Fx ) ├ ∃y ∀x (x ∈y ≡Fx ) ├ ∀F ∃y ∀x (x ∈y ≡ Fx )) ├ Instantiate the quantified variable F to the concept [λzz ∉ z], to yield: ∃y∀x (x ∈y ≡[λzz ∉ z ]x ) By λ -Conversion, equivalent to: ∃y∀x (x ∈y ≡x ∉x ) 前・本節に挙げたのは何れも、経験的な意味を持たない、しかし文法には厳密に従う例である。 これは形式化され た論理体系が経験的、それどころか数学的な意味すら十全に捉え得ていないことを示す。 3 2.7 How the Paradox is Engendered Extending second-order logic by Basic Law V requires a situation where the domain of concepts has to be strict- ly larger than the domain of extensions while at the same time the latter has to be as large as the former. Various views resulting from philosophical diagonosis include : [0] Extensional view of concepts: F ＝G ≡ ∀x (Fx ≡Gx ) The Existence of Extensions Principle correlates each concept F with an extension ϵF. Extension ϵF Concept F Basic Law V ← Basic Law V → Basic Law V : ϵ F ＝ ϵ G ≡ ∀ x ( F x ≡ G x ) [1] Basic Law V の両方向の単射性： V ← : ∀x (Fx ≡Gx ) → ϵF ＝ ϵG ϵF ≠ ϵG → ¬∀x (Fx ≡Gx ) [0] : ¬∀x (Fx ≡Gx ) → F ≠G ϵF ≠ ϵG → F ≠G V → : ϵF ＝ ϵG → ∀x (Fx ≡Gx ) ¬∀x (Fx ≡Gx ) → ϵF ≠ ϵG [0] : F ≠G → ¬∀x (Fx ≡Gx ) F ≠G → ϵF ≠ ϵG There exist models of second-order logic with the Comprehension Principle, but without Basic Law V, in which the domain of concepts is not larger than the domain of objects [2]. The requirement that there be more concepts than extensions is imposed jointly by the Comprehension Principle for Concepts and the new significance this principle takes on in the presence of Basic Law V. Basic Law V → requires that there be at least as many ex- tensions as there are concepts. Thus, the addition of Basic Law V to Frege’s system forces the domain of concepts to be larger than the domain of objects (and so larger than the domain of extensions), due to the end- less cycle of new concepts that arise in connection with the new extensions contributed by Basic Law V. 上記、特に下線の件は、extension の全体が concept の集合に単射される、しかし後者には Russell 文のような、 決して確定した外延に対応しない概念も現れる、という事実を指しているのであろうか？ とすれば、逆説の発生の 分析が当の逆説の存在に依存することになり、この論証は、それが哲学上のものであり、論理・数学的厳密の埒外 に在るとしても、奇妙な感を免れない。 あるいは、概念の領域が、x (x ＝a ) の外延としての各対象 {a } の全体の、 単射 V → による像、従ってその部分である外延の集合の像を含む一方、その任意の部分集合 A もまた、概念 x (x ∈A )) の外延を与える（Comprehension Principle）から、冪集合をも含み、対象領域は概念領域の真部分集 合を成す、と言いたいのか？ 何れにせよ、この意味不明確な、錯綜した文言が解明しようとしている、逆説的事態 の根底には、対象全体の領域を集合として扱い、外延の定義と対象化の、際限・制約の無い再帰的適用が有る。 “endless cycle of new concepts that arise in connection with the new extensions contributed by Basic Law V” と は、そのような、素朴集合論的操作濫用のことなのであろう。 The traditional view to repair the Fregean theory of extensionsis is to restrict either Basic Law V or the Com- prehension Principle .On the other hand, to abandon second-order logic (and the Comprehension Principle) alto- gether.