June 26, 2021 Off By Felso

The final phase of Frege’s project involves deriving the laws of arithmetic from the axioms that Frege accepted as logical. While making these derivations, Frege applied to a law that he called Fundamental Law V. We can express this law as follows:

The value-fields of a function f(x) and another function g(x) are one and the same if and only if f(x) = g(x) for all x’s.

A special case of Basic Law V is applicable to predicates and their extents. If {x|Fx} denotes the extent of the predicate F, that is, it contains all the things that satisfy the predicate F. In this case, Basic Law V can be restated as follows: “The predicates F and G have the same coverage if and only if Fx <-> Gx for all x” (Here the symbol “<->” indicates the reciprocal condition). In other words, “the set of Fs and the set of Gs are one and the same if and only if every F is G and every G is F”.

Frege accepted this law as logical and self-evident and tried to show that from it Hume’s principle of equinumeracy and Peano’s arithmetic axioms could be derived. The derivation of the axioms of arithmetic from Basic Law V is today referred to as Frege’s Theorem.

As Frege prepares to publish the second volume of the Grundgesetze, Russell writes to him, stating that a paradox can be derived from Basic Law V. Russell shows that “the set of sets that are not members of themselves” can be defined in Frege’s system, but it can be shown that such a set can be both a member and not a member of itself, thus a contradiction arises. In this case, Frege’s system includes a contradiction and becomes inconsistent. Frege acknowledges the problem and writes an Appendix as a solution. The solution proposes an amendment to Basic Law V. Frege begins his appendix with these lines: “Nothing more unfortunate happens to someone who writes on science than when one of the foundations of his work is shaken after his work is completed. This is the situation I am in, with a letter from Mr. Bertrand Russell just as the printing of this volume is nearing completion.” (Russell’s letter and Frege’s reply were published in Heijenoort (1967).) It later turns out that Frege’s solution was not sufficient.

Frege’s work on logic is recognized over time by references to the works of Bertrand Russell (1872 – 1970), Rudolf Carnap (1891-1970), and Ludwig Wittgenstein (1889-1951).

The first paradox that Russell discovered and named after him is the class paradox of classes that are not members of themselves. For example, “the human class itself is not a ‘human’.” If a class is a member of itself, it has the property that qualifies it, so it is not a member of itself.

If it is not a member of itself, it does not have the characteristic that characterizes it. Therefore, it is a member of itself. In both cases there is a contradiction.

On the same model, paradoxes such as that of “I lie” Cretan Epimenides can be constructed. These paradoxes directly brought into question the principles of logic. The so-called “mathematics crisis of foundations” emerged, as logic could no longer provide a solid foundation.

Russell discovered in 1908 that paradoxes arose from a vicious circle in which a whole was recognized as a member of itself. To avoid this, “it was sufficient to adhere to the principle of the vicious circle, which states that no totality can be members definable in terms of that whole.” This necessitated the determination of the elements of a unity by themselves, prioritizing this unity.

Rusell’s theory of types is aimed at solving this problem. Accordingly, it can be explained by the hierarchy of mutually exclusive classes (class of individuals, class of classes of individuals, class of classes of individuals, etc.), predicates (predicates of individuals, predicates of individuals, predicates of individuals, etc.) and types of propositions.

The Rissell paradox (the paradox of the set of all sets) vanishes, as the question of whether a class is a member of itself loses all meaning. The same is true for the predicate paradox, since no predicate can be predicated on its own. As for the liar paradox, it is likewise resolved, since the proposition “I am lying” cannot be applied to itself.

However, a new difficulty arises at this stage: meaningfulness; Every formula is defined by a type that creates its own significance. For example, the function “x is long L(x}” has meaning only in terms of individual values, not function values. Significant regular formulas le-L(x) provide a philosophically fruitful distinction between meaningless formulas -L (later, Rudolf Carnap (will be discussed by) and different language levels, which will be rejected by Wittgenstein.