Friedrich Ludwig Gottlob Frege (b. 1848, d. 1925) was a Germanmathematician, logician, and philosopher who worked at the Universityof Jena. Frege essentially reconceived the discipline of logic byconstructing a formal system which, in effect, constituted the first‘predicate calculus’. In this formal system, Fregedeveloped an analysis of quantified statements and formalized thenotion of a ‘proof’ in terms that are still acceptedtoday. Frege then demonstrated that one could use his system toresolve theoretical mathematical statements in terms of simplerlogical and mathematical notions. One of the axioms that Frege lateradded to his system, in the attempt to derive significant parts ofmathematics from logic, proved to be inconsistent. Nevertheless, hisdefinitions (e.g., of thepredecessor relation and of theconcept ofnatural number) and methods (e.g., for derivingthe axioms of number theory) constituted a significant advance. Toground his views about the relationship of logic and mathematics,Frege conceived a comprehensive philosophy of language that manyphilosophers still find insightful, though recent scholarship suggeststhat Frege borrowed a significant number of elements in his philosophyof language from the Stoics. Moreover, his lifelong project, ofshowing that mathematics was reducible to logic, was notsuccessful.

• 1. Frege’s Life and Influences
• 2. Frege’s Logic and Philosophy of Mathematics
  • 2.1 The Basis of Frege’s Term Logic and Predicate Calculus
  • 2.2 Complex Statements and Generality
  • 2.3 Proof and Definition
  • 2.4 Courses-of-Values, Extensions, and Proposed Mathematical Foundations
  • 2.5 The Analysis of Statements of Number
  • 2.6 Natural Numbers
  • 2.7 Frege’s Conception of Logic
• 3. Frege’s Philosophy of Language
  • 3.1 Frege’s Puzzles
  • 3.2 Frege’s Theory of Sense and Denotation
  • 3.3 Frege’s Philosophy of Language in Context
• Bibliography
  • A. Primary Sources
  • B. Secondary Sources
• Academic Tools
• Other Internet Resources
• Related Entries

1. Frege’s Life and Influences

According to thecurriculum vitae that the 26-year old Fregefiled in 1874 with hisHabilitationsschrift, he was born onNovember 8, 1848 in Wismar, a town then in Mecklenburg-Schwerin butnow in Mecklenburg-Vorpommern. His father, Alexander, a headmaster ofa secondary school for girls, and his mother, Auguste (neeBialloblotzky), brought him up in the Lutheran faith. Frege attendedthe localGymnasium for 15 years, and after graduation in 1869,entered the University of Jena (see Frege 1874, translation inMcGuinness (ed.) 1984, 92).

At Jena, Frege attended lectures by Ernst Karl Abbe, whosubsequently became Frege’s mentor and who had a significantintellectual and personal influence on Frege’s life. Fregetransferred to the University of Göttingen in 1871, and two yearslater, in 1873, was awarded a Ph.D. in mathematics, having written adissertation under Ernst Schering titledÜber einegeometrische Darstellung der imaginären Gebilde in der Ebene(“On a Geometrical Representation of Imaginary Forms in thePlane”). Frege explains the project in his thesis as follows:“By a geometrical representation of imaginary forms in the planewe understand accordingly a kind of correlation in virtue of whichevery real or imaginary element of the plane has a real, intuitiveelement corresponding to it” (Frege 1873, translation inMcGuinness (ed.) 1984, 3). Here, by ‘imaginaryforms’, Frege is referring to imaginary points, imaginary curvesand lines, etc. Interestingly, one section of the thesis concerns therepresentation of complex numbers by magnitudes of angles in theplane.

In 1874, Frege completed hisHabilitationsschrift,entitledRechnungsmethoden,die sich auf eine Erweiterungdes Grössenbegriffes gründen (“Methods ofCalculation Based on an Extension of the Concept of Quantity”).Immediately after submitting this thesis, the good offices ofAbbe led Frege to become aPrivatdozent (Lecturer) atthe University of Jena. Library records from the University of Jenaestablish that, over the next 5 years, Frege checked out texts inmechanics, analysis, geometry, Abelian functions, and ellipticalfunctions (Kreiser 1984, 21).^[1] No doubt, many of these texts helped him toprepare the lectures he is listed as giving by the University of Jenacourse bulletin, for these lectures are on topics that often match thetexts, i.e., analytic geometry, elliptical and Abelian functions,algebraic analysis, functions of complex variables, etc. (Kratzsch1979).^[2]

This course of Frege’s reading and lectures during the period of1874–1879 dovetailed quite naturally with the interests hedisplayed in hisHabilitationsschrift. The ‘extensionof the concept of quantity’ referred to in the title concernsthe fact that our understanding of quantities (e.g., lengths,surfaces, etc.) has to be extended in the context of complexnumbers.^[3] Hesays, right at the beginning of this work:

According to the old conception, length appears as something materialwhich fills the straight line between its end points and at the sametime prevents another thing from penetrating into its space by itsrigidity. In adding quantities, we are therefore forced to place onequantity against another. Something similar holds for surfaces andsolid contents. The introduction of negative quantities made a dent inthis conception, and imaginary quantities made it completelyimpossible. Now all that matters is the point of origin and the endpoint – the idea of filling the space has been completelylost. All that has remained is certain general properties of addition,which now emerge as the essential characteristic marks of quantity.The concept has thus gradually freed itself from intuition and madeitself independent. This is quite unobjectionable, especially sinceits earlier intuitive character was at bottom mere appearance.Bounded straight lines and planes enclosed by curves can certainly beintuited, but what is quantitative about them, what is common tolengths and surfaces, escapes our intuition. … There isaccordingly a noteworthy difference between geometry and arithmetic inthe way in which their fundamental principles are grounded. Theelements of all geometrical constructions are intuitions, and geometryrefers to intuition as the source of its axioms. Since the object ofarithmetic does not have an intuitive character, its fundamentalpropositions cannot stem from intuition… (Frege 1874,translation in McGuinness (ed.) 1984, 56)

Here we can see the beginning of two lifelong interests of Frege,namely, (1) in how concepts and definitions developed for one domainfare when applied in a wider domain, and (2) in the contrast betweenlegitimate appeals to intuition in geometry and illegitimate appealsto intuition in the development of pure number theory. Indeed, somerecent scholars have (a) shown how Frege’s work in logic was informedin part by his understanding of the analogies and disanalogies betweengeometry and number theory (Wilson 1992), and (b) shown that Frege wasintimately familiar with the division among late 19th centurymathematicians doing complex analysis who split over whether it isbetter to use the analytic methods of Weierstrass or the intuitivegeometric methods of Riemann (Tappenden 2006). Weierstrass’s 1872paper, describing a real-valued function that is continuouseverywhere but differentiable nowhere,^[4] was well known and provided an example of an ungraphable functionsthat places limits on intuition. Yet, at the same time, Frege clearlyaccepted Riemann’s practice and methods derived from taking functionsas fundamental, as opposed to Weierstrass’s focus on functions thatcan be represented or analyzed in terms of other mathematical objects(e.g., complex power series).

In 1879, Frege published his first bookBegriffsschrift, eineder arithmetischen nachgebildete Formelsprache des reinen Denkens(Concept Notation: A formula language of pure thought, modelledupon that of arithmetic) and was promotedtoaußerordentlicher Professor (Extraordinarius Professor)at Jena. Although theBegriffsschrift constituted a majoradvance in logic, it was neither widely understood nor well-received.Some scholars have suggested that this was due to the facts that thenotation was 2-dimensional instead of linear and that he didn’t buildupon the work of others but rather presented something radically new(e.g., Mendelsohn 2005, 2). Though our discussion below is framedprimarily with an eye towards the system Frege developed in histwo-volume work of 1893/1903 (Grundgesetze der Arithmetik),the principal elements of this later system can be found already intheBegriffsschrift of 1879.

Frege’s next really significant work was his second book,Die Grundlagen der Arithmetik: eine logisch mathematischeUntersuchung über den Begriff der Zahl, published in1884. Frege begins this work with criticisms of previous attempts todefine the concept of number, and then offers his ownanalysis. TheGrundlagen contains a variety of insights stilldiscussed today, such as: (a) the claim that a statement of number(e.g., ‘there are eight planets’) is a higher-orderassertion about a concept (see Section 2.5 below); (b) his famousContext Principle (“never ask for the meaning of a word inisolation, but only in the context of a proposition”), and (c)the formulation of a principle (now called ‘Hume’sPrinciple’ in the secondary literature) that asserts theequivalence of the claim “the number ofFs is equal tothe number ofGs” with the claim that “there is aone-to-one correspondence between the objects falling underFand the objects falling underG” (see Section 2.5 below).More generally, Frege provides in theGrundlagen anon-technical philosophical justification and outline of the ideasthat he was to develop technically in his two-volumeworkGrundgesetze der Arithmetik (1893/1903).

In the years 1891–1892, Frege gave more thought to thephilosophy of language that would help ground his philosophy ofmathematics. He published three of his most well-known papers,‘Function and Concept’ (1891), ‘On Sense andReference’ (1892a), and ‘On Concept and Object’(1892b) in this period. These works were eventually supplemented bythree other papers on the philosophy of language, from a later period,‘The Thought’ (1918a), ‘Negation’ (1918b), and‘Compound Thoughts’ (1923). As we shall see in Section3.3 below, these works have to be considered in light of recentscholarship providing evidence that Frege heavily borrowed from Stoicphilosophy. Though Frege always motivated and recast (often inrigorous terms) ideas that he absorbed from others, the documentationconcerning the origins of many of his ideas in the Stoic corpus has tobe taken seriously.

In 1893, he published the first volume of the technical workpreviously mentioned,Grundgesetze der Arithmetik. In 1896,he was promoted toordentlicher Honorarprofessor (regularhonorary professor). Six years later (on June 16, 1902), as he waspreparing the proofs of the second volume oftheGrundgesetze, he received a letter from Bertrand Russell,informing him that one could derive a contradiction in the system hehad developed in the first volume. Russell’s letter frames theparadox first in terms of the predicateP = ‘being apredicate which cannot be predicated of itself’, and then interms of the class of all those classes that are not members ofthemselves. Frege, in the Appendix to the second volume, rephrasedthe paradox in terms of his own system.

Frege never fully recovered from the fatal flaw discovered in thefoundations of hisGrundgesetze. His attempts at salvagingthe work by restricting Basic Law V were not successful. However, hecontinued teaching at Jena, and from 1903–1917, he published sixpapers, including ‘What is a Function?’ (1904) and‘On the Foundations of Geometry’ (Frege 1903b and 1906).In the latter, Frege criticized Hilbert’s understanding and use of theaxiomatic method (see the entry on theFrege-Hilbert controversy). Fromthis time period, we have the lecture notes that Rudolf Carnap took asa student in two of his courses (see Reck and Awodey 2004). In 1917,he retired from the University of Jena.

In the last phase of his life, from 1917–1925, Fregepublished only the papers mentioned earlier (1918a, 1918b, 1923) anddeveloped some unpublished fragments of philosophical works.Unfortunately, his last years saw him become more than justpolitically conservative – his diary for a brief period in 1924show sympathies for fascism and anti-Semitism (see Frege 1924 [1996],translated by R. Mendelsohn). He died on July 26, 1925, in BadKleinen (now in Mecklenburg-Vorpommern).

