Frege-numbers + Begriffsschrift revisited

Transcription

1 History of logic: from Frege to Gödel 26th September 2017

2 Grundlagen der Arithmetik: some philosophical issues Introduction:

3 Grundlagen der Arithmetik: some philosophical issues Introduction: "In this investigation I have adhered to the following fundamental principles:

4 Grundlagen der Arithmetik: some philosophical issues Introduction: "In this investigation I have adhered to the following fundamental principles: There must be a sharp separation of the psychological from the logical, the subjective from the objective;

5 Grundlagen der Arithmetik: some philosophical issues Introduction: "In this investigation I have adhered to the following fundamental principles: There must be a sharp separation of the psychological from the logical, the subjective from the objective; The meaning of a word must be asked for in the context of a proposition, not in isolation;

6 Grundlagen der Arithmetik: some philosophical issues Introduction: "In this investigation I have adhered to the following fundamental principles: There must be a sharp separation of the psychological from the logical, the subjective from the objective; The meaning of a word must be asked for in the context of a proposition, not in isolation; The distinction between concept and object must be kept in mind."

7 Grundlagen der Arithmetik: some philosophical issues Introduction: "In this investigation I have adhered to the following fundamental principles: There must be a sharp separation of the psychological from the logical, the subjective from the objective; The meaning of a word must be asked for in the context of a proposition, not in isolation; The distinction between concept and object must be kept in mind." Numbers are neither physical nor mental objects.

8 Grundlagen der Arithmetik: some philosophical issues Introduction: "In this investigation I have adhered to the following fundamental principles: There must be a sharp separation of the psychological from the logical, the subjective from the objective; The meaning of a word must be asked for in the context of a proposition, not in isolation; The distinction between concept and object must be kept in mind." Numbers are neither physical nor mental objects. But how is it possible to know about numbers and their properties?

9 Numerical quantifiers

10 Numerical quantifiers Numerals used to settle the number of certain objects are second-level concepts.

11 Numerical quantifiers Numerals used to settle the number of certain objects belonging under a certain concept are second-level concepts.

12 Numerical quantifiers Numerals used to settle the number of certain objects belonging under a certain concept are second-level concepts. The Fregean use of the numeral n: The number n belongs to the concept F

13 Numerical quantifiers Numerals used to settle the number of certain objects belonging under a certain concept are second-level concepts. The Fregean use of the numeral n: The number n belongs to the concept F "... the number 0 belongs to a concept if, whatever a may be, the proposition holds universally that a does not fall under that concept. " 0 x(fx) x Fx

14 Numerical quantifiers Numerals used to settle the number of certain objects belonging under a certain concept are second-level concepts. The Fregean use of the numeral n: The number n belongs to the concept F "... the number 0 belongs to a concept if, whatever a may be, the proposition holds universally that a does not fall under that concept. " 0 x(fx) x Fx "the number (n + 1) belongs to the concept F if there is an object a falling under F such mat the number n belongs to the concept falling under F, but not a."

15 Numerical quantifiers Numerals used to settle the number of certain objects belonging under a certain concept are second-level concepts. The Fregean use of the numeral n: The number n belongs to the concept F "... the number 0 belongs to a concept if, whatever a may be, the proposition holds universally that a does not fall under that concept. " 0 x(fx) x Fx "the number (n + 1) belongs to the concept F if there is an object a falling under F such mat the number n belongs to the concept falling under F, but not a." n+1 x(fx) a(fa n x(fx x a))

16 The Julius Caesar-problem: numbers as objects

17 The Julius Caesar-problem: numbers as objects "... we can never... decide by means of our definitions whether the number Julius Caesar belongs to a concept, or whether that well-known conqueror of Gaul is a number or not."

18 The Julius Caesar-problem: numbers as objects "... we can never... decide by means of our definitions whether the number Julius Caesar belongs to a concept, or whether that well-known conqueror of Gaul is a number or not." Background assumption: numbers are objects, moreover, unique objects.

