# Logic and Arithmetic: Volume 1. Natural Numbers

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• Frege's Logic, Theorem, and Foundations for Arithmetic;
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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. Thus, if given certain facts about the number of objects falling under the certain concepts, the definition of predecessor correctly predicts that Precedes 1,2. Frege next defines the relational concept x is an ancestor of y in the R-series. The intuitive idea is easily grasped if we consider the relation x is the father of y.

Suppose that a is the father of b , that b is the father of c , and that c is the father of d. In what follows, we sometimes introduce other such abbreviations. In formal terms:. However, Clinton's brother is not one of Chelsea's forefathers, since he fails to be her father, her grandfather, or any of the other links in the chain of fathers from which Chelsea descended. The reader should consider what happens when R is taken to be the relation precedes. The general definition of the weak ancestral of R yields the following facts, many of which correspond to theorems in Gg : [ 10 ].

Frege's definition of natural number requires one more preliminary definition. It may be recalled that Frege identified the number 0 as the cardinal number of the concept being non-self-identical. That is:. Since the logic of identity guarantees that no object fails to be self-identical, nothing falls under the concept being non-self-identical.

Had one of Frege's explicit definitions of the cardinal numbers worked as he had intended, the number 0 would, in effect, be identified with the extension of all extensions of concepts under which nothing falls. It is straightforward to prove the following Lemma Concerning Zero from this definition of Frege's definition of the concept natural number can now be stated in terms of the weak-ancestral of Predecessor:.

Indeed, the natural numbers are precisely the finite cardinals. In formal terms, Frege's definition becomes:. In what follows, we shall sometimes use the variables m , n , and o to range over the natural numbers. In this section, we reconstruct the proof of this theorem which can be extracted from Frege's work using the definitions and theorems assembled so far.

Some of the steps in this proof can be found in Gl. See the Appendix to Boolos for a reconstruction. Our reconstruction follows Frege's Gg in spirit and in most details, but we have tried to simplify the presentation in several places. For a more strict description of Frege's Gg proof, the reader is referred to Heck The following should help prepare the reader for Heck's excellent essay.

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Hence, by the definition of number, 0 is a number. It seems that Frege never actually identified this fact explicitly in Gl or labeled this fact as a numbered Theorem in Gg I. It is possible that he thought it was too obvious to mention. It is also a simple consequence of the foregoing that 0 doesn't succeed any number. This can be represented formally as follows:. Proof : Assume, for reductio , that some object, say b , is such that Precedes b ,0. So nothing precedes 0. Since nothing precedes 0, no natural number precedes 0.

The fact that no two numbers have the same successor is somewhat more difficult to prove cf. We may formulate this theorem as follows, with m , n , and o as restricted variables ranging over the natural numbers:. In other words, this theorem asserts that predecessor is a one-to-one relation on the natural numbers.

To prove this theorem, it suffices to prove that predecessor is a one-to-one relation full stop. One can prove that predecessor is one-to-one from Hume's Principle, with the help of the following Equinumerosity Lemma, the proof of which is rather long and involved. The Equinumerosity Lemma asserts that when F and G are equinumerous, x falls under F , and y falls under G , then the concept object falling under F other than x is equinumerous to the concept object falling under G other than y.

The picture is something like this:. Then we have:. Now we can prove that Predecessor is a one-to-one relation from this Lemma and Hume's Principle cf. Proof : Assume that both a and b are precedessors of c. By the definition of predecessor, we know that there are concepts and objects P , Q , d , and e , such that:. So, if Predecessor is a one-to-one relation, it is a one-to-one relation on the natural numbers.

Therefore, no two numbers have the same successor. This completes the proof of Theorem 3. It is important to mention here that not only is Predecessor a one-to-one relation, it is also a function:. The fact that Predecessor is a function will play a part in the proof that every number has a successor.

Then we may state the Principle of Mathematical Induction as follows: if a 0 falls under F and b F is hereditary on the natural numbers, then every natural number falls under F. Frege actually proves the Principle of Mathematical Induction from a more general principle that governs any R -series whatsoever. We will call the latter the General Principle of Induction. It asserts that whenever a falls under F , and F is hereditary on the R -series beginning with a , then every member of that R -series falls under F.

Here is a definition:. In other words, F is hereditary on the members of the R -series beginning with a just in case every adjacent pair x and y in this series with x bearing R to y is such that y falls under F whenever x falls under F. Now given this definition, we can reformulate the General Principle of Induction more strictly as:.

Frege's proves this claim by making an insightful appeal to his Rule of Substitution. We may sketch the proof strategy as follows. Assume that the antecedent of the General Principle of Induction holds for an arbitrarily chosen concept, say P.