2. Frege’s Logic and Philosophy of Mathematics

Frege provided a foundation for the modern discipline of logic bydeveloping a more perspicuous method of formally representing thelogic of thoughts and inferences. He did this by developing: (a) asystem allowing one to study inferences formally, (b) an analysis ofcomplex sentences and quantifier phrases that showed an underlyingunity to certain classes of inferences, (c) an analysisofproof anddefinition, (d) a theory of extensionswhich, though seriously flawed, offered an intriguing picture of thefoundations of mathematics, (e) an analysis of statements about number(i.e., of answers to the question ‘How many?’), (f)definitions and proofs of some of the basic axioms of number theoryfrom a limited set of logically primitive concepts and axioms, and (g)a conception of logic as a discipline which has some compellingfeatures. We discuss these developments in the followingsubsections. However, it should be noted that, for those who areencountering it for the first time, we have simplified thepresentation of Frege’s logic in a significant number of ways.For those already familiar with the notation and who want to see amore precise presentation, see the entry onFrege’s logic.

2.1 The Basis of Frege’s Term Logic and Predicate Calculus

In an attempt to realize Leibniz’s ideas for a universal formallanguage and a rational calculus, Frege developed a formal notationfor regimenting thought and reasoning. Though this notation was firstoutlined in hisBegriffsschrift (1879), the most maturestatement of Frege’s system was in his 2-volumeGrundgesetze derArithmetik (1893/1903). Frege’s two systems are bestcharacterized as term logics, since all of the complete expressionsare denoting terms. Frege analyzed ordinary predication in thesesystems, and so they can also be conceived as predicate calculi. Apredicate calculus is a formal system (a formal language and a methodof proof) in which one can represent valid inferences amongpredications, i.e., among statements in which properties arepredicated of objects.

In this subsection, we shall examine the most basic elements ofFrege’s 1893/1903 term logic and predicate calculus. These are thestatements involving function applications and the simple predicationswhich fall out as a special case.

2.1.1 The Basis of Frege’s Term Logic

In Frege’s term logic, all of the terms and well-formed formulasare denoting expressions. These include: (a) simple names of objects,like ‘\(2\)’ and ‘\(\pi\)’, (b) complex termswhich denote objects, like ‘\(2^2\)’ and ‘\(3 +1\)’, and (c) sentences (which are also complex terms). Thecomplex terms in (b) and (c) are formed with the help of‘incomplete expressions’ which signify functions, such asthe unary squaring function ‘\((\:)^2\)’ and the binaryaddition function ‘\((\:) + (\:)\)’. In these functionalexpressions, ‘\((\:)\)’ is used as a placeholder for whatFrege called thearguments of the function; the placeholderreveals that the expressions signifying function are, on Frege’sview, incomplete and stand in contrast to complete expressions such asthose in (a), (b), and (c). (Though Frege thought it inappropriate tocall the incomplete expressions that signify functions‘names’, we shall sometimes do so in what follows, thoughthe reader should be warned that Frege had reasons for not followingthis practice.) Thus, a mathematical expression such as‘\(2^2\)’ denotes the result of applying thefunction \((\:)^2\) to the number \(2\) as argument, namely, thenumber \(4\). Similarly, the expression ‘\(7 + 1\)’ denotes theresult of applying the binary function \(+((\:),(\:))\) to thenumbers \(7\) and \(1\) as arguments, in that order.

Even the sentences of Frege’s mature logical system are(complex) denoting terms; they are terms thatdenotetruth-values. Frege distinguished two truth-values,The True and The False, which he took to be objects. The basicsentences of Frege’s system are constructed using the expression‘\((\:) = (\:)\)’, which signifies a binary function thatmaps a pair of objectsx andy to The True ifx is identical toy andmapsx andy to The False otherwise. A sentence suchas ‘\(2^2 = 4\)’ therefore denotes the truth-value TheTrue, while the sentence ‘\(2^2 = 6\)’ denotes TheFalse.

An important class of these identity statements are statements of theform‘\(f(x) = y\)’,where \(f(\:)\) is anyunary function(i.e., function of a single variable), \(x\) is the argument of thefunction, and \(f(x)\) is the value of the function for the argument\(x\). Similarly, \(f(x,y) = z\) is an identity statement involving abinary function of two variables. And so on, for functions ofmore than two variables.

If we replace a complete name appearing in a sentence by aplaceholder, the result is an incomplete expression that signifies aspecial kind of function which Frege called aconcept.Concepts are functions which map every argument to one of thetruth-values. Thus, ‘\((\:) \gt 2\)’ denotes the conceptbeing greater than \(2\), which maps every object greaterthan \(2\) to The True and maps every other object to TheFalse. Similarly, ‘\((\:)^2 = 4\)’ denotes theconceptthat which when squared is identical to \(4\). Fregewould say that any object that a concept maps to The Truefallsunder the concept. Thus, the number \(2\) falls under theconceptthat which when squared is identical to \(4\). Inwhat follows, we use lower-case expressions like \(f(\:)\) to talkgenerally about functions, and upper-case expressions like \(F(\:)\)to talk more specifically about those functions which areconcepts.

Frege supposed that a mathematical claim such as ‘\(2\) isprime’ should be formally represented as‘\(P(2)\)’. The verb phrase ‘is prime’ isthereby analyzed as denoting the concept \(P(\:)\) which mapsprimes to The True and everything else to The False. Thus, a simplepredication like ‘\(2\) is prime’ becomesanalyzed in Frege’s system as a special case of functionalapplication.

2.1.2 The Predicate Calculus Within Frege’s Term Logic

The preceding analysis of simple mathematical predications led Fregeto extend the applicability of this system to the representation ofnon-mathematical thoughts and predications. This move formed the basisof the modern predicate calculus. Frege analyzed a non-mathematicalpredicate like ‘is happy’ as signifying a function of onevariable which maps its arguments to a truth-value. Thus, ‘ishappy’ denotes a concept which can be represented in the formalsystem as ‘\(H(\:)\)’. \(H(\:)\) maps those argumentswhich are happy to The True, and maps everything else to TheFalse. The sentence ‘John is happy’, represented as‘\(H(j)\)’, is thereby analyzed as: the object denoted by‘John’ falls under the concept signified by‘\((\:)\) is happy’. Thus, a simple predication isanalyzed in terms of falling under a concept, which in turn, isanalyzed in terms of functions which map their arguments to truthvalues. By contrast, in the modern predicate calculus, this last stepof analyzing predication in terms of functions is not assumed;predication is seen as more fundamental than functional application.The sentence ‘John is happy’ is formally represented as‘\(Hj\)’, where this is a basic form of predication(‘the object \(j\) instantiates or exemplifies the property\(H\)’). In the modern predicate calculus, functionalapplication is analyzable in terms of predication, as we shall soonsee.

In Frege’s analysis, the verb phrase ‘loves’signifies a binary function of two variables: \(L((\:),(\:))\). Thisfunction takes a pair of arguments \(x\) and \(y\) and maps them toThe True if \(x\) loves \(y\) and maps all other pairs of arguments toThe False. Although it is a descendent of Frege’s system, themodern predicate calculus analyzesloves as a two-placerelation (written: \(Lxy\)) rather than a function; some objects standin the relation and others do not. The difference betweenFrege’s understanding of predication and the one manifested bythe modern predicate calculus is simply this: in the modern predicatecalculus, relations are taken as basic, and functions are defined as aspecial case of relation, namely, those relations \(R\) such that forany objects \(x\), \(y\), and \(z\), if \(Rxy\) and \(Rxz\), then\(y=z\). By contrast, Frege took functions to be more basic thanrelations. His logic is based on functional application rather thanpredication; a binary relation is analyzed as a binary function thatmaps a pair of arguments to a truth-value. Thus, a 3-place relationlikegives would be analyzed in Frege’s logic as afunction that maps arguments \(x\), \(y\), and \(z\) to an appropriatetruth-value depending on whether \(x\) gives \(y\) to \(z\); the4-place relationbuys would be analyzed as a function thatmaps the arguments \(x\), \(y\), \(z\), and \(u\) to an appropriatetruth-value depending on whether \(x\) buys \(y\) from \(z\) foramount \(u\); etc.

2.2 Complex Statements and Generality

So far, we have been discussing Frege’s analysis of‘atomic’ statements. To complete the basic logicalrepresentation of thoughts, Frege added notation for representing morecomplex statements (such as negated and conditional statements) andstatements of generality (those involving the expressions‘every’ and ‘some’). Though we no longer usehis notation for representing complex and general statements, it isimportant to see how the notation in Frege’s term logic alreadycontained all the expressive power of the modern predicate calculus.

There are four special functional expressions which are used inFrege’s system to express complex and general statements:

Intuitive Significance	Functional Expression	The Function It Signifies
Statement		The function which maps The True to The True and maps all otherobjects to The False; used to express the thought that the argument ofthe function is a true statement.
Negation		The function which maps The True to The False and maps all otherobjects to The True
Conditional		The function which maps a pair of objects to The False if the first(i.e., named in the bottom branch) is The True and the secondisn’t The True, and maps all other pairs of objects to The True
Generality		The second-level function which maps a first-level conceptΦ to The True if Φ maps every objectto The True; otherwise it maps Φ to The False.

The best way to understand this notation is by way of some tables,which show some specific examples of statements and how those arerendered in Frege’s notation and in the modern predicatecalculus.

2.2.1 Truth-functional Connectives

The first table shows how Frege’s logic can express thetruth-functional connectives such as not, if-then, and, or, andif-and-only-if.

Example	Frege’s Notation	Modern Notation
John is happy		\(Hj\)
It is not the case that John is happy		\(\neg Hj\)
If the sun is shining, then John is happy		\(Ss \to Hj\)
The sun is shining and John is happy		\(Ss \amp Hj\)
Either the sun is shining or John is happy		\(Ss \lor Hj\)
The sun is shining if and only if John is happy		\(Ss \equiv Hj\)

As one can see, Frege didn’t use the primitive connectives‘and’, ‘or’, or ‘if and only if’,but always used canonical equivalent forms defined in terms ofnegations and conditionals. Note the last row of the table —when Frege wants to assert that two conditions are materiallyequivalent, he uses the identity sign, since this says that theydenote the same truth-value. In the modern sentential calculus, thebiconditional does something equivalent, for a statement of the form\(\phi \equiv \psi\) is true whenever \(\phi\) and \(\psi\) are bothtrue or both false. The only difference is, in the modern sententialcalculus \(\phi\) and \(\psi\) are not construed as terms denotingtruth-values, but rather as sentences having truth conditions. Ofcourse, Frege could, in his notation, use the sentence ‘\((\phi\to \psi) \amp (\psi \to \phi)\)’ to assert \(\phi \equiv\psi\).

2.2.2 Quantified Statements

The table below compares statements of generality in Frege’s notationand in the modern predicate calculus. Frege used a special typeface(Gothic) for variables in general statements.

