Frege’s basic terminology: logic includes all denoting expressions. Its elements include simple/complex denoting objects such as “π,” or “John,” and sentences;
What does “falling under a concept” mean? For Frege, concepts are functions which map every argument to a truth value (The True or The False). For example, the sentence “( ) > 2” denotes the concept “being greater than 2.” The verb phrase “is prime” is a function P( ) that maps all primes to The True and others to The False. “Falling under a concept” means a simple predication. Given the sentence “John is happy,” “John” falls under the concept “( ) is happy.” (from SEP)
the extensions of a concept F are objects which F maps onto “the true.” Objects fall under concepts, s.t. sentences are true; extensions refer to the objects that produces true statements.
note: this component of predication in logic also underlies Kant’s epistemology.
To define a number, one first needs to define equality. Number symbols are invented because there is such a need for designating them whenever concepts are equinumerous. Frege’s definition of equinumerosity (gleichzahlig, translated into the simple “equality” in Austin’s translation) follows from Leibniz’s: “things are the same as each other, of which one can be substituted for the other without loss of truth.”
Frege first offers a possible (but wrong) way of defining 0 and 1: 0 belongs to the concept that does not have any object falling under it; 1 belongs to the concept F, if: if the statement that a does not fall under F is not true universally, and if a and b both fall under F, a = b. Then by induction one can define (n + 1) in the following way: if there is an a falling under F, and if n belongs to the concept “falling under F, but not a.”
This is not feasible because it is not clear what “belonging to a concept” means. Also there is no way to prove that the number thus obtained is unique. We cannot prove if a and b belongs to the same concept, a = b. In fact one has only specified the usage of “n belongs to” but not the number itself. One cannot judge if “Julius Caesar” belongs to a concept. One cannot recognize the same number again. Number thus defined is only a property of a concept.
Therefore, the first step toward defining a number is defining equinumerosity, so that in different propositional contexts, the same number can be recognized.
How to define equality
Equality is defined as the equality of extensions, i.e., objects that map concepts to The True.
Definition: the extension of the concept “equal to the concept F” is identical with the extension of the concept “equal to the concept G” if and only if the number that belongs to F and G are the same.
More specifically, this means that every object that falls under the concept F stand in relation φ with every object that falls under the concept G. In other words, for every a falling under F, a stands in relation φ with an object b falling under G; for every b falling under G, there is an a falling under F such that φ(a) = b. This is a 1-1 correlation. If concepts F and G are equinumerous, then the extension to “equinumerous to F” and “equinumerous to G” are the same—we say the same number belongs to F and G. Because numbers are defined as extensions, they are objects. But the extensions to “equinumerous to concept F” would also be concepts. Therefore, numbers are concepts used as objects in propositional functions, as mentioned in the footnote on page 80.
Therefore, equality is a relation between sets, that is between objects falling under two different concepts. Equality or equinumerosity is recognized and 1-1 correspondence is a mechanical process, resembling the waiter lying down forks to knives without being aware. This would allow Frege to establish the relation of equality as a relation independent of the human mind.
How to define 0 and 1
0 is the extension of the concept “equinumerous to the concept ‘not identical to itself’”—there is no object falling under the concept “not identical to itself” so that 0 is the extension (all concepts) that has no object falling under it.
Here the specification of the concept “not identical to itself” is somewhat arbitrary, but it is convenient because we know that there is definitely such a concept which would make “( ) is equinumerous to the concept ‘not identical to itself’” a true statement.
now we know that 0 exists. So we can specify the concept “identical with 0”—we know that 0 is identical with 0. So there is one object, 0, that falls under the concept “identical with 0.”
1 is the extension of the concept “equinumerous to the concept ‘identical with 0’”—so the set of all objects/concepts that are equinumerous to “identical with 0” which has one object falling under it.
How to prove that 1 follows 0
define “n follows m” as: if the number n belongs to the concept F, i.e., m is the extension of “equinumerous to F,” and given an object x falling under the concept F, then the number that m belongs to “falling under F but not x,” i.e., m is the extension of “equinumeros to ‘falling under F but not x,’” then we say that n follows m.
does 1 follow 0? 1 belongs to the concept “identical with 0.” There is an object falling under “identical with 0,” the object 0, and so the concept “identical with 0 but not 0” has no object falling under it. This concept is equinumerous to “not identical with itself” and 0 belongs to this concept. Therefore, 1 follows 0.
how to define the rest of the natural numbers?
define the natural number series in such a way that every number except 0 follows directly after a number.
1. φ(x) falls under F
2. if d falls under F, it is universally true that every φ (d) falls under F
then we know that y = φ(x) follows x in the φ series.
starting from x, we would transfer our attention one by one to objects that stand in relation φ to x.
Q: How does one transfer the “n follows m” to the relation φ?
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