1 24.400 Proseminar in philosophy I Fall 2003 Frege’s Grundlagen We can define (in second-order logic) ⎡There are exactly as many Φs as Ψs⎤ as: ⎡∃R(∀x(Φx ⊃ ∃1y(Ψy & Rxy)) & ∀y(Ψy ⊃ ∃1x(Φx &Rxy)))⎤ which we will abbreviate as ⎡Φ ≈ Ψ⎤. (See §72.) For any one-place predicate letter or predicate variable Φ, ⎡#Φ⎤ is a singular term (intended to be the formal counterpart of ⎡the number of Φs⎤, or, in Frege’s jargon, ⎡the number belonging to the concept Φ⎤. For convenience, we treat ⎡[ζ: α]⎤, where α is a formula whose only free first-order variable is ζ, as a one-place predicate letter that applies to an object just in case that object satisfies α; thus ‘[x: x=x]’ is a predicate letter that applies to everything that is self-identical, viz. everything. Then we may state “Hume’s Principle” (§73) thus: (HP) ∀F∀G(#F = #G ↔ F ≈ G) Frege assumed that for every concept C (whether a concept of objects, like the concept moon of Jupiter; or a concept of concepts, like the concept equinumerous with the concept moon of Jupiter; or…) there is an object, the extension of C (something like the set of the things that fall under C). If α is some formula with only the variable ξ (of any order) free, then we can write ⎡{ξ | α}⎤ to denote the extension of the concept expressed by the formula α. Thus ‘{x | x=x}’ denotes the extension of the (first level) concept is self-identical; and ‘{F | ∃x∃yFx & Fy & x ≠ y}’ denotes the extension of the (second level) concept is instantiated at least twice.2 So we can write Frege’s definition of ⎡#Φ⎤ (§68) as ⎡{X | X ≈ Φ}⎤. The extensions of concepts C1 and C2 are identical just in case the same things fall under C1 and C2. In particular, considering just second-level concepts (i.e. concepts under which fall concepts under which fall objects), Frege supposed that every instance of the following axiom schema is true: (V) {X | A(X)} = {X | B(X)} ↔ ∀X(A(X) ↔ B(X)) (To get an instance of (V), replace ‘X’ with a second-order variable, and replace ‘A(X)’ by a formula with only that variable free; ditto for ‘B(X)’.) The proof of Hume’s principle (§68) Right-to-left. Suppose that F ≈ G, and that for some H, H ≈ F. As Frege informally argues, ≈ is transitive. So H ≈ G. Suppose that F ≈ G, and that for some H, H ≈ G. As ≈ is symmetric, G ≈ F. By transitivity, H ≈ F. So: ∀X(X ≈ F ↔ X ≈ G) By (V): {X | X ≈ F} = {X | X ≈ G} ↔ ∀X(X ≈ F ↔ X ≈ G) And so: {X | X ≈ F} = {X | X ≈ G} By the definition of ⎡#F⎤: #F = #G Left-to-right (fn. 1). Suppose that #F = #G. Then, by the definition of ⎡#F⎤, {X | X ≈ F} = {X | X ≈ G}. By (V): ∀X(X ≈ F ↔ X ≈ G)3 In particular: F ≈ F ↔ F ≈ G Since ≈ is reflexive, F ≈ F; so: F ≈ G Unfortunately (V) (a special case of Axiom V of the Grundgesetze) is, as Russell pointed out, inconsistent. Definitions1. ‘0’ is defined as ‘#[x: x≠x]’ (§74) 2. ‘1’ is defined as ‘#[x: x=0]’ (§77) 3. ⎡Snm⎤ (⎡n is the successor of m⎤) is defined as: ⎡∃F∃x(Fx & #F = n & #[y: Fy & y≠x] = m)⎤ ( §76) 4. ⎡Nx⎤ (⎡x is a finite Number⎤) is defined as: ⎡∀F((∀x(Sx0 ⊃ Fx) & ∀y(Fy ⊃ ∀z(Szy ⊃ Fz)) ⊃ Fx) v x = 0⎤ ( §83) (that is: ⎡x bears the ancestral of S to 0, or x = 0⎤; the ancestral is defined in §79.) Second-order Peano Arithmetic PA2 has three primitive bits of vocabulary (a name ‘0’, a one-place predicate ‘N’, and a two-place predicate ‘S’) and five axioms: N0 [zero is a number]4 ∀x(Nx ⊃∃1y(Ny & Syx)) [every number has a unique successor] ∀x(Nx ⊃ ~S0x) [zero is not the successor of any number] ∀x∀y∀z(Sxy & Sxz ⊃ y=z) [no two things have the same successor] ∀F((F0 & ∀x((Nx & Fx) ⊃ ∀y(Syx ⊃ Fy))) ⊃ ∀x(Nx ⊃ Fx)) [If zero falls under a concept, and if, for every number falling under it, the successor of that number also falls under it, then every number falls under the concept] Some interesting facts Second-order Peano Arithmetic is derivable in second-order logic with a single non-logical axiom: Hume’s principle. Hume’s Principle is consistent (it has a model whose domain consists of the natural numbers). More cautiously, if PA2 is inconsistent, so is Hume’s Principle. (It is not doubted that PA2 is consistent.) Fregean contextual definitions, like (V) and (HP), are tricky things. An example due to Boolos: there is a contextual definition that is consistent, but the conjunction of it with (HP) is inconsistent. All this and more can be found in various parts of Boolos’ Logic, Logic, and Logic, a book which you should own in any case. The book that started the debate about the status of (HP) is Wright’s Frege’s Conception of Numbers as Objects. A recent collection of work by Wright and Hale on the neo-Fregean program in the philosophy of mathematics is The Reason’s Proper Study (OUP