Example	Frege Notation	Modern Notation
Everything is mortal		\(\forall xMx\)
Something is mortal		\(\neg \forall x\neg Mx\) i.e., \(\exists xMx\)
Nothing is mortal		\(\forall x \neg Mx\) i.e., \(\neg \existsxMx\)
Every person is mortal		\(\forall x(Px \to Mx)\)
Some person is mortal		\(\neg\forall x(Px \to \neg Mx)\) i.e.,\(\exists x(Px \amp Mx)\)
No person is mortal		\(\forall x(Px \to \neg Mx)\) i.e., \(\neg \exists x(Px \amp Mx)\)
All and only persons are mortal		\(\forall x(Px \equiv Mx)\)

Note the last line. Here again, Frege uses the identity sign to helpstate the material equivalence of two concepts. He can do this becausematerially equivalent concepts \(F\) and \(G\) are such that \(F\)maps an object \(x\) to The True whenever \(G\) maps \(x\) to TheTrue; i.e., for all arguments \(x\), \(F\) and \(G\) map \(x\) to thesame truth-value.

In the modern predicate calculus, the symbols‘\(\forall\)’ (‘every’) and‘\(\exists\)’ (‘some’) are called the‘universal’ and ‘existential’ quantifier,respectively, and the variable ‘\(x\)’ in the sentence‘\(\forall xMx\)’ is called a ‘quantifiedvariable’, or ‘variable bound by the quantifier’. Wewill follow this practice of calling statements involving one of thesequantifier phrases ‘quantified statements’. As one can seefrom the table above, Frege didn’t use an existentialquantifier. He was aware that a statement of the form ‘\(\existsx\phi \)’ could always be defined as ‘\(\neg \forall x\neg \phi\)’, where \(\phi\) is any formula.

It is important to mention here that the predicate calculusformulable in Frege’s logic is a ‘second-order’ predicatecalculus. This means it allows quantification over functions as well asquantification over objects; i.e., statements of the form ‘Everyfunction \(f\) is such that …’ and ‘Some function\(f\) is such that …’ are allowed. Thus, the statement‘objectsa andb fall under the sameconcepts’ would be written as follows in Frege’notation:

and in the modern second-order predicate calculus, we write thisas:

\(\forall F (Fa \equiv Fb) \)

Readers interested in learning more about Frege’s notation canconsult Beaney (1997, Appendix 2), Furth (1967), Reck & Awodey(2004, 26–34), and Cook (2013). In what follows, however, weshall continue to use the notation of the modern predicate calculusinstead of Frege’s notation. In particular, we adopt thefollowing conventions:

1. We shall often use ‘\(Fx\)’ instead of‘\(F(x)\)’ to represent the fact that \(x\) falls underthe concept \(F\); we use ‘\(Rxy\)’ instead of‘\(R(x,y)\)’ to represent the fact that \(x\) stands inthe relation \(R\) to \(y\); etc.
2. Instead of using expressions with placeholders, such as‘\((\:) = (\:)\)’ and ‘\(P(\:)\)’, to signifyfunctions and concepts, we shall simply use ‘\(=\)’ and‘\(P\)’.
3. When one replaces one of the complete names in a sentence by avariable, the resulting expression will be called anopensentence or anopen formula. Thus, whereas ‘\(3 <2\)’ is a sentence, ‘\(3 < x\)’ is an open sentence;and whereas ‘\(Hj\)’ is a formal sentence that might beused to represent ‘John is happy’, the expression‘\(Hx\)’ is an open formula which might be rendered‘\(x\) is happy’ in natural language.
4. Finally, we shall on occasion employ the Greek symbol \(\phi\) asa metavariable ranging over formal sentences, which may or may not beopen. Thus, ‘\(\phi (a)\)’ will be used to indicate anysentence (simple or complex) in which the name ‘\(a\)’appears; ‘\(\phi(a)\)’ isnot to be understood asFrege-notation for a function \(\phi\) applied to argument \(a\).Similarly, ‘\(\phi(x)\)’ will be used to indicate an opensentence in which the variable \(x\) may or may not be free, not afunction of \(x\).

Again, it should be noted that the foregoing has been a simplifiedpresentation of Frege’s logic, for those who are encountering it forthe first time. With these basics in hand, see the entry onFrege’s logic for a more precisepresentation.

2.2.3 Frege’s Logic of Quantification

Frege’s functional analysis of predication coupled with hisunderstanding of generality freed him from the limitations of the‘subject-predicate’ analysis of ordinary language sentencesthat formed the basis of Aristotelian logic and it made it possible forhim to develop a more general treatment of inferences involving‘every’ and ‘some’. In traditional Aristotelianlogic, the subject of a sentence and the direct object of a verb arenot on a logical par. The rules governing the inferences betweenstatements with different but related subject terms are different fromthe rules governing the inferences between statements with differentbut related verb complements. For example, in Aristotelian logic, therule which permits the valid inference from ‘John lovesMary’ to ‘Something loves Mary’ is different from therule which permits the valid inference from ‘John lovesMary’ to ‘John loves something’. The rule governingthe first inference is a rule which applies only to subject termswhereas the rule governing the second inference governs reasoningwithin the predicate, and thus applies only to the transitive verbcomplements (i.e., direct objects). In Aristotelian logic, theseinferences have nothing in common.

In Frege’s logic, however, a single rule governs both theinference from ‘John loves Mary’ to ‘Something lovesMary’ and the inference from ‘John loves Mary’ to‘John loves something’. That’s because the subjectJohn and the direct object Mary are both considered on a logical par,as arguments of the functionloves. In effect, Frege saw nological difference between the subject ‘John’ and thedirect object ‘Mary’. What is logically important is that‘loves’ denotes a function of two arguments. No matterwhether the quantified expression ‘something’ appears assubject (‘Something loves Mary’) or within a predicate(‘John loves something’), it is to be resolved in the sameway. In effect, Frege treated these quantified expressions asvariable-binding operators. The variable-binding operator ‘some\(x\) is such that’ can bind the variable ‘\(x\)’ inthe open sentence ‘\(x\) loves Mary’ as well as thevariable ‘\(x\)’ in the open sentence ‘John loves\(x\)’. Thus, Frege analyzed the above inferences in thefollowing general way:

• John loves Mary. Therefore, some \(x\) is such that \(x\)loves Mary.
• John loves Mary. Therefore, some \(x\) is such that John loves\(x\).

Both inferences are instances of a single valid inference rule. Tosee this more clearly, here are the formal representations of the aboveinformal arguments:

• \(Ljm\) ∴ \(\exists x (Lxm)\)
• \(Ljm\) ∴ \(\exists x (Ljx)\)

The logical axiom which licenses both inferences has the form:

\(Ra_1 \ldots a_i \ldots a_n \to \exists x(Ra_1 \ldots x \ldotsa_n)\),

where \(R\) is a relation that can take \(n\) arguments, \(a_1,\ldots , a_n\) are any constants (names), and \(1 \leq i \leqn\). This logical axiom tells us that from a simple predicationinvolving an \(n\)-place relation, one can existentially generalize onany argument, and validly derive a existential statement.

Indeed, this axiom can be made even more general. If \(\phi (a)\) isany statement (formula) in which a constant (name) \(a\) appears, and\(\phi (x)\) is the result of replacing one or more occurrences of\(a\) by \(x\), then the following is a logical axiom:

\(\phi (a) \to \exists x \phi (x)\)

The inferences which start with the premise ‘John lovesMary’, displayed above, both appeal to this axiom forjustification. This axiom is actually derivable as a theorem fromFrege’s Basic Law IIa (1893, §47). Basic Law IIa isformulated as \(\forall x\phi (x) \to \phi (a)\), where \(\phi(a)\) isthe result of substituting \(a\) for one or more variables \(x\) boundby the quantifier (though see below for a more careful discussion ofthis axiom). So the axiom displayed above for the existentialquantifier can be derived from IIa using the rules governingconditionals, negation, and the definition of \(\exists x\phi\)discussed above.

There is one other consequence of Frege’s logic ofquantification that should be mentioned. Frege took claims of the form\(\exists x\phi\) to be existence claims. He suggested thatexistence is not a concept under which objects fall butrather a second-level concept under which first-level concepts fall. Aconcept \(F\) falls under this second-level concept just in case \(F\)maps at least one object to The True. So the claim ‘Martiansdon’t exist’ is analyzed as an assertion about theconceptmartian, namely, that nothing falls under it. Fregetherefore tookexistence to be that second-level conceptwhich maps a first-level concept \(F\) to The True just in case\(\exists xFx\). Some philosophers have thought that this analysisvalidates Kant’s view thatexistence is not a (real)predicate.

2.3 Proof and Definition

2.3.1 Proof

Frege’s system (i.e., his term logic/predicate calculus) consistedof a language and an apparatus for proving statements. The latterconsisted of a set of logical axioms (statements considered to betruths of logic) and a set of rules of inference that lay out theconditions under which certain statements of the language may becorrectly inferred from others. Frege made a point of showing how everystep in a proof of a proposition was justified either in terms of oneof the axioms or in terms of one of the rules of inference or justifiedby a theorem or derived rule that had already been proved.

Thus, as part of his formal system, Frege developed a strictunderstanding of a ‘proof’. In essence, he defined a proofto be any finite sequence of statements such that each statement in thesequence either is an axiom or follows from previous members by a validrule of inference. Thus, a proof of a theorem of logic, say φ, istherefore any finite sequence of statements (with φ the finalstatement in the sequence) such that each member of the sequence: (a)is one of the logical axioms of the formal system, or (b) follows fromprevious members of the sequence by a rule of inference. These areessentially the definitions that logicians still use today.

2.3.2 Definition

Frege was extremely careful about the proper description anddefinition of logical and mathematical concepts. He developed powerfuland insightful criticisms of mathematical work which did not meet hisstandards for clarity. For example, he criticized mathematicians whodefined a variable to be a number that varies rather than an expressionof language which can vary as to which determinate number it may takeas a value.

More importantly, however, Frege was the first to claim that aproperly formed definition had to have two important metatheoreticalproperties. Let us call the new, defined symbol introduced in adefinition thedefiniendum, and the term that is used todefine the new term thedefiniens. Then Frege was the firstto suggest that proper definitions have to be botheliminable(a definendum must always be replaceable by its definiens in anyformula in which the former occurs) andconservative (adefinition should not make it possible to prove new relationshipsamong formulas that were formerly unprovable). Concerning one of hisdefinitions in theBegriffsschrift (§24), Fregewrites:

We can do without the notation introduced by this sentence, and hence without the sentence itself as its definition; nothing follows from the sentence that could not also be inferred without it. Our sole purpose in introducing such definitions is to bring about an extrinsic simplification by stipulating an abbreviation.

Frege later criticized those mathematicians who developed‘piecemeal’ definitions or ‘creative’definitions. In theGrundgesetze der Arithmetik, II (1903,Sections 56–67) Frege criticized the practice of defining aconcept on a given range of objects and later redefining the concepton a wider, more inclusive range of objects. Frequently, this‘piecemeal’ style of definition led to conflict, since theredefined concept did not always reduce to the original concept whenone restricts the range to the original class of objects. In that samework (1903, Sections 139–147), Frege criticized the mathematicalpractice of introducing notation to name (unique) entities withoutfirst proving that there exist (unique) such entities. He pointed outthat such ‘creative definitions’ were simply unjustified.Creative definitions fail to be conservative, as this was explainedabove.

2.4 Courses-of-Values, Extensions, and Proposed Mathematical Foundations

2.4.1 Courses-of-Values and Extensions

Frege’s ontology consisted of two fundamentally different types ofentities, namely, functions and objects (1891, 1892b, 1904). Functionsare in some sense ‘unsaturated’; i.e., they are the kind ofthing which take objects as arguments and map those arguments to avalue. This distinguishes them from objects. As we’ve seen, the domainof objects included two special objects, namely, the truth-values TheTrue and The False.