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That is, assume:. We then simply have to show Pb. Recall that Fact 7 is:. This is a theorem of logic containing the free variables x , y , and F. Frege instantiates x and y to a and b , respectively. This is where Frege used his Rule of Substitution. The concept being instantiated for F is the concept member of the R-series beginning with a and which falls under P. Thus, if the antecedent can be established, the proof is done.

## Mathematical Symbols

However, this claim can be established straightforwardly from things we know to be true and, in particular, from facts contained in the antecedent of the Principle we are trying to prove, which we assumed as part of our conditional proof. The reader is encouraged to complete the proof as an exercise. For those who would like to check their work, we give the complete Proof of the General Principle of Induction here.

Now to derive Principle of Mathematical Induction from the General Principle of Induction, we formulate the instance of the latter in which a is 0 and R is Precedes :. Frege uses the Principle of Mathematical Induction to prove that every number has a successor in the natural numbers. We may formulate the theorem as follows:. To understand Frege's strategy for proving this theorem, recall that the weak ancestral of the Predecessor relation, i. Frege then considers the concept member of the predecessor-series ending with n , i.

Frege then shows, by induction, that every natural number n precedes the number of the concept member of the predecessor-series ending with n. That is, Frege proves that every number has a successor by proving the following Lemma on Successors by induction:. This asserts that every number n precedes the number of numbers in the predecessor series ending with n. Frege can establish Theorem 5 by proving the Lemma on Successors and by showing that the successor of a natural number is itself a natural number.

Although we haven't yet defined the natural numbers following 0, the following intuitive sequence is driving Frege's strategy:. For example, the third member of this sequence is true because there are 3 natural numbers 0, 1, and 2 that are less than or equal to 2; so the number 2 precedes the number of numbers less than or equal to 2. Frege's strategy is to show that the general claim, that n precedes the number of numbers less than or equal to n , holds for every natural number.

So, given this intuitive understanding of the Lemma on Successors, Frege has a good strategy for proving that every number has a successor. Now to prove the Lemma on Successors by induction, we need to reconfigure this Lemma to a form which can be used as the consequent of the Principle of Induction; i. This is the concept: being an object y which precedes the number of the concept: member of the predecessor series ending in y.

The result is therefore something that we may take as having been proved:. Since the consequent is the Lemma on Successors, Frege can prove this Lemma by proving both that 0 falls under Q cf. Gg I , Theorem and that Q is hereditary on the natural numbers cf. Gg I , Theorem :. Proof that Q is hereditary on the natural numbers. Given this proof of the Lemma on Successors, Theorem 5 is not far away. The Lemma on Successors shows that every number precedes some cardinal number of the form F. We still have to show that such successor cardinals are natural numbers. That is, it still remains to be shown that if a number n precedes something y , then y is a natural number:.

Proof : Suppose that Precedes n , a. Theorem 5 now follows from the Lemma on Successors and the fact that successors of natural numbers are natural numbers. With the proof of Theorem 5, we have completed the proof of Frege's Theorem. Before we turn to the last section of this entry, it is worth mentioning the mathematical significance of this theorem. From Frege's Theorem, one can derive arithmetic.

It is an immediate consequence of the functionality of Predecessor that every number has a unique successor. That means we can define the successor function:. These definitions constitute the foundations of arithmetic.

## Should Children Learn Math by Starting With Counting?

Frege has insightfully isolated a group of basic laws in which they may be grounded. Readers interested in how these results are affected when Hume's Principle is combined with predicative second-order logic should consult Linnebo Frege's Theorem is an elegant derivation of the basic laws of arithmetic which can be carried out independently of the portion of Frege's system which led to inconsistency. We discuss the reasons for his attitude, among other things, in what follows. A discussion of the philosophical questions surrounding Frege's Theorem should begin with some statement of how Frege conceived of his own project when writing Begr , Gl , and Gg.

It seems clear that epistemological considerations in part motivated Frege's work on the foundations of mathematics. If Frege could show that the basic laws of number theory are derivable from analytic truths of logic, then he could argue that we need only appeal to the faculty of understanding as opposed to some faculty of intuition to explain our knowledge of the truths of arithmetic. Frege's goal then stands in contrast to the Kantian view of the exact mathematical sciences, according to which general principles of reasoning must be supplemented by a faculty of intuition if we are to achieve mathematical knowledge.

The Kantian model here is that of geometry; Kant thought that our intuitions of figures and constructions played an essential role in the demonstrations of geometrical theorems. In Frege's own time, the achievements of Frege's contemporaries Pasch, Pieri and Hilbert showed that such intuitions were not essential. Frege's strategy then was to show that no appeal to intuition is required for the derivation of the theorems of number theory.

This in turn required that he show that the latter are derivable using only rules of inference, axioms, and definitions that are purely analytic principles of logic. Here is what Frege says:. Since it is only in the context of a proposition that words have any meaning, our problem becomes this: To define the sense of a proposition in which a number word occurs. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. In the present book, this shall be confirmed, by the derivation of the simplest laws of Numbers by logical means alone.

