Vuzshura References  Boolos, G. Academic Tools How to cite this entry. If we assemble these truths into a conjunction and apply existential generalization in the appropriate places, the result is the definiens of the definition of predecessor instantiated to the numbers 1 and 2. Frege realized that though we may identify this sequence of numbers with the natural numbers, such a sequence is simply a list: His contributions to the philosophy of language include:.
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Basic Law V also correctly implies the Principle of Extensionality. This principle asserts that if two extensions have the same members, they are identical. Frege quickly added an Appendix to the second volume, describing two distinct ways of deriving a contradiction from Basic Law V. He also suggested a way of repairing Law V, but Quine later showed that such a repair was disastrous, since it would force the domain of objects to contain at most one object.
In the next subsections, we describe the two ways of deriving a contradiction from Basic Law V that Frege described in the Appendix to Gg. The first establishes the contradiction directly, without any special definitions. We can represent his reasoning as follows. The contradiction now goes as follows. Philosophers have diagnosed the inconsistency in various ways, and it is safe to say that the matter is still somewhat controversial. In this subsection, we discuss only the basic elements of the problem.
With this extensional view of concepts in mind, we can see how a paradox is engendered. Each direction of Basic Law V requires that this correlation have certain properties. We shall see, for example, that the right-to-left direction of Basic Law V i. This means that the correlation between concepts and extensions that Basic Law V sets up must be a function — no concept gets correlated with two distinct extensions though for all Va tells us, distinct concepts might get correlated with the same extension.
However, the left-to-right direction of Basic Law V i. So, the correlation that Basic Law V sets up between concepts and extensions will have to be one-to-one; i. Since every concept is correlated with some extension, there have to be at least as many extensions as there are concepts.
But the problem is that second-order logic with Basic Law V as a whole requires that there be more concepts than extensions. 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. The Comprehension Principle for Concepts asserts the existence of a concept for every condition on objects expressible in the language.
Now although it may seem that this principle, in and of itself, forces the domain of concepts to be larger than the domain of objects, it is a model-theoretic fact that there are models of second-order logic with the Comprehension Principle for Concepts but without Basic Law V in which the domain of concepts is not larger than the domain of objects.
However, as we saw in the last paragraph, Vb requires that there be at least as many extensions as there are concepts. Thus, the addition of Basic Law V to second-order logic implies an impossible situation in which the domain of concepts has to be strictly larger than the domain of extensions while at the same time the domain of extensions has to be as large as the domain of concepts. Recently, there has been a lot of interest in discovering ways of repairing the Fregean theory of extensions.
The traditional view is that one must either restrict Basic Law V or restrict the Comprehension Principle for Concepts. Recently, Boolos , developed one of the more interesting suggestions for revising Basic Law V without abandoning second-order logic and its comprehension principle for concepts.
On the other hand, there have been many suggestions for restricting the Comprehension Principle for Concepts. The most severe of these is to abandon second-order logic and the Comprehension Principle for Concepts altogether. Parsons and Burgess They investigate systems of second-order logic which have been extended by Basic Law V but in which the Comprehension Principle for Concepts is restricted in some way.
We will not discuss the above research further in the present entry, for none of these alternatives have achieved a clear consensus. As noted in the Introduction, Frege validly proved a rather deep fact about the natural numbers notwithstanding the inconsistency of Basic Law V. But this fact went unnoticed for many years. Though Geach claimed such a derivation was possible, C. We will do this in two stages. To explain this idea, Frege noted that one and the same external phenomenon can be counted in different ways; for example, a certain external phenomenon could be counted as 1 army, 5 divisions, 25 regiments, companies, platoons, or 24, people.
Each way of counting the external phenomenon corresponds to a manner of its conception. Frege offers both an implicit i. However, before discussing this principle, the reader should convince him- or herself of the following four facts: 1 that the material equivalence of two concepts implies their equinumerosity, 2 that equinumerosity is reflexive, 3 that equinumerosity is symmetric, and 4 that equinumerosity is transitive. In formal terms, the following facts are provable: Facts About Equinumerosity: 1.
In some cases, it is easy to identify the relation in question. We will examine these derivations in the following sections. Both are biconditionals asserting the equivalence of an identity among singular terms the left-side condition with an equivalence relation on concepts the right-side condition. Indeed, both correlate concepts with certain objects. See the independent work of Geach , —7 , Hodes , , Burgess and Hazen They are included here for those who wish to have a more complete understanding of what Frege in fact attempted to do.
Frege in fact identifies the cardinal number 2 with this extension, for it contains all and only those concepts under which two objects fall. Similarly, Frege identifies the cardinal number 0 with the extension consisting of all those first-level concepts under which no object falls; this extension would include such concepts as unicorn, centaur, prime number between 3 and 5, etc. We know that Basic Law V does not offer such a coherent conception.
In other words, the proof relies on a kind of higher-order version of the Law of Extensions described above , the ordinary version of which we know to be a consequence of Basic Law V. In Gg, extensions do not contain concepts as members but rather objects.
So Frege had to find another way to express the explicit definition described in the previous subsection. His technique was to let extensions go proxy for their corresponding concepts. Instead, he derives both directions separately without combining them or indicating that the two directions should be conceived as a biconditional.
In the next section, we go through the proof. Frege realized that though we may identify this sequence of numbers with the natural numbers, such a sequence is simply a list: it does not constitute a definition of a concept e.
No two natural numbers have the same successor. Principle of Mathematical Induction Every natural number has a successor. Moreover, Frege recognized the need to employ the Principle of Mathematical Induction in the proof that every number has a successor.
One cannot prove the claim that every number has a successor simply by producing the sequence of expressions for cardinal numbers e. All such a sequence demonstrates is that for every expression listed in the sequence, one can define an expression of the appropriate form to follow it in the sequence. This is not the same as proving that every natural number has a successor. In what follows, we sometimes introduce other such abbreviations.
Grundgesetze der Arithmetik, 2 Bde.
Frege’s Theorem and Foundations for Arithmetic