In his work of 1893/1903, Frege attempted to expand the domain ofobjects by systematically associating, with each function \(f\), anobject which he calledthe course-of-values of \(f\). Thecourse-of-values of a function is a record of the value of the functionfor each argument. The principle Frege used to systematizecourses-of-values is Basic Law V (1893/§20;):

The course-of-values of the concept \(f\) is identical tothe course-of-values of the concept \(g\) if and only if \(f\) and\(g\) agree on the value of every argument (i.e., if and only if forevery object \(x\), \(f(x) = g(x)\)).

Frege used the a Greek epsilon with a smooth breathing mark above itas part of the notation for signifying the course-of-values ofthe function \(f\):

\(\stackrel{,}{\varepsilon}\!\!f(\varepsilon)\)

where the first occurrence of the Greek \(\varepsilon\) (with thesmooth breathing mark above it) is a ‘variable-bindingoperator’ which we might read as ‘the course-of-valuesof’. To avoid the appearance of variable clash, we may also usea Greek \(\alpha\) (also with a smooth breathing mark above) as avariable-binding operator. Using this notation, Frege formallyrepresented Basic Law V in his system as:

Basic Law V
\(\stackrel{,}{\varepsilon}\!\!f(\varepsilon) =\, \stackrel{,}{\alpha}\!g(\alpha) \equiv \forall x(f(x) = g(x))\)

(Actually, Frege used an identity sign instead of the biconditionalas the main connective of this principle, for reasons describedabove.)

Frege called the course-of-values of a concept \(F\) itsextension. The extension of a concept \(F\) records justthose objects which \(F\) maps to The True. Thus Basic Law V appliesequally well to the extensions of concepts. Let‘\(\phi(x)\)’ be an open sentence of any complexity withthe free variable \(x\) (the variable \(x\) may have more than oneoccurrence in \(\phi(x)\), but for simplicity, assume it has only oneoccurrence). Then Frege would use the expression:

\(\stackrel{,}{\varepsilon}\!\! \phi (\varepsilon)\),

where the second epsilon replaces \(x\) in \(\phi (x)\), to denote theextension of the concept \(\phi\). Where ‘\(n\)’ is thename of an object, Frege could define ‘object \(n\) is anelement of the extension of the concept \(\phi\)’ in thefollowing simple terms: ‘the concept \(\phi\) maps \(n\) to TheTrue’, i.e., \(\phi(n)\). For example, the number \(3\) is anelement of the extension of the conceptodd number greater than \(2\) if and only if this conceptmaps \(3\) to The True.

Unfortunately, Basic Law V implies a contradiction, and this waspointed out to Frege by Bertrand Russell just as the second volume oftheGrundgesetze was going to press. Russell recognized thatsome extensions are elements of themselves and some are not; theextension of the conceptextension is an element of itself,since that concept would map its own extension to The True. Theextension of the conceptspoon is not an element of itself,because that concept would map its own extension to The False (sinceextensions aren’t spoons). But now what about theconceptextension which is not an element of itself? Let\(E\) represent this concept and let \(e\) name the extension of\(E\). Is \(e\) an element of itself? Well, \(e\) is an element ofitself if and only if \(E\) maps \(e\) to The True (by the definitionof ‘element of’ given at the end of the previousparagraph, where \(e\) is the extension of the concept \(E\)). But\(E\) maps \(e\) to The True if and only if \(e\) is an extensionwhich is not an element of itself, i.e., if and only if \(e\) is notan element of itself. We have thus reasoned that \(e\) is an elementof itself if and only if it is not, showing the incoherency inFrege’s conception of an extension.

Further discussion of this problem can be found in the entry onRussell’s Paradox, and a more completeexplanation of how the paradox arises in Frege’s system is presented inthe entry onFrege’s theorem and foundations for arithmetic.

2.4.2 Proposed Foundation for Mathematics

Before he became aware of Russell’s paradox, Frege attempted toconstruct a logical foundation for mathematics. Using the logicalsystem containing Basic Law V (1893/1903), he attempted to demonstratethe truth of the philosophical thesis known aslogicism, i.e.,the idea not only that mathematical concepts can be defined in terms ofpurely logical concepts but also that mathematical principles can bederived from the laws of logic alone. But given that the crucialdefinitions of mathematical concepts were stated in terms ofextensions, the inconsistency in Basic Law V undermined Frege’s attemptto establish the thesis of logicism. Few philosophers today believethat mathematics can be reduced to logic in the way Frege had in mind.Mathematical theories such as set theory seem to require somenon-logical concepts (such as set membership) which cannot be definedin terms of logical concepts, at least when axiomatized by certainpowerful non-logical axioms (such as the proper axioms ofZermelo-Fraenkel set theory). Despite the fact that a contradictioninvalidated a part of his system, the intricate theoretical web ofdefinitions and proofs developed in theGrundgesetzenevertheless offered philosophical logicians an intriguing conceptualframework. The ideas ofBertrand Russell andAlfred North Whitehead inPrincipia Mathematica owea huge debt to the work found in Frege’sGrundgesetze.

Despite Frege’s failure to provide a coherent systematization of thenotion of an extension, we shall make use of the notion in what followsto explain Frege’s theory of numbers and analysis of number statements,without assuming Basic Law V. It suffices to use ourinformal understanding of the notion, though extensions can berehabilitated in various ways, either axiomatically as in modern settheory or as in the various ways logicians have found to weakenFrege’s system. (See T. Parsons 1987, Burgess 1998, Heck 1996, Wehmeier1999, Ferreira & Wehmeier 2002, Fine 2002, Anderson & Zalta2004, Ferreira 2005, and Antonelli & May 2005.)

2.5 The Analysis of Statements of Number

In what has come to be regarded as a seminal treatise,DieGrundlagen der Arithmetik (1884), Frege began work on the idea ofderiving some of the basic principles of arithmetic from what hethought were more fundamental logical principles and logical concepts.Philosophers today still find that work insightful. The leading ideais that a statement of number, such as ‘There are eightplanets’ and ‘There are two authors ofPrincipiaMathematica’, is really a statement about a concept. Fregerealized that one and the same physical phenomena could beconceptualized in different ways, and that answers to the question‘How many?’ only make sense once a concept \(F\) issupplied. Thus, one and the same physical entity might beconceptualized as consisting of \(1\) army, \(5\) divisions, \(20\)regiments, \(100\) companies, etc., and so the question ‘Howmany?’ only becomes legitimate once one supplies the conceptbeing counted, such asarmy,division,regiment, orcompany (1884, §46).

Using this insight, Frege took true statements like ‘There areeight planets’ and ‘There are two authors ofPrincipiaMathematica’ to be “second level” claims aboutthe conceptsplanet andauthor of Principia Mathematica,respectively. In the second case, the second level claim asserts thatthe first-level conceptbeing an author of PrincipiaMathematica falls under the second-level conceptbeing aconcept under which two objects fall. This sounds circular, sinceit looks like we have analyzed

There are two authors ofPrincipia Mathematica,

which involves the concepttwo, as:

The conceptbeing an author of Principia Mathematica fallsunder the conceptbeing a concept under which two objectsfall,

which also involves the concepttwo. But despiteappearances, there is no circularity, since Frege analyzes thesecond-order conceptbeing a concept under which two objectsfall without appealing to the concepttwo. He did this bydefining ‘\(F\) is a concept under which two objectsfall’, in purely logical terms, as any concept \(F\) thatsatisfies the following condition:

There are distinct things \(x\) and \(y\) that fall underthe concept \(F\) and anything else that falls under the concept \(F\)is identical to either \(x\) or \(y\).

In the notation of the modern predicate calculus, this is formalizedas:

\(\exists x \exists y (x\! \neq\! y \amp Fx \amp Fy \amp\forall z(Fz \to z\! =\! x \lor z\! =\! y))\)

Note that the conceptbeing an author of PrincipiaMathematica satisfies this condition, since there are distinctobjects \(x\) and \(y\), namely, Bertrand Russell and AlfredNorth Whitehead, who authoredPrincipia Mathematica and whoare such that anything else authoringPrincipia Mathematica isidentical to one of them. In this way, Frege analyzed a statement ofnumber (‘there are two authors ofPrincipiaMathematica’) as higher-order logical statements aboutconcepts.

Frege then took his analysis one step further. He noticed that eachof the conditions in the following sequence of conditions defined aclass of ‘equinumerous’ concepts, where‘F’ in each case is a variable ranging overconcepts:

Notice that if concepts \(P\) and \(Q\) are both conceptswhich satisfy one of these conditions, then there is a one-to-onecorrespondence between the objects which fall under \(P\) and theobjects which fall under \(Q\). That is, if any of the aboveconditions accurately describes both \(P\) and \(Q\), thenevery object falling under \(P\) can be paired with a unique anddistinct object falling under \(Q\) and, under this pairing, everyobject falling under \(Q\) gets paired with some unique anddistinct object falling under \(P\). (By the logician’sunderstanding of the phrase ‘every’, this last claim evenapplies to those concepts \(P\) and \(Q\) which satisfyCondition (0).) Frege would call such \(P\) and \(Q\)equinumerous concepts (1884, §72). Indeed, for eachcondition defined above, the concepts that satisfy the condition areall pairwise equinumerous to one another.

With this notion of equinumerosity, Frege defined ‘the number ofthe concept \(F\)’ to be the extension consisting of allthe concepts that are equinumerous with \(F\) (1884, §68).To get started, Frege then defined Zero to be the number of theconceptbeing non-self-identical (1884, §74). If we usethe notation \(\#F\) to representthe number of the concept F,and use the \(\lambda\)-notation \([\lambda x \: \phi]\) to name thecomplex conceptbeing an object x such that \(\phi\), Frege’sdefinition of Zero becomes:

\(0 =_\mathit{df} \#[\lambda x \: x \neq x]\)

Thus, the number 0 becomes defined as the extension of all theconcepts equinumerous to the conceptnot beingself-identical. This extension contains all the concepts thatsatisfy Condition (0) above, and so the number of all such concepts is0. For example, the number of the conceptbeing a squarecircle is \(0\), since nothing falls under it. Similarly, onecould define the number \(1\) as the extension consisting of all theconcepts that satisfy Condition (1) above, and define the number \(2\)as the extension of all the concepts that satisfy Condition (2) above,and so on. But though this would define a sequence of entities whichare numbers, this procedure doesn’t actually define theconceptnatural number (finite number). Frege,however, had a deep idea about how to do this.

2.6 Natural Numbers

In order to define the concept ofnatural number, Frege firstdefined, for every \(2\)-place relation \(R\), the general concept‘\(x\) is an ancestor of \(y\) in the \(R\)-series’. Thisnew relation is called ‘the ancestral of the relation\(R\)’. The ancestral of the relation \(R\) was first defined inFrege’sBegriffsschrift (1879, §26, Proposition76; 1884, §79). The intuitive idea is easily grasped if weconsider 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 \(d\). Then Frege’s definition of‘\(x\) is an ancestor of \(y\) in the fatherhood-series’ensures that \(a\) is an ancestor of \(b\), \(c\), and \(d\) in thisseries, that \(b\) is an ancestor of \(c\) and \(d\) in this series,and that \(c\) is an ancestor of \(d\) in this series.