19 The Julius Caesar-problem: numbers as objects "... we can never... decide by means of our definitions whether the number Julius Caesar belongs to a concept, or whether that well-known conqueror of Gaul is a number or not." Background assumption: numbers are objects, moreover, unique objects. Numbers are objects that are not given by intuition; they are given by reason (pace Kant).

20 The Julius Caesar-problem: numbers as objects "... we can never... decide by means of our definitions whether the number Julius Caesar belongs to a concept, or whether that well-known conqueror of Gaul is a number or not." Background assumption: numbers are objects, moreover, unique objects. Numbers are objects that are not given by intuition; they are given by reason (pace Kant). Their objectivity lies solely in the fact that there are true propositions about them.

21 The Julius Caesar-problem: numbers as objects "... we can never... decide by means of our definitions whether the number Julius Caesar belongs to a concept, or whether that well-known conqueror of Gaul is a number or not." Background assumption: numbers are objects, moreover, unique objects. Numbers are objects that are not given by intuition; they are given by reason (pace Kant). Their objectivity lies solely in the fact that there are true propositions about them. We should posit the recognition conditions for numbers; that s all that we can and should do.

22 The Hume Principle

23 The Hume Principle When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal (quoted from Hume)

24 The Hume Principle When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal (quoted from Hume) #F = #G f( x(fx 1 y(f(x, y) Gy) y(gy 1 x(fx f(x, y)))

25 The Hume Principle When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal (quoted from Hume) #F = #G f( x(fx 1 y(f(x, y) Gy) y(gy 1 x(fx f(x, y))) Short for the right side: F G, F is equinumerous with G.

26 The Hume Principle When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal (quoted from Hume) #F = #G f( x(fx 1 y(f(x, y) Gy) y(gy 1 x(fx f(x, y))) Short for the right side: F G, F is equinumerous with G. An abstraction principle (based on an equivalence relation).

27 The Hume Principle When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal (quoted from Hume) #F = #G f( x(fx 1 y(f(x, y) Gy) y(gy 1 x(fx f(x, y))) Short for the right side: F G, F is equinumerous with G. An abstraction principle (based on an equivalence relation). Equivalence classes are proper classes.

28 The Hume Principle When two numbers are so combined, as that the one has always a unit answering to every unit of the other, we pronounce them equal (quoted from Hume) #F = #G f( x(fx 1 y(f(x, y) Gy) y(gy 1 x(fx f(x, y))) Short for the right side: F G, F is equinumerous with G. An abstraction principle (based on an equivalence relation). Equivalence classes are proper classes. The principle is impredicative: the domains of quantification for x and y contain the objects to be defined.

29 Frege numbers

30 Frege numbers The number belonging to the concept F: #F = Ĝ(F G)

31 Frege numbers The number belonging to the concept F: #F = Ĝ(F G) Ĝ: the extension of the concept G

32 Frege numbers The number belonging to the concept F: #F = Ĝ(F G) Ĝ: the extension of the concept G n is a number (solution to the Julius Caesar problem): n F(n = #F)

33 Frege numbers The number belonging to the concept F: #F = Ĝ(F G) Ĝ: the extension of the concept G n is a number (solution to the Julius Caesar problem): n F(n = #F) 0 is the number belonging to the concept not equal with itself : 0 = #(λx(x x))

34 Frege numbers The number belonging to the concept F: #F = Ĝ(F G) Ĝ: the extension of the concept G n is a number (solution to the Julius Caesar problem): n F(n = #F) 0 is the number belonging to the concept not equal with itself : 0 = #(λx(x x)) λxa(x): the concept described by the open sentence A(x).