By what means are we justified in recognizing numbers as objects? Even if this problem is not solved to the degree I thought it was when I wrote this volume, still I do not doubt that the way to the solution has been found. The basic problem for Frege's strategy, however, is that for his logicist project to succeed, his system must at some point include either as an axiom or theorem statements that explicitly assert the existence of certain kinds of abstract entities and it is not obvious how to justify the claim that we know such explicit existential statements. Given our description of his system, it should be clear that Frege's logical system includes existence claims for the following entities:.

Although Frege attempted to reduce the latter two kinds of entities truth-values and numbers to extensions, the fact is that the existence of concepts and extensions are implied by his Rule of Substitution and Basic Law V, respectively. Logic, it is often argued, should be free of such existence assumptions. A Kantian might well complain both that explicit existence claims seem to be synthetic rather than analytic i. If so, then some other faculty such as intuition might still be needed to account for our knowledge of the existence claims of arithmetic. Boolos was the first to note that the Rule of Substitution causes a problem of this kind for Frege's program, since it is equivalent to a quite liberal existential claim, namely, the Comprehension Principle for Concepts.

Boolos suggests a defense for Frege with respect to this particular aspect of his logic, namely, to reinterpret by paraphrasing the second-order quantifiers so as to avoid commitment to concepts. See Boolos for the details. Boolos's suggestion, however, is one which would require Frege to abandon his realist theory of concepts. Although Frege wouldn't quite put it this way, we have seen that his system treats open formulas with free object variables as if they denoted concepts. The use of such notation faces the same epistemological puzzles that Frege's Rule of Substitution faces.

The question of how we obtain knowledge of such principles is still an open question in philosophy. It is an important question to address, since Frege's most insightful definitions are cast using quantifiers ranging over concepts and relations e. In contemporary philosophy, this question is still poignant, since many philosophers do accept that properties and relations of various sorts exist. These entities are the contemporary analogues of Frege's concepts.

The question for Frege's project, then, is why should we accept as a law of logic a statement that implies the existence of individuals and a a correlation of this kind? Frege recognized that Basic Law V could be doubted but held it was a law of logic:. Moreover, he thought that an appeal to extensions would answer one of the questions that motivated his work:.

His idea here seems to be that since Basic Law V is supposed to be purely analytic or true in virtue of the meanings of its terms, we apprehend a pair of extensions whenever we truly judge that concepts F and G are materially equivalent. Some philosophers argue that Frege would have been correct to argue in just this way had Basic Law V been consistent. They argue that Basic Law V or consistent principles having the same logical form justifies reference to the entities described in the left-side condition by grounding such reference in the truth of the right-side condition.

But this, of course, raises an obvious problem. To justify reference to extensions, we must first justify the claim that those extensions exist. It is not clear that the claim that concepts are materially equivalent can justify such an existence claim. But given Frege's view that Basic Law V is analytic, it seems that he must hold that the right-side condition implies the corresponding left-side condition as a matter of meaning. This view, however, runs up against the following argument.

Suppose the right hand condition implies the left-side condition as a matter of meaning. That is, suppose that R implies L as a matter of meaning:. Now note that L itself can be analyzed, from a logical point of view. But if R implies L as a matter of meaning, and L implies D as a matter of meaning, then R implies D as a matter of meaning. This seems doubtful.

The material equivalence of F and G does not imply the existence claim D as a matter of meaning, whatever notion of meaning is involved. Below, it will be adapted to show that the right-to-left direction of Hume's Principle is not analytic. See Boolos , — , for reasons why Vb and the left-to-right direction of Hume's Principle are not analytic. The moral to be drawn here is that the modern Fregean must attempt to explain our knowledge of existence claims for abstract objects such as extensions head on , and not try to justify them indirectly, by attempting to justify claims that imply such existence claims.

We might agree that there must be logical objects of some sort if logic is to have a subject matter, but if Frege is to achieve his goal of showing that our knowledge of arithmetic is free of intuition, then the logical knowledge with which he identifies arithmetical knowledge must be either be purely analytic or shown otherwise to be free of intuition. We'll return to this theme in the final subsection. Given that the proof of Frege's Theorem makes no appeal to Basic Law V, some philosophers have argued Frege's best strategy for achieving his goal is to replace Basic Law V with Hume's Principle and argue that Hume's Principle is an analytic principle of logic.

The claim that Hume's Principle is an analytic principle of logic is subject to the same problem just posed for Basic Law V. The equinumerosity of F and G does not, as a matter of meaning, imply identity claims that entail the existence of numbers. Concerning this definition, Frege says:. Now trouble for Hume's Principle begins to arise when we recognize that it is a contextual definition that has the same logical form as this definition for directions.