More generally, if given a series of facts of the form \(aRb\),\(bRc\), \(cRd\), and so on, Frege showed how to define the relation\(R^*\), i.e.,x is an ancestor of y in the R-series, whichFrege referred to as: \(y\) follows \(x\) in the \(R\)-series. Toexploit this definition in the case of natural numbers, Frege had todefine both the relation \(x\)precedes \(y\) and theancestral of this relation, namely, \(x\)is an ancestor of\(y\)in the predecessor-series. He first defined therelational concept \(x\)precedes \(y\) as follows (1884,§76):

\(x\)precedes \(y\) iff there is a concept \(F\) andan object \(z\) such that:

• \(z\) falls under \(F\),
• \(y\) is the (cardinal) number of the concept \(F\),and
• \(x\) is the (cardinal) number of the conceptobject fallingunder \(F\)other than \(z\)

If we again use the notation \(\#F\) to denote the number of \(F\)sand the \(\lambda\)-notation \([\lambda u \: \phi]\) to name thecomplex conceptbeing an object \(u\)such that\(\phi\), Frege’s definition becomes:

\( \mathit{Precedes}(x,y) =_\mathit{df} \exists F \exists z (Fz \,\amp\, y\! =\! \#F \,\amp\, x\! =\! \#[\lambda u \: Fu \amp u\! \neq\! z]) \)

To see the intuitive idea behind this definition, consider how thedefinition is satisfied in the case of the number 1 preceding thenumber \(2\): there is a concept \(F\) (e.g., let \(F\) =being an author of Principia Mathematica) and an object \(z\)(e.g., let \(z\) = Alfred North Whitehead) such that:

• Whitehead falls under the conceptauthor of PrincipiaMathematica,
• \(2\) is the (cardinal) number of the conceptauthor ofPrincipia Mathematica, and
• \(1\) is the (cardinal) number of the conceptauthor ofPrincipia Mathematica other than Whitehead.

Note that the last conjunct is true because there is exactly 1object (namely, Bertrand Russell) that falls under the conceptauthor of Principia Mathematica other than Whitehead.

Thus, Frege has a definition ofprecedes which applies to theordered pairs \(\langle 0,1\rangle\), \(\langle 1,2\rangle \),\(\langle 2,3\rangle\), … . Frege then defined theancestral of this relation, namely, \(x\)is an ancestor of\(y\)in the predecessor-series, or\(\mathit{Precedes}^*\). Though the exact definition will not be givenhere, we note that it has the following consequence: from the factsthat \(10\) precedes \(11\) and \(11\) precedes \(12\), it followsthat \(10\)precedes\(^*\) \(12\) in thepredecessor-series. Note, however, that although\(10\)precedes\(^*\) \(12\), \(10\) does notprecede \(12\), for the notion ofprecedes is thatofimmediately precedes. Note also that by defining theancestral of the precedence relation, Frege had in effect defined\(x\) < \(y\) relative to the predecessor-series.

Recall that Frege defined the number \(0\) as the number of the conceptbeing non-self-identical, and that \(0\) thereby becomesidentified with the extension of all concepts which fail to beexemplified. Using this definition, Frege then defined (1884,§83)natural number as follows:

\(x\) is anumber  =_df  either \(x\! =\! 0\) or \(0\) is an ancestor of \(x\) in the predecessor-series

which we might represent formally as:

\( \mathit{Number}(x) =_\mathit{df} x\! =\! 0 ∨ \mathit{Precedes}^*(0,x)\)

In other words, a natural number is any member of thepredecessor-series beginning with 0.

Using this definition as a basis, Frege later derived many importanttheorems of number theory. Philosophers appreciated the importance ofthis work only relatively recently (C. Parsons 1965, Smiley 1981,Wright 1983, and Boolos 1987, 1990, 1995). Wright 1983 in particularshowed how the Dedekind/Peano axioms for number might be derived fromone of the consistent principles that Frege discussed in 1884, nowknown as Hume’s Principle (“The number of \(F\)s is equalto the number of \(G\)s if and only if there is a one-to-onecorrespondence between the \(F\)s and the \(G\)s”). It wasrecently shown by R. Heck [1993] that, despite the logicalinconsistency in the system of Frege 1893/1903, Frege himself validlyderived the Dedekind/Peano axioms from Hume’sPrinciple. Although Frege used Basic Law V (which yields aninconsistency when added to his second-order logic) to establishHume’s Principle, once Hume’s Principle was established,the subsequent derivations of the Dedekind/Peano axioms make nofurther essential appeals to Basic Law V. Following the lead ofGeorge Boolos, philosophers today call the derivation of theDedekind/Peano Axioms from Hume’s Principle ‘Frege’sTheorem’. For a comprehensive introduction to the subtle andcomplex logical reasoning involved in this theorem, see the entryFrege’s theorem and foundations for arithmetic.

2.7 Frege’s Conception of Logic

Before receiving the famous letter from Bertrand Russell informing himof the inconsistency in his system, Frege thought that he had shownthat arithmetic is reducible to the truths of logic. It is recognizedtoday, however, that at best Frege showed that arithmetic is reducibleto second-order logic extended only by Hume’s Principle. Somephilosophers think Hume’s Principle is analytically true (i.e., truein virtue of the very meanings of its words), while others resist theclaim, and there is an interesting debate on this issue in theliterature (see, e.g., Boolos 1997, Wright 1999).

However, for the purposes of this introduction to Frege’s work,there are prior questions on which it is more important tofocus. Whereas Frege thought that the truths of arithmetic arederivable from analytic truths of logic, Kant thought arithmeticprinciples are synthetic, in which case they wouldn’t be derivablefrom analytic truths. Their different conceptions of logic helps toexplain why these two philosophers came to such different conclusions.In this section, we therefore turn to the following questions:

1. How did Frege’s conception of logic differ from Kant’s?In particular:
   • What resources (or laws) did Kant and Frege both consider to be logical?
   • Did Kant and Frege agree about the content and subject matter of logic?
2. How did Frege’s conception of logic differ from that oflater logicians?

An answer to the first question sets the stage for answering thesecond.

2.7.1 How Frege’s Conception of Logic Differed from Kant’s

One of the most important differences between Kant and Frege concernsthe resources available to logic. Kant’s logic is limited to (a)Aristotelian term logic with a simple theory of disjunctive andhypothetical propositions, and (b) representing inclusion relationsamong concepts (MacFarlane 2002, 26). By contrast, Frege’s logicincludes (a) a term-forming operator \(\sm{\varepsilon}\), whichallows one to form the singular term\(\sm{\varepsilon}f(\varepsilon)\) from the function expression\(f(\xi)\), and (b) a Rule of Substitution, which allows one tosubstitute complex open formulas for free second-order variables intheorems of logic but also allows one to define, and assert theexistence of, complex concepts, including concepts defined in terms ofquantifiers over concepts. We’ll discuss both of theseresources below, but first, the discussion needs some context.

The differences concerning the resources available to logic revolvearound a key issue, namely, whether the additional resources Fregeassigns to logic require an appeal to non-logical constructions,specifically to a faculty of ‘intuition’, that is, anextralogical source which presents our minds with phenomena aboutwhich judgments can be formed. (Recall the discussion above aboutFrege’s early interest in appeals to intuition.) The debateover which resources do and do not require an appeal to intuition isan important one. Frege continued a trend started by Bolzano (1817),who eliminated the appeal to intuition in the proof of theIntermediate Value Theorem in the calculus (which in its simplest formasserts that a continuous function having both positive and negativevalues must cross the origin). Bolzano proved this theorem from thedefinition of continuity, which had recently been given in termssimilar to the definition of a limit (see Coffa 1991, 27). A Kantianmight simply draw a graph of a continuous function which takes valuesabove and below the origin, and thereby ‘demonstrate’ thatsuch a function must cross the origin. But appeal to a graph involvesan appeal to intuition, and both Bolzano and Frege saw such appeals tointuition as potentially introducing logical gaps into a proof. Thereare reasons to be suspicious about such appeals: (1) there arefunctions which we can’t graph or otherwise construct forpresentation to our intuitive faculty, e.g., the function \(f\) whichmaps rational numbers to \(0\) and irrational numbers to \(1\), or thefunctions noted by Weierstrass, which are everywhere continuous butnowhere differentiable; (2) once we take certain intuitive notions andformalize them in terms of explicit definitions, the formal definitionmight imply counterintuitive results; and (3) the rules of inferencefrom statements to constructions and back are not always clear.

Frege dedicated himself to the idea of eliminating appeals tointuition in the proofs of the basic propositions of arithmetic. Heexplicitly remarked upon this fact in a number of works throughout hiscareer (1879, Preface/5, Part III/§23; 1884, §§62, 87;1893, §0; and 1903, Appendix). Thus, he would deny Kant’sdictum (A51 [B75]), ‘Without sensibility, no object would begiven to us’, and claim instead that \(0\) and \(1\) are objectsbut that they ‘can’t be given to us in sensation’(1884, 101). Frege’s view is that we can understand or graspthem as objects if we (a) define them as extensions of concepts, and(b) show that singular terms of the form\(\sm{\varepsilon}f(\varepsilon)\) can be axiomatized by an analyticproposition. (The latter was to be accomplished by Basic Law V, and sothe collapse of this law in light of Russell’s paradoxundermined this part of his plan for avoiding appeals tointuition.)

Moreover, philosophers have questioned whether Frege’s Rule ofSubstitution (Grundgesetze I, 1893, §48, item 9) alsorequired an appeal to intuition. The Rule of Substitution allows oneto substitute complex formulas for free second-order variables inlogical theorems to produce new logical theorems. Boolos argued (1985,336–338) that since Frege’s Rule of Substitution isequivalent to a Comprehension Principle for Concepts, it isextralogical in character. The Comprehension Principle for Conceptsasserts \(\exists F \forall x(Fx \equiv \phi)\), provided \(\phi\)doesn’t have a free variable \(F\); the proviso blocks theinstance \(\exists F\forall x(Fx \equiv \neg Fx)\), from which one canquickly derive a contradiction. Thus, the Comprehension Principle forConcepts asserts the existence of a concept corresponding to everyexpressible condition on objects. From Kant’s point of view,such existence claims were thought to be synthetic and in need ofjustification by the faculty of intuition. So, although it was one ofFrege’s goals to avoid appeals to the faculty of intuition,there is a question as to whether his system of second-order logic(minus Basic Law V), which implies a principle asserting the existenceof a wide range of concepts, really is limited in its scope to purelylogical laws of an analytic nature.

If we now put aside their differences about logic’s resourcesand the appeal to intuition, there are other ways in which the Kantianand Fregean conceptions of logic differ. Both MacFarlane (2002) andLinnebo (2003) point out that one of Kant’s central views aboutlogic is that its axioms and theorems are purely formal in nature(i.e., abstracted from all semantic content and concerned only withtheforms of judgments) and are applicable across all thephysical and mathematical sciences (1781 [1787], A55 [B79], A56 [B80],A70 [B95]; and 1800, 15). And Kant takes the laws of logic to benormative and prescriptive (something one can get wrong), and not justdescriptive (1800, 16); they provideconstitutive norms ofthought (MacFarlane 2002, 35; Tolley 2008). Indeed, Linnebo takes twotheses, that logic is formal and provides laws that are constitutivenorms of thought, to be distinctive of Kant’s conception(Linnebo 2003, 240).

