This is not the same as proving that every natural number has a successor. 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 next defines the relation x is an ancestor of y in the R-series.
In what follows, we sometimes introduce other such abbreviations. Moreover, precedes is a witness to Fact 2: Frege identified the number 0 as the number of the concept being non-self-identical. Since the logic of identity guarantees that no object is non-self-identical, nothing falls under the concept being non-self-identical. It is straightforward to prove the following Lemma Concerning Zero from this definition of Indeed, the natural numbers are precisely the finite cardinals.
Some of the steps in this proof can be found in Gl. See the Appendix to Boolos for a reconstruction. The statement that zero is a natural number is an immediate consequence of the definition of natural number:. It seems that Frege never actually identified this fact explicitly in Gl or labeled this fact as a numbered Theorem in Gg I. This can be represented formally as follows:.
The fact that no two natural numbers have the same successor is somewhat more difficult to prove cf. 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. The picture is something like this:. 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 functional relation:. Predecessor is a Functional Relation: Then we may state the Principle of Mathematical Induction as follows: Principle of Mathematical Induction: We will call the latter the General Principle of Induction.
Here is a definition:. Now given this definition, we can reformulate the General Principle of Induction more strictly as:.
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General Principle of Induction: We may sketch the proof strategy as follows. Recall that Fact 7 is:. Thus, if the antecedent can be established, the proof is done. 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:. Proof of the General Principle of Induction. Frege uses the Principle of Mathematical Induction to prove that every natural number has a successor that's a natural number. We may formulate the theorem as follows:. Frege then considers the concept member of the predecessor-series ending with n , i.
That is, Frege proves that every natural number has a successor by proving the following Lemma on Successorsby induction:. 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. 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.
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 Mathematical Induction; i. This is the concept: The result is therefore something that has been proved and that we therefore know to be true:. Gg I , Theorem Given this proof of the Lemma on Successors, Theorem 5 is not far away.
We still have to show that such successor cardinals are natural numbers. Successors of Natural Numbers are Natural Numbers: Theorem 5 now follows from the Lemma on Successors and the fact that successors of natural numbers are natural numbers. Before we turn to the last section of this entry, it is worth mentioning the mathematical significance of this theorem.
It is an immediate consequence Theorem 5 and the fact that Predecessor is a functional relation that every number has a unique successor. These definitions constitute the foundations of arithmetic. We discuss the thinking behind this attitude, and other things, in what follows. 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.
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. 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. Given the above discussion, it should be clear that Frege at some point in Gg endorsed existence claims, either directly in his formalism or in his metalanguage, 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 derivable from his Rule of Substitution and Basic Law V, respectively. In light of these existence claims, a Kantian might well suggest not only that explicit existence claims are synthetic rather than analytic i.
If the Kantian is right, then some other faculty such as intuition might still be needed to account for our knowledge of the existence claims of arithmetic. 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. The question of how we obtain knowledge of such principles is still an open question in philosophy. In contemporary philosophy, this question is still poignant, since many philosophers do accept that properties and relations of various sorts exist.
Moreover, he thought that an appeal to extensions would answer one of the questions that motivated his work:. Now it is unclear why Frege thought that he could answer the question posed here by saying that we apprehend numbers as the extensions of concepts. Some philosophers do argue that certain consistent principles having the same logical form as Basic Law V are analytic, and that such principles justify reference to the entities described in the left-side condition by grounding such reference in the truth of the right-side condition.
To hold that Basic Law V is analytic, it seems that one must hold that the right-side condition implies the corresponding left-side condition as a matter of meaning. 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 conclusion can be questioned: In other words, the suggestion that Va i. The moral to be drawn here is that, even if Basic Law V were consistent, it is not exactly clear how its right side analytically implies the existence of extensions. In the end, we may need some other way of justifying our knowledge of principles like Basic Law V, that imply the existence of abstract objects — the justification discussed so far seems to contain a gap.