35 Frege numbers, continuation

36 Frege numbers, continuation m is an immediate follower of n in the sequence of numbers: mpn F G a(m = #G n = #F Ga x(fx (Ga x a)))

37 Frege numbers, continuation m is an immediate follower of n in the sequence of numbers: mpn F G a(m = #G n = #F Ga x(fx (Ga x a))) n is a finite (or natural) number iff it follows 0 in the P-sequence of numbers (see Begriffsschrift, ch. 3): n 0P n

38 Theorems about Frege numbers

39 Theorems about Frege numbers The Hume principle holds.

40 Theorems about Frege numbers The Hume principle holds. The number belonging to the concept equal with 0 is an immediate follower of 0: (#λx(x = 0))P0

41 Theorems about Frege numbers The Hume principle holds. The number belonging to the concept equal with 0 is an immediate follower of 0: P is one-to-one. (#λx(x = 0))P0

42 Theorems about Frege numbers The Hume principle holds. The number belonging to the concept equal with 0 is an immediate follower of 0: P is one-to-one. (#λx(x = 0))P0 Every number except of 0 is an immediate follower of some number: n(( n n 0) m(npm))

43 Theorems about Frege numbers The Hume principle holds. The number belonging to the concept equal with 0 is an immediate follower of 0: P is one-to-one. (#λx(x = 0))P0 Every number except of 0 is an immediate follower of some number: n(( n n 0) m(npm)) If n is a natural number, then the number belonging to the concept belonging to the P-sequence beginning with 0 and finishing with n is an immediate follower of n and it is different from n.

44 Theorems about Frege numbers The Hume principle holds. The number belonging to the concept equal with 0 is an immediate follower of 0: P is one-to-one. (#λx(x = 0))P0 Every number except of 0 is an immediate follower of some number: n(( n n 0) m(npm)) If n is a natural number, then the number belonging to the concept belonging to the P-sequence beginning with 0 and finishing with n is an immediate follower of n and it is different from n. "Frege s Theorem": the Frege numbers satisfy the Peano axioms.

45 Function and Concept: semantical values

46 Function and Concept: semantical values Truth values: True and False, ideal objects.

47 Function and Concept: semantical values Truth values: True and False, ideal objects. Complete expressions of the language: names and sentences. They denote objects (sentences denote truth values).

48 Function and Concept: semantical values Truth values: True and False, ideal objects. Complete expressions of the language: names and sentences. They denote objects (sentences denote truth values). Incomplete expressions of the language refer to functions. An expression G with one empty place refers to the function that for each expression F which can fill the empty place renders the denotatum of the completed expression G(F) to the reference of F.

49 Function and Concept: semantical values Truth values: True and False, ideal objects. Complete expressions of the language: names and sentences. They denote objects (sentences denote truth values). Incomplete expressions of the language refer to functions. An expression G with one empty place refers to the function that for each expression F which can fill the empty place renders the denotatum of the completed expression G(F) to the reference of F. Each function has a value range. Two functions have the same value range iff they give the same output value for identical input values.

50 Function and Concept: semantical values Truth values: True and False, ideal objects. Complete expressions of the language: names and sentences. They denote objects (sentences denote truth values). Incomplete expressions of the language refer to functions. An expression G with one empty place refers to the function that for each expression F which can fill the empty place renders the denotatum of the completed expression G(F) to the reference of F. Each function has a value range. Two functions have the same value range iff they give the same output value for identical input values. Concepts are functions whose values are all and only truth values. Extension of a concept is its value range.

51 Function and Concept: semantical values Truth values: True and False, ideal objects. Complete expressions of the language: names and sentences. They denote objects (sentences denote truth values). Incomplete expressions of the language refer to functions. An expression G with one empty place refers to the function that for each expression F which can fill the empty place renders the denotatum of the completed expression G(F) to the reference of F. Each function has a value range. Two functions have the same value range iff they give the same output value for identical input values. Concepts are functions whose values are all and only truth values. Extension of a concept is its value range. Logical connectives are (truth) function expressions, and the horizontal stroke ("content stroke") of the Begriffsschrift refers to a truth function, too.