By contrast, Frege rejects the idea that logic is a purely formalenterprise (MacFarlane 2002, 29; Linnebo 2003, 243). He took logic tohave its own unique subject matter, which included not only factsabout concepts (concerning negation, subsumption, etc.) and identity,but also facts about relations (e.g., their properties andancestrals). Frege (1906, 428 [1984, 338]) says:

Just as the concept point belongs to geometry, so logic, too, has itsown concepts and relations; and it is only in virtue of this that itcan have a content. Toward what is thus proper to it, its relation isnot at all formal. No science is completely formal; but evengravitational mechanics is formal to a certain degree, in so far asoptical and chemical properties are all the same to it. … Tologic, for example, there belong the following: negation, identity,subsumption, subordination of concepts.

And, of course, as we’ve seen, Frege supposed that there is a domainof special logical objects (courses of values), among whichhe defined, and – until confronted by Russell’s paradox –took himself to have proved facts about,extensionsandnatural numbers (1884, 1893/1903). Logic, then, is notpurely formal, from Frege’s point of view, but rather can providesubstantive knowledge of concepts and objects.

There is some question, however, as to the extent to which Frege tooklogic to provide constitutive norms of thought. Linnebo suggests thatFrege eventually rejected this idea. Though he offers variousarguments for thinking that Frege moved away from the constitutivitythesis, his [Linnebo’s] main argument concerns the fact that Fregewanted to position Basic Law V as a logical claim, but that Basic LawV doesn’t seem to be a constitutive norm of thought (Linnebo 2003,247). This is a persuasive reason, though it does make one wonderwhat Frege could have meant, in the second volumeofGrundgesetze (§147), when he said, concerning BasicLaw V:

If there are logical objects at all … then there must also be ameans of apprehending, or recognizing them. This … is performed… by the fundamental law of logic that permits thetransformation of an equality holding generally\([\forall x(f(x)\! =\! g(x))]\) into anequation\([\sm{\varepsilon}f(\varepsilon) = \sm{\alpha}g(\alpha)]\).[Author’s note: the equations in brackets were added for the sake ofclarity.]

Given Frege’s commitment to logical objects as part of thecontent of logic, the above passage suggests that he might haveregarded the law which transformed an equality holding generally intoan equation as a constitutive norm of thought.

But many Frege scholars are convinced that Frege took the laws oflogic to provide constitutive norms of thought (MacFarlane 2002,Taschek 2008, Steinberger 2017). MacFarlane, in particular, arguesthat Kant and Frege may have agreed that one of the most importantcharacteristics of logic is its generality, and that this generalityconsists in the fact that it provides normative rules andprescriptions. He notes that “[t]he generality of logic, forFrege as for Kant, is anormative generality: logic isgeneral in the sense that it provides constitutive norms forthoughtas such, regardless of its subject matter”(2002, 35). So, though they may differ as to which principles arelogical, there may be at least one point of reconciliation concerninghow Kant and Frege conceived of logic.

2.7.2 How Frege’s Conception Differed from Later Logicians

Given the constraints of the present entry, we shall not attempt todiscuss this question in any detail; instead, we present only thebarest of outlines. After all, modern logicians and philosophers oflogic have not yet come to agreement about the proper conception oflogic. Many have a conception of logic that is yet different fromboth Kant’s and Frege’s, one that was, to some extent,anticipated by Bolzano, namely, that logical concepts and laws remaininvariant under reinterpretation of the non-logical constants or underpermutations of the domain of quantification. But since this modernconception is still a matter of debate, it may be that elements ofFrege’s conception will yet play a role in our understanding ofwhat logic is.

It is important to recognize just how much Frege took himself to befocusing on thecontent, as opposed to theform, ofthoughts. His concern to more precisely represent the content ofthoughts is stated explicitly in an 1882 lecture before Jena’sSociety for Medicine and Natural Science, where he distinguished his1879 system from Boole’s logic by saying:

I was not trying to present an abstract logic in formulas;I was trying to express contents in an exacter and more perspicuousmanner than is possible in words, by using written symbols. I wastrying, in fact, to create a “linguacharacteristica” in the Leibnizian sense, not a mere“calculus ratiocinator”—not that I do notrecognize such a deductive calculus as a necessary constituent of aBegriffsschrift. [Frege 1882, V.H. Dudman (trans.)1968]

So Frege was not just trying to develop an abstract reasoning systemfor the precise derivations of theorems from axioms (see vanHeijenoort 1967 for discussion). Frege was at least as interested informalizing thecontent of reasoning as he was in formulatingthe rules for deriving a given thought from some group ofthoughts. Frege wouldnot have regarded the logical axioms ofhis formal systems as axiomschemata, i.e., as ametalinguistic sentence patterns whose instances (i.e., the sentencesof the object language that match the pattern) are axioms (seeGoldfarb 2001 for discussion). Nor would he have agreed that thelogical axioms of his system were uninterpreted sentences. His uneasewith the modern conception of an uninterpreted formal system wasexpressed in his reaction to Hilbert’sFoundations ofGeometry (Frege 1906, 384 [1984, 315]):

The word ‘interpretation’ is objectionable, for whenproperly expressed, a thought leaves no room for differentinterpretations. We have seen that ambiguity simply has to be rejected…

Instead, Frege thought that his logical axioms are (a) fundamentaltruths governing negation, conditionalization, quantification,identity, and description, and (b) principles from which other suchfundamental truths could be derived. Indeed, even though Fregesometimes introduces methods for abbreviating these truths, he takesgreat pains to insist that these abbreviations are to be understood interms of the full content being expressed. For example, he summarizesthe law of universal instantiation (Basic Law IIa) in 1893/§47using a formula that we would write nowadays as:

\(\forall xFx \to Fy\)

But closer inspection of §20, where this principle is firstdiscussed, and of §17, where he introduces some notationalconventions used in §20, makes it clear that the above isshorthand for:

\(\forall F \forall y(\forall xFx \to Fy)\)

This latter is a sentence; it is not a schema, nor an open formulawith a free variable, nor an uninterpreted sentence. Rather itasserts something that Frege takes to be fundamental law, namely: forany property \(F\) and for any object \(y\), if everythingfalls under \(F\), then \(y\) falls under \(F\).

Frege’s understanding of, and attitude towards, the formulas ofhis formal language goes a long way towards explaining why hisposition, in the now famous debate with Hilbert about the status ofthe axioms in a formal system, is not an unreasonable one. At firstglance, it looks as if Frege has mistakenly challenged Hilbert’smethod of relative interpretability, whereby one can prove theconsistency and independence of axiom systems by re-intepreting, andthereby reducing, them to systems assumed to beconsistent. Frege’s objections about what exactly has beenestablished by these relative consistency proofs may seem misguided toa modern ear. But since Frege and Hilbert understood the notionsofconsistency andindependence differently, theydidn’t always directly engage with the other’s ideas.Blanchette nicely shows, both in the entry on theFrege-Hilbert controversy and in herbook (2012, Ch. 5), that if the notions of consistency andindependence are understood Hilbert’s way, then Hilbert’smethods do establish what he says they do, but that if these notionsare understood Frege’s way, they don’t. The reader should pursuethese works for a more detailed explanation and nuanced discussion ofthe disagreement.

One can appreciate how Frege and Hilbert might have failed to engagewith one another by considering a simple analogy. Consider theinference from “\(x\) had a nightmare” to “\(x\) hada dream” and ask the question, is the latter a logicalconsequence of the former? If one examines the inferencepurelyformally, as Hilbert might, then the sentences have the form‘\(Fx\)’ and ‘\(Gx\)’ and the questionbecomes, does \(Fx\) logically imply \(Gx\)? The answer for Hilbertwould be ‘No’, because one caninterpret‘\(F\)’ and ‘\(G\)’ in such a way that theinference fails, e.g., just assign the standard meaning of‘dream’ to ‘\(F\)’ and assign the standardmeaning of ‘nightmare’ to ‘\(G\)’, i.e.,interpret ‘\(F\)’ as the propertyhaving a dreamand interpret ‘\(G\)’ as the propertyhaving anightmare. This shows that, from this purely formal point ofview, “\(x\) had a nightmare” doesn’t logicallyimply “\(x\) had a dream”. (From this purely formalviewpoint, one additionally needs the premise \(\forall y(Fy \to Gy)\)to infer \(Gx\) from \(Fx\).)

Although Frege had a formal system in which the open sentence \(Fx\)doesn’t logically imply \(Gx\), he takes logical consequence tobe a relation amongthoughts. He wouldn’t answer thequestion, of whether one thought is a logical consequence of another,solely by looking at the form of the sentences that express them, butrather by looking at the content of those sentences. Given the meaning(content) that ‘nightmare’ and ‘dream’ in facthave, “\(x\) had a dream” is a logical consequence of“\(x\) had a nightmare”, forhaving a nightmare,i.e.,having a bad dream, logically implieshaving adream. So, for Frege, this would clearly be a case of logicalconsequence.

This analogy might help one to see how Frege and Hilbert might differin their approach to questions of consistency and interpretation. OnFrege’s view, the consistency of a group of axioms depends oncontent, and if the form of these axioms, under logical analysis,sufficiently captures their content, this consistency will beinherited by their formal representations as well. Of course, inproving consistency, Hilbert was concerned primarily to determinewhether an axiom system entailed a contradiction having the form\(\phi \amp \neg \phi\). So, given thisformal goal,Hilbert’s methods are useful and immune to criticism.

But this brings us to one final issue that is crucial to Frege’sconception of logic, namely, the extent to which his formalrepresentations capture the content of the claims beinganalyzed. This issue is relevant because Frege’s primary tool foranalyzing the content of a mathematical or philosophical claim is byway of representing the content in a system thataxiomatizesthe fundamental concepts that are needed for the analysis. This issueis the subject of the first half of Blanchette 2012. To see what is atstake, we vary the example from the one used in Blanchette 2012(24). Frege would represent the arithmetical law:

No (natural) number precedes zero

in the first instance, as:

\( \neg \exists x(\mathit{Number}(x) \amp \mathit{Precedes}(x,0)) \)

Then if we substitute Frege’s definitions of\(\mathit{Number}(x)\), \(\mathit{Precedes}(x,y)\), and \(0\), asdescribed in Sections 2.5 and 2.6 above, his representation of thearithmetical law becomes:

\(\neg \exists x((x\! =\! 0 \lor \mathit{Precedes}^*(0,x)) \amp \\ \ \ \ \exists F \exists z(Fz \amp 0\! =\! \#F \amp x\! =\! \#[\lambda u \: Fu \amp u \! \neq\! z])) \)

Though the formal representation could be taken further, if we expandthe definitions of \(\mathit{Precedes}^*\), \(\#F\), and \(\#[\lambdau \: Fu \amp u \!\neq\! z]\), enough has been said to pose thequestion: why think that by deriving the formal representation frommore fundamental principles, Frege has derived the arithmetic law thatno natural number precedes Zero? This question is tackled in somedetail in in the early part of Blanchette 2012, which investigatesFrege’s understanding of conceptual analysis. Her answer(Chapter 4) is that the formal representation of the arithmetic lawhas to be (self-evidently) logically equivalent to a good analysis ofthe original. If it is, then notwithstanding Frege’s failedreduction of numbers to extensions, a derivation of the formalrepresentation from more general logical laws of the kind representedin his system would have in fact achieved the goal of reducingarithmetic to logic. The reader is directed to her work for discussionof this important point.