It is central to Frege's view that the numbers are objects , and so he believes that it is incumbent upon him to say which objects they are. In Gl , Frege solves the problem by giving his explicit definition of numbers in terms of extensions. Unfortunately, this is only a stopgap measure, for when Frege later systematizes extensions in Gg , Basic Law V has the same logical form as Hume's Principle and the above contextual definition of directions.

Even if Frege somehow could have successfully restricted the quantifiers of Gg to avoid the Julius Caesar problem, he would no longer have been able to extend his system to include names of ordinary non-logical objects. That means his logical system could not be used for the analysis of ordinary language. But it was just the analysis of ordinary language that led Frege to his insight that a statement of number is an assertion about a concept.

Even when we replace the inconsistent Basic Law V with the powerful Hume's Principle, Frege's work still leaves two questions unanswered: 1 How do we know that numbers exist? The first question arises because Hume's Principle doesn't seem to be a purely analytic truth of logic; if neither Hume's Principle nor the existential claim that numbers exist is analytically true, by what faculty do we come to know the truth of the existential claim?

The second question arises because Frege's work offers no general condition under which we can identify an arbitrarily chosen object x with a given number such as the number of planets; how then can Frege claim to have precisely specified which objects the numbers are within the domain of all logical and non-logical objects?

So questions about the very existence and identity of numbers still plague Frege's work. These two questions arise because of a limitation in the logical form of these Fregean biconditional principles such as Hume's Principle and Basic Law V. These contextual definitions combine two jobs which modern logicians now typically accomplish with separate principles.

The latter should specify identity conditions for logical objects in terms of their most salient characteristic, one which distinguishes them from other objects. Such an identity principle would then be more specific than the global identity principle for all objects Leibniz's Law which asserts that if objects x and y fall under the same concepts, they are identical.

By way of example, consider modern set theory. Zermelo set theory Z has a distinctive comprehension principle for sets:. Note that the second principle offers identity conditions in terms of the most salient features of sets, namely, the fact that they, unlike other objects, have members. The identity conditions for objects which aren't sets, then, can be the standard principle that identifies objects whenever they fall under the same concepts. This leads us naturally to a very general principle of identity for any objects whatever:.

Now, if something is given to us as a set and we ask whether it is identical with an arbitrarily chosen object x , this specifies a clear condition that settles the matter. The only questions that remain for the theory Z concern its existence principle: Do we know that the comprehension principle is true, and if so, how? The question of existence is thus laid bare.

We do not approach it by attempting to justify a principle that implies the existence of sets via definite descriptions which we don't yet know to be well-defined. In his classic essays and , Boolos appears to recommend this very procedure of using separate existence and identity principles.

Though Boolos doesn't explicitly formulate an identity principle to complement Numbers, it seems clear that the following principle would offer identity conditions in terms of the most distinctive feature of numbers:. It is then straightforward to formulate a general principle of identity, as we did in the case of the set theory Z:. It openly faces the epistemological questions head-on: Do we know that Numbers is true, and if so, how?

This is where philosophers need to concentrate their energies. By replacing Fregean biconditionals such as Hume's Principle with separate existence and identity principles, we reduce two problems to one and and isolate the real problem for Fregean foundations of arithmetic, namely, the problem of giving an epistemological justification of distinctive existence claims e. For anything like Frege's program to succeed, it must at some point assert as an axiom or theorem the existence of logical objects of some kind.

Those separate existence claims should be the focus of attention, for they are the point at which logic and metaphysics dovetail. The theory of logical objects, if carried out without any mathematical primitives, might in fact be best understood as a theory where logic and metaphysics dovetail.

A proper epistemology for such a theory should offer some epistemological justification of the separate existence claims that are theorems of that theory. That is a moral that could be drawn from Frege's work. I was motivated to write the present entry after reading an early draft of an essay by William Demopoulos.

The draft was eventually published as Demopoulos and Clark Demopoulos kindly allowed me to quote certain passages from that early draft in the footnotes to the present entry. I am also indebted to Roberto Torretti, who carefully read this piece and identified numerous infelicities; to Franz Fritsche, who noticed a quantifier transposition error in Fact 2 about the strong ancestral, to Seyed N.

Finally, I am indebted to Jerzy Hanusek for pointing out that Existence of Extensions principle can be derived more simply in Frege's system directly from the classical logic of identity.

Logic and Arithmetic: Volume 1. Natural Numbers
Logic and Arithmetic: Volume 1. Natural Numbers
Logic and Arithmetic: Volume 1. Natural Numbers
Logic and Arithmetic: Volume 1. Natural Numbers
Logic and Arithmetic: Volume 1. Natural Numbers
Logic and Arithmetic: Volume 1. Natural Numbers

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