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 at some point he has to address the question of how we can know that numbers exist. A reason must be given as to why the claim:. Concerning this definition, Frege says:. In Gl , Frege solves the problem by giving his explicit definition of numbers in terms of extensions. 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 apply his system by extending it 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. The second question arises because the Julius Caesar problem applies to Hume's Principle; without a solution to that problem, Frege can't claim to have precisely specified which objects the numbers are, so as to delineate them within the domain of all logical and non-logical objects? Marco Ruffino - - Grazer Philosophische Studien Danielle Macbeth - - Harvard University Press.
Heck - - Oxford University Press. Mark Textor - - Routledge. Logic and the Inexpressible in Frege and Heidegger. Edward Witherspoon - - Journal of the History of Philosophy 40 1: Added to PP index Total downloads 4, of 2,, Recent downloads 6 months 12 40, of 2,, How can I increase my downloads? Sign in to use this feature. This article has no associated abstract. No keywords specified fix it. Gottlob Frege in 20th Century Philosophy categorize this paper. History of Western Philosophy. However, because the senses of these expressions are different--in 1 the object is presented the same way twice, and in 2 it is presented in two different ways--it is informative to learn of 2.
While the truth of an identity statement involves only the references of the component expressions, the informativity of such statements involves additionally the way in which those references are determined, i. So far we have only considered the distinction as it applies to expressions that name some object including abstract objects, such as numbers. For Frege, the distinction applies also to other sorts of expressions and even whole sentences or propositions. At some points, Frege even suggests that the sense of a whole proposition is composed of the senses of the component expressions.
In the example considered in the previous paragraph, it was seen that the truth-value of the identity claim depends on the references of the component expressions, while the informativity of what was understood by the identity claim depends on the senses. For this and other reasons, Frege concluded that the reference of an entire proposition is its truth-value, either the True or the False.
The sense of a complete proposition is what it is we understand when we understand a proposition, which Frege calls "a thought" Gedanke. Just as the sense of a name of an object determines how that object is presented, the sense of a proposition determines a method of determination for a truth-value. However, in other works, Frege makes it quite clear that the distinction can also be applied to "incomplete expressions", which include functional expressions and grammatical predicates.
These expressions are incomplete in the sense that they contain an "empty space", which, when filled, yields either a complex name referring to an object, or a complete proposition. Thus, the incomplete expression "the square root of " contains a blank spot, which, when completed by an expression referring to a number, yields a complex expression also referring to a number, e. The incomplete expression, " is a planet" contains an empty place, which, when filled with a name, yields a complete proposition.
According to Frege, the references of these incomplete expressions are not objects but functions. The reference of the expression "square root of " is thus a function, which takes numbers as arguments and yields numbers as values. The situation may appear somewhat different in the case of grammatical predicates. However, because Frege holds that complete propositions, like names, have objects as their references, and in particular, the truth-values the True or the False, he is able to treat predicates also as having functions as their references.
In particular, they are functions mapping objects onto truth-values.
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The expression, " is a planet" has as its reference a function that yields as value the True when saturated by an object such as Saturn or Venus, but the False when saturated by a person or the number three. It is clear that functions are to be understood as the references of incomplete expressions, but what of the senses of such expressions? Here, Frege tells us relatively little save that they exist.
There is some amount of controversy among interpreters of Frege as to how they should be understood. It suffices here to note that just as the same object e. While "identity", as Frege uses the term, is a relation holding only between objects, Frege believes that there is a relation similar to identity that holds between functions just in case they always share the same value for every argument. Since all and only those things that have hearts have kidneys, strictly speaking, the concepts denoted by the expressions " has a heart", and " has a kidney" are one and the same.
Clearly, however, these expressions do not present that concept in the same way. For Frege, these expressions would have different senses but the same reference. Frege also tells us that it is the incomplete nature of these senses that provides the "glue" holding together the thoughts of which they form a part. Frege also uses the distinction to solve what appears to be a difficulty with Leibniz's law with regard to identity. This law was stated by Leibniz as, "those things are the same of which one can be substituted for another without loss of truth," a sentiment with which Frege was in full agreement.
As Frege understands this, it means that if two expressions have the same reference, they should be able to replace each other within any proposition without changing the truth-value of that proposition. Normally, this poses no problem. However, there seem to be some serious counterexamples to this principle. We know for example that "the morning star" and "the evening star" have the same customary reference.
However, it is not always true that they can replace one another without changing the truth of a sentence. For example, if we consider the propositions:. If we assume that Gottlob does not know that the morning star is the same heavenly body as the evening star, 5 may be true while 6 false or vice versa.
Frege meets this challenge to Leibniz's law by making a distinction between what he calls the primary and secondary references of expressions. Frege suggests that when expressions appear in certain unusual contexts, they have as their references what is customarily their senses. In such cases, the expressions are said to have their secondary references. Typically, such cases involve what Frege calls "indirect speech" or " oratio obliqua ", as in the case of statements of beliefs, thoughts, desires and other so-called "propositional attitudes", such as the examples of 5 and 6.
However, expressions also have their secondary references for reasons which should already be apparent in contexts such as "it is informative that Let us consider the examples of 5 and 6 more closely. To Frege's mind, these statements do not deal directly with the morning star and the evening star itself. Rather, they involve a relation between a believer and a thought believed.
Thoughts, as we have seen, are the senses of complete propositions. Beliefs depend for their make-up on how certain objects and concepts are presented, not only on the objects and concepts themselves. The truth of belief claims, therefore, will depend not on the customary references of the component expressions of the stated belief, but their senses. Since the truth-value of the whole belief claim is the reference of that belief claim, and the reference of any proposition, for Frege, depends on the references of its component expressions, we are lead to the conclusion that the typical senses of expressions that appear in oratio obliqua are in fact the references of those expressions when they appear in that context.
Such contexts can be referred to as "oblique contexts", contexts in which the reference of an expression is shifted from its customary reference to its customary sense. In this way, Frege is able to actually retain his commitment in Leibniz's law. The expressions "the morning star" and "the evening star" have the same primary reference, and in any non-oblique context, they can replace each other without changing the truth-value of the proposition. However, since the senses of these expressions are not the same, they cannot replace each other in oblique contexts, because in such contexts, their references are non-identical.
Frege ascribes to senses and thoughts objective existence. In his mind, they are objects every bit as real as tables and chairs. Their existence is not dependent on language or the mind. Instead, they are said to exist in a timeless "third realm" of sense, existing apart from both the mental and the physical. Frege concludes this because, although senses are obviously not physical entities, their existence likewise does not depend on any one person's psychology. A thought, for example, has a truth-value regardless of whether or not anyone believes it and even whether or not anyone has grasped it at all.
Moreover, senses are interpersonal. Different people are able to grasp the same senses and same thoughts and communicate them, and it is even possible for expressions in different languages to express the same sense or thought. Frege concludes that they are abstract objects, incapable of full causal interaction with the physical world. They are actual only in the very limited sense that they can have an effect on those who grasp them, but are themselves incapable of being changed or acted upon. They are neither created by our uses of language or acts of thinking, nor destroyed by their cessation.
Unfortunately, Frege does not tell us very much about exactly how these abstract objects pick out or present their references. Exactly what is it that makes a sense a "way of determining" or "mode of presenting" a reference? In the wake of Russell's theory of descriptions, a Fregean sense is often interpreted as a set of descriptive information or criteria that picks out its reference in virtue of the reference alone satisfying or fitting that descriptive information.
In giving examples, Frege implies that a person might attach to the name "Aristotle" the sense the pupil of Plato and teacher of Alexander the Great. This sense picks out Aristotle the person because he alone matches this description. Here, care must be taken to avoid misunderstanding. The sense of the name "Aristotle" is not the words "the pupil of Plato and teacher of Alexander the Great"; to repeat, senses are not linguistic items. It is rather that the sense consists in some set of descriptive information, and this information is best described by a descriptive phrase of this form.
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The property of being the pupil of Plato and teacher of Alexander is unique to Aristotle, and thus, it may be in virtue of associating this information with the name "Aristotle" that this name may be used to refer to Aristotle. As certain commentators have noted, it is not even necessary that the sense of the name be expressible by some descriptive phrase , because the descriptive information or properties in virtue of which the reference is determined may not be directly nameable in any natural language.
From this standpoint, it is easy to understand how there might be senses that do not pick out any reference. Names such as "Romulus" or "Odysseus", and phrases such as "the least rapidly converging series" or "the present King of France" express senses, insofar as they lay out criteria that things would have to satisfy if they were to be the references of these expressions. However, there are no things which do in fact satisfy these criteria. Therefore, these expressions are meaningful, but do not have references.
Because the sense of a whole proposition is determined by the senses of the parts, and the reference of a whole proposition is determined by the parts, Frege claims that propositions in which such expressions appear are able to express thoughts, but are neither true nor false, because no references are determined for them.
This interpretation of the nature of senses makes Frege a forerunner to what has since been come to be known as the "descriptivist" theory of meaning and reference in the philosophy of language. The view that the sense of a proper name such as "Aristotle" could be descriptive information as simple as the pupil of Plato and teacher of Alexander the Great , however, has been harshly criticized by many philosophers, and perhaps most notably by Saul Kripke.
Kripke points out that this would make a claim such as "Aristotle taught Alexander" seem to be a necessary and analytic truth, which it does not appear to be. Moreover, he claims that many of us seem to be able to use a name to refer to an individual even if we are unaware of any properties uniquely held by that individual. For example, many of us don't know enough about the physicist Richard Feynman to be able to identify a property differentiating him from other prominent physicists such as Murray Gell-Mann, but we still seem to be able to refer to Feynman with the name "Feynman".
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John Searle, Michael Dummett and others, however, have proposed ways of expanding or altering Frege's notion of a sense to circumvent Kripke's worries. This has lead to a very important debate in the philosophy of language, which, unfortunately, we cannot fully discuss here.
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Gottlob Frege — Gottlob Frege was a German logician, mathematician and philosopher who played a crucial role in the emergence of modern logic and analytic philosophy. Contributions to Logic Trained as a mathematician, Frege's interests in logic grew out of his interests in the foundations of arithmetic. Contributions to the Philosophy of Mathematics Frege was an ardent proponent of logicism, the view that the truths of arithmetic are logical truths.
However, we must bear in mind that the propositions: If we consider the two claims: For example, if we consider the propositions: References and Further Reading a. Translated as "Reply to Thomae's Holiday Causerie. Translated as "On Concept and Object. Edited by Michael Beaney. Edited by Peter Geach and Max Black. Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens.
Harvard University Press, Also as Conceptual Notation and Related Articles.
Edited and translated by Terrell W. Oxford University Press, Translated as "On Mr. Peano's Conceptual Notation and My Own. Translated as "On Formal Theories of Arithmetic. Translated as "Function and Concept. Translated as "Compound Thoughts. Translated in part as The Basic Laws of Arithmetic: Exposition of the System. Edited and translated by Montgomery Furth. University of California Press, Translated as "On the Foundations of Geometry. Translated by Eike-Henner W.
Yale University Press, Translated as The Foundations of Arithmetic: Translated as Posthumous Writings.
- Gottlob Frege (Stanford Encyclopedia of Philosophy)!
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Translated by Peter Long and Roger White. University of Chicago Press, Translated as "Whole Numbers. University of Jena, Translated in CP Review of Philosophie der Arithmetik , by Edmund Husserl. Translated as "On Sense and Meaning. As "On Sinn and Bedeutung. And as "On Sense and Reference. Translated as "On the Law of Inertia. Translated as "Renewed Proof of the Impossibility of Mr. Translated as "What is a Function? Translated as Philosophical and Mathematical Correspondence. Translated by Hans Kaal. Important Secondary Works Angelelli, Ignacio. Studies on Gottlob Frege and Traditional Philosophy.
Frege's Theory of Judgment. Frege's Theory of Sense and Reference. Cambridge University Press, Essays in Honor of Henry M. Sheffer, edited by P. Liberal Arts Press, An Introduction to His Philosophy. Barnes and Noble, Frege and Other Philosophers. The Interpretation of Frege's Philosophy. Reflections on Frege's Philosophy. Northwestern University Press, Haaparanta, Leila and Jaakko Hintikka, eds.
University of Illinois Press, The Metaphysics of Gottlob Frege.