3. Frege’s Philosophy of Language

While pursuing his investigations into mathematics and logic (andquite possibly, in order to ground those investigations), Frege wasled to develop a philosophy of language. His philosophy of languagehas had just as much, if not more, impact than his contributions tologic and mathematics. However, Bobzien (2021) offers compellingdocumentation suggesting that significant elements of Frege’sphilosophy of language were adapted from Stoic logic. In thissection, therefore, we first rehearse a key element of Frege’sphilosophy of language, namely, the distinction betweensenseanddenotation. To motivate this distinction, we go throughthe two key puzzles Frege attempted to solve (Section 3.1) and thenexamine how the distinction between the sense and denotation of terms(and sentences) solve these puzzles (Section 3.2). Finally, we attemptto put Frege’s philosophy of language in context (Section 3.3), inlight of new research arguing for the significant influence that(discussions of) Stoic texts had on his ideas.

3.1 Frege’s Puzzles

Frege’s seminal paper in the philosophy of language is‘Über Sinn und Bedeutung’ (‘On Sense andReference’, 1892a). In this paper, Frege considered two puzzlesabout language and noticed, in each case, that one cannot account forthe meaningfulness or logical behavior of certain sentences simply onthe basis of the denotations of the terms (names and descriptions) inthe sentence. One puzzle concerned identity statements and the otherconcerned sentences with subordinate clauses such as propositionalattitude reports. To solve these puzzles, Frege suggested that theterms of a language have both a sense and a denotation, i.e., that atleast two semantic relations are required to explain the significanceor meaning of the terms of a language. This idea has inspired researchin the field for over a century and we discuss it in what follows.(See Heck and May 2006 for further discussion of Frege’scontribution to the philosophy of language.)

3.1.1 Frege’s Puzzle About Identity Statements

Here are some examples of identity statements:

\(117+136 = 253\).
The morning star is identical to the evening star.
Mark Twain is Samuel Clemens.
Bill is Debbie’s father.

Frege believed that these statements all have the form‘\(a=b\)’, where ‘\(a\)’ and‘\(b\)’ are either names or descriptions thatdenote individuals. He naturally assumed that a sentence ofthe form ‘\(a=b\)’ is true if and only if the object \(a\)just is (identical to) the object \(b\). For example, the sentence‘\(117+136 = 253\)’ is true if and only if the number \(117+136\)just is the number \(253\). And the statement ‘Mark Twain is SamuelClemens’ is true if and only if the person Mark Twain just isthe person Samuel Clemens.

But Frege noticed (1892a) that this account of truth can’t beall there is to the meaning of identity statements. The statement‘\(a=a\)’ has a cognitive significance (or meaning) thatmust be different from the cognitive significance of‘\(a=b\)’. We can learn that ‘Mark Twain = MarkTwain’ is true simply by inspecting it; but we can’t learnthe truth of ‘Mark Twain = Samuel Clemens’ simply byinspecting it — you have to examine the world to see whether thetwo persons are the same. Similarly, whereas you can learn that‘\(117+136 = 117+136\)’ and ‘the morning star isidentical to the morning star’ are true simply by inspection,you can’t learn the truth of ‘\(117+136 = 253\)’ and‘the morning star is identical to the evening star’ simplyby inspection. In the latter cases, you have to do some arithmeticalwork or astronomical investigation to learn the truth of theseidentity claims. Now the problem becomes clear: the meaning of‘\(a=a\)’ clearly differs from the meaning of‘\(a=b\)’, but given the account of the truth described inthe previous paragraph, these two identity statements appear to havethe same meaning whenever they are true! For example, ‘MarkTwain = Mark Twain’ is true just in case: the person Mark Twain isidentical with the person Mark Twain. And ‘Mark Twain = SamuelClemens’ is true just in case: the person Mark Twain isidentical with the person Samuel Clemens. But given that Mark Twainjust is Samuel Clemens, these two cases are the same case, and thatdoesn’t explain the difference in meaning between the twoidentity sentences. And something similar applies to all the otherexamples of identity statements having the forms‘\(a=a\)’ and ‘\(a=b\)’.

So the puzzle Frege discovered is: how do we account for thedifference in cognitive significance between ‘\(a=b\)’ and‘\(a=a\)’ when they are true?

3.1.2 Frege’s Puzzle About Propositional Attitude Reports

Frege is generally credited with identifying the following puzzleabout propositional attitude reports, even though he didn’t quitedescribe the puzzle in the terms used below. A propositional attitudeis a psychological relation between a person and a proposition. Belief,desire, intention, discovery, knowledge, etc., are all psychologicalrelationships between persons, on the one hand, and propositions, onthe other. When we report the propositional attitudes of others, thesereports all have a similar logical form:

\(x\) believes that \(p\)
\(x\) desires that \(p\)
\(x\) intends that \(p\)
\(x\) discovered that \(p\)
\(x\) knows that \(p\)

If we replace the variable ‘\(x\)’ by the name of a personand replace the variable ‘\(p\)’ with a sentence thatdescribes the propositional object of their attitude, we get specificattitude reports. So by replacing ‘\(x\)’ by‘John’ and ‘\(p\)’ by ‘Mark TwainwroteHuckleberry Finn’ in the first example, theresult would be the following specific belief report:

John believes that Mark Twain wroteHuckleberry Finn.

To see the problem posed by the analysis of propositional attitudereports, consider what appears to be a simple principle of reasoning,namely, the Principle of Identity Substitution (this is not to beconfused with the Rule of Substitution discussed earlier). If a name,say \(n\), appears in a true sentence \(S\), and the identity sentence\(n=m\) is true, then the Principle of Identity Substitution tells usthat the substitution of the name \(m\) for the name \(n\) in \(S\)does not affect the truth of \(S\). For example, let \(S\) be the truesentence ‘Mark Twain was an author’, let\(n\) be the name ‘Mark Twain’, and let \(m\) bethe name ‘Samuel Clemens’. Then since the identitysentence ‘Mark Twain = Samuel Clemens’ is true, we cansubstitute ‘Samuel Clemens’ for ‘Mark Twain’without affecting the truth of the sentence. And indeed, the resultingsentence ‘Samuel Clemens was an author’ is true. In otherwords, the following argument is valid:

Mark Twain was an author.
Mark Twain = Samuel Clemens.
Therefore, Samuel Clemens was an author.

Similarly, the following argument is valid.

\(4 \gt 3\)
\(4 = 8/2\)
Therefore, \(8/2 \gt 3\)

In general, then, the Principle of Identity Substitution seems to takethe following form, where \(S\) is a sentence, \(n\) and \(m\) arenames, and \(S(n)\) differs from \(S(m)\) only by the factthat at least one occurrence of \(m\) replaces \(n\):

From \(S(n)\) and \(n=m\), infer \(S(m)\).

This principle seems to capture the idea that if we say somethingtrue about an object, then even if we change the name by which we referto that object, we should still be saying something true about thatobject.

But Frege, in effect, noticed the following counterexample to thePrinciple of Identity Substitution. Consider the followingargument:

John believes that Mark Twain wroteHuckleberry Finn.
Mark Twain = Samuel Clemens.
Therefore, John believes that Samuel Clemens wroteHuckleberryFinn.

This argument is not valid. There are circumstances in which thepremises are true and the conclusion false. We have already describedsuch circumstances, namely, one in which John learns the name‘Mark Twain’ by readingHuckleberry Finn butlearns the name ‘Samuel Clemens’ in the context of learningabout 19th century American authors (without learning that the name‘Mark Twain’ was a pseudonym for Samuel Clemens). John maynot believe that Samuel Clemens wroteHuckleberryFinn. The premises of the above argument, therefore, do notlogically entail the conclusion. So the Principle of IdentitySubstitution appears to break down in the context of propositionalattitude reports. The puzzle, then, is to say what causes theprinciple to fail in these contexts. Why aren’t we still sayingsomething true about the man in question if all we have done ischanged the name by which we refer to him?

3.2 Frege’s Theory of Sense and Denotation

To explain these puzzles, Frege suggested (1892a) that in addition tohaving a denotation, names and descriptions also express asense.^[5] The sense of an expression accounts for itscognitive significance—it is the way by which one conceives ofthe denotation of the term. The expressions ‘\(4\)’ and‘\(8/2\)’ have the same denotation but expressdifferent senses, different ways of conceiving the same number. Thedescriptions ‘the morning star’ and ‘the eveningstar’ denote the same planet, namely Venus, but expressdifferent ways of conceiving of Venus and so have differentsenses. The name ‘Pegasus’ and the description ‘themost powerful Greek god’ both have a sense (and their senses aredistinct), but neither has a denotation. However, even though thenames ‘Mark Twain’ and ‘Samuel Clemens’ denotethe same individual, they express different senses. (See May 2006b fora nice discussion of the question of whether Frege believed that thesense of a name varies from person to person.) Using the distinctionbetween sense and denotation, Frege can account for the difference incognitive significance between identity statements of the form‘\(a=a\)’ and those of the form‘\(a=b\)’. Since the sense of ‘\(a\)’ differsfrom the sense of ‘\(b\)’, the components of the sense of‘\(a=a\)’ and the sense of ‘\(a=b\)’ aredifferent. Frege can claim that the sense of the whole expression isdifferent in the two cases. Since the sense of an expression accountsfor its cognitive significance, Frege has an explanation of thedifference in cognitive significance between ‘\(a=a\)’ and‘\(a=b\)’, and thus a solution to the first puzzle.

Moreover, Frege proposed that when a term (name or description)follows a propositional attitude verb, it no longer denotes what itordinarily denotes. Instead, Frege claims that in such contexts, aterm denotes its ordinary sense. This explains why the Principle ofIdentity Substitution fails for terms following the propositionalattitude verbs in propositional attitude reports. The Principleasserts that truth is preserved when we substitute one name foranother having the same denotation. But, according to Frege’stheory, the names ‘Mark Twain’ and ‘SamuelClemens’ denote different senses when they occur in thefollowing sentences:

John believes that Mark Twain wroteHuckleberry Finn.
John believes that Samuel Clemens wroteHuckleberry Finn.

If they don’t denote the same object, then there is no reason tothink that substitution of one name for another would preservetruth.

Frege developed the theory of sense and denotation into athoroughgoing philosophy of language. This philosophy can beexplained, at least in outline, by considering a simple sentence suchas ‘John loves Mary’. In Frege’s view, the words‘John’ and ‘Mary’ in this sentence are names,the expression ‘loves’ signifies a function, and,moreover, the sentence as a whole is a complex name. Each of theseexpressions has both a sense and a denotation. The sense anddenotation of the names are basic; but sense and denotation of thesentence as a whole can be described in terms of the sense anddenotation of the names and the way in which those words are arrangedin the sentence alongside the expression ‘loves’ (thoughsee Heck & May 2011 for discussion on whether the sense of apredicate maps the sense of a name to a thought). Let us refer to thedenotation and sense of the words as follows:

\(d[j]\) refers to the denotation of the name ‘John’.
\(d[m]\) refers to the denotation of the name‘Mary’.
\(d[L]\) refers to the denotation of the expression‘loves’.
\(s[j]\) refers to the sense of the name‘John’.
\(s[m]\) refers to the sense of the name‘Mary’.
\(s[L]\) refers to the sense of the expression‘loves’.

We now work toward a theoretical description of the denotation of thesentence as a whole. On Frege’s view, \(d[j]\) and\(d[m]\) are the real individuals John and Mary,respectively. \(d[L]\) is a function that maps\(d[m]\) (i.e., Mary) to the function ( )loves Mary. This latter function serves as the denotation ofthe predicate ‘loves Mary’ and we can use the notation\(d[Lm]\) to refer to it semantically. Now the function \(d[Lm]\) maps\(d[j]\) (i.e., John) to the denotation of the sentence ‘Johnloves Mary’. Let us refer to the denotation of the sentence as\(d[jLm]\). Frege identifies the denotation of a sentence as one ofthe two truth values. Because \(d[Lm]\) maps objects to truth values,it is a concept. Thus, \(d[jLm]\) is the truth value The True if Johnfalls under the concept \(d[Lm]\); otherwise it is the truth value TheFalse. So, on Frege’s view, the sentence ‘John lovesMary’ names a truth value.^[6]

The sentence ‘John loves Mary’ also expresses a sense.Its sense may be described as follows. Although Frege doesn’tappear to have explicitly said so, his work suggests that \(s[L]\)(the sense of the expression ‘loves’) is a function. Thisfunction would map \(s[m]\) (the sense of the name ‘Mary’)to the sense of the predicate ‘loves Mary’. Let us referto the sense of ‘loves Mary’ as \(s[Lm]\). Now again,Frege’s work seems to imply that we should regard \(s[Lm]\) as afunction which maps \(s[j]\) (the sense of the name‘John’) to the sense of the whole sentence. Let us callthe sense of the entire sentence \(s[jLm]\).^[7] Frege calls the sense of a sentence athought, and whereas there are only two truth values, hesupposes that there are an infinite number of thoughts.

With this description of language, Frege can give a general account ofthe difference in the cognitive significance between identitystatements of the form ‘\(a=a\)’ and‘\(a=b\)’. The cognitive significance is not accounted forat the level of denotation. On Frege’s view, the sentences‘\(4=8/2\)’ and ‘\(4=4\)’ both denotethe same truth value. The function \((\:)=(\:)\) maps \(4\) and\(8/2\) to The True, i.e., maps \(4\) and \(4\) to TheTrue. So \(d[4=8/2]\) is identical to \(d[4=4]\); they areboth The True. However, the two sentences in question expressdifferent thoughts. That is because \(s[4]\) is different from\(s[8/2]\). So the thought \(s[4=8/2]\) is distinctfrom the thought \(s[4=4]\). Similarly, ‘Mark Twain = MarkTwain’ and ‘Mark Twain = Samuel Clemens’ denote thesame truth value. However, given that \(s\)[Mark Twain] is distinctfrom \(s\)[Samuel Clemens], Frege would claim that the thought\(s\)[Mark Twain = Mark Twain] is distinct from the thought\(s\)[Mark Twain = Samuel Clemens].

Furthermore, recall that Frege proposed that terms followingpropositional attitude verbs denote not their ordinary denotations butrather the senses they ordinarily express. In fact, in the followingpropositional attitude report, not only do the words ‘MarkTwain’, ‘wrote’ and ‘Huckleberry Finn’ denote their ordinary senses, but the entire sentence‘Mark Twain wroteHuckleberry Finn’ also denotesits ordinary sense (namely, a thought):

John believes that Mark Twain wroteHuckleberry Finn.

Frege, therefore, would analyze this attitude report as follows:‘believes that’ denotes a function that maps the denotationof the sentence ‘Mark Twain wroteHuckleberryFinn’ to a concept. In this case, however, the denotation ofthe sentence ‘Mark Twain wroteHuckleberry Finn’is not a truth value but rather a thought. The thought it denotes isdifferent from the thought denoted by ‘Samuel Clemens wroteHuckleberry Finn’ in the following propositionalattitude report:

John believes that Samuel Clemens wroteHuckleberry Finn.

Since the thought denoted by ‘Samuel Clemens wroteHuckleberry Finn’ in this context differs from thethought denoted by ‘Mark Twain wroteHuckleberryFinn’ in the same context, the concept denoted by‘believes that Mark Twain wroteHuckleberry Finn’is a different concept from the one denoted by ‘believes thatSamuel Clemens wroteHuckleberry Finn’. One mayconsistently suppose that the concept denoted by the former predicatemaps John to The True whereas the concept denoted by the latterpredicate does not. Frege’s analysis therefore preserves ourintuition that John can believe that Mark Twain wroteHuckleberryFinn without believing that Samuel Clemens did. It also preservesthe Principle of Identity Substitution—the fact that one cannotsubstitute ‘Samuel Clemens’ for ‘Mark Twain’when these names occur after propositional attitude verbs does notconstitute evidence against the Principle. For if Frege is right,names do not have their usual denotation when they occur in thesecontexts.

3.3 Frege’s Philosophy of Language in Context

In an important paper, Bobzien (2021) develops an inductively strongargument to the conclusion that Frege adopted ideas by the Stoics, asthe latter were described in Carl Prantl’s four volumeHistoryof Western Logic (Geschichte der Logik im Abendland,1855–1870) and in Diogenes Laertius’Lives of thePhilosophers (Vitae Philosophorum, 2nd century CE). TheStoic ideas described in these two works originate with Zeno ofCitium, Cleanthes, Chrysippus of Soli, Diogenes of Babylon, etc., andso both works are ‘secondary’ literature. The relevantreports in Diogenes’Lives (which are largely quoted inPrantl) are among the closest we have to the source material for Stoicwritings on language and logic, while Section VI of Volume 1 ofPrantl’sGeschichte is a 95-page summary of Stoicviews. Bobzien notes how widely read Prantl's work was, and howFrege's training left him well-qualified to understand the many quotedpassages in Greek and Latin in his work. She then compiles evidence tothe conclusion that Frege borrowed heavily (and without attribution)from these works. This evidence is cumulative, for there is no smokinggun. We can't point to any dog-eared copies with marginalia inFrege’s own hand, or any copies of works by Prantl and Diogenesthat can be traced to Frege’s library, or any records of Fregechecking out the works by Prantl and Diogenes from the Jena Universitylibrary.^[8]

But in the 50-page Section III of her paper, Bobzien constructsnumerous tables of comparison that place passages from the Stoic andFregean corpora side-by-side and interlaces them with observations andcareful scholarship that connect the most salient (transliterated)Greek phrases with their German counterparts. The overall effect is akind ofres ipsa loquitur – the thing speaks for itself– and places the ball squarely in the court of those who wouldattempt to deny the identities and similarities. The evidence doesn'tchallenge the insight and ingenuity that Frege brought to bear whenheformalized the language and logic of mathematics (andcertain constructions of natural language) in terms of the 19thcentury understanding of functions, and so the formal representationspresented earlier in this entry were something that wasnotborrowed from the Stoics. And though Frege independently motivated theideas he absorbed from the Stoics, Bobzien shows how many aspects ofFrege’s underlying approach to language is directly founded onideas from fifteen hundred years ago and so are not as new asoriginally thought.

It remains to get a sense of the range of the elements in common, someof which were pointed out by others in previous research.^[9]At the most general level, Bobzien compares the ways that the Stoicsand Frege relate semantic ‘content’ (Greeklektav. FregeanSinn) to the linguistic expression of thatcontent. Both distinguish the ‘incomplete’(ellipē v.ungesättigt) content ofpredicates from the complete content of sentences, and there areparallels in what the Stoics and Frege have to say about complete‘assertible’ contents (axiōmatav.Gedanken), especially in connection with the fact thatthese contents are the primary bearers of truth and falsity. Bobzienthen compares the two philosophies with regard to other kinds ofcomplete contents, such as those expressed by commands, sentencequestions, expressions of emotion, word questions, and indexicals. Ineach case, Frege’s descriptions of the content presented bythese linguistic expressions closely follow those of the Stoics.Another group of similarities arises in her comparison of thedoctrines regarding complex contents produced with propositionalconnectives, such as those expressed by sentences with negations(including contradictories, double negations, etc.), conjunctions,disjunctions, conditionals, and sentences with ‘because’.A comparison of the two approaches to quantification and universalityconcludes the paper.

By producing so many passages in parallel between the Stoic andFregean corpora, the details and evidence accumulate, thereby becomingincreasingly persuasive. Any student of Frege who wants to understandthe full range of his views on language will have to come to gripswith how those views are related to Stoic philosophy. Though Bobzien dulynotes the others who had previously remarked on some of thesimilarities (see againfootnote 9), her paper of2021 is a good place to start, since the evidence assembled there isso comprehensive.

Bibliography

A. Primary Sources

Frege’s Complete Corpus

Chronological Catalog of Frege’s Work

Works by Frege Cited in this Entry

Two useful source books for translations of Frege’s Writings are:

• P. Geach and M. Black (eds. and trans.), 1980,Translationsfrom the Philosophical Writings of Gottlob Frege, Oxford:Blackwell, third edition.
• B. McGuinness (ed.), 1984,Collected Papers on Mathematics, Logic, and Philosophy, Oxford: Blackwell.

Many of the works listed below are translated and collected in the above.

B. Secondary Sources

• Anderson, D., and Zalta, E., 2004, “Frege, Boolos, andLogical Objects”,Journal of Philosophical Logic, 33(1):1–26.
• Angelelli, I., 1967,Studies on Gottlob Frege andTraditional Philosophy, Dordrecht: D. Reidel.
• Antonelli, A., and May, R., 2005, “Frege’s OtherProgram”,Notre Dame Journal of Formal Logic, 46(1):1–17.
• Beaney, M., 1996,Frege: Making Sense, London:Duckworth.
• –––, 1997,The Frege Reader, Oxford:Blackwell
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• Die Grundlagen der Arithmetik, (528 KB PDF file),original German text.
• Side-by-side translation (English)/transcription (German) of Frege’s paperÜber die wissenschaftliche Berechtigungeiner Begriffsschrift, translated as “On the Scientific Justification of aConcept Script,”inBorderless Philosophy, 2 (2019): 76–94.
• MacTutor History of Mathematics Archive
• Metaphysics Research Lab Web Page on Frege
• Frege, Gottlob, by Kevin Klement (U. Massachusetts/Amherst), in theInternetEncyclopedia of Philosophy.

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Acknowledgments

I would like to thank the following people: Kai Wehmeier, whosecareful eye as a logician and Frege scholar caught several passageswhere I had bent the truth past the breaking point; Emily Bender, whopointed out that I hadn’t observed the distinction betweenrelative and subordinate clauses in discussing Frege’s analysisof belief reports; Paul Oppenheimer, for making some suggestions thatimproved the diction and clarity in a couple of sentences, and for asuggestion for improvement to Section 3.2; Wolfgang Kienzler, forsuggesting several important improvements to the main text (he andConden Chao both sent corrections and suggestions fortheChronological Catalog of Frege’sWork); Patricia Blanchette and Richard Zach for reading over, andproviding constructive comments on, the reworked Section 2.7(“Frege’s Conception of Logic”), which formed partof the update in late 2019; and Susanne Bobzien, for reading andcommenting on the first draft of the new (as of 2022) Section